6th - iecmsa · mahir rasulov..... 3 on some applications ... some results for matrix...

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6TH INTERNATIONAL EURASIAN CONFERENCE

ON MATHEMATICAL SCIENCES

AND APPLICATIONS

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FOREWORD

Following five very successful previous conferences (IECMSA-2012, Prishtine,

Kosovo, IECMSA-2013, Sarajevo, Bosnia and Herzegovina, IECMSA-2014, Vienna, Austria, and

IECMSA-2015, Athens, Greece, IECMSA-2016, Belgrade, Serbia) now the “6th International

Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)’’ has been held

in Budapest, Hungary on August 15-18, 2017 .

This conference is supported by Sakarya University, Kocaeli University, Namik Kemal

University, Amasya University, Bilecik Seyh Edebali University, Baku State University, Institute

of Applied Mathematics, Turkic World Mathematical Society, and International Balkan

University.

The major goal of the series of IECMSA is to facilitate bridge building across the disparate

fields within Mathematical Sciences and their Applications. Since 2012, these conferences have been

held regularly every year and featured many distinguished speakers from the across the globe, with

talks focusing on algebra, analysis, applied mathematics, geometry, mathematics education,

statistics, topology, physics, economics or computer sciences.

The scientific program of IECMSA-2017 features invited talks, followed by contributed oral

and poster presentations in parallel sessions. Scientific topics cover the latest developments in all

subfields of mathematics. The abstracts of all presentations have been substituted in this book.

Also, the electronic version of the abstracts of all presentations can be found in the Conference

Abstracts Book at www.iecmsa.org

http://www.iecmsa.org/

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I wish to thank all members of scientific committee and sponsors for their continued support

to the IECMSA. And finally, I would like to sincerely thank all participants of IECMSA-2017 for

contributing to this great meeting in many different ways. I believe and hope that each of them will

get the maximum benefit from the conference.

Welcome to Budapest!

Prof. Dr. Murat TOSUN

Chairman

On behalf of the Organizing Committee

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HONORARY COMMITTEE

Prof. Dr. Muzaffer ELMAS (Rector of Sakarya University)

Prof. Dr. Sadettin HÜLAGÜ (Rector of Kocaeli University)

Prof. Dr. İbrahim TAŞ (Rector of Bilecik Seyh Edebali University)

Prof. Dr. Osman ŞİMŞEK (Rector of Namık Kemal University)

Prof. Dr. Metin ORBAY (Rector of Amasya University)

Prof. Dr. İsmail KOCAYUSUFOĞLU (Rector of International Balkan University)

Prof. Dr. H. Hilmi HACISALİHOĞLU (Honorary President of TWMS

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SCIENTIFIC COMMITTEE

Prof. Dr. Abdel Salhi University of Essex

Prof. Dr. Bayram Sahin Ege University

Prof. Dr. Cengizhan Murathan Uludag University

Prof. Dr. Ding-Xuan Zhou Zhejiang University

Prof. Dr. Emine Mısırlı Ege University

Prof. Dr. Etibar Penakhov Baku State University

Prof. Dr. Fikret A. Aliev Baku State University

Prof. Dr. Gennady A. Leonov Saint-Petersburg State University

Prof. Dr. Gerald Beer California State University

Prof. Dr. Hans Peter Kunzi University of Cape Town

Prof. Dr. Hellmuth Stachel Vienna Technical University

Prof. Dr. İsmail Ekincioglu Dumlupınar University

Prof. Dr. Ioan Rasa Technical University of Cluj-Napoca

Prof. Dr. Josef Slapal Brno University of Technology

Prof. Dr. Maria Isabel Garrido Complutense University of Madrid

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SCIENTIFIC COMMITTEE

Prof. Dr. Mehmet Ali Güngör Sakarya University

Prof. Dr. Mo Sal Moslehian Ferdowsi University of Mashhad

Prof. Dr. Musatafa Çalışkan Gazi University

Prof. Dr. Mohammad Mursaleen Aligarh Muslim University

Prof. Dr. Naime Ekici Çukurova University

Prof. Dr. Nuri Kuruoğlu İstanbul Gelişim University

Prof. Dr. Nesip Aktan Necmettin Erbakan University

Prof. Dr. Neville Robbins San Francisco State University

Prof. Dr. Sadık Keleş İnönü University

Prof. Dr. Siamak Yassemi University of Tehran

Prof. Dr. Sujatha Ramdorai University of British Columbia

Prof. Dr. Sidney A. Morris Federation University of Australia

Prof. Dr. Varga Kalantarov Koç University

Prof. Dr. Vladimir Touraev Indiana University

Prof. Dr. Wolfgang Sproessig Technische Uni. Bergakademie Freiber

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ORGANIZING COMMITTEE

GENERAL COORDINATOR:

Prof. Dr. Murat Tosun Sakarya University

VICE-GENERAL COORDINATOR:

Prof. Dr. Ljubisa Kočinac Nis University

VICE-GENERAL COORDINATOR:

Prof. Dr. István Juhász Alfred Renyi Institute of Mathematics

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CHAPTERS

INVITED TALKS………………………………...……………………………………………………………... 2

ALGEBRA………………………………………………………………………………… 13

ANALYSIS…………………………………………………………………………….. 34

APPLIED MATHEMATICS…………………………………………………. 93

DISCRETE MATHEMATICS…………………………………………………... 186

GEOMETRY…………………………………………………………………………… 189

MATHEMATICS EDUCATION………………………………………… 230

STATISTICS……………………………………………………………………………… 265

TOPOLOGY…………………………………………………………………………….. 272

THE OTHER AREAS…………………………………………………………. 285

POSTERS……………………………………………………………………………………….. 301

PARTICIPIANTS……………………………………………………………………….. 256


Prof. Dr. Arunabha Chanda Jadavpur University

Prof. Dr. Cristina Flaut Ovidius University

Prof. Dr. Cihan Özgür Balikesir University

Prof. Dr. Debasis Giri Haldia Institute of Technology

Prof. Dr. Kadri Arslan Uludağ University

Prof. Dr. Kailash C. Madan Ahlia University

Prof. Dr. Kazım İlarslan Kırıkkale University

Prof. Dr. Laura Ventura Universita Degli Studi Di Padova

Prof. Dr. Levent Kula Ahi Evran University

Prof. Dr. Mahmut Ergüt Namık Kemal University

Prof. Dr. Mao-Ting Chien Soochow University

Prof. Dr. Moiz ud Din Khan COMSATS Institute of Information Technology

Prof. Dr. Muhammad Saeed Khan COMSATS Institute of Information Technology

Prof. Dr. Nejat Ekmekçi Ankara University

Prof. Dr. Nirmal C. Sacheti Sultan Qaboos University

Prof. Dr. Pallath Chandran Sultan Qaboos University

Prof. Dr. Pranesh Kumar University of Northern British Columbia

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Prof. Dr. Ram N. Mohapatra University of Central Florida

Prof. Dr. Slavica Ivelić Bradanović University of Split

Prof. Dr. Vichian Laohakosol Kasetsart University

Prof. Dr. Victor Jimenez Lopez Universidad De Murcia

Prof. Dr. Victor Martinez-Luaces Universidad De La República

Prof. Dr. Yue Kuen Kwok Hong Kong University

Assoc. Prof. Dr. Emrah Evren Kara Duzce University

Assoc. Prof. Dr. İsmet Altıntaş Sakarya University

Assoc. Prof. Dr. Soley Ersoy Sakarya University

Assoc. Prof. Dr. Şenol Dost Hacettepe University

Assist. Prof. Dr. Ayse Zeynep Azak Sakarya University

Assist. Prof. Dr. Mahmut Akyiğit Sakarya University

Assist. Prof. Dr. Murat Sarduvan Sakarya University

Assist. Prof. Dr. Hidayet Hüda Kösal Sakarya University

Assist. Prof. Dr. Tülay Erişir Erzincan University

Dr. Handan Akyiğit Sakarya Universitesi

Lecturer Furkan Aydın Kahramanmaraş Sütçü imam University

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CONTENTS

Invited Talks

Resolvability of Topological Spaces 2

István JUHÁSZ ........................................................................................................................... 1

Boundedness properties of function spaces 2

Ljubiša D. R. KOČINAC and L’ubica HOLÁ ............................................................................ 2

Finite Differences Method of the Riemann Type Problem for 2D Conservative Laws in a Class

of Discontinuous Functions 3

Mahir RASULOV ........................................................................................................................ 3

On Some Applications of Measures of Noncompactness 4

Mohammad MURSALEEN ......................................................................................................... 4

Mathematical Literacy and a New Classification Proposal for Mathematical Literacy

Problems 6

Murat ALTUN ............................................................................................................................ 6

Fixed point property for some Alexandroff topological spaces with T0-separation axiom and

its applications 8

Sang-Eon HAN ........................................................................................................................... 8

The Role of Convergence Theory in Mathematics 9

Szymon DOLECKI ..................................................................................................................... 9

From Dual Bodies to the Kneser-Poulsen Conjecture 11

Karoly BEZDEK ...................................................................................................................... 11

Algebra

Some Results for Matrix Representation of Fibonacci Octonions 13

Adnan KARATAŞ and Serpil HALICI ...................................................................................... 13

On Special Elements of Split Octonions 14

Adnan KARATAŞ and Serpil HALICI ...................................................................................... 14

Numerical solutions of mKdV equation via modified cubic B-spline based Differential

Quadrature Method 16

Ali BAŞHAN ............................................................................................................................. 16

New Bounds for the Levels and Sublevels of Algebras Obtained by the Cayley-Dickson

Process 17

Cristina FLAUT........................................................................................................................ 17

Some Results on Generalized Derivations and (,) Lie Ideals 19

Evrim GÜVEN .......................................................................................................................... 19

A Note on the Sequence of Balancing Numbers 20

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Gül Karadeniz GÖZERİ ........................................................................................................... 20

Graph Operations of Randic Index 21

Gülistan Kaya GÖK and Şerife BÜYÜKKÖSE ........................................................................ 21

On Codes Written by Matrices Lexicographically Ordered 22

Mustafa ÖZKAN and Figen ÖKE ............................................................................................. 22

About Quasi-Cyclic Codes over the Field 23

Mustafa ÖZKAN and Figen ÖKE ............................................................................................. 23

On Certain Type of Band Matrices with the Generalized Fibonacci Numbers 24

Neşe ÖMÜR and Cemile Duygu ÇOLAK ................................................................................. 24

On Generalization of Quaternions 25

Serpil HALICI and Adnan KARATAŞ ...................................................................................... 25

Generalized Quaternions and Dual Fibonacci Octonions 27

Serpil HALICI and Adnan KARATAŞ ...................................................................................... 27

Formulæ for Two Weighted Binomial Identities with the Falling Factorials 29

Sibel Koparal, Neşe Ömür and Emrah Kılıç ............................................................................ 29

S3–graded iso(1,2) 30

Yasemen UÇAN , Reşat KÖŞKER and Ramazan TEKERCİOĞLU……………………………... 30

On Multiplicative Generalized Derivations in 3-Prime Near-Rings 32

Zeliha BEDİR and Öznur GÖLBAŞI 2..................................................................................... 32

Analysis

f -Lacunary statistical convergence of order for double sequences 34

Abdulkadir KARAKAŞ, Hıfsı ALTINOK and Yavuz ALTIN ..................................................... 34

Some Special Values of Vertices of Trees on the Suborbital Graphs 35

Ali Hikmet DEĞER and Ümmügülsün AKBABA ..................................................................... 35

On Some Special Values of Fibonacci and Lucas Sequence with the Suborbital Graphs 36

Ali Hikmet DEĞER and Ümmügülsün AKBABA ..................................................................... 36

A Note on the Ambarzumyan's Theorem 37

Alp Arslan KIRAÇ .................................................................................................................... 37

Polars in Locally Convex Cones 37

A. RANJBARI ........................................................................................................................... 37

A Sufficient Condition for Complex-Valued Summable Functions to be Absolutely Continuous

a.e. 40

Alp Arslan KIRAÇ .................................................................................................................... 40

On Quasi Subordinations for Analytic and Bi-Univalent Function Class 42

Arzu AKGÜL ............................................................................................................................ 42

Quadratic Time-Frequency Representations on Mixed Lorentz Type Modulation Spaces 44

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Ayşe SANDIKÇI ....................................................................................................................... 44

Bilinear Multipliers on nM P,Q 45

Ayşe SANDIKÇI ....................................................................................................................... 45

Reguler Equivalents of a Measure Space 46

Bahaettin CENGİZ and Banu GÜNTÜRK ............................................................................... 46

Hilbert Matrix on Spaces of Holomorphic Functions 47

Boban KARAPETROVIĆ .......................................................................................................... 47

Some Relation on the Poly-Genocchi Numbers and Polynomials with a q-Parameter 48

Burak KURT ............................................................................................................................. 48

The Fixed Point of the Sum of Nonlinear Operators in WC-Banach Algebras Relative to the

Weak Topology 49

Cesim TEMELand Musa ÇAKIR .............................................................................................. 49

Investigating the Center of L-weakly and M-weakly Compact Operators 50

E. BAYRAM, A.W. WICKSTEAD ............................................................................................. 50

The Generating Function of ( , )p q -Bernstein Polynomials and Their Properties Based on

( , )p q -Calculus 51

Erkan AGYUZ and Mehmet ACIKGOZ ................................................................................... 51

Characterizations of Matrix and Compact Operators on the Space 𝑬𝒓𝜽𝒑 52

Fadime GÖKÇE and Mehmet Ali SARIGÖL ........................................................................... 52

Matrix and Compact Operators on the Absolute Fibonacci Spaces 54

Fadime GÖKÇE and Mehmet Ali SARIGÖL ........................................................................... 54

The Theory of 𝒏-Scales 56

Furkan Semih DÜNDAR .......................................................................................................... 56

Some Specific Properties of Concave Functions Defined by the Generalized Srivastava-Attiya

Operator 57

Hasan BAYRAM and Şahsene ALTINKAYA ............................................................................ 57

A Note On Weighted Composition Operators on Besov-Type Spaces 59

Hamid VAEZI .......................................................................................................................... 59

Generalized Lupaş Operators 59

Hatice Gül İNCE İLARSLAN, Ali ARAL and Gülen BAŞCANBAZ-TUNCA ........................... 59

Statistical Convergence of order (β,γ) for Sequences of Fuzzy Numbers 60

Hıfsı ALTINOK and Mikail ET ................................................................................................. 60

Statistical Convergence of order β for Double Sequences of Fuzzy Numbers Defined by a

Modulus Function 61

Hıfsı ALTINOK, Yavuz ALTIN and Mahmut IŞIK.................................................................... 61

Δnm-lacunary Statistical Convergence of Order α 62

Hıfsı ALTINOK , Mikail ET and Mahmut IŞIK ....................................................................... 62

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Generalized Statistical Convergence of order β for Sequences of Fuzzy Numbers 63

Hıfsı ALTINOK, Abdulkadir KARAKAŞ and Yavuz ALTIN ..................................................... 63

A New Application of Quasi Power Increasing Sequences 64

Hikmet SEYHAN ÖZARSLAN .................................................................................................. 64

Almost Increasing Sequences and Their New Applications 65

Hikmet SEYHAN ÖZARSLAN and Bağdagül KARTAL ........................................................... 65

Wavelet-Type Transforms Generated by Some Semigroups and Their Applications 66

İlham A. ALIEV ........................................................................................................................ 66

Higher Order Fractional Boundary Value Problems with Integral Boundary Conditions 67

İsmail YASLAN ......................................................................................................................... 67

Boundedness of Products of Weighted Composition Operators and Differentiation Operators

between Weighted Bergman Spaces and Weighted Zygmund Spaces of Analytic Functions 68

Jasbir S. MANHAS ................................................................................................................... 68

The Vitali Convergence Theorem for Nonlinear Integrals 69

Jun KAWABE ........................................................................................................................... 69

Berezin Number Inequalities and Engliš Algebras 70

M. GÜRDAL, M.B. HUBAN and M.T. GARAYEV ................................................................... 70

Hadamard Type Inequalities for m-Convex and (α,m)-Convex Functions via Fractional

Integrals 71

Merve Avcı ARDIÇ, Alper EKİNCİ, Ahmet Ocak AKDEMİR and M. Emin ÖZDEMİR ......... 71

On Wijsman Asymptotically Deferred Statistical Equivalence of Order α for Set Sequences 73

Mikail ET, Hıfsı ALTINOK and Rifat ÇOLAK ........................................................................ 73

On Infinite Bernoulli Matrices 74

Murat KİRİŞCİ ......................................................................................................................... 74

Curious Bounds for the Gamma Function 75

Necdet BATIR ........................................................................................................................... 75

Some Fixed-Circle Theorems and Discontinuity at Fixed Circle 77

Nihal Yılmaz ÖZGÜR and Nihal TAŞ ...................................................................................... 77

Bergman Projections on the Weighted Harmonic Bloch Spaces on the Ball, Atomic

Decompositions and Gleason’s Problem 78

Ömer Faruk DOĞAN ............................................................................................................... 78

Majorization Problem on a Subclass of Analytic Functions 80

Öznur ÖZKAN KILIÇ, Osman ALTINTAŞ ............................................................................... 80

A Note on the Extended Type Riemann-Liouville Fractional Derivative Operator 81

Praveen AGARWAL, İ. Onur KIYMAZ2, Shilpi JAIN3 and Ayşegül ÇETİNKAYA4 ................. 81

On the Harmonic Averages of Sequences of Fuzzy Numbers 83

Rahmet SAVAŞ and Sefa Anıl SEZER ...................................................................................... 83

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Malliavin Calculus of Bismut Type for an Operator of Order Four on a Lie Group 84

Rémi LÉANDRE ....................................................................................................................... 84

On Convergence Properties of Sequences of k-Positive Operators in the Subspace of Analytic

Functions 85

Seda KARATEKE and Tülin COŞKUN .................................................................................... 85

Some Operator Inequalities Related to Means and Entropies 88

Shigeru FURUICHI .................................................................................................................. 88

On a Subclass of Harmonic Univalent Functions Involving a Linear Operator 88

Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................... 88

On a Coefficient Problem for a Subclass of Bi-Univalent Functions Defined by Using a q-

Derivative Operator 90

Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................... 90

On Asymptotically f-Statistical Equivalent Set Sequences 92

Sukran KONCA and Mehmet KUCUKASLAN ......................................................................... 92

Reverse Inequalities for the Berezin Numbers of Operators 93

U. YAMANCI, M. GÜRDAL and M. T. GARAYEV .................................................................. 93

On the (p,q)-Analogue of Poly-Bernoulli Polynomials 94

Veli KURT ................................................................................................................................ 94

Applied Mathematics

An Affine Scaling Method Using a Class of Differential Barrier Functions 96

Abdessamad BARBARA ............................................................................................................ 96

Hyperbolic Tangent Solution to the Conformable Time Fractional Zakharov-Kuznetsov

Equation in 3D Space 97

Alper KORKMAZ and Ozlem Ersoy HEPSON ........................................................................ 97

Asymptotic Properties of the Lasota Equation in Various Functional Spaces 98

Anna POSKROBKO and Antoni Leon DAWIDOWICZ ........................................................... 98

Iterative Solutions to the Systems of Linear Differential Equations 99

Arzu GÜLEROĞLU .................................................................................................................. 99

A Mathematical Model on Coral-Population Dynamics 100

Arzu ÜNAL ............................................................................................................................. 100

The Class of 𝑳 ∩ 𝑫 and Its Application to Renewal Reward Process 101

Asli Bektaş KAMIŞLIK, Tülay KESEMEN, Tahir KHANIYEV3 ............................................ 101

Mathematical Modeling for Real Life Problems 103

Aysegul SAGLAM ARSLAN .................................................................................................... 103

Higher Order Trigonometric B-Spline Algorithms to the Solution of Coupled Burgers’

Equation 104

Aysun Tok ONARCAN and Ozlem ERSOY HEPSON ............................................................ 104

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Numerical Solution of Nonlinear Parabolic Type Equation in the Class of Generalized

Functions with Degeneration 105

Bahaddin SINSOYSAL, Hakan BALand Kubra YILMAZ ....................................................... 105

A Finite Differences Scheme for Solving the System of Differential Equations of Non-

Isentropic Flow in a Class of Discontinuous Functions 106

Bahaddin SINSOYSAL, Hasan CARFI and Nazife KOC ....................................................... 106

A Finite Difference Method for Solving Generalized FitzHugh-Nagumo Equation 107

Bilge İNAN ............................................................................................................................. 107

Numerical Solutions of Boundary Value Problems for Nonlinear Delay Differential Equations 108

Bülent YILMAZ and Volkan YAMAN2 .................................................................................... 108

The Analytical Solutions of the Space-Time Fractional Modified Kawahara Equation 109

Dilek Varol BAYRAM, Sevil ÇULHA, Ayşegül DAŞCIOĞLU ............................................... 109

An Approximation Method for the Fractional Linear Fredholm Integrodifferential Equations

by Laguerre Polynomials 110

Dilek VAROL BAYRAM and Ayşegül DAŞCIOĞLU .............................................................. 110

Convergence of Backward Semi-Lagrangian Scheme with BDF2 for Two-Dimensional

Burgers’ Equation 111

Dojin KIM, Soyoon BAK, Philsu KIM, ................................................................................... 111

Compressed Sensing with Cyclic-S Hadamard Matrix for TeraHertz Imaging Applications 113

Esra Şengün ERMEYDAN and İlyas ÇANKAYA ................................................................... 113

Symbolic Regression: An Application to the Development of Models for the Surface Tension 115

Eva T. López SANJUÁN and María Isabel Parra ARÉVALO ............................................... 115

On the Stability Analysis of Sturm-Liouville Operator with Jump Conditions 119

Etibar S. PANAKHOV and Ahu ERCAN ........................................................................ 119116

Inverse Nodal Problem for p-Laplacian Diffusion Equation with Polynomially Dependent

Spectral Parameter 119

Etibar S. PANAKHOV , Emrah YILMAZ and Tuba GULSEN ....................................... 119117

On the Stress Distribution in the Elastic Body with a Locally Curved Triple Walled Carbon

Nanotube 119

Fatma COBAN and Reşat KOSKER ...................................................................................... 119

Construction of Exact Solutions to Partial Differential Equations via CRE Method 120

Filiz TAŞCAN and Arzu AKBULUT ....................................................................................... 120

Numerical Solution of an Inverse Problem for the Stokes Equations 121

Gullazata DAIRBAYEVA ........................................................................................................ 121

Existence of Solution for the Problem with a Concentrated Source in a Subdiffusive Medium 122

H. Terence LIU ....................................................................................................................... 122

Numerical Solution of the Conformable Fractional Diffusion Equation 123

Handan Çerdik YASLAN ........................................................................................................ 123

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Oscillation Criteria Fractional Difference Equations with Mixed Nonlinearities ................ 124

Hande BÜYÜKÇAVUŞOĞLU, Mustafa Kemal YILDIZ and Tuğba YALÇIN UZUN............ 124

Free Vibration of a Tapered Euler-Bernoulli Beam with 3D Tip Mass by Multi-Step

Differential Transformation Method 125

Hilal Doğanay KATI and Hakan GÖKDAĞ .......................................................................... 125

Image Segmentation with Fan-shaped Gradients 127

Hosook KIM and Hyoungseok KIM ....................................................................................... 127

The Computational and Experimental Studies on (E)-4-Methyl-N'-(1-(Pyridin-2-Yl)

Ethylidene) Benzenesulfonohydrazide 128

Hümeyra BATI, Erbil Murat AYDIN, Murat ÇINARLI, Nezihe ÇALIŞKAN ......................... 128

An Application of Integration Interpolation Method to Cauchy Type Singular Integral

Equation System 129

Hüseyin OĞUZ and Elçin YUSUFOĞLU .............................................................................. 129

A New Analysis for Fractional Order Ebola Model with Non-Local and Non-Singular Kernel 130

Ilknur KOCA .......................................................................................................................... 130

Linearization of the State Equation of a Nonlinearly Elastic Material of a Circular Composite

Shaft for Estimation of Natural Frequencies of Torsional Vibrations 131

I.A. TARASYUK and A.S. KRAVCHUK ................................................................................. 131

Parameter Dependent Generalized Sylvester and T-Sylvester Equations 133

Ivana KUZMANOVIĆ and Ninoslav TRUHAR ...................................................................... 133

Generalization of the Linear Momentum Operator in Quantum Mechanics: A Position-

Dependent Effective Mass Approach 134

J. J. PEÑA, J. MORALES, J. GARCÍA-RAVELO, Leonardo SALINAS ................................ 134

On Trace Inequalities for Generalized Quasi-Metric Adjusted Skew Informations 135

Kenjiro YANAGI ..................................................................................................................... 135

The Solutions of Special Second Order Ordinary Differential Equations Using Symmetries 137

Kısmet KASAPOĞLU ............................................................................................................. 137

Schrödinger Equation with Variable Effective Mass: Linear and Staggered Mass

Distributions 138

Leonardo SALINAS, H. LUNA-GARCÍA, J. GARCÍA-RAVELO, J. J. PEÑA ........................ 138

Weak Solutions to Interdiffusion Models with Vegard Rule 139

Lucjan SAPA, Bogusław BOŻEK and Marek DANIELEWSKI .............................................. 139

Self-Adjoint Extensions of Differential Operators with Potentials-Point Interactions 141

Manaf Dzh. MANAFOV ......................................................................................................... 141

Detecting a Hyperbolic Quadratic Eigenvalue Problem Using a Subspace Algorithm 142

Marija MILOLOŽA PANDUR ............................................................................................... 142

A Generalization of Midpoint Type Inequality of m-Logarithmically Convex Functions 143

Mehmet Eyüp KİRİŞ ............................................................................................................... 143

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Some Approximation Results for Lupaş-Kantorovich-Schurer Operators Based on (p, q )-

Calculus 146

Melek SOFYALIOĞLU and Kadir KANAT ........................................................................... 146

Conformable Fractional Klein-Gordon and Cahn-Allen Equations 147

Meryem ODABASI ................................................................................................................. 147

On the Finite Element Approximation in the Maximum Norm of Elliptic Quasi-Variational

Inequalities with Nonlinear Source Terms 148

Messaoud BOULBRACHENE ................................................................................................ 148

Exponentially Cusped Prismatic Shells in the 0N Approximation of I. Vekua’s Hierarchical

Models 149

Miranda GABELAIA .............................................................................................................. 149

Bounds for Blow up Time in Nonlinear Pseudo-Parabolic Equations 150

Müge MEYVACI ..................................................................................................................... 150

Oscillation Criteria for Higher Order Fractional Difference Equations with Nonlinearities 152

Mustafa Kemal YILDIZ , Ebru YILMAZ and Umut Mutlu ÖZKAN ....................................... 152

Lyapunov Type Inequalities for nth-order Three point BVP Differential Equations with Mixed

Nonlinearities 154

Naceri MOSTEPHA and Abdullah OZBEKLER .................................................................... 154

3D FEM Analysis of Buckling of a Piezoelectric Rectangular Thick Plate 155

Nazmiye YAHNIOGLU and Fatih AYLIKCI .......................................................................... 155

Identities for the double L22-integral transform 156

Neşe DERNEK, Fatih AYLIKÇI ............................................................................................. 156

Simplified Hirota Method for a Nonlinear Partial Differential Equation 158

Ömer ÜNSAL, Ahmet BEKIR, Murat KOPARAN .................................................................. 158

Extended Transformed Rational Function Method for Some (3+1) Dimensional Nonlinear

Partial Differential Equations 160

Ömer ÜNSAL, Ahmet BEKIR, Murat KOPARAN .................................................................. 160

Some Results on Cooperative Grey Games 162

Osman PALANCI, Rana S.M. BAKRI, Marwah Imad Najm NAJM, Sirma Zeynep

ALPARSLAN GOK ................................................................................................................. 162

A Cubic Subdomain Galerkin Method over the Geometrically Graded Mesh to the Singularly

Perturbed Problem 163

Ozlem Ersoy HEPSON and Idris DAG .................................................................................. 163

Optimal Control of Distributed Systems in Problems of Quartz Optical Fiber Production 164

Pervadchuk V.P., Vladimirova D.B., Gordeeva I.V. .............................................................. 164

On the Kinetics of the Dimer-Dimer Reactions over Supported Catalysts 166

Pranas KATAUSKIS ............................................................................................................... 166

The Importance of Algebraic Geometry and Cusp Singularities for Phase Transition

Dynamics of Neural Populations in Cerebral Cortex 167

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R Murat DEMİRER , Oya DEMİRER .................................................................................... 167

Stress Distribution in the Infinite Elastic Body with Two Neighboring Periodical Curved

Carbon Nanotube Located on Different and Parallel Planes 168

Reşat KÖŞKER and İsmail GÜLTEN ..................................................................................... 168

Discrete Fractional Solutions of a Legendre Equation 170

Resat YILMAZER ................................................................................................................... 170

Discrete Fractional Solutions of a Associated Laguerre Equation 171

Resat YILMAZER and Erdal BAS .......................................................................................... 171

Asymptotic Stability of an Evolutionary Nonlinear Boltzmann-Type Equation 172

Roksana BRODNICKA and Henryk GACKI .......................................................................... 172

Hybrid Interval Algorithms for the Simultaneous Inclusion of Polynomial Complex Zeros: An

Experimental Comparison 174

Roseleine Neves MACHADO and Luiz Guerreiro LOPES .................................................... 174

An Investigation on the Free Vibration of a Sphere with Inhomogeneous Initial Stresses 175

S. D. AKBAROV, , N. YAHNIOGLU and H. H. GULIYEV ..................................................... 175

GDP and Efficiency of Russian Economy 176

Sergey M. BORODACHEV .................................................................................................... 176

Stabilization of Discrete Systems by High Order Compensators 177

Şerife YILMAZ İRTEM, Taner BÜYÜKKÖROĞLU, Vakif DZHAFAROV ............................ 177

Numerical Investigation of a Steady Flow of an Incompressible Pseudoplastic Fluid in a Lid

Driven Cavity 179

Serpil ŞAHİN and Hüseyin DEMİR ....................................................................................... 179

Numerical Comparision of Newtonian and Dilatant Fluids in an Enclosed Cavity Region 180

Serpil ŞAHİN and Hüseyin DEMİR ....................................................................................... 180

New Hybrid Conjugate Gradient Method as a Convex Combination of HS and FR Conjugate

Gradient Methods 181

Snezana S. DJORDJEVIC ...................................................................................................... 181

A New Formula for Higher Order Derivatives of Type xfdx

dx

n

k

and Its Applications 182

Telhat ÖZDOĞAN, Melek ERASLAN, Metin ORBAY............................................................ 182

Asymptotic Behaviour for Functional Three-Dimensional Navier-Stokes-Voigt Equations 183

T. CARABALLO and Antonio M. Márquez DURÁN .............................................................. 183

Mixing Problems: Associated Matrices and Stability of Linear ODE Systems 184

Victor MARTINEZ-LUACES .................................................................................................. 184

Backward Semi-Lagrangian Scheme for Guiding Center Problems 186

Xiangfan PIAO, Dojin KIM and Philsu KIM ......................................................................... 186

A Comparison of Multi-step and Multi-stage Method 187

Yonghyeon JEON and Sunyoung BU, Philsu KIM ................................................................. 187

xix

Green's Functions For Problems Simulating Potential Fields in Thin-Walled Assemblies of

Irregular Configuration 188

Yu. A. MELNIKOV and V. N. BORODIN............................................................................... 188

The Hurwitz’s Formula and Boundary Value Problems for Second-Order Equations 190

Yu. E. ANIKONOV, İsmet GÖLGELEYEN and Mustafa YILDIZ .......................................... 190

Discrete Mathematics

Secret Sharing Schemes Based on Extension Fields 192

Selda ÇALKAVUR .................................................................................................................. 192

Geometry

Pseudo Spherical Indicatrix Elastic Curves 195

Ahmet YÜCESAN Gözde ÖZKAN TÜKEL and Tunahan TURHAN ...................................... 195

Problems of g- lifts 196

Arif SALIMOV and Rabia CAKAN ......................................................................................... 196

Position Vectors of Curves with Respect to Darboux Frame in the Galilen 3-Space 197

Buket CEYLAN DİRİŞEN and Tevfik ŞAHİN ......................................................................... 197

Polynomial Helices in 𝑬𝒏 198

Bülent ALTUNKAYA and Levent KULA ................................................................................ 198

On Normal Magnetic Curves in 3-Dimensional Heisenberg Group H3 199

Cihan ÖZGÜR ........................................................................................................................ 199

On Chen Invariants for Submanifolds of Riemannian Product Manifolds 200

Erol KILIÇ, Mehmet GÜLBAHAR and Sadık KELEŞ ........................................................... 200

Special Curves in Finsler Space 201

Fatma ATEŞ, Zehra ÖZDEMİR, F. Nejat EKMEKCİ and Mustafa ÇALIŞKAN ................... 201

A Differential Calculus on Rh(1|2) 202

Fatma BULUT ........................................................................................................................ 202

Application Example of the Processing of the Cubes in the Geometry Lesson in Elementary

Schools 203

Gönül TÜRKAN DEMİR, Keziban ORBAY and Emine ALTUNAY ŞAM .............................. 203

A study on Elastic Strips 205

Gözde ÖZKAN TÜKELAhmet YÜCESAN .............................................................................. 205

Involutes of Order k of a Space-like Curve in 4

1IE 206

Günay ÖZTÜRK, İlim KİŞİ2, Sezgin BÜYÜKKÜTÜK and Kadri ARSLAN .......................... 206

Similarity Geometry of Null Cartan Curves 207

Hakan ŞİMŞEK ...................................................................................................................... 207

xx

Semisimilarity and Consemisimilarity for matrices over a Commutative Quaternion 209

Hidayet Hüda KÖSAL , Mahmut AKYİĞİT and Murat TOSUN ............................................ 209

On an Almost Contact Metric Manifold with a Type of Semi-symmetric Non-metric

Connection 210

Hulya Bagdatli YILMAZ ......................................................................................................... 210

On Invariants of a System of Vectors in Minkowski Spacetime 212

İdris ÖREN ............................................................................................................................. 212

Determining by Control Points of Null Bézier Curves with Degree 2 and 3 Curves in 3-

Dimensional Minkowski Spacetime 214

İdris ÖREN ............................................................................................................................. 214

The Elliptic Matrices Associated with Eliptic Biquaternions and De-Moivre’s Formula for

These Matrices 215

Murat Tosun and Kahraman Esen Özen ................................................................................ 215

New Results for General Helix in Euclidean 3-Space 216

Kazım İLARSLAN ................................................................................................................... 216

On Cardan Position for the Lorentzian Plane Motion of a Rigid Body 217

Kemal EREN, Soley ERSOY and Mahmut ERGÜT ................................................................ 217

Some Characterizations for Screen Isotropic Leaves on Lightlike Hypersurfaces 219

Mehmet GÜLBAHAR, Sadık KELEŞ and Erol KILIÇ ........................................................... 219

Special Smarandache Curves with Respect to Darboux Frame in the Galilean 3-Space 220

Merve OKUR and Tevfik ŞAHİN ............................................................................................ 220

Elliptical Motion on a Given Ellipsoid 221

Mustafa ÖZDEMİR ................................................................................................................ 221

Ruled Surfaces in Contact Geometry 223

Nural YÜKSEL, Murat Kemal KARACAN, Hasibe İKİZ and Çağlar Zeki ODABAŞI .......... 223

A Note on Spherical Orthotomic of Lorentzian Spherical Spacelike Curves 224

Önder Gökmen YILDIZ .......................................................................................................... 224

Local Diffeomorphic Image of Spherical Antiorthotomic of Lorentzian Spherical Timelike

Curves 225

Önder Gökmen YILDIZ .......................................................................................................... 225

The Kinetic Energy Formula for the Closed Planar Homothetic Motions in Complex Plane 226

Önder ŞENER, Ayhan TUTAR and Serdar SOYLU ............................................................... 226

New Developments in Lightlike Submanifolds Theory 227

Sadık KELEŞ, Erol KILIÇ and Mehmet GÜLBAHAR ........................................................... 227

Holditch Type Theorem for Kinetic Energy of Projective Curve Under the 1-Parameter

Spatial Motion 228

Serdar SOYLU , Ayhan TUTAR and Önder ŞENER .............................................................. 228

xxi

On the Casual Characteristics of Higher Order Bertand Mate, Involute Curve, Mannheim of

a Non-Null Curve in IL³ 230

Şeyda KILIÇOĞLU and Süleyman ŞENYURT ....................................................................... 230

On the Second Order Involute of a Space-like Curve with Time-like Binormal in IL³ 232

Şeyda KILIÇOĞLU and Süleyman ŞENYURT ....................................................................... 232

On Biharmonic Hypersurfaces in Semi-Euclidean Spaces 233

Sibel SEVİNÇ, Gülşah AYDIN ŞEKERCİ and A. Ceylan ÇÖKEN ........................................ 233

Timelike and Spacelike Split Quaternion Ruled Surfaces in Minkowski 3-space 234

Kıvanç KARAKAŞ , Mesut ALTINOK and Levent KULA ...................................................... 234

Mathematics Education

Parameterized Examination in Econometrics 236

Anna MALINOVA, Vesselin KYURKCHIEV and Georgi SPASOV ....................................... 236

The Effect of Differentiated Instruction on Mathematical Attitudes of Students 238

Ayten Pinar BAL and Onur EKİNCİ ...................................................................................... 238

Investigation of the Test Anxieties of 8th Grade Students 239

Bülent ALTUNKAYA, Kamile ŞANLI KULA and Cahit AYTEKİN ........................................ 239

Examining Pre-service Teachers’ Perceptions and Evaluations about Student-Invented

Strategies 241

Büşra KARTAL and Yasemin KIYMAZ .................................................................................. 241

Investigation of Parent Expectations of Middle School Students from Mathematics Education* 242

Cahit AYTEKİN, Serdal BALTACI, Bülent ALTUNKAYA3, ................................................... 242

The Impact of the Probability Explorer Software on Preparing Probability Teaching Plans of

Elementary Mathematics Teacher Candidates* 244

Cahit AYTEKİN, Serdal BALTACI, Yasemin KIYMAZ3, Avni Yıldız4 .................................... 244

Proof Evaluation: Mathematicians and Prospective Mathematics Teachers’ Perspectives 246

Esra Selcen YAKICI-TOPBAŞ, Elçin EMRE-AKDOĞAN, Fatma Nur AKTAŞ, Gönül

YAZGAN-SAĞ and Ziya ARGÜN ........................................................................................... 246

New Method of Grading Exams based on Computational Statistics 248

Evgeny GERSHIKOV ............................................................................................................. 248

Difference Equations in Economics 250

Jelena STANOJEVIC and Katarina KUKIC .......................................................................... 250

Factors Affecting Scoring Open-Ended Questions 251

Kübra AKKAYA and Selahattin ARSLAN .............................................................................. 251

Development Process of In-Service Training Intended for Teachers to Perform Teaching of

Mathematics with Computer Algebra Systems 252

Mehmet Alper ARDIÇ and Tevfik İŞLEYEN .......................................................................... 252

xxii

Examination of the Activities about Enciphering Done with Fifth Grade Students and Their

Views about the Activities 253

Muhammet ŞAHAL and Ahmet Şükrü ÖZDEMİR ................................................................ 253

The Concept of Zero: A Comparison of Students with Learning Disability vs Non-Learning

Disability 254

N. Dilşad GÜVEN and Esra Selcen YAKICI TOPBAŞ and Ziya ARGÜN ............................. 254

Preservice Mathematics Teachers’ Mathematizing Skills in The Process of Solving a Real Life

Problem 256

Nuray Çalışkan DEDEOĞLU ................................................................................................ 256

Learning Implementations about Cooperative Learning Method: A Case Study in Türkiye 258

Perihan DINC ARTUT and Ayten Pınar BAL ........................................................................ 258

Investigation of Number Sense Strategies Used By the 8th Grade Students in Turkey 260

Perihan DINC ARTUT and Zübeyde ER ................................................................................ 260

Examining Teacher Candidates’ Thinking Skills 262

Perihan DİNÇ ARTUTand Ayten Pınar BAL ......................................................................... 262

Difficulties of High School Students in Complex Numbers 264

Selahattin ARSLAN ................................................................................................................ 264

Investigation of Mathematical Viewpoints of Primary School Teacher Program Students 265

Tuğba BARAN KAYA and Ahmet IŞIK ................................................................................... 265

A Framework Suggestion for the Analyze of the Solving Process of a Geometrical

Construction Problem 266

Tuğçe KOZAKLI ÜLGER, Işıl BOZKURT and Murat ALTUN ............................................. 266

Active Learning in Flipped College Algebra Class 268

Violeta VASILEVSKA ............................................................................................................. 268

Statistics

New Method for Obtaining of Weighted Distributions 271

Božidar V. POPOVİĆ ............................................................................................................. 271

Transmuted Exponential Power Distribution and its Distributional Properties 273

Buğra SARAÇOĞLU .............................................................................................................. 273

A New Bivariate Archimedean Copula Based on a Transformed Generator Function 274

Çiğdem TOPÇU GÜLÖKSÜZ, Nuri ÇELİK .......................................................................... 274

The Comparison of Population Means under Unequal Variances 275

Neriman AKDAM, M. Fedai KAYA2 and Buğra SARAÇOĞLU3 ........................................... 275

A New Lifetime Distribution 276

Nuri CELİK and Cigdem TOPCU GULOKSUZ .................................................................... 276

Topology

xxiii

A Topological Study of Harmony and Counterpoint in Music Using Quotient Orbifolds 278

Aditya SIVAKUMAR and Dmitri TYMOCZKO ...................................................................... 278

New Functors from Fuzzy Normed Spaces Category 279

Deniz Pınar SUNAOĞLU and Erdal GÜNER ...................................................................... 279

Soft Metric Spaces 280

Ebru Aydoğdu, Abdülkadir Aygünoğlu and Halis Aygün ...................................................... 280

A New Property between Compactness and Completeness in Generalized Metric Spaces 281

Emrah Evren KARA and Merve İLKHAN .............................................................................. 281

Extensions of T_0-quasi-metrics 282

Filiz YILDIZ and Hans-Peter KÜNZİ .................................................................................... 282

Some Categorical Approaches to the Relations between Inverse Systems of Bitopological and

Ditopological Spaces 283

Filiz YILDIZ ........................................................................................................................... 283

Normal Product Adjacency for Simplicial Homology Groups of Digital Images 284

Gülseli BURAK ...................................................................................................................... 284

Bracket Polynomials of Torus Links as Fibonacci Polynomials 285

Kemal TAŞKÖPRÜ, İsmet ALTINTAŞ and Merve BEYAZTAŞ.............................................. 285

Controllability of Affine Control Systems on Solvable Lie Groups 286

Memet KULE .......................................................................................................................... 286

Some Effective Results Related to the New Statistical Cauchy Sequence 288

Merve İLKHAN and Emrah Evren KARA .............................................................................. 288

Products of Hopfian Manifolds and Their Shape Fibrators' Properties 289

Violeta VASILEVSKA ............................................................................................................. 289

Soft AB-Sets and Soft αAB-Sets in Soft Topological Spaces 291

Zehra Güzel ERGÜL and Naime TOZLU .............................................................................. 291

The Other Areas

Approximation by Certain Linear Positive Operators of Three Variables 294

Afşin Kürşat GAZANFER ....................................................................................................... 294

Extension of Silver-Meal Algorithm with Variable Demand and Delivery Time to Cope with

Bullwhip Effect in Multi-Echelon Supply Chains 295

Halil İbrahim CEBECİ and Doğan ÜNAL ............................................................................. 295

Consumer and Producer Surpluses of the Quadratic Demand and Supply Functions by Using

Trapezoidal Fuzzy Numbers and Signed Distance 297

İsmail ÖZCAN and Salih AYTAR ........................................................................................... 297

On Creating Tessellations with Using Transformational Geometry 298

Kübra ÖZLÜ DEĞER and Ali Hikmet DEĞER ..................................................................... 298

xxiv

Relationship between Electricity Consumption and Economic Growth in some Developing

Countries: MS-VECM Analysis 299

Melike BİLDİRİCİ , Fazıl KAYIKÇI ...................................................................................... 299

Test of validity of Baltic Dry index with MS-ARMA and SET-ARMA models 300

Melike BİLDİRİCİ , Fazıl KAYIKÇI and Işıl Şahin ONAT .................................................... 300

Chaotic Examination of Turkish Financial Market 301

Melike E. BİLDİRİCİ , Fulya ÖZAKSOY and Bahri SONÜSTÜN ........................................ 301

Chaotic Structure of Oil Price 302

Melike E. BİLDİRİCİ , Fulya ÖZAKSOY .............................................................................. 302

Optimization of MANET Routing Tables with Plant Propagation Algorithm 303

Mevlüt ERSOY and Tuncay YİĞİT ......................................................................................... 303

The Integrability and the Zero-Hopf Bifurcation of the Three-Dimensional Lotka-Volterra

Systems 305

Rizgar H. SALIH .................................................................................................................... 305

Posters

Estimates of Initial Coefficients of a New Analytic and Bi-Univalent Function Class Defined

by Integral Operator Involving Polylogarithm Function 307

Arzu AKGÜL .......................................................................................................................... 307

Finding Initial Coefficients for a Class of Bi-Univalent Functions Given by q-Derivative

Operators 310

Arzu AKGÜL .......................................................................................................................... 310

Dual-Hyperbolic Fibonacci and Lucas Numbers 313

Arzu CİHAN, Ayşe Zeynep AZAK and Mehmet Ali GÜNGÖR .............................................. 313

Toeplitz Operators with Symbols in Some Function Spaces 315

Ayşe SANDIKÇI ..................................................................................................................... 315

A Further Generalization of Gamma, Beta and Hypergeometric Functions 316

Ayşegül ÇETINKAYA, İ. Onur KIYMAZ, Praveen AGARWAL and Shilpi JAIN ................... 316

Teachers’ Perceptions About Written Examination Preparation Process 318

Ayten Pınar BAL and Fatma SADIK ...................................................................................... 318

The Analysis of Mistakes and Solution Strategies Applıed in Algebraic Verbal Problems 319

Ayten Pınar BAL and Ahmet KARACAOĞLU ....................................................................... 319

Examination of Burnout and Life Satisfaction of Nurses 320

Emrah GÜRLEK and Kamile ŞANLI KULA and Mehmet YETİŞ and Aysu YETİŞ ............... 320

Ruled Surface Pair Generated by Darboux Vectors of a Curve and Its Natural Lift in 𝑰𝑹𝟏𝟑 322

Evren ERGÜN , Mustafa ÇALIŞKAN and Keziban ORBAY .................................................. 322

Caratheodory’s Theorem in B-1-convexity 323

Gabil ADILOV and Ilknur YESILCE ..................................................................................... 323

xxv

The Simulation of the Order Parameter Probability Distribution for the Four Dimensional

Ising Model on the Creutz Cellular Automaton with the Linear Dimensions L=24, 26 and 28 324

Ganimet MÜLAZIMOĞLU KIZILIRMAK ............................................................................. 324

Some Numerical Experiments for a Sharper Version of Hölder Inequality 325

Gültekin TINAZTEPE ............................................................................................................. 325

Conformable variational iteration method for solving the time-fractional Fornberg-Whitham

equation 326

Handan Çerdik YASLAN ........................................................................................................ 326

A Further Generalization of Fractional Operators 328

İ. Onur KIYMAZ, Ayşegül ÇETİNKAYA1, Praveen AGARWAL and Shilpi JAIN3 ................ 328

Higher-Order Multi-Point Fractional Boundary Value Problems 330

İsmail YASLAN ....................................................................................................................... 330

Elliptic Biquaternion Algebra 331

Kahraman Esen ÖZEN and Murat TOSUN ........................................................................... 331

Beliefs about Mathematical Problem Solving of University Students 332

Kamile ŞANLI KULA and Ezgi ÇAĞATAY İN ....................................................................... 332

Investigation of Structural Phase Transformation in Iridium Dioxide (IrO2) under High

Pressure 333

Köksal KIZILIRMAK, Ganimet M. KIZILIRMAK and Hülya ÖZTÜRK................................ 333

Helixes on Clifford Surfaces in a Hyperbolic Space of Positive Curvature 335

Lyudmila ROMAKINA ............................................................................................................ 335

A New Generalization of Whittaker Function and its Properties 336

M. Baki YAĞBASAN, Ayşegül ÇETİNKAYA1 and İ. Onur KIYMAZ1 .................................... 336

Classification of Homothetical Hypersurfaces and Its Applications to Production Functions in

Economics 338

Mahmut ERGUT and Muhittin Evren AYDIN ...................................................................... 338

Generation of Pseudo-Random Numbers from Given Probabilistic Distribution with the Use

of Chaotic Maps 339

Marcin LAWNIK .................................................................................................................... 339

A New Approach on Slant Curves in three Dimensional Lie Groups 340

Osman Zeki OKUYUCU, Caner DEĞIRMEN and İsmail GÖK ............................................ 340

Analysis of Antimicrobial Resistance among Clinical Isolates from Denizli 341

Selma KIRAÇ, Dilek KESKIN and Muradiye YARAR ............................................................ 341

An Approach to Volterra and Fredholm Integral Equations by Homotopy Perturbation

Method 343

Serpil ŞAHİN .......................................................................................................................... 343

Composition Formulae Associated Fractional Integral Operator with the Multi-Index Mittag-

Leffler Functions 344

Shilpi JAIN, Praveen AGARWAL, Ayşegül ÇETINKAYA and İ. Onur KIYMAZ ................... 344

xxvi

On a General Subclass of Univalent Functions Based on the q-Derivative Operator 346

Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................. 346

Estimates of Coefficients for a Subclass of Bi-Univalent Functions by Making Use of Faber

Polynomial Expansions 348


On a Subclass of Univalent Functions Defined by Chebyshev Polynomials 350


Coefficient Estimates for Analytic Bi-Bazilevi cFunctions of Order and Type 351


Singularities of the Darboux Ruled Surface of a Space Curve in the Pseudo-Galilean Space 353

Tevfik ŞAHİN .......................................................................................................................... 353

On the Hyperbolic Spinors and Split Quaternions 354

Tülay ERİŞİR and Mehmet Ali GÜNGÖR.............................................................................. 354

An Introduction to Fibonaci, Lucas and Generalized Fibonacci Commutative Quaternions 355

Ömer TETIK and Mahmut AKYIĞIT ..................................................................................... 355

6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)

0

INVITED

TALKS


1

Resolvability of Topological Spaces

István Juhász1

Abstract. A topological space X is called λ-resolvable, where λ is a (finite or

infinite) cardinal, if X contains λ many pairwise disjoint dense subsets. X is maximally

resolvable if it is X -resolvable, where

min : ,X G G open G

The expectation is that “nice” spaces should be maximally resolvable, as verified e.g.

by the well-known facts that both metric and linearly ordered spaces, as well as compact

Hausdorff spaces, are maximally resolvable. There is, however, a countable regular (hence

“nice”) space with no isolated points that is not even 2-resolvable.

In this talk we present resolvability results about spaces that are more general than

the above. On one hand, we consider the class of monotonically normal spaces that includes

both metric and linearly ordered spaces, on the other we consider spaces with properties that

are more general than compactness, namely Lindel¨ofness, countable compactness, and

pseudocompactness.

1 Alfred Renyi Institute of Mathematics, Hungary, [email protected]

http://www.renyi.hu/~juhasz/


2

Boundedness Properties of Function Spaces

Ljubiša D. R. KOČINAC1 and L’ubica HOLÁ2

Abstract. Let X be a Tychonoff space, (Y,d) a metric space, and C(X,Y) be the set of

continuous mappings from X to Y. There is a variety of natural uniformities and the

corresponding topologies on this set: the topology of pointwise convergence, the compact-

open topology, the topology of uniform convergence, the m-topology, the graph topology, and

others. We investigate several boundedness properties of these uniform spaces related to the

classical covering properties by Menger, Hurewicz, Rothberger and Gerlits-Nagy.

When Y is the real line R with the standard metric, then we write C(X) instead of

C(X,R). The set C(X) equipped with each of the mentioned topologies is a topological group.

In this case we consider boundedness properties of these Hausdorff topological groups with

the operation of pointwise addition.

Keywords. M-bounded, H-bounded, R-bounded, GN-bounded.

AMS 2010. 46E10, 54C35, 54D20, 54E15, 54H11.

References

[1] Babinkostova, L., Kočinac, Lj.D.R., Scheepers, M., Combinatorics of open covers (XI):

Menger- and Rothberger-bounded groups, Topology Appl., 154, 1269--1280, 2007.

[2] Di Maio, G., Holá, L'., Holy, D., McCoy, R.A., Topologies on the space of continuous

functions, Topology Appl., 86, 105-122, 1998.

[3] Holá, L'., Zsilinszky, L., Completeness properties of the graph topology, Topology Proc.,

46, 1-14, 2015.

[4] Kočinac, Lj.D.R., Selection principles in uniform spaces, Note Mat., 22, 127-139,

2003/2004.

1University of Niš, Niš, Serbia, [email protected] 2Slovak Academy of Sciences, Bratislava, Slovakia, [email protected]


3

Finite Differences Method of the Riemann Type Problem for 2D Conservative Laws in a

Class of Discontinuous Functions

Mahir RASULOV1

Abstract. In this study a new method for finding the numerical solution of the Cauchy

problem with a discontinuous initial profile for the nonlinear 2D dimensional scalar

conservation laws is suggested. Firstly, some properties of the weak solution of the linearized

equation are investigated. Taking these properties into consideration, we introduce an

auxiliary problem since it has some advantages over the main problem, and it is equivalent to

the main problem in a definite sense. The proposed auxiliary problems allows us to develop

effective finite differences schemes for finding the solution with higher accuracy, such that

the obtained solution expresses all of the physical properties of the problem under

investigation. Using of the solution of the introduced auxiliary problem the method for

obtaining the location of shock appearing in the solution of main problem and it is evolution

is investigated. Moreover, using the auxiliary problem we can write the higher order

numerical scheme with respect to the time variable.

Keywords. Scalar conservation laws, auxiliary problem, exact and numerical solution

in a class of discontinuous functions

AMS 2010. 53A40, 20M15.

1 Beykent University,, Istanbul, Turkey, [email protected]


4

On Some Applications of Measures of Noncompactness

Mohammad MURSALEEN1

Abstract. In this talk, we present a brief survey of theory and applications of measures

of noncompactness. The classical measures of noncompactness are discussed and their

properties are compared. The approaches for constructing measure of noncompactness in a

general metric or linear space are described, along with the classical results for existence of

fixed point for condensing operators. Also several generalization of classical results are

mentioned and their applications in various problems of analysis such as linear equation,

differential equations, integral equations and common solutions of equations are discussed.

The most effective way in the characterization of compact operators between the

Banach spaces is applying the Hausdorff measure of noncompactness. In this chapter, we

present some identities or estimates for the operator norms and the Hausdorff measures of

noncompactness of certain operators given by infinite matrices that map an arbitrary BK-

space into the sequence space 0 , ,c c l and 1.l Many linear compact operators may be

represented as matrix operators in sequence spaces or integral operators in function spaces

[1].

We apply the technique of measures of noncompactness to the theory of infinite

systems of differential equations in some Banach sequence space 0 , , pc c l (1 p ).

Infinite systems of ordinary differential equations describe numerous world real problems

which can be encountered in the theory of branching processes, the theory of neural nets, the

theory of dissociation of polymers and so on. Let us also mention that several problems

investigated in mechanics lead to infinite systems of differential equations. Moreover, infinite

systems of differential equations can be also used in solving some problems for parabolic dif-

differential equations investigated via semidiscretization. We adopt the technique of measures

of noncompactness to the theory of infinite systems of differential equations. Particularly, we

are going to present a few existence results for infinite systems of differential equations

formulated with the help of convenient and handy conditions. We study of the solvability of

the infinite systems of differential equations in some classical Banach sequence spaces.

References

1 Aligarh Muslim University, India, [email protected]


5

[1] Banas, J. and Mursaleen, M., Sequence Spaces and Measures of Noncompactness with

Applications to Differential and Integral Equations, Springer, 2014.


6

Mathematical Literacy and a New Classification Proposal for Mathematical Literacy

Problems

Murat ALTUN1

Abstract. The improvement of mathematical literacy level remains one of the

fundamental problems of primary education. Mathematical literacy is defined as “an

individual’s capacity to identify and understand the role that mathematics plays in the world,

make well-founded judgements and use and engage with mathematics in ways that meet the

needs of that individual’s life as a constructive, concerned and reflective citizen. In a short

term, mathematical literacy is defined as “an individual’s capacity to formulate, employ and

interpret mathematics in a variety of contexts”. Assessments for identifying mathematical

literacy levels are conducted by allowing students to use their mathematical competencies

through addressing them as contextual, conceptual, and operational problems. Countries need

to know students’ difficulties at solving mathematical literacy problems to be able to take the

necessary precautions.

Four different types of mathematical literacy classification have been employed so far:

Subject areas, capabilities, competency clusters and process skills. However, there are some

problems in assessments, so, there is a need to make a new classification in order to inhibit the

uncertainty in assessments. In this study, a new classification has been identified and

suggested.

In this study, middle school students (8th grade) were asked mathematical literacy

questions. The data obtained from the student responses were subjected to factor analysis. A

six-factor structure was obtained at the end of the analysis. The obtained factors were found to

have adequate variance in explaining the students’ mathematical literacy. These factors were

named as making algorithmic operations, mastering rich mathematical content, mathematical

inference, developing mathematical proposals and interpreting a developed proposal,

understanding the mathematical equivalence of real world situations, and understanding the

counterpart of mathematical language in life. The data analysis revealed that the students

failed in the factors of mathematical inference, developing mathematical proposals and

interpreting a developed proposal, and understanding the mathematical equivalence of real

world situations. The results of this study can help and guide future researches for analyzing

1 Uludag University, Bursa, Turkey, : [email protected]


7

mathematical literacy achievement, developing a mathematical literacy scale, developing

mathematics curricula and organizing its instruction.


8

Fixed point property for some Alexandroff topological spaces with T0-separation axiom

and its applications

Sang-Eon Han1

Abstract. In this talk we study the fixed point property (FPP for short) and the almost

fixed point property (AFPP for brevity) for non-Hausdorff topological spaces (or digital

spaces). Besides, we refer to some applicatoions of this approach. More precisely, we focus

on the following approaches.

(1) FPP for digital images motivated by the Rosenfeld model [1]

(2) AFPP for digital images motivated by the Rosenfeld model [1]

(3) FPP for Khalimsky topological spaces [3], [5]

(4) FPP for Marcus-Wyse topological spaces [2], [4]

(5) FPP for Alexandroff spaces with T0-separation axiom

(6) Order theoretical approach

Keywords. Fixed point property, Khalimsky topological space, Alexandroff space.

AMS 2010. 54A10, 68U10.

References

[1] Han, S.-E., Banach fixed point theorem from the viewpoint of digital topology, Journal of

Nonlinear Sciences and Applications 9(3) 895-905, 2016.

[2] Han, S.-E., Almost fixed point property for digital spaces associated with Marcus-Wyse

topological spaces, Journal of Nonlinear Sciences and Applications 10, 34-47, 2017.

[3] Han, S.-E., Fixed point property for digital spaces, Journal of Nonlinear Sciences and

Applications 10, 2510-2523, 2017.

[4] Han, S.-E., The FPP of an M-retract and its applications, Topology and its Applications,

Online First press, 2017.

[5] Han, S.-E., The fixed point property of the smallest open neighborhood of the Khalimsky

topological plane, Filomat, Online First press, 2017.

1 Chonbuk National University, Jeonju, Korea, [email protected]


9

The Role of Convergence Theory in Mathematics

Szymon Dolecki 1

Abstract. A convergence on a set X is a relation between filers F on X and

elements x of X

limx F

such that F G implies lim lim ,F G and lim 2 :Xx F x F for each .x X

Every topology can be seen as a convergence given by

lim ,x F N x F

where N x is the filter of all neighborhoods of x with respect to .

Actually, the study of topologies is equivalent to the study of certain families of filters; on one

hand, each topology is determined by its neighborhoods, on the other, each filter H on X

gives rise either to a topology on ,X or to a topology on X where X so that H

is the neighborhood filter either of some elements of X or of .

The emergence of convergence theory was triggered by a realization that some natural and

important phenomena correspond to non-topological convergences, for instance, the

restrictions of topologies to sequential filters, some convergences occurring in measure

theory, the natural convergence on the space of continuous maps, and the dual convergences

related to the Arens-Mackey duality in functional analysis.

In [1] Choquet discovers that certain hyperspaces are non-topological, but are always

pseudotopological. The introduction of pseudotopologies2, an exponential category including

the (non-exponential) category of topologies, may be considered as an actual starting point of

convergence theory. Pseudotopologies are a completion of topologies in a similar way as

complex numbers of real numbers.

These new perspectives enable one to view many topological properties and classes of spaces

and maps as solutions of functorial inequalities in the category of convergences. This is the

case of sundry variants of quotient maps; they turn out to be the quotients with respect to

certain subcategories of convergences.

1 Mathematical Institute of Burgundy, Dijon, France, [email protected] 2 With continuous maps as morphisms.


10

Variants of compactness appear as a single property with respect to various subcategories of

convergences.

Compactness and completeness turn out to be degrees of a single unifying quality.

One can see ([2]) that the convergence-theoretic framework is simpler, more natural, far-

reaching and powerful than that of topologies.

References

[1] Choquet G., Convergences, Ann. Univ. Grenoble, 23, 55-112, 1947-48.

[2] Dolecki, S., and Mynard, F., Convergence foundations of topology, Word Scientific, 2016.


11

From Dual Bodies to the Kneser-Poulsen Conjecture

Karoly Bezdek 1

Abstract. The Kneser–Poulsen Conjecture (1955) states that if the centers of a family

of N congruent balls in Euclidean d-space are contracted, then the volume of the intersection

does not decrease. The first half of my talk will give a summary of the status of this long-

standing conjecture in geometry. The second half will discuss the following latest

development. A uniform contraction is a contraction where all the pairwise distances in the

first set of centers are larger than all the pairwise distances in the second set of centers. I will

present a proof of the Kneser- Poulsen conjecture for uniform contractions whenever N is

sufficiently large (depending only on d) in Euclidean, spherical as well as hyperbolic d-space

for all 1d . The method of proof is centered around a Blaschke-Santalo type inequality for

dual bodies.

1 University of Calgary, Canada, [email protected]


12

ALGEBRA


13

Some Results for Matrix Representation of Fibonacci Octonions

Adnan KARATAŞ1 and Serpil HALICI2

Abstract. In this study, we investigate non-standard matrix representation of octonion

algebra. This representation allows us to calculate octonion multiplication without knowing

the octonion multiplication table. Also, we use mentioned matrix representation for Fibonacci

octonions and we obtain two different Cassini identities related with the matrix representation.

Keywords. Matrix representation, Fibonacci numbers, octonions.

AMS 2010. 11B39, 17A20

References

[1] Smith, Jonathan D. H. An introduction to quasigroups and their representations, CRC

Press, 2006.

[2] Lounesto, P., Clifford algebras and spinors, Clifford Algebras and Their Applications in

Mathematical Physics. Springer Netherlands, 25-37, 1986.

[3] Koshy, T., Fibonacci and lucas numbers with applications, A Wiley-Interscience

Publication, 2001.

[4] Keçilioğlu, O., and Akkus, I., The fibonacci octonions, Advances in Applied Clifford

Algebras 25.1, 151-158, 2015.

[5] Halici, S., and Karataş, A., Some matrix representations of fibonacci quaternions and

octonions, Advances in Applied Clifford Algebras: 1-10.

[6] Günaydin, M., and Gürsey F., Quark structure and octonions, Journal of Mathematical

Physics 14.11, 1651-1667, 1973.

[7] Gürsey, F., and Chia-Hsiung T., On the role of division, Jordan and related algebras in

particle physics, World Scientific, 1996.

[8] Baez, J., The octonions, Bull. of the American Mathematical Society 39.2, 145-205, 2002.

1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Pamukkale University, Denizli, TURKEY, [email protected]


14

On Special Elements of Split Octonions

Adnan KARATAŞ1 and Serpil HALICI2

Abstract. In this study, we focus on split octonion algebras over finite fields and

investigate non trivial idempotent and nilpotent elements. We present formulas which gives

idempotent and nilpotent elements according to chosen finite fields. Also, we obtain some

fundamental properties including them. Furthermore, we show that for some subalgebras of

octonions our formulas compatible with earlier studies in this subject.

Keywords. Idempotent, nilpotent, octonions.

AMS 2010. 16N40, 17A20.

References

[1] Aristidou, M.; Demetre, A., idempotent elements in quaternion rings over Zp,

International Journal of Algebra, 6.5-8, 249-254, 2012.

[2] Aristidou, M.; Demetre, A. A note on nilpotent elements in quaternion rings over Zp,

International Journal of Algebra, 6.14, 663-666, 2012.

[3] Baez, J. The Octonions, Bulletin of the American Math. Soc., 39.2, 145-205, 2002.

[4] Schafer, R. D., An introduction to nonassociative algebras, Vol. 22, Courier Corporation,

1966.

[5] Ward, J. P., Quaternions and cayley numbers, Algebra and Applications. Vol. 403,

Springer Science and Business Media, 2012.

[6] Miguel, C. J.; Serodio, R., On the structure of quaternion rings over Zp, International

Journal of Algebra, 5.27, 1313-1325, 2011.

[7] Conway, J. H.; Smith, D. A., On quaternions and octonions: their geometry, arithmetic,

and symmetry, Bulletin of the American Mathematical Society, 42, 2, 229-243, 2003.



15

[8] Okubo, S., Introduction to octonion and other non-associative algebras in physics, Vol.2,

Cambridge University Press, 1995.

[9] Smith, J. Dh., An Introduction to quasigroups and their representations, CRC Press, 2006.

[10] Dickson, L. E., On quaternions and their generalization and the history of the eight

square theorem, Annals of Mathematics, 155-171,1919..


16

Numerical solutions of mKdV equation via modified cubic B-spline based Differential

Quadrature Method

Ali BAŞHAN1

Abstract. In this study, an effective approach is used to obtain numerical solutions of

the third order non-linear modified Korteweg-de Vries (mKdV) equation by using differential

quadrature method based on modified cubic B-spline functions. Single soliton, interaction of

double solitons, evolution of solitons, and Train of the solitons test problems have been

studied. To check the accuracy of the method, error norms L2, and L∞ and the three lowest

invariants I1, I2 and I3 are calculated, and compared with some earlier works. Stability analysis

is also investigated.

Keywords. Differential quadrature method, mKdV equation, Strong stability-

preserving Runge-Kutta method.

AMS 2010. 65D07, 65M99, 65L06, 65L07.

References

[1] Ablowitz, M. J. and Clarkson, P. A., Solitons, nonlinear evolution equations and inverse

Scattering, Cambridge University Press, Cambridge, 1991.

[2] Ablowitz, M. J. and Segur, H., Solitons and inverse acattering transform, SIAM,

Philadelphia, 1981.

[3] Başhan, A., Karakoç, S. B. G., Geyikli, T., Approximation of the KdVB equation by the

quintic B-spline differential quadrature method, Kuwait Journal of Science, 42(2),67-92,

2015.

[4] Başhan, A., Uçar, Y., Yağmurlu, N. M., Esen, A. Numerical solution of the complex

modified korteweg-de vries equation by DQM, Journal of Physics: Conference Series 766

(2016) 012028 doi:10.1088/1742-6596/766/1/012028.

1 Bulent Ecevit University, Zonguldak, TURKEY, [email protected]


17

New Bounds for the Levels and Sublevels of Algebras Obtained by the Cayley-Dickson

Process

Cristina FLAUT1

Abstract. In this talk, we present some new bounds for the level and sublevel of

algebras obtained by the Cayley-Dickson process when their level and sublevel are greater

than dimension of the algebras.

References

[1] Brown, R. B., On generalized Cayley-Dickson algebras, Pacific J. of Math., 20,3, 415-

422, 1967.

[2] Dai, Z. D., Lam, T. Y., Peng, C. K., Levels in algebra and topology, Bull. Amer. Math.

Soc., 3, 845-848, 1980.

[3] Elman, R., Lam, T. Y., Pfister forms and K-theory of fields, Journal of Algebra 23, 181–

213, 1972.

[4] Flaut, C., Isotropy of some quadratic forms and its applications on levels and sublevels of

algebras, J. Math. Sci. Adv. Appl., 12, 2, 97-117, 2011.

[5] Flaut, C., Levels and sublevels of algebras obtained by the Cayley–Dickson process, Ann.

Mat. Pura Appl., 192, 6, 1099-1114, 2013.

[6] Hoffman, D. W., Isotropy of quadratic forms over the function field of a quadric, Math. Z,

220, 3, 461-476, 1995.

[7] Hoffman, D. W., Levels of quaternion algebras, Archiv der Mathematik, 90, 5, 401-411,

2008.

[8] Karpenko, N.A., Merkurjev, A.S., Essential dimension of quadratics, Inventiones

Mathematicae, 153, 361-372, 2003.

1 Ovidius University, Constanta, ROMANIA, [email protected] ; [email protected],

http://cristinaflaut.wikispaces.com/

mailto:[email protected]


http://cristinaflaut.wikispaces.com/


18

[9] Knebusch, M., Generic splitting of quadratic forms I, Proc. London Math. Soc. 33, 65-93,

1976.

[9] Laghribi A., Mammone P., On the level of a quaternion algebra, Comm. Algebra, 29, 4,

1821-1828, 2001.

[10] Lewis, D. W., Levels and sublevels of division algebras, Proc. Roy. Irish Acad. Sect. A,

87, 1, 103-106, 1987.

[11] Lewis, D. W., Levels of quaternion algebras, Rocky Mountain J, Math. 19, 787-792,

1989.

[12] O’ Shea, J., New values for the levels and sublevels of composition algebras, preprint.

[13] O’ Shea, J., Levels and sublevels of composition algebras, Indag. Mathem., 18, 1, 147-

159, 2007.

[14] O’ Shea, J., Bounds on the levels of composition algebras, Mathematical Proceedings of

the Royal Irish Academy 110A, 1, 21-30, 2010.

[15] O’ Shea, J., Sums of squares in certain quaternion and octonion algebras, C.R. Acad.

Sci. Paris Sér. I Math, 349, 239-242, 2011.

[16] Pfister, A., Zur Darstellung von-I als Summe von quadraten in einem Korper, J. London

Math. Soc. 40, 159-165, 1965.

[17] Pumplün, S., Sums of squares in octonion algebras, Proc. Amer. Math. Soc., 133, 3143-

3152, 2005.

[18] Schafer, R. D., An Introduction to Nonassociative Algebras, Academic Press, New-York,

1966.

[19] Schafer, R. D., On the algebras formed by the Cayley-Dickson process, Amer. J. Math.,

76, 435-446, 1954.

[20] Scharlau,W., Quadratic and Hermitian Forms, Springer Verlag, 1985.

[21] Tignol, J.-P., Vast, N., Representation de -1 comme somme de carré dans certain

algèbres de quaternions, C.R. Acad. Sci. Paris Sér. Math. 305, 13, 583-586, 1987.


19

Some Results on Generalized Derivations and (,) Lie Ideals

Evrim GÜVEN1

Abstract. Let R be a prime ring with characteristic not 2 and , , , , , µ,

automorphisms of R. Let h:RR be a nonzero left-generalized (, )-derivation, a,bR and

V≠0 a left (, )-Lie ideal of R. The main object in this article is to study the situations.

(1) h(I)C,µ(V), (2) ah(I)C,µ(V), (or h(I)aC,µ(V)), (3) h(V)=0, (4) h(V)a=0 or

ah(V)=0.

Keywords. Prime rings, generalized derivation, Lie Ideal.

AMS 2010. 16N60, 16W25.

References

[1] Aydın, N., Kaya, K., Some generalizations in prime rings with (𝜎,𝜏)-derivation, Doga-Tr.

J. of Math. 16, 1992.

[2] Bresar, M. On the distance of the compositions of two derivations to the generalized

derivations, Glasgow Math. J. 33, 80-93, 1991.

[3] Chang, J. C., On the identity h(x)=af(x)+g(x)b, Taiwanese J. Math, 7(1), 103-113, 2003.

[4] Güven, E., Kaya, K. and Soytürk, M., Some results on (𝜎,𝜏)-lie ideals, Math. J. Okayama

Univ. 49, 2007.

[5] Güven, E. One sided (𝜎,𝜏)-lie Ideals and generalized derivations in prime rings,5th

International Eurasian Conference on Mathematical Sciences and Applications (IECMSA),

Belgrade SERBIA, 2016.

[6] Kaya, K and Aydin, N, Some results on generalized lie Ideals, A Scienti.c Journal Issued

by Jordan University for Women, Vol. 3, No. 1, 1999.

[7] Kaya, K., (𝜎, 𝜏)-Lie ideals in prime rings. An Univ Timisoara Ser Stiinte Math. 30, 1992.

[8] Mayne, J. H., Centralizing mappings of prime rings, Canad. Math. Bull, 27, 122-126,

1984.

[9] Park, K.-H. and Jung, Y.-S., Some result concerning (,)-derivations on prime rings, J.

Korea Soc. Math. Educ. Ser. B: Pure Appl. Math.,Volume 10, Number 4, 2003.

1 Kocaeli University, Kocaeli, TURKEY, [email protected]


20

A Note on the Sequence of Balancing Numbers

Gül Karadeniz GÖZERİ1

Abstract. In this work, we consider balancing numbers which are the square roots of

the square triangular numbers. The definition of the sequence of balancing numbers, Binet

formula for this sequence and usefull techniques are presented. The purpose of this work is to

provide some new results on the sequence of balancing numbers.

Keywords. Balancing numbers, triangular numbers, recurrence relation.

AMS 2010. 11B37, 11B39, 11B83.

References

[1] Behera A., Panda G. K., On the square roots of triangular numbers, The Fibonacci

Quarterly, 37, 2, 98-105, 1999.

[2] Castillo R. C., A survey on triangular number, factorial and some associated numbers,

Indian Journal of Science and Technology, 9, 41, 1-7, 2016.

[3] Catarino P., Campos H., Vasco P., On some identities for balancing and cobalancing

numbers, Annales Mathematicae et Informaticae, 45, 11-24, 2015.

[4] Garge A. S., Shirali S.A., Triangular numbers, Resonance, 17,7, 672-81, 2012.

[5] Hoggatt V. E., Bicknell M., Triangular numbers, Fibonacci Quarterly, 12, 221-30, 1974.

[6] Olajos P., Properties of balancing, cobalancing and generalized balancing numbers,

Annales Mathematicae et Informaticae, 37, 125-138, 2010.

[7] Panda G.K., Some fascinating properties of balancing numbers, Proceedings of the

Eleventh International Conference on Fibonacci Numbers and their Applications, Congr.

Numer. 194 , 185-189, 2009.

1Istanbul University, Istanbul, TURKEY, [email protected]


21

Graph Operations of Randic Index

Gülistan Kaya GÖK1 and Şerife BÜYÜKKÖSE2

Abstract. The Randic index 𝑅(𝐺) of a graph 𝐺 whose vertex 𝑖 and 𝑗 has degree 𝑑𝑖 and

𝑑𝑗 respectively is defined by sum of the 1

√𝑑𝑖𝑑𝑗 if the vertices i and j are adjacent. We introduce

the weighted Randic index, we establish basic mathematical properties of Randic index and

we investigate its behavior under some graph operations in this paper.

Keywords. Randic index, Weighted graph, Graph operations.

AMS 2010. 05C22, 05C50.

References

[1] Alizadeh, Y., Iranmanesh, A., Doslic, T., Additively weighted harary index of some

composite graphs, Discrete Mathematics, 313, 26-34, 2013.

[2] Gutman, I., Furtula, B.(Eds.), Recent results in the theory of randic index, Univ.,

Kragujevac, 2008.

[3] Gutman, I., Furtula, B., Bozkurt, S., B., On the randic energy, Linear Algebra and its

Applications, 442, 50-57, 2014.

[4] Liu, H., Lu, M., Tian, F., On the randic index, Journal of Mathematical Chemistry, 38.3,

2005.

[5] Randic, M., On history of the randic index and emerging hostility toward chemical graph

theory, MATCH Commun. Math. Comput. Chem., 59, 5-124, 2008.

[6] Vasudev, C., Graph theory with applications, (1), New Delhi/ Indian: New Age

International Publishers, 4-5, 21, 56-57, 2006.

1 Hakkari University, Hakkari, TURKEY, [email protected] 2 Gazi University, Ankara, TURKEY, [email protected]


22

On Codes Written by Matrices Lexicographically Ordered

Mustafa ÖZKAN1 and Figen ÖKE2

Abstract. In this study, the paper obtained by authors M. Özkan and F. Öke [1] is

extended the ring 23

[ ]u

u . Certain matrices lexicographically ordered are written using

the elements of 23

[ ]u

u . The relations between the codes generate by these matrices and

Hadamard codes are described.

Key Words. Lexicographically ordered, Hadamard Codes , Codes over Ring.

AMS 2010.94B05 , 94B15.

References

[1] Özkan M. and Öke F., A relation between Hadamard codes and some special codes over

2 2u , App Mathematics and Inf. Sci. vol. 10 no 2, pp 701-704,2016.

[2] Özkan M. and Öke F. Codes defined via especial matrices over the ring and Hadamard

codes, Mathematical Sciences and Applications E-Notes, Volume 5, No :1, 93-98,2017.

[3] Krotov, D. S., 4-linear perfect codes, Diskretn. Anal. Issled. Oper. Ser.1.Vol. 7, 4,78–

90, 2000.

[4] Özkan M. and Öke F. Construction of Hadamard codes with rings, International

Workshop on Mathematical Methods in Engineering 2017 , Ankara-TURKEY 27- 29.04.2017

[5] Vermani, L. R., Elements of Algebraic Coding Theory, Chapman Hall , India., 1996.

[6] Udomkavanich P., Jitman, S., On the gray image of ( mu1 )-cyclic codes

...k k k

m

p p pu u , Int. J. Contemp. Math. Sciences, Vol.26, 4, 1265-1272, 2009.

1 Trakya University Edirne, TURKEY, [email protected] 2 Trakya University, Edirne, TURKEY, [email protected]




23

About Quasi-Cyclic Codes over the Field

Mustafa ÖZKAN1 and Figen ÖKE2

Abstract. In this paper quasi-cyclic codes over the field p are studied and they are classified

according to their lengths for each prime number p , using the theories obtained before us for 2p

and 3p .

Key Words. Gray Map, Quasi-cyclic Codes , Finite Field.

AMS 2010. 94B15 , 94B60.

References

[1] Özkan M. and Öke F., Some Special Codes Over 2

3 3 3 3v u u , Mathematical Sciences

and Applications E-Notes. vol. 4 no 1, pp 40-44, 2016.

[2] Özkan M. and Öke F., Gray images of (1 + v)-constacyclic codes over a particular ring, Palestine

Journal of Mathematics, Vol. 6, pp : 241-245.,2017

[3] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error Correcting Codes, (North- Holland

Publishing Company, 1977)

[4] S.Karadeniz, B.Yıldız, On (1+ v) -constacyclic codes 2 2 2 2u v uv , Journal of the

Franklin Institude , 348, 2625-2632, 2011

[5] S. Roman, Coding and Information Theory, (Springer Verlag, 1992)

1 Trakya University, Edirne, TURKEY, [email protected] 2 Trakya University, Edirne, TURKEY, [email protected]




24

On Certain Type of Band Matrices with the Generalized Fibonacci Numbers

Neşe ÖMÜR1 and Cemile Duygu ÇOLAK2

Abstract. In this study, considering the technique used in [2], we show their arbitrary

powers for band matrices, whose entries are expressed in the terms of generalized Fibonacci

numbers tnU and tnV , where 0n , 0t .

Keywords. The generalized Fibonacci numbers, band matrices.

AMS 2010. 11B39, 15A15, 15A36.

References

[1] Kılıç, E., Stanica, P., Factorizations and representations of second order linear

recurrences with indices in arithmetic progressions, Bulletin of the Mexican Mathematical

Society, 15, 1, 23-36, 2009.

[2] Hirasaka, M., Komatsu, T., On computation of arbitrary integer Powers for certain type of

band matrices with Fibonacci numbers, Journal of Combinatorics and Number Theory, 7, 1,

65-78, 2015.

[3] Rimas, J., On computing of arbitrary positive integer Powers for one type of symmetric

tridiagonal matrices of even order-I, Applied Mathematics and Computation, 168, 783-787,

2005.

1 Kocaeli University, Kocaeli, TURKEY, [email protected] 2 Kocaeli University, Kocaeli, TURKEY, [email protected]


25

On Generalization of Quaternions

Serpil HALICI1 and Adnan KARATAŞ 2

Abstract. In this study, generalized Horadam quaternions are defined. These

quaternions generalize the well-known quaternions such as Fibonacci, Lucas, Pell, Jacobsthal,

Pell-Lucas, Jacobsthal-Lucas quaternions. For these new quaternions, we obtain the Binet

formulas that generalize the Binet formulas of all special quaternions.

Keywords: Quaternions, Horadam Sequences, Generalized Fibonacci Sequence.

AMS: 11B37, 11B39, 11B83, 16G37.

References

[1] Catarino, P., A note on h (x)− fibonacci quaternion polynomials, Chaos, Solitons&

Fractals 77, 1-5, 2015.

[2] Çimen, C. Bolat and Ipek A., On Pell quaternions and Pell-Lucas quaternions, Advances

in Applied Clifford Algebras 26.1, 39-51, 2016.

[3] Flaut, C., and Savin, D., Quaternion algebras and generalized fibonacci–lucas

quaternions, Advances in Applied Clifford Algebras 25.4, 853-862, 2015.

[4] Halici, S., On fibonacci quaternions, Advances in Applied Clifford Algebras 22.2, 321-

327, 2012.

[5] Haukkanen, Pentti., A note on Horadam's sequence, Fib. Quart.. 40.4, 358-361, 2002.

[6] Horadam, A. F., Complex fibonacci numbers and fibonacci quaternions, The American

Mathematical Monthly 70.3, 289-291, 1963.

[7] Horadam, A. F., A generalized fibonacci sequence, The American Math. Mont. 68.5, 455-

459, 1961.

[8] Horadam, A. F., Complex fibonacci numbers and fibonacci quaternions, The American

Mathematical Monthly 70.3, 289-291, 1963.



26

[9] Horadam, A. F., Special properties of the sequence Wn (a, b; p,q), Fib. Quart.5(5), 424-

434, 1967.

[10] Horadam, A. F., Quaternion recurrence relation, Ulam Quarterly 2.2, 23-33, 1993.

[11] Horzum, T. and E. G. Kocer., On some properties of Horadam polynomials, Int. Math.

Forum. Vol. 4. No. 25. 2009.

[12] Larcombe, P., O. Bagdasar, and E. Fennessey., Horadam sequences: a survey, Bulletin

of the ICA 67, 49-72, 2013.

[13] Lounesto, P., Clifford algebras and spinors, vol. 286. Cambridge university press,

Cambridge, 2001.

[14] Polatli, E., and Seyhun K., On quaternions with generalized fibonacci and lucas number

components, Advances in Difference Equations 1, 2015

[15] Swamy, M. N. S., On generalized fibonacci quaternions, Fibonacci Quart.11(5), 547-

549, 1973.

[16] Szynal-Liana, Anetta, and Iwona Włoch., The pell quaternions and the pell octonions,

Advances in Applied Clifford Algebras 26.1, 435-440, 2016.

[17] Szynal-Liana, A., and Iwona W., A note on jacobsthal quaternions, Advances in Applied

Clifford Algebras 26, 441-447, 2016.


27

Generalized Quaternions and Dual Fibonacci Octonions

Serpil HALICI1 and Adnan KARATAŞ2

Abstract. In this study, we investigate the dual Fibonacci quaternions and then we

define the dual Fibonacci octonions. We give some algebraic properties of these octonions.

Moreover, we give the Binet formulas and the generating functions of them.

Key words: Recurrence Relations, Quaternions, Octonions .

2000 Mathematics Subject Classification. 11B37, 11B39, 20G20.

References

[1] Horadam, A. F., Complex fibonacci numbers and fibonacci quaternions, Amer.Math.

Monthly, 70, 289,291, 1963.

[2] Horadam, A. F., Quaternion recurrence relations, Ulam Quat. 2, 23-33, 1993.

[3] Iakin, A. L., Generalized Quaternions of higher order, The Fib. Quarterly,15, 343-346,

1977.

[4] Iakin, A. L., Generalized quaternions with quaternion components, The Fib.Quarterly, 15,

350-352, 1977.

[5] Iyer, M. R., A note on fibonacci quaternions, The Fib. Quarterly, 3,225-229, 1969.

[6] Swamy, M. N. S., On generalized fibonacci quaternions, The Fib. Quarterly,5, 547-550,

1973.

[7] Koshy, T., Fibonacci and lucas numbers with applications, A Wiley-Interscience

publication, U.S.A, 2001.

[8] Sangwine, S. J., Ell, T. A. and Bihan, N. L., Fundamental represantations and algebraic

properties of biquaternions or complexified quaternions, Adv. Appl. Clifford Algebras, 21,

607-636, 2011.



28

[9] Ward, J. P., Quaternions and cayley numbers: algebra and applications, volume 403 of

Mathematics and Its Applications. Kluwer, Dordrecht, 1997.

[10] Kuipers, J. B., Quaternions and rotation sequences, Princeton University, Press,

Princeton, New Jersey, 1999.

[11] Kantor, I. L. and Solodnikov, A. S., Hypercomplex numbers, an elementary introduction

to algebras, Springer-Verlag, New York, 1989.

[12] Halıcı, S., On complex fibonacci quaternions, Advances in Applied Clifford Algebras,

105-112 pp., 2013.


29

Formulæ for Two Weighted Binomial Identities with the Falling Factorials

Sibel Koparal1, Neşe Ömür2 and Emrah Kılıç3

Abstract. In this study, we will give closed formulæ for weighted and alternating weighted

binomial sums with the generalized Fibonacci and Lucas numbers including both falling

factorials and powers of indices. Furthermore we will derive closed formulæ for weighted

binomial sums including odd powers of the generalized Fibonacci and Lucas numbers.

Keywords. Binary linear recurrences, binomial sums, closed formula, operator.

AMS 2010. 05A19, 11B37, 11B39.

References

[1] Khan, M. and Kwong, H., Some binomial identities associated with the generalized

natural number sequence, The Fibonacci Quarterly, 49, 1, 57-65, 2011.

[2] Kılıç, E., Ulutaş, Y.T. and Ömür, N., Formulas for weighted binomial sums with the

powers of terms of binary recurrences, Miskolc Math. Notes, 13, 1, 53-65, 2012.

[3] Kılıç, E., Ömür, N. and Ulutaş, Y.T., Alternating sums of the powers of Fibonacci and

Lucas numbers, Miskolc Math. Notes, 12, 1, 87-103, 2011.

1 Kocaeli University, Kocaeli, TURKEY, [email protected] 2 Kocaeli University, Kocaeli, TURKEY, [email protected] 3 TOBB University, Ankara, TURKEY,[email protected]


30

S3–graded iso(1,2)

Yasemen UÇAN1 , Reşat KÖŞKER2 and Ramazan TEKERCİOĞLU3

Abstract. Lie algebras and groups have an important role in physics, engineering and

applied mathematics. For example, a great majority of symmetries are expressed by Poincare

symmetries [1]. Inspired by this, we intend to construct Fractional supersymmetric (1,2)iso

algebra in this study. There are other approaches for Fractional Supersymmetry in the

literature [2, 3, 4]. However, in our study, we obtained 3S –graded (1,2)iso algebra using the

method in [4], which is consistent with the co-algebra structure. Firstly, we have introduced

3

1,2(1,2) (1,2)ISO SO R group, which is the semi-direct product of (1,2)SO and 3

1,2R

groups in three dimensional space. Here, (1,2)SO group consists of matrices preserving the

quadratic form 2 2 2 2

1 2 3x x x R in Pseudo Euclid space and 3

1,2R translations group in 3IR

space [5]. Then, to arrive 3S - graded (1,2)iso algebra one adds new elements Q to

generators jX of the corresponding (1,2)U iso algebra where , j

jQ Q Q b X . We

denoted this algebra with 3 (1,2)NU iso .

Keywords. Poincaré symmetries, Fractional Supersymmetry, Semi-direct product.

References

[1] Goze, M., Raush deTraunbenberg, M. and Tanasa, A J. Math. Phys., 48, 093507-1-

093507-24, 2007.

[2] Ahn, C., Bernard, D. and Leclair, A. Nucl. Phys. B, 346, 409-439, 1990.

[3] Ahmedov, H. and Dayi, O. F. Mod. Phys. Lett. A, 15, 1801-18111, 2000.

[4] Ahmedov, H., Yildiz, A., and Ucan, Y., J. Phys. A:Math.Gen., 34, 6413-6423, 2001.

1 Yildiz Technical University, İstanbul, TURKEY, [email protected] 2 Yildiz Technical University, İstanbul, TURKEY, [email protected] 3 Yildiz Technical University, İstanbul, TURKEY, [email protected]


31

[5] Vilenkin, N. Ya. And Klimyk, A. U., Representations of Lie Groups and Special

Functions, Kluwer Academic, Dordrecht, Netherlands, 1991.


32

On Multiplicative Generalized Derivations in 3-Prime Near-Rings

Zeliha BEDİR 1 and Öznur GÖLBAŞI 2

Abstract. An additively written group equipped with a binary operation

, such that and for all

is called a left near-ring. Any ring is a near-ring, hence a near-ring is a generalization of a

ring. A near-ring is said to be prime if implies or . An additive

map 𝑑: 𝑁 → 𝑁 is called a derivation if 𝑑(𝑥𝑦) = 𝑥𝑑(𝑦) + 𝑑(𝑥)𝑦 holds for all 𝑥, 𝑦 ∈ 𝑁. In the

present study, we deal with the generalizations of the notion of derivation on 3-prime near-

ring 𝑁.

Keywords. Near-ring, derivation, multiplicative generalized derivation

AMS 2010. 16Y30, 16W25.

References

[1] Bell, H. and Mason, G., On derivations in near rings, near rings and near Öelds,

NorthHolland Mathematical Studies, 137, 31-35, 1987.

[2] Bell, H. E., On derivations in near-rings II, Kluwer Academic Pub. Math. Appl., Dordr.,

426, 191-197, 1997.

[4] Daif, M. N. , Bell, H. E., Remarks on derivations on semiprime rings, Int. J. Math. Math.

Sci., 15(1), 205-206, 1992.

[5] Daif, M. N. and Tammam El-Sayiad, M. S., Multiplicative generalized derivations which

are additive, East-west J. Math., 9 (1), 31-37, 2007.

[6] Bedir, Z., Gölbaşı, Ö, Notes on prime near rings with multiplicative derivation,

Cumhuriyet University Faculty of Science, Science Journal (CSJ), Vol. 38, No. 2,355-363,

2017.

[8] Kamal, A. M. and Al-Shaalan, K. H., Existence of derivations on near-rings, Math.

Slovaca, 63, no:3, 431-438, 2013.


33

ANALYSIS


34

f -Lacunary statistical convergence of order for double sequences

Abdulkadir KARAKAŞ1, Hıfsı ALTINOK2 and Yavuz ALTIN3

Abstract. New concepts of f -statistical convergence for double sequences of order

and strongly f -Cesàro summability for double sequences of order are introduced for

sequences of (complex or real) numbers. Futhermore we give the relationship between the

spaces 2

,0( , )w f , 2 ( , )w f and

2

, ( , )w f . Furthermore we expressed that the properties of

the strong f -Cesàro summability of order which is related to strong f -Cesàro

summability of order . Also some relations between f -statistical convergence of order

and strongly f -Cesàro summability of order are given.

Keywords. Double sequences, Statistical convergence, Cesàro summability.

AMS 2010. 40A05, 40C05, 46A45.

References

[1] Aizpuru, A., Listán-García, M. C.; Rambla-Barreno, F. Density by moduli and statistical

convergence, Quaest. Math., 37, 4, 525-530, 2014.

[2] Aizpuru, A., Listán-García, M. C.; Rambla-Barreno, F. Double density by moduli and

statistical convergence, Bull. Belg. Math. Soc. Simon Stevin., 19, 4, 663-673, 2012.

[3] Altin, Y., Altinok, H.; Çolak, R., Statistical convergence of order for difference

sequences,. Quaest. Math. 38, 4, 505-514, 2015.

[4] Bhardwaj, Vinod K., Dhawan, Shweta, f-statistical convergence of order and strong

Cesàro summability of order with respect to a modulus, J. Inequal. Appl., 332 pp.14,

2015.

[5] Bhardwaj, Vinod K., Dhawan, Shweta. Density by moduli and lacunary statistical

convergence. Abstr. Appl. Anal., Art. ID 9365037, 11 pp, 2016.

1 Siirt University, Siirt, TURKEY, [email protected] 2 Fırat University, Elazığ, TURKEY, [email protected] 3 Fırat University, Elazığ, TURKEY, [email protected]


35

Some Special Values of Vertices of Trees on the Suborbital Graphs

Ali Hikmet DEĞER1 and Ümmügülsün AKBABA2

Abstract. In this research some special vertices of paths of minimal length on

suborbital graphs are investigated. Using edge conditions for these types of graphs,

relationship with the values of vertices on graphs and some elements of special sequence

values are also described.

Keywords. Suborbital graphs, Modular group, Continued fractions.

AMS 2010. 20H10, 20H05, 05C05, 05C20.

References

[1] Jones, G. A., Singerman, D., Wicks, K., The modular group and generalized farey

graphs, London Math. Soc. Lect. Note Ser., Cambridge Univ. Press, Vol. 160, pp. 316-338,

1991.

[2] Sims, C.C., Graphs and finite permutation groups, Math. Zeitsch., Vol. 95, pp. 76-86,

1967.

[3] Akbas, M., On suborbital graphs for the modular group, Bull. London Math. Soc. 33,

647-652, 2001.

[4] Değer, A. H., Beşenk, M., Güler, B. Ö., On Suborbital graphs and related continued

fractions, Applied Mathematics and Computation, 218, 3, 746-750, 2011.

[5] Değer, A. H., Vertices of paths of minimal lengths on suborbital graphs, Filomat, 31, 3,

913-923, 2017.

[6] Sarma, R., Kushwaha, S., Krishnan, R., Continued fractions arising from 𝐹1,2, Journal of

Number Theory, Vol. 154, pp. 179-200, 2015.

1 Karadeniz Technical University, Trabzon, TURKEY, [email protected] 2 Karadeniz Technical University, Trabzon, TURKEY, [email protected]


36

On Some Special Values of Fibonacci and Lucas Sequence with the Suborbital Graphs

Ali Hikmet DEĞER1 and Ümmügülsün AKBABA2

Abstract. Using the action of the modular group and a special equivalence relation on

the extended rational set, Jones, Singerman and Wicks [1] defined the suborbital graphs.

Modular group acts transitively and imprimitively on this set. So suborbital graphs have

vertices from the infty block from equivalence relation and stabilizer of this block is the

congruence subgroup of the modular group.

In the present study vertices of these suborbital graphs and relationships with

Fibonacci and Lucas sequences are investigated.

Keywords. Suborbital graphs, Modular group, Fibonacci sequence, Lucas sequence.

AMS 2010. 20H10, 20H05, 05C05, 05C20.

References

[1] Jones, G. A., Singerman, D., Wicks, K., The modular group and generalized farey

graphs, London Math. Soc. Lect. Note Ser., Cambridge Univ. Press, Vol. 160, pp. 316-338,

1991.

[2] Akbas, M., On suborbital graphs for the modular group, Bull. London Math. Soc. 33,

647-652, 2001.

[3] Değer, A. H., Beşenk, M., Güler, B. Ö., On suborbital graphs and related continued

fractions, Applied Mathematics and Computation, 218, 3, 746-750, 2011.

[4] Değer, A. H., Vertices of paths of minimal lengths on suborbital graphs, Filomat, 31, 3,

913-923, 2017.

[5] Değer A.H., Imprimitive action of the normalizer of Gamma_0(N) and suborbital graphs,

AIP Conf. Proc., vol.1676, pp.1-4, 2015.

[6] Beşenk M., Değer A.H., Güler B.Ö., An application on suborbital graphs, AIP Conf.

Proc., vol.1470, pp.187-190, 2012.



37

A Note on the Ambarzumyan's Theorem

Alp Arslan KIRAÇ1

Abstract. The classical Ambarzumyan’s theorem was proved under a more general

inverse assertion. Then, Yurko proved that it is enough to specify only the first eigenvalue

rather than the whole spectrum. We weaken slightly the Yurko’s conditions on the first

eigenvalue.

Keywords. Inverse spectral problems, Hill operator, Ambarzumyan theorem.

AMS 2010. 34A55, 34B30, 34L05, 47E05, 34B09.

References

[1] Ambarzumian, V., Über eine Frage der Eigenwerttheorie. Zeitschrift für Physik, 53, 690–

695, 1929.

[2] Borg, G., Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe bestimmung der

differentialgleichung durch die eigenwerte. Acta Math. 78, 1–96, 1946.

[3] Cheng, Y. H., Wang, T.E., Wu, C.J., A note on eigenvalue asymptotics for Hill’s equation.

Appl. Math. Lett. 23(9), 1013–1015, 2010.

[4] Chern, H. H., Lawb, C.K., Wang, H.J., Corrigendum to Extension of Ambarzumyan’s

theorem to general boundary conditions. J. Math. Anal. Appl. 309, 764–768, 2005.

[5] Chern, H. H., Shen, C.L., On the n-dimensional Ambarzumyan’s theorem. Inverse

Problems 13, 1, 15–18, 1997.

[6] Freiling, G., Yurko, V.A., Inverse Sturm-Liouville Problems and Their Applications.

NOVA Science Publishers, New York, 2001.

[7] Hochstadt, H., Lieberman, B., An inverse sturm-liouville problem with mixed given data.

SIAM J. Appl. Math. 34, 676–680, 1978.

[8] Levitan, B. M., Gasymov, M.G., Determination of a differential equation by two of its

spectra. Usp. Mat. Nauk 19, 3–63, 1964.

[9] Kıraç, A. A., On the Ambarzumyan’s theorem for the quasi-periodic problem.

Anal.Math.Phys. 6, 297–300, 2016.

1 Pamukkale University, Denizli, TURKEY, [email protected]


38

[10] Veliev, O. A., Kıraç, A. A., On the nonself-adjoint differential operators with the

quasiperiodic boundary conditions. International Mathematical Forum 2, 35, 1703–1715,

2007.

[11] Yang, C. F., Huang, Z.Y., Yang, X. P., Ambarzumyan’s theorems for vectorial sturm-

liouville systems with coupled boundary conditions. Taiwanese J. Math. 14, 4, 1429–1437,

2010.

[12] Yurko, V. A., On Ambarzumyan-type theorems, Applied Mathematics Letters 26, 506–

509, 2013.


39

Polars in Locally Convex Cones

A. Ranjbari1 and S. Jafarizad2

Abstract. The theory of locally convex cones uses an order theoretical structure on a

cone. We consider three types of polarity from [4] and [7] and we investigate some properties

of them. Also, we prove some relations between these polars.

Keywords. Locally convex cones, polar, duality.

AMS 2010. 46A03, 46A20, 46A40

References

[1] Ayaseh, D., Ranjbari, A., Bornological Convergence in Locally Convex Cones, Mediterr.

J. Math., 13 (2016) 1921-1931.

[2] Ayaseh, D., Ranjbari, A., Bornological locally convex cones, Matematiche (Catania) 69 (2)

(2014) 267-284.

[3] Ayaseh, D., Ranjbari, A., Locally convex quotient lattice cones, Math. Nachr. 287 (10)

(2014) 1083-1092.

[4] Plotkin, G.D., A Domain-Theoretic Banach-Alaoglu Theorem, Math. Struct. Compute.

Sci. 16 (2) (2006) 299-313.

[5] Ranjbari, A., Saiflu, H., Projective and inductive limits in locally convex cones, J. Math.

Anal. Appl. 332 (2) (2007)1097 -1108.

[6] Ranjbari, A., Strict inductive limits in locally convex cones, Positivity 15 (3) (2011) 465-

471.

[7] Roth, W., Operator-valued measures and integrals for cone-valued functions, Vol. 1964

of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009.

1 University of Tabriz, Tabriz, Iran, [email protected] (Speaker) 2 University of Tabriz, Tabriz, Iran, [email protected]




40

A Sufficient Condition for Complex-Valued Summable Functions to be Absolutely

Continuous a.e.

Alp Arslan KIRAÇ1

Abstract. In this study, we consider Gasymov potantial. We prove that if a complex-

valued summable potential has the Fourier coefficients q(n)= 0 for n ≤ 0, then the potential

of Hill’s equation is absolutely continuous, a.e.

Keywords. Schrödinger operator, smoothness of potential, gap lengths.

AMS 2010. 47E05 , 58F19

References

[1] ] Hochstadt, H., Estimates on the stability intervals for the Hill’s equation, Proc. Amer.

Math. Soc. 14, 930–932, 1963.

[2] Marchenko, V. A., Sturm–Liouville operators and applications. Oper. Theory Adv. Appl.,

Vol. 22, Birkhauser, 1986.

[3] Magnus, W., Winkler, S., Hill’s Equation, Interscience, 1969.

[4] Marchenko, V.A., Ostrovskii, I. V., Characterization of the spectrum of Hill’s operator,

Mat. Sb. 97, 1975, 540–606; English transl.: Math. USSR-Sb. 26 (175).

[5] J. Pöschel, E. Trubowitz, Inverse Spectral Theory, Academic Press, 1987.

[6] Trubowitz, E., The inverse problem for periodic potentials, CPAM 30, 321–342, 1977.

[7] Kappeler, T. and Mityagin, B., Estimates for periodic and Dirichlet eigenvalues of the

Schrödinger operator, SIAM J. Math. Anal. 33, 113–152, 2001.

[8] Djakov, P., Mityagin, B., Smoothness of Schrödinger operator potential in the case of

Gevrey type asymptotics of the gaps. J. Funct. Anal. 195, 89–128, 2002.

[9] Gasymov, M. G., Spectral analysis of a class of second order nonselfadjoint differential

operators, Funksional. Anal. i Prilozhen. 14, 14–19, 1980.



41

[10] Sansuc, J. J. and Tkachenko, V., Spectral properties of non-selfadjoint Hill’s operators

with smooth potentials. In: A. Boutet de Monvel and V. Marchenko (eds.), Algebraic and

Geometric Methods in Mathematical Physics. 371–385, Kluwer, 1996.

[11] Tkachenko, V. A., Spectral analysis of nonselfadjoint Hill operatör, Doklady AN SSSR,

322, 2, 248-252, 1992.

[12] Tkachenko, V., Characterization of Hill operators with analytic potentials. Integral

Equations Operator Theory 41, 360–380, 2001.

[13] Djakov, P., Mityagin, B., Spectral triangles of Schrödinger operators with complex

potentials. Selecta Math. (N.S.) 9, 495–528, 2003.

[14] Pöschel, J., Hill’s potentials in weighted Sobolev spaces and their spectral gaps, Math.

Ann. 349:433–458, 2001.

[15] Kıraç, A. A., Inverse problems associated with the Hill operator. Electron. J. Diff. Equ.

41, 1–12, 2016.


42

On Quasi Subordinations for Analytic and Bi-Univalent Function Class

Arzu AKGÜL1

Abstract: In this study, certain subclasses of analytic bi-univalent functions are

defined and established bounds for the coefficients for this subclass. Also several related

classes are considered and connections to earlier known results are made.

Keywords: Bi-univalent functions, bi-starlike functions, coefficient estimates.

Mathematics Subject Classification: 2010: 30C45, 30C50.

References

[1] Ali, R. M., Lee, S. K., Ravichandran, V. and Supramaniam, S., Coefficient estimates for

bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25, 344-351,2012.

[2] Brannan, D.A., Taha, T.S., On some classes of bi-univalent functions, in: S.M. Mazhar, A.

Hamoui, N.S. Faour (Eds.), Math. Anal. and Appl., Kuwait; February 18.21, 1985, in: KFAS

Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp.

53.60. see also Studia Univ. Babe¸s-Bolyai Math. 31 (2) (1986) 70.77.

[3] Brannan, D. A. and Clunie, J. G., Aspects of comtemporary complex analysis,

(Proceedings of the NATO Advanced Study Instute Held at University of Durham:July 1-20,

1979). New York: Academic Press, 1980.

[4] Frasin, B. A. and Aouf, M. K., New subclasses of bi-univalent functions, Applied

Mathematics Letters, 24, 1569-1573, 2011.

[5] Deniz, E., Certain subclasses of bi-univalent functions satisfying subordinate conditions,

Journal of Classical analysis, 2, 1, 40-60, 2012.

[6] Jahangiri, J. M. and Hamidi, S. G., Coefficient estimates for certain classes of bi-univalent

functions, Int. J. Math. Math. Sci., ArticleID 190560, 4, 2013.

[7] Lee, S. Y., Quasi-subordinate functions and coefficient conjectures, J. Korean Math. Soc.,

12, no. 1, 43.50, 1975.



43

[8] Kumar, S. S., Kumar, V. and Ravichandran, V., Estimates for the initial coefficients of bi-

univalent functions, Tamsui Oxf. J. Inf. and Math. Sci. 29, 4, 487-504, 2013.

[9] Levin, M., On a coefficient problem for bi-univalent functions, Proceeding of the

American Mathematical Society, 18, 63-68, 1967.

[10] Ma, W. C. and Minda, D., A unified treatment of some special classes of univalent

functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157.169,

Conf. Proc. Lecture Notes Anal. I Int. Press, Cambridge, MA.

[11] Netanyahu, E., The minimal distance of the image boundary from the orijin and the

second coefficient of a univalent function in j, Archive for Rational Mechanics and Analysis,

32, 100-112, 1969.

[12] Patil, A. B. and Naik, U. H., Estimates on initial coe¢ cients of certain subclass es of bi-

univalent functions associated with quas-subordination, Global Journal of Mathematical

Analysis, 5, 2, 6-10, 2017.

[13] Ch. Pommerenke, Univalent functions, Vandenhoeck and Rupercht, Göttingen, 1975.

[14] Ren, F. Y., Owa, S. and Fukui, S., Some inequalities on quasisubordinate functions, Bull.

Aust. Math. Soc., 43, 2, 317-324, 1991.

[15] Robertso, M. S., Quasi-subordination and coefficient conjecture, Bull. Amer. Math.

Soc., 76, 1.9, 1970.

[16] Srivastava, H. M., Mishra, A. K., and Gochhayat, P., Certain subclasses of analytic and

bi-univalent functions, Applied Mathematics Letters, 23, 10, 1188-1192, 2010.

[17] Srivastava, H. M., Joshi, S. B., Joshi, S. S., Pawar, H., Coefficient estimates for certain

subclasses of meromorphically bi-univalent functions, Palest. J. Math., 5 (Special Issue: 1),

250-258, 2016.

[18] Xu, Q.-H.,Gui, Y.-C. and Srivastava, H. M., Coincident estimates for a certain subclass

of analytic and biunivalent functions, Appl. Math. Lett. 25, 990.994, 2012.


44

Quadratic Time-Frequency Representations on Mixed Lorentz Type Modulation Spaces

Ayşe SANDIKÇI1

Abstract. In this work we investigate the continuity properties of a class of time

frequency representations which is based on the short-time Fourier transform on the Lorentz

mixed normed modulation space dM P,Q which is the set of all tempered distributions

df S such that the short-time Fourier transform gV f of f is in the Lorentz mixed norm

space 2dL P,Q , where dg S is a non-zero window function, 1 21 P p ,p

and 1 21 Q q ,q . Some key references are given below.

Keywords. Lorentz mixed normed space, Lorentz mixed normed modulation space,

time frequency representations.

AMS 2010. 42B10, 47B38.

References

[1] Boggiatto, P., De Donno, G. and Oliaro, A., Time-frequency representations of Wigner

type and pseudodiffererential operators, Trans. Amer. Math. Soc., 362, 4955-4981, 2010.

[2] Cordero, E. and Gröchenig, K., Time-frequency analysis of localization operators, J.

Funct. Anal., 205, 1, 107-131, 2003.

[3] Fernandez, D. L., Lorentz spaces, with mixed norms, J. Funct. Anal., 25, 128-146, 1, 1977.

[4] Gröchenig, K., Foundation of time-frequency analysis. Birkhäuser, Boston, ISBN 0-8176-

4022-3, 2001.

[5] Hunt, R.A., On L(p,q) spaces, Extrait de L'Enseignement Mathematique, T.XII, fasc.4,

249-276, 1966.

[6] Sandıkçı, A., On Lorentz mixed normed modulation spaces, J. Pseudo-Differ. Oper. Appl.,

3, 263-281, 2012.

[7] Sandıkçı, A., Boundedness of localization operators on Lorentz mixed normed modulation

spaces, Journal of Inequalities and Applications, 2014, 430, 2014.

1 Ondokuz Mayis University, Samsun, TURKEY, [email protected]


45

Bilinear Multipliers on nM P,Q

Ayşe SANDIKÇI1

Abstract. Let m be a locally integrable function on 2n . If

2n

2πi ξ η,x

mˆ ˆB f ,g x f ξ g η m ξ,η e dξdη

defines a bounded bilinear operator from n n

1 1 2 2M P ,Q M P ,Q to n

3 3M P ,Q

m is said to be a bilinear multiplier on n

of type 1 1 2 2P ,Q ;P ,Q , where nM P,Q

is

the set of all tempered distributions nf S such that the short-time Fourier transform

gV f of f is in the Lorentz mixed norm space 2nL P,Q . The study of the some properties

of these spaces is investigated and give some examples. Some key references are given below.

Keywords. Bilinear multiplier, Lorentz mixed normed modulation space.

AMS 2010. 42A45, 42B15.

References

[1] Blasco, O., Notes on the spaces of bilinear multipliers, Revista de la Union Matematica

Argentina, 50(2), 2337, 2009.

[2] Blasco, O., Bilinear multipliers and transference, International Journal of Mathematics

and Mathematical Sciences, 4, 545-554, 2005

[3] Leeuw, K., On Lp multipliers, Annals of Mathematics, Second Series, Vol.81, No.2, 364-

379, 1965.

[4] Sandıkçı, A., On Lorentz mixed normed modulation spaces, J. PseudoDi⁄er. Oper. Appl. 3,

263-281, 2012.

[5] Villarroya, F., Bilinear multipliers on Lorentz spaces, Czechoslovak Mathematical

Journal, 58(133), 1045-1057, 2008



46

Reguler Equivalents of a Measure Space

Bahaettin CENGİZ1 and Banu GÜNTÜRK2

Abstract. Integration in Lebesgue sense with respect to a regular Borel measure has

considerable advantages over integration with respect to an arbitrary positive measure, and it

becomes even more advantageous if the regular measure is also perfect. In this article, for a

given finite positive measure space (𝑋, 𝒜, 𝜇) we determine all locally compact Hausdorff

spaces 𝑌 on which there is a regular Borel measure ν with support 𝑌 such that the measure

algebras 𝒜/𝜇 and ℬ/𝜈 are isometric, (and consequently their respective 𝐿𝑝 spaces are

isometric) where ℬ denotes the Borel algebra of 𝑌. And also some partial results are obtained

in the case of an infinite measure space. As a bonus, one other thing we prove in this paper is

that for an arbitrary measure 𝜇, the space 𝐿∞(𝜇) is the topological dual of 𝐿1(𝜇) if and only if

its maximal ideal space is extremally disconnected.

Keywords. Equivalent measures, maximal ideal space, perfect measure, Hyperstonean

space.

AMS 2010. 28A25, 28C15, 46J10, 28A60.

1 Baskent University, Ankara, TURKEY, [email protected] 2 Baskent University, Ankara, TURKEY, [email protected]


47

Hilbert Matrix on Spaces of Holomorphic Functions

Boban KARAPETROVIĆ1

Abstract. We prove that the Hilbert matrix operator 𝐻 is bounded on the Besov space

𝐻𝜈𝑝,𝑞,𝛼

if and only if 0 < 𝜅𝑝,𝛼,𝜈 < 1, where 𝜅𝑝,𝛼,𝜈 = 𝜈 − 𝛼 −1

𝑝+ 1. In particular, 𝐻 is

bounded on the Bergman space 𝐴𝑝,𝛼 if and only if 1 < 𝛼 + 2 < 𝑝 and is bounded on the

Dirichlet space 𝐷𝛼𝑝 = 𝐴1

𝑝,𝛼 if and only if max−1, 𝑝 − 2 < 𝛼 < 2𝑝 − 2 (see [6]). In [4] it

was shown that the norm of the Hilbert matrix operator 𝐻 on the Bergman space 𝐴𝑝 is equal

𝜋

sin2𝜋

𝑝

, when 4 ≤ 𝑝 < ∞, and it was also conjectured that ‖𝐻‖𝐴𝑝→𝐴𝑝 =𝜋

sin2𝜋

𝑝

, when 2 < 𝑝 < 4.

Following [1] we prove this conjecture.

Keywords. Hilbert matrix, Bergman spaces, Besov spaces, Dirichlet spaces.

AMS 2010. 47B35, 47B38,30H20, 30H25.

References

[1] Božin, V., Karapetrović, B., Norm of the Hilbert matrix on Bergman spaces, submitted for

publication, 2017.

[2] Diamantopoulos, E., Hilbert matrix on Bergman spaces, Illinois Journal of

Mathematics,48, no. 3, 1067-1078, 2004.

[3] Diamantopoulos, E., Siskakis, A. G., Composition operators and the Hilbert matrix,

Studia Mathematica, 140, no. 2, 191-198, 2000.

[4] Dostanić, M., Jevtić, M., Vukotić, D., Norm of the Hilbert matrix on the Bergman and

Hardy spaces and theorem of Nehari type, J. Funct. Anal., 254, 2800-2815, 2008.

[5] Galanopoulos, P., Girela, D., Peláez, J. A., Siskakis, A. G., Generalized Hilbert operators,

Ann. Acad. Sci. Fenn. Math., 39, 231-258, 2014.

[6] Jevtić, M., Karapetrović, B., Hilbert matrix on spaces of Bergman-type, J. Math. Anal.

Appl., 453, 241-254, 2017.

1 University of Belgrade, Faculty of Mathematics, Belgrade, SERBIA, [email protected]


48

Some Relation on the Poly-Genocchi Numbers and Polynomials with a q-Parameter

Burak KURT1

Abstract. Recently, Cenkci and Komatsu defined and investigated poly-Bernoulli

numbers and polynomials with a q-parameter. They proved some relations between these

polynomials and weighted Stirling numbers of the first and second kind. Also, they gave some

relations between poly-Bernoulli polynomials a q-parameter and the poly-Cauchy

polynomials of the first and the second kind with a q-parameter. Komatsu generalized the

poly-Cauchy numbers with a q-parameter.

In this work, we define poly-Genocchi numbers and polynomials with a q-parameter.

We give a recurrence relation for the poly-Genocchi polynomials. Also, we prove some

relations and closed formulae for the poly-Genocchi numbers with a q-parameter.

Keywords. Genocchi numbers and polynomials, Bernoulli numbers and polynomials,

The Stirling numbers of the second kind, Weighted Stirling numbers of the second kind..

AMS 2010. 11B68; 11B73; 11B75.

References

[1] Arakawa, T, Kaneko M., On poly-Bernoulli numbers, Commentarıı Mathematic Univ.

Sanct. Pauli, 48, 159-167, 1999.

[2] Bayad, A, Hamahata Y., Polylogarithms and poly-Bernoulli polynomials, Kyushu. J.

Math., 65, 15-24, 2011.

[3] Jolany, H, Corcino, R. B., Explicit formula for generalization of poly-Bernoulli numbers

and polynomials with a, b, c parameters, J. of classical Analysis, 6(2), 119-135, 2015.

[4] Kaneko, M., Poly-Bernoulli numbers, Journal de Théorie des Nombres de Bardeox, 9,

221-228, 1997.

[5] Kamono, K., A formula for multi-poly-Bernoulli numbers of negative index, Kyush. J.

Math., 67, 29-37, 2013.

1 Akdeniz University, Antalya, TURKEY, [email protected]


49

The Fixed Point of the Sum of Nonlinear Operators in WC-Banach Algebras Relative to

the Weak Topology

Cesim TEMEL1and Musa ÇAKIR2

Abstract. In this work we aim to investigate the existence of solutions of

Krasnoselskii type equations for weak topology in WC-Banach algebras. We also provide

some results of Schauder and Banach fixed point theorems in WC-Banach algebras for use in

the new variant of Krasnoselskii fixed theorem in WC-Banach algebras.

Keywords. Fixed point theorem, Krasnoselskii theorem, weakly sequentially

continuous operator, weakly compact operator, WC-Banach algebras.

AMS 2010. 47H10, 47H08, 47H09.

References

[1] Arino, O., Gautier, S. , Pento, J. P., A fixed point theorem for sequentially continuous

mapping with application to ordinary differential equations, Functional Ekvac. 27 (3), 273-

279, 1984.

[2] Banas, J., Taoudi, M. A., Fixed points and solutions of operator equations for the weak

topology in Banach algebras, Taiwanese J. Math. 3, 871-893, 2014.

[3] Ben Amar, A., Chouayekh, S., Jeribi, A., Fixed point theory in a new class of Banach

algebras and application, Afr. Math. 24, 705-724, 2013.

[4] Jeribi, A., Krichen, B., Mefteh, B., Fixed point theory in WC-Banach algebras, Turk. J.

Math.40, 283-291, 2016.

[5] Krasnoselskii, M. A., Some problems of nonlinear analyasis, Amer. Math. Soc. Trans. 10

(2), 345-409, 1958.

[6] Smart, D.R., Fixed points theorems, Cambridge University Press, London, New York

1974.

1Yuzuncu Yil University, Van, TURKEY, [email protected] 2Yuzuncu Yil University, Van, TURKEY, [email protected]


50

Investigating the Center of L-weakly and M-weakly Compact Operators

E. BAYRAM1, A.W. WICKSTEAD2

Abstract. An operator T on an ordered vector space E , with positive cone E , is

called central if it is bounded by a multiple of the identity operator, i.e., there exists some

scalar 0 such that - x Tx x holds for all x E . The collection of all central

operators on E is called the centre of E and denoted by Z E . Amongst many works on the

centre, especially in the context of vector lattices, there have been attempts to identify the

centre of various spaces of regular operators. Some isometric results were given by Wickstead

showing that if E and F are Banach lattices then there is an embedding of the algebraic tensor

product Z E Z F into ,rZ L E F which is an isometry when Z E Z F is given

the injective tensor product norm. He also gives some density results regarding this, and

similar, embeddings. Also, some related results are obtained for spaces of compact and

weakly compact operators.

In this study we extend and modified known results concerning the centre of spaces of

regular (resp. weakly compact or compact) operators between two Banach lattices to the

setting of L-weakly compact and M-weakly compact operators. We also show that the L-

weakly compact, M-weakly compact and compact operators lying in the centre of a Banach

lattice coincide.

Keywords: L-weakly compact operator, M-weakly compact operator, Banach lattice, Centre

of ordered vector space.

Acknowledgement: This research was supported by The Scientific and Technological

Research Council of Turkey (TUBITAK) within the context of 2219-Post Doctoral

Fellowship Program and by the Research Foundation of Namik Kemal University (Project

No. NKUBAP.01.GA.17.108)

1 Namik Kemal University, Tekirdag, TURKEY, [email protected] 2 Queen's University, Belfast, NORTHERN IRELAND, [email protected]


51

The Generating Function of ( , )p q -Bernstein Polynomials and Their Properties Based

on ( , )p q -Calculus

Erkan AGYUZ1 and Mehmet ACIKGOZ2

Abstract. In the present paper, we have proposed a new generating function for

Bernstein polynomials based on ( , )p q -calculus. From these generating functions, we derive

some new properties and results for ( , )p q - Bernstein polynomials.

Keywords. ( , )p q -calculus, ( , )p q -Bernstein polynomials, Generating functions.

AMS 2010. 05A30, 11B68.

References

[1] Acıkgoz, M., Araci, S., On the generating function of the Bernstein polynomials, in

Proceedings of the 8th International Conference of Numerical Analysis and Applied

Mathematics (ICNAAM.10),AIP,Rhodes,Greece, March 2010.

[2] Bernstein, S., Demonstration du théoréme de Weierstrass fondeé sur la calcul des

probabilités, Communications of the Mathematical Society of Charkow. Séries 2 , vol. 13, pp.

1.2, 1912-1913.

[3] Mursaleen, M, Ansari, K.J., Khan, A., On (p; q)-analogue of Bernstein operators, Appl.

Math. Comput. 266, 874-882, 2015.

[4] Goldman R., Simeonov P., Simsek Y., Generating Functions for the q-Bernstein Bases,

SIAM Journal on Discrete Mathematics, 28(3), 1009.1025, 2014.

[5] Simsek Y., Acikgoz M., .A new generating function of q-Bernstein-type polynomials and

their interpolation function, Abstract and Applied Analysis, Article ID 769095, 12 pages,

2010.

1 Gaziantep University, Gaziantep, TURKEY, [email protected] 2 Gaziantep University, Gaziantep, TURKEY, [email protected]


52

Characterizations of Matrix and Compact Operators on the Space |𝑬𝒓𝜽|𝒑

Fadime GÖKÇE1 and Mehmet Ali SARIGÖL2

Abstract: In this study, we introduce the new spaces |𝐸𝑟𝜃|

𝑝 and |𝐸𝑟

𝜃|𝑝,∞

as the

sets of all series summable by the methods |𝐸𝑟, 𝜃𝑛|𝑝 for 1 ≤ 𝑝 < ∞ and |𝐸𝑟, 𝜃𝑛|∞

for 𝑝 = ∞ and then, we mention briefly some topological and algebraic properties of

these spaces. Furthermore, we characterize certain matrix and compact operators on those

spaces and also determine their norms and Hausdroff meausures of noncompactness.

Key Words and Phrases: Absolute Euler summability, matrix transformations,

sequence spaces, inclusion relations, norms, Hausdroff meausure of noncompactness

2010 Mathematics Subject Classification: 40C05,40D25,40F05,46A45

References

[1] Altay, B., Başar, F. and Mursaleen, M., On the Euler sequence spaces which include

the spaces 𝑙𝑝 and 𝑙∞ II, Nonlinear Anal., 65, no. 3, 707—717, 2006.

[2] Altay, B., Başar, F. and Mursaleen, M., On the Euler sequence spaces which include

the spaces 𝑙𝑝 and 𝑙∞ I”, Inform. Sci, 176, no.10 1450-1462, 2005.

[3] Boos, J. and Cass, P., Classical and modern methods in summability, Oxford

University Press, New York, 2000.I. Daubechies, Comm. Pure Appl. Math. 41, 909, 1988.

[4] Mursaleen, R. M., Applied Summability Methods, Springer, 2013.

[5] Sarıgöl, M.A., Spaces of Series Summable by Absolute Cesaro and Matrix

Operators, Comm. Math Appl., 7, no. 1, 11-22, 2016.

[6] Sarıgöl, M.A., Extension of Mazhar's theorem on summability factors, Kuwait J.

Sci., 42, no. 3, 28-35, 2015.






53

[7] Sarıgöl, M.A., Matrix transformations on fields of absolute weighted mean

summability, Studia Sci. Math. Hungar., 48, no. 3, 331-341, 2011.

[8] Sarıgöl, M.A., On the local properties of factored Fourier series, Appl. Math.Comp.,

216, no.11, 3386-3390, 2010.

[9] Stieglitz, M., Tietz, H., Matrix transformationen von Folgenraumen Eine

Ergebnisübersicht, Mathematische Zeitschrift, 154, no. 1, 1-16, 1977.

[10] Wilansky, A., Summability Through Functional Analysis”, Mathematics Studies. 85.

North Holland, Amsterdam, 1984.


54

Matrix and Compact Operators on the Absolute Fibonacci Spaces

Fadime GÖKÇE1 and Mehmet Ali SARIGÖL2

Abstract: In the present study, we introduce the absolute Fibonacci space

|𝐹𝑢|𝑘 where 𝑢 is a sequence of positive numbers and 1 ≤ 𝑘 < ∞. Then, we give some

inclusion relations, investigate topological and algebraic structure such as BK-space, 𝛼-,𝛽-, 𝛾

-duals and base. Further, we characterize certain matrix and compact operators on these

spaces, also determine their norms and Hausdroff meausures of noncompactness.

Key Words and Phrases: Absolute summability,Fibonacci numbers, matrix

transformations, sequence spaces, norms, Hausdroff meausure of noncompactness

2010 Mathematics Subject Classification: 40C05,40D25,40F05,46A45

References

[1] Boos, J.and Cass, P ., Classical and modern methods in summability, Oxford

University Press, New York, 2000.I. Daubechies, Comm. Pure Appl. Math. 41, 909, 1988.

[2] Kara, E. E., & Ilkhan, M., Some properties of generalized Fibonacci sequence spaces,

Linear and Multilinear Algebra, 64(11), 2208-2223, 2016.

[3] Kara, E. E., Some topological and geometrical properties of new Banach sequence

spaces. Journal of Inequalities and Applications, 2013(1), 38, 2013.

[4] Malkowsky, E. and Rakocevic, V., On matrix domains of triangles, Appl. Math.

Comp., 189, no. 2, 1146-1163, 2007.

[5] Malkowsky, E. and Rakocevic, V., An introduction into the theory of sequence space

and measures of noncompactness, Zb. Rad.(Beogr), 9, no. 17, 2000.

[6] Sarıgöl, M.A., Spaces of Series Summable by Absolute Cesaro and Matrix

Operators, Comm. Math Appl., 7, no. 1, 11-22, 2016.






55

[7] Sarıgöl, M.A., Extension of Mazhar's theorem on summability factors, Kuwait J.

Sci., 42, no. 3, 28-35, 2015.

[8] Sarıgöl, M.A., On the local properties of factored Fourier series, Appl. Math.Comp.,

216, no.11, 3386-3390, 2010.

[9] Stieglitz, M. and Tietz, H., Matrix transformationen von Folgenraumen Eine

Ergebnisübersicht, Mathematische Zeitschrift, 154, no. 1, 1-16, 1977.

[10] Wilansky, A. , Summability Through Functional Analysis, Mathematics Studies. 85.

North Holland, Amsterdam, 1984.


56

The Theory of 𝒏-Scales

Furkan Semih DÜNDAR1

Abstract. We provide a theory of n-scales previously called as n dimensional time scales. In

previous approaches to the theory of time scales by Bohner and Guseinov, multi-dimensional

scales were taken as product space of two time scales. n-scales make the mathematical

structure more flexible and appropriate to real world applications in physics and related fields.

Here we define an n-scale as an arbitrary closed subset of R^n. Modified forward and

backward jump operators, Δ-derivatives and multiple integrals on n-scales are defined.

1 Boğaziçi University. İstanbul, TURKEY, [email protected]


57

Some Specific Properties of Concave Functions Defined by the Generalized Srivastava-

Attiya Operator

Hasan BAYRAM1 and Şahsene ALTINKAYA2

Abstract. In this paper we investigate a class Ψ𝜇,𝑏𝑚,𝑘𝐶0(𝛼) of concave functions by

using the generalized Srivastava-Attiya operator. Such as, coefficient inequalities, distortion

bounds.

Keywords. Hadamard product, concave functions, distortion theorem, Hurwitz-Lerch

Zeta functions, multiplier transformation, Srivastava-Attiya operator.

AMS 2010. 30C45, 30C50.

References

[1] Attiya, A. A., Hakami, A. H., Some subordination results associated with generalized

Srivastava-Attiya operator, Adv. Difference Equ., 105, 2013.

[2] Choi, J. and H. M. Srivastava, Certain families of series associated with the Hurwitz-

Lerch Zeta function, Appl. Math. Comput. 170, 399—409, 2005.

[3] Lopez C. Ferreira and J. L. Lopez, Asymptotic expansions of the Hurwitz-Lerch Zeta

function, J. Math. Anal. Appl., 298, 210—224, 2004.

[4] Murugusundaramoorthy, G., Coefficient estimate of bi-Bazilevic function defined by

Srivastava-Attiya operator, 69, 2, 45-56, 2014.

[5] Srivastava H. M., Some Fox-Wright generalized hypergeometric functions and associated

families of convolution operators, Appl. Anal. Discrete Math., 1, 56—71, 2007.

1 Uludağ University, Bursa, TURKEY, [email protected] 2 Uludağ University, Bursa, TURKEY, [email protected]


58

A Note On Weighted Composition Operators on Besov-Type Spaces

Hamid Vaezi1 and Sima Houdfar 2

Abstract. Let D be the open unit disc in the complex plane C. Denote by

by H(D) the class of all analytic functions on D. Given an analytic self map ᵩ

and analytic function ᴪ in the unit disc D, the weighted composition operator

ᴪ Cᵩf on H(D) is defined by

ᴪ Cᵩf (z) =ᴪ(z) f (ᵩ (z)) , zϵD.

In this article we study the boundedness and compactness of the

Weighted composition operators between Besov- type spaces, by using

the Carleson measure.

Keywords Weighted composition operator, Besovtype space, Carleson measure

AMS 2010. 47B33, 30H25.

References

[1] Colonna, F., Li, S., , Weighted composition operators from the Besove spaces into the

Bloch spaces, Bull Malays. Math. Sci. Soc., 36, 1027-1039, 2013.

[2] Cowen, C. C., Maccluer B. D., Composition operators on spaces of analytic functions,

Studies in Advanced Math., CRC Press, Boca Raton, 1995.

[3] Hassanlou, M., Vaezi, H., Wang, M., Weighted composition operators on weak vector

valued Bergman spaces and Hardy spaces, J. Math. Anal., 9, 35-43, 2015.

[4] Shapiro, J. H., Composition operators and classical function theory, Springer Verlag,

NewYork, 1993.

1 University of Tabriz, Tabriz Iran, [email protected] 2 University of Tabriz, Tabriz Iran, [email protected]



59

Generalized Lupaş Operators

Hatice Gül İNCE İLARSLAN1, Ali ARAL2 and Gülen BAŞCANBAZ-TUNCA3

Abstract. In this work, by taking a continuously differentiable, increasing and

unbounded function ρ, we consider an extension of the Lupaş operator Ln in the form

Ln (f∘ρ⁻¹)∘ρ for convenient functions f on [0,∞). We give weighted approximation,

Voronovskaya type theorem, quantitative estimates for the local approximation.

Keywords. Generalized Lupaş operator; Weighted approximation; Voronovskaya type

theorem.

AMS 2010. 41A36, 41A25.

References

[1] Agratini, O., On a sequence of linear and positive operators. Facta Univ. Ser. Math.

Inform. No. 14, 41—48, 1999.

[2] Aral, A., Inoan, D. and Raşa, I., On the generalized Szász-Mirakyan operators. Results

Math. 65, no. 3-4, 441-452, 2014.

[3] Gadjiev A. D., Aral A., The estimates of approximation by using a new type of weighted

modulus of continuity. Comp. Math. Appl., 54, 127-135, 2007.

[4] Lupas¸ A., The approximation by some positive linear operators. In: Proceedings of the

International Dortmund Meeting on Approximation Theory (M.W. Müller et al.,eds.),Ak

ademie Verlag, Berlin, 201-229, 1995.

1 Gazi University, Ankara, TURKEY, [email protected] 2 Kırıkkale University, Kırıkkale, TURKEY, [email protected] 3 Ankara University, Ankara, TURKEY, [email protected]


60

Statistical Convergence of order (β,γ) for Sequences of Fuzzy Numbers

Hıfsı ALTINOK1 and Mikail ET2

Abstract. In this paper, we introduce the concept of statistical convergence of order

(β,γ) and strongly p-Cesàro summable of order (β,γ) for sequences of fuzzy numbers and give

some inclusion relations between them.

Keywords. Sequence of fuzzy numbers, statistical convergence, Cesàro summability.

AMS 2010. 40A05; 40A25; 40A30; 40C05; 03E72.

References

[1] Çolak, R. Statistical convergence of order α, Modern Methods in Analysis and Its

Applications, New Delhi, India: Anamaya Pub, 121—129, 2010.

[2] Et, M. and Sengul, H. Some Cesaro-Type Summability Spaces of Order alpha and

Lacunary Statistical Convergence of Order alpha, FILOMAT, 28, 8, 1593-1602, 2014.

[3] Fast, H. Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.

[4] Matloka, M. Sequences of fuzzy numbers, BUSEFAL 28, 28-37, 1986.

[5] Nuray, F. and Savaş, E. Statistical convergence of sequences of fuzzy real numbers, Math.

Slovaca 45, 3, 269-273, 1995.

[6] Savas, E. A note on sequence of fuzzy numbers, Inform. Sci. 124, 1,4, 297-300, 2000.

[7] Schoenberg, IJ. The integrability of certain functions and related summability methods,

Amer. Math. Monthly 66, 361-375, 1959.

1 Firat University, Elazığ, TURKEY, [email protected] 2 Firat University, Elazığ, TURKEY, [email protected]


61

Statistical Convergence of order β for Double Sequences of Fuzzy Numbers Defined by a

Modulus Function

Hıfsı ALTINOK1, Yavuz ALTIN2 and Mahmut IŞIK3

Abstract. In the present paper, we extend the notions of statistically convergence of

order β and strong Cesàro summability of order β for sequences of fuzzy numbers and

introduce the notions of f-statistically convergence of order β and strong Cesàro summability

of order β for β∈(0,1] with respect to an unbounded modulus function f for double sequences

of fuzzy numbers and give some inclusion theorems.

Keywords. Statistical convergence, Cesàro summability, Modulus function

AMS 2010. 40A05; 40A25; 40A30; 40C05; 03E72.

References

[1] Aizpuru, A.; Listan-Garcia, M. C. and Rambla-Barreno, F. Density by moduli and

statistical convergence. Quaest. Math. 37, 525-530, 2014.

[2] Bhardwaj, V. K. and Dhawan, S. f-statistical convergence of order α and strong Cesaro

summability of order α with respect to a modulus, J. Inequal. Appl. 2015:332 DOI

10.1186/s13660-015-0850-x, 2015.

[3] Colak, R. and Altin, Y. Statistical convergence of double sequences of order α, J. Funct.

Space. Appl. 2013.

[4] Et, M. and Sengul, H. Some Cesaro-Type Summability Spaces of Order alpha and

Lacunary Statistical Convergence of Order alpha, FILOMAT, 28, 8, 1593-1602, 2014.

[5] Fast, H. Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.


[7] Savaş, E. A note on double sequences of fuzzy numbers, Tr. J. of Mathematics, 20, 175—

178, 1996.

1 Firat University, Elazığ, TURKEY, [email protected] 2 Firat University, Elazığ, TURKEY, [email protected] 3 Firat University, Elazığ, TURKEY, [email protected]


62

Δnm-lacunary Statistical Convergence of Order α

Hıfsı ALTINOK 1, Mikail ET2 and Mahmut IŞIK 3

Abstract. The purpose of this work is to introduce the concepts of Δnm-lacunary

statistical convergence of order α and Δnm -lacunary strongly convergence of order α. We

establish some connections between Δnm -lacunary strongly convergence of order α and Δn

m -

lacunary statistical convergence of order α. It is shown that if a sequence is Δnm -lacunary

strongly convergent of order α then it is Δnm -lacunary statistically convergent of order α.

Keywords. Difference sequence, statistical convergence, lacunary sequence.

AMS 2010. 40A05, 40C05, 46A45.

References

[1] Çolak, R., Statistical convergence of order α, Modern Methods in Analysis and Its

Applications, NewDelhi, India: Anamaya Pub, 121—129, 2010.

[2] Et, M. and Colak, R. : On some generalized difference sequence spaces, Soochow J. Math.

21, 4, 377-386, 1995.

[3] Freedman, A. R. Sember, J. J. and Raphael, M. : Some Cesàro-type summability spaces,

Proc. Lond. Math. Soc. 37 (1978), 508-520.

[6] Fridy, J. A. : On the statistical convergence, Analysis 5 (1985), 301 - 313.

[5] Fridy, J. A. and Orhan, C. : Lacunary statistical convergence, Pacific J. Math. 160 (1993),

43-51.

[4] Kizmaz, H. : On certain sequence spaces, Canadian Math. Bull. 24 (1981 ), 169-176.

[5] Tripathy, B. C. ; Esi, A. and Tripathy, B. K. On a new type of generalized difference

Cesaro Sequence spaces, Soochow J. Math. 31:3 (2005), 333-340.

1 Fırat University, Elazığ, TURKEY, [email protected] 2 Fırat University, Elazığ, TURKEY, [email protected] 3 Fırat University, Elazığ, TURKEY, [email protected]


63

Generalized Statistical Convergence of order β for Sequences of Fuzzy Numbers

Hıfsı ALTINOK1, Abdulkadir KARAKAŞ2 and Yavuz ALTIN3

Abstract. In the present paper, we introduce the concepts of ∆m-statistical

convergence of order β for sequences of fuzzy numbers and strongly ∆m-summable statistical

convergence of order β for sequences of fuzzy numbers by using a modulus function f and

taking supremum on metric d for 0 < β ≤ 1 and give some inclusion relations between them.

Keywords. Sequence of fuzzy numbers, statistical convergence, difference sequence.

AMS 2010. 40A05; 40A25; 40A30; 40C05; 03E72.

References


Applications, New Delhi, India: Anamaya Pub, 121-129, 2010.

[2] Et, M. and Çolak, R. On some generalized difference sequence spaces, Soochow J. Math.

21, 4, 377-386, 1995.

[3] Kızmaz, H. On certain sequence spaces, Canad. Math. Bull. 24, 2, 169-176, 1981.


[5] Nuray, F. and Savaş, E. Statistical convergence of sequences of fuzzy real numbers, Math.

Slovaca 45, 3, 269-273, 1995.

[6] Savas, E. A note on sequence of fuzzy numbers, Inform. Sci. 124, 1,4, 297-300, 2000.

[7] Schoenberg, IJ. The integrability of certain functions and related summability methods,


[8] Srivastava, P. D. and Mohanta, S. Statistical convergence of generalized difference

sequence space of fuzzy numbers. Taiwanese J. Math. 17, 5, 1659-1676, 2013.

1 Firat University, Elazığ, TURKEY, [email protected] 2 Siirt University, Elazığ, TURKEY, [email protected] 3 Firat University, Elazığ, TURKEY, [email protected]


64

A New Application of Quasi Power Increasing Sequences

Hikmet SEYHAN ÖZARSLAN1

Abstract. In the present paper, we prove a general theorem dealing with absolute

matrix summability methods of infinite series by using the concept of quasi power

increasing sequence instead of almost increasing sequence. Some important known theorems

are deduced from our theorems.

Keywords. Riesz mean, summability factor, absolute matrix summability, quasi

power increasing sequences, infinite series, Hölder inequality, Minkowski inequality.

AMS 2010. 26D15, 40D15, 40F05, 40G99.

References

[1] Bari, N. K., Stečkin, S. B., Best approximations and differential properties of two

conjugate functions, Trudy Moskov. Mat. Obšč., 5, 483-522, 1956 (in Russian).

[2] Bor, H., On two summability methods, Math. Proc. Cambridge Philos. Soc., 97, 147-149,

1985.

[3] Bor, H., On absolute Riesz summability factors, Adv. Stud. Contemp. Math. (Pusan), 3(2)

23-29 , 2001.

[4] Hardy, G. H., Divergent Series, Oxford University Press, Oxford, 1949.

[5] Leindler, L., A new application of quasi power increasing sequences, Publ. Math.

Debrecen., 58, 791-796, 2001 .

[6] Özarslan, H. S., Kandefer, T., On the relative strength of two absolute summability

methods, J. Comput. Anal. Appl., 11, 576-583, 2009.

[7] Özarslan, H. S., A new application of absolute matrix summability, C. R. Acad. Bulgare

Sci., 68, 967-972, 2015.

[8] Sulaiman, W. T., Inclusion theorems for absolute matrix summability methods of an

infinite series. IV, Indian J. Pure Appl. Math., 34 (11), 1547-1557, 2003.

This work was supported by Research Fund of the Erciyes University, Project Number: FDK-2017-6945 1 Erciyes University, Kayseri, TURKEY, [email protected]



65

Almost Increasing Sequences and Their New Applications

Hikmet SEYHAN ÖZARSLAN1 and Bağdagül KARTAL2

Abstract. In [3], Bor has proved a theorem concerning absolute summability of

infinite series. In the present paper, this known theorem has been generalized to absolute

matrix summability by using matrix transformations and almost increasing sequences. Some

results have also been obtained.

Keywords. Matrix transformations, absolute matrix summability, almost increasing

sequences, infinite series, Hölder inequality, Minkowski inequality.

AMS 2010. 26D15, 40D15, 40F05, 40G99.

References

[1] Bari, N. K., Stečkin, S. B., Best approximations and differential properties of two

conjugate functions, Trudy Moskov. Mat. Obšč., 5, 483-522, 1956 (in Russian).

[2] Bor, H., On two summability methods, Math. Proc. Cambridge Philos. Soc., 97, 147-149,

1985.

[3] Bor, H., A note on absolute Riesz summability factors, Math. Inequal. Appl., 10, 619-625,

2007.

[4] Hardy, G. H., Divergent Series, Oxford University Press, Oxford, 1949.

[5] Mazhar, S. M., A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica.,

25, 233-242, 1997.

[6] Özarslan, H. S., A new application of almost increasing sequences, Miskolc Math. Notes.,

14, 201-208, 2013.

[7] Özarslan, H. S., A new application of generalized almost increasing sequences, Bull.

Math. Anal. Appl., 8, 9-15, 2016.

[8] Özarslan, H. S., Karakaş, A., A new result on the almost increasing sequences, J. Comp.

Anal. Appl., 22, 989-998, 2017.

This work was supported by Research Fund of the Erciyes University, Project Number: FDK-2017-6945 1 Erciyes University, Kayseri, TURKEY, [email protected] 2 Erciyes University, Kayseri, TURKEY, [email protected]




66

Wavelet-Type Transforms Generated by Some Semigroups and Their Applications

İlham A. ALIEV1

Abstract. We introduce wavelet-type transforms generated by some “wavelet

measures” and so-called “modified beta-semigroups”. These semigroups are natural

generalizations of the Gauss-Weierstrass and Abel-Poisson semigroups associated with the

Laplace-Bessel differential operator. We introduce also a notion of bi-parametric potential-

type operators, which generalize the Bessel and Flett potentials associated with the Laplace-

Bessel differential operator. Explicit inversion formulae for these bi-parametric potential-type

operators are obtained by making use of the aforementioned wavelet-type transforms.

Keywords. Generalized Bessel potentials, wavelet transforms, Gauss-Weierstrass

semigroup, Poisson semigroup, inversion formulas

AMS 2010. Primary 26A33; Secondary 42C40, 44A35, 45P05

1 Akdeniz University, Antalya, TURKEY; [email protected]


67

Higher Order Fractional Boundary Value Problems with Integral Boundary Conditions

İsmail YASLAN1

Abstract. In this paper, we investigate the existence of positive solutions for higher

order multi-point nonlinear fractional boundary value problems with integral boundary

conditions. We establish the criteria for the existence of at least one, two and three positive

solutions for higher order m-point nonlinear fractional boundary value problems with integral

boundary conditions by using a result from the theory of fixed point index, Avery-Henderson

fixed point theorem and the Legget-Williams fixed point theorem, respectively.

Keywords. Boundary value problems, cone, fixed point theorems, positive solutions,

Riemann-Liouville fractional derivative, integral boundary conditions

AMS 2010. 26A33, 34B10, 34B18.

References

[1] Anco, S., Wald, R., Does there exist a sensible quantum theory of an algebravalued scalar

field, Phys. Rev., 39, 2297-2307, 1989.

[2] Avery, R. I., Henderson, J., Two positive fixed points of nonlinear operators on ordered

Banach spaces, Comm. Appl. Nonlinear Anal., 8, 27-36, 2001.

[3] Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Academic Press,

San Diego, 1988.

[4] Lan, K. Q., Multiple positive solutions of semilinear differential equations with

singularities, J. London Math. Soc., 63, 690-704, 2001.

[5] Legget, R.W., Williams, L.R., Multiple positive fixed points of nonlinear operators on

ordered Banach space, Indiana Univ. Math. J., 28, 673-688, 1979.

[6] Sabatier, O.P., Agrawal, J.A., Machado, T., Advances in Fractional Calculus, Springer,

Dordrecht, The Netherlands, 2007.

[7] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integral and Derivatives: Theory

and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.



68

Boundedness of Products of Weighted Composition Operators and Differentiation

Operators between Weighted Bergman Spaces and Weighted Zygmund Spaces of

Analytic Functions

Jasbir S. MANHAS1

Abstract. Let v and w be weights on the unit disc D. Let Av,p(D) be the weighted

Bergman space of analytic functions and Zw(D) be the weighted Zygmund space of analytic

functions. In this paper, we investigate the analytic mappings φ : D →D and ψ :D→₵ which

characterize the boundedness of products of weighted composition operators and

differentiation operators 𝐷𝑊𝜓,φ and 𝑊𝜓,φ𝐷 between the weighted Bergman spaces Av,p(D)

and Zygmund spaces Zw(D).

Keywords. Weighted Bergman Spaces, Weighted Zygmund Spaces, Weighted

Composition Operators, Differentiation Operators, Bounded Operators.

AMS 2010. 47B38, 47B33.

1 Sultan Qaboos University, Muscat, OMAN, [email protected]


69

The Vitali Convergence Theorem for Nonlinear Integrals

Jun KAWABE1

Abstract. The Choquet [1] integral and the Sugeno integral [5] for a measurable

function with respect to a nonadditive (also called monotone) measure may be considered as

nonlinear functionals and are widely used in application areas such as decision theory under

uncertainty, game theory, data mining and others. For those functionals, their continuity

corresponds to the convergence theorem of integrals, which means that the limit of the

integrals of a sequence of functions is the integral of the limit function. Thus many attempts

have been made to formulate the counterparts of the monotone, the bounded, and the

dominated convergence theorems for the Choquet, the Šipoš [2], the Sugeno, and the Shilkret

integrals [3].

The purpose of this talk is to present the Vitali convergence theorem for such

nonlinear integrals. A key ingredient is a perturbation of integral that manages the small

change of the integral value arising as a result of adding small amounts to a measure and an

integrand [2].

Keywords. Nonadditive measure, nonlinear integral, Vitali convergence theorem.

AMS 2010. 28A25, 28E10.

References

[1] Choquet, G., Theory of capacities, Ann. Inst. Fourier (Grenoble), 5, 131-295, 1953-1954.

[2] Kawabe, J., A unified approach to the monotone convergence theorem for nonlinear

integrals, Fuzzy Sets Syst., 304, 1-19, 2016.

[3] Shilkret, N., Maxitive measure and integration, Indag. Math., 33, 109-116, 1971.

[4] Šipoš, J., Integral with respect to a pre-measure, Math. Slovaca, 29, 141-155, 1979.

[5] Sugeno, M., Theory of fuzzy integrals and its applications, Ph.D. dissertation, Tokyo

Institute of Technology, Tokyo, 1974.

1 Shinshu University, Nagano, JAPAN, [email protected]


70

Berezin Number Inequalities and Engliš Algebras

M. GÜRDAL1, M.B. HUBAN2 and M.T. GARAYEV3

Abstract. In this paper, we consider the following natural question: when the power

inequality nn AberAber holds for any integer ?1n We solve this question in some

particular case.

Acknowledgement: This work is supported by TUBA through Young Scientist

Award Program (TUBA-GEBIP/2015).

Keywords. Berezin symbol, Berezin number, Engliš Algebras.

AMS 2010. 47A63, 47B35.

References

[1] Berezin, F.A., Covariant and contravariant symbols for operators, Math. USSR-Izv., 6,

1117-1151, 1972.

[2] Dragomir, S.S., A survey of some recent inequalities for the norm and numerical radius of

operators in Hilbert spaces, Banach J. Math. Anal., 1, 154-175, 2007.

[3] Engliś, M., Toeplitz operators and the Berezin transform on H², Linear Alg. Appl.,

223/224, 171-204, 1995.

[4] Garayev, M.T., Gürdal, M., Huban, M.B., Reproducing kernels, Engliš algebras and some

applications, Studia Mathematica, 232(2), 113-141, 2016.

[5] Hardy, G., Littlewood, J.E., Polya, G., Inequalities, 2 nd ed. Cambridge University Press,

Cambridge, 1967.

[6] Saitoh, S., Sawano, Y., Theory of reproducing kernels and applications, Springer, 2016.

1 Suleyman Demirel University, Isparta, TURKEY, [email protected] 2 Suleyman Demirel University, Isparta, TURKEY, [email protected] 3 King Saud University, Riyadh, SAUDİ ARABİA, [email protected]


71

Hadamard Type Inequalities for m-Convex and (α,m)-Convex Functions via Fractional

Integrals

Merve Avcı ARDIÇ1, Alper EKİNCİ2, Ahmet Ocak AKDEMİR3 and M. Emin ÖZDEMİR4

Abstract. In this paper, we established some new Hadamard-type integral inequalities

for functions whose derivatives of absolute values are m-convex and (α,m)-convex functions

via Riemann-Liouville fractional integrals.

Keywords. m-convex functions, (α,m)-convex functions, Riemann-Liouville

fractional integral.

AMS 2010. 26A51, 26D15.

References

[1] Bakula, M. K., Özdemir, M. E., Pečarić, J., Hadamard type inequalities for m-convex and

(α,m)-convex functions, J. Inequal. Pure Appl. Math. 9, Article 96, 2008.

[2] Bakula, M. K., Pečarić, J., Ribičić, M., Companion inequalities to Jensen's inequality for

m-convex and (α,m)-convex functions, J. Inequal. Pure Appl. Math. 7, Article 194, 2006.

[3] Dragomir, S.S., On some new inequalities of Hermite-Hadamard type for m-convex

functions, Tamkang J. Math., 3 (1) 2002.

[4] Dragomir, S.S., Agarwal, R.P., Two inequalities for differentiable mappings and

applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett.,

11 (5), 91-95,1998.

[5] Dragomir, S.S., Toader, G.H., Some inequalities for m-convex functions, Studia Univ.

Babeş-Bolyai, Math., 38 (1), 21-28, 1993.

[6] Gorenflo, R., Mainardi, F., Fractional calculus: integral and differential equations of

fractional order, Springer Verlag, Wien, 223-276, 1997.

1 Adıyaman University, Adıyaman, TURKEY, [email protected] 2 Ağrı İbrahim Çeçen University, Ağrı, TURKEY, [email protected] 3 Ağrı İbrahim Çeçen University, Ağrı, TURKEY, [email protected] 4 Uludağ University, Bursa, TURKEY, [email protected]


72

[7] Gradshteyn, I.S., Ryzhik, I.M., Table of Integrals, Series, and Products, 7th ed., Academic

Press, Elsevier Inc., 2007.

[8] Miheşan, V.G., A generalization of the convexity, Seminar of Functional Equations,

Approx. and Convex, Cluj-Napoca (Romania) , 1993.

[9] Miller S., Ross, B., An introduction to the Fractional Calculus and Fractional Differential

Equations, John Wiley and Sons, USA, p.2, 1993.

[10] Özdemir, M.E., Avcı, M., Kavurmacı, H., Hermite-Hadamard-type inequalities via

(α,m)-convexity, Comput. Math. Appl., 61, 2614-2620, 2011.

[11] Özdemir, M.E., Avcı, M., Set, E., On some inequalities of Hermite--Hadamard type via

m-convexity, Appl. Math. Lett. 23 (9), 1065-1070, 2010.

[12] Özdemir, M.E, Kavurmacı, H., Set, E., Ostrowski's type inequalities for (α,m)-convex

functions, Kyungpook Math. J. 50, 371-378, 2010.

[13] Özdemir, M. E., Set, E., Sarıkaya, M. Z., Some new Hadamard's type inequalities for co-

ordinated m-convex and (α,m)-convex functions, Hacettepe J. of. Math. and Statistics., 40,

219-229, 2011.

[14] Podlubni, I., Fractional Differential Equations, Academic Press, San Diego, 1999.

[15] Toader, G. H., Some generalisations of the convexity, Proc. Colloq. Approx. Optim, 329-

338, 1984.


73

On Wijsman Asymptotically Deferred Statistical Equivalence of Order α for Set

Sequences

Mikail ET1, Hıfsı ALTINOK 2 and Rifat ÇOLAK3

Abstract: In this study we introduce and examine the concepts of Wijsman

asymptotically deferred statistical equivalence of order α and Wijsman strong r-deferred

Cesàro asymptotically equivalence of order α for set sequences. Also, we give some relations

connected to these concepts.

Keywords. Asymptotically Statistical Equivalence, Deferred Statistical Convergence;

AMS 2010. 40A05, 40C05, 46A45.

References

[1] Agnew, R. P. On deferred Cesàro means, Ann. of Math. (2) 33(3), 413—421, 1932.


Applications, NewDelhi, India: Anamaya Pub, 121—129, 2010.

[3] Küçükaslan, M. and Yılmaztürk, M. On deferred statistical convergence of

sequences. Kyungpook Math. J. 56, no. 2, 357—366, 2016.

[4] Marouf, M. Asymptotic equivalence and summability. Int. J. Math. Math. Sci., 16, 755-

762, 1993.

[5] Patterson, R. F. On asymptotically statistical equivalent sequences. Demonstratio Math.

36(1), 149—153, 2003.

[6] Ulusu, U. and Nuray, F. On asymptotically lacunary statistical equivalent set sequences, J.

Math. 2013, Art. ID 310438, 5, 2013

[7] Yılmaztürk, M. and Küçükaslan, M. On strongly deferred Cesàro summability and

deferred statistical convergence of the sequences, Bitlis Eren Univ. J. Sci. and Technol. 3, 22-

25, 2011.

1 Fırat University, Elazığ, TURKEY, [email protected] 2 Fırat University, Elazığ, TURKEY, [email protected] 3 Fırat University, Elazığ, TURKEY, [email protected]


74

On Infinite Bernoulli Matrices

Murat KİRİŞCİ1

Abstract. In this work, we study some properties of infinite Bernoulli matrices.

Further, we investigate relations between infinite Bernoulli matrices and some infinite

matrices such as Fibonacci, Pascal and special matrices.

Infinite Bernoulli polynomials matrix B(x) and Bernoulli matrix B are investigated.

Linearity as an operator, regularity and invertibility of the matrix B are showed. The examples

of the matrices B(x) and B are given.

Keywords. Bernoulli matrix, Bernoulli numbers, Pascal matrix, Fibonacci matrix

AMS 2010. Primary 11B68, Secondary 11C08, 05A10.

References

[1] G.S. Call, Pascal's matrices, Amer. Math. Monthly, 100 (1993), 372-376, 1993.

[2] Gi-Snac, Cheon, A Note on the Bernoulli and Euler Polynomials, Appl. Math. Letters, 16,

365-368, 2003.

[3] E.E. Kara, M. Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear

and Multilinear Algebra, 39(2), 217-230, 2016.

[4] M. Kirisci, Fibonacci Statistical Convergence and Korovkin type Approximation

Theorems, (to appear).

[5] R. E. Powel, S.M. Shah, Summability Theory and Applications, Prentice-Hall, 1988.

[6] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math,

154, 1622-1632, 2006.

1 Istanbul University, Istanbul, TURKEY, [email protected]; [email protected]




75

Curious Bounds for the Gamma Function

Necdet BATIR1

Abstract. We improve some inequalities proved in [1] and establish the following

new inequalities for the gamma function.

4/1

2

4/1

2

18

1

32)1(...049653963.0

32

x

xexxx

xex xxxx

ve

1

2exp)1(

)1(logexp

xxx

x

xx .

Here )(/)(')( xxx is the digamma function.

Keywords. Gamma function, digamma function, inequalities.

AMS 2010. 33B15.

References

[1] Alzer, H. and Batir, N., Monotonicity properties of the gamma function, Appl. Math.

Letters, 20, 778-781, 2007.

[2] Batir, N., Inequalities for the gamma function, Arch. Math.(Basel), (91), 554-563, 2008.

[3] Batir, N., Very accurate approximations for the factorial function, J. Math. Inequal., 3,

335-344, 2010.

[4] Borwein, J. M. and Borwein, P., Pi and the AGM, John Wiles and Sons, 1987.

[5] Chen, C-P. and Tong, W-W., Sharp inequalities and asymptotic expansions for the gamma

function, J. Number Theory, 160, 418-431, 2016.

[6] Chen, C-P. and Liu, J-Y., Inequalities and asymptotic expansions for the gamma function,

J. Number Theory, 149, 313-326, 2015.

[7] Marsden, J. E., Basic Complex Analysis, W. H. Freeman and Company, San Fransisco,

1973.

1 Nevşehir HBV University, Nevşehir, TURKEY, [email protected]


76

[8] Mortici, C., New approximations of the gamma function in terms of the digamma function,

Appl. Math. Letters, 23, 97–100, 2010.


77

Some Fixed-Circle Theorems and Discontinuity at Fixed Circle

Nihal Yılmaz ÖZGÜR1 and Nihal TAŞ 2

Abstract. In this talk, we give some existence and uniqueness theorems for fixed

circles of self-mappings on a metric space with some illustrative examples. Recently, real-

valued neural networks with discontinuous activation functions have been a great importance

in practice. Hence we give some new results for discontinuity at fixed circle on a metric

space.

Keywords. Fixed circle, discontinuity, activation function, metric space.

AMS 2010. 54E40, 47H09.

References

[1] Bisht, R. K., Pant, R. P., A remark on discontinuity at fixed point, J. Math. Anal. Appl.

445, 1239-1242, 2017.

[2] Bisht, R. K., Pant, R. P., Contractive definitions and discontinuity at fixed point, Appl.

Gen. Topol. 18, 1, 173-182, 2017.

[3] Forti, M., Nistri, P., Global convergence of neural networks with discontinuous neuron

activations, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50, 11, 1421-1435, 2003.

[4] Özdemir, N., İskender, B.B., Özgür, N.Y., Complex valued neural network with Möbius

activation function, Commun. Nonlinear Sci. Numer. Simul. 16, 12, 4698-4703, 2011.

[5] Özgür, N. Y., Taş, N., Some fixed-circle theorems on metric spaces, arXiv:1703.00771

[math.MG].

[6] Taş, N., Özgür, N.Y., A new contribution to discontinuity at fixed point, arXiv:1705.03699

[math.MG].

[7] Wang, L. L., Chen, T. P., Multistability of neural networks with Mexican-hat-type

activation functions, IEEE Trans. Neural Netw. Learn. Syst. 23, 11, 1816-1826, 2012.

1 Balikesir University, Balikesir, TURKEY, [email protected] 2 Balikesir University, Balikesir, TURKEY, [email protected]


78

Bergman Projections on the Weighted Harmonic Bloch Spaces on the Ball, Atomic

Decompositions and Gleason’s Problem

Ömer Faruk DOĞAN 1

Abstract. We study the properties of one parameter family of weighted harmonic

Bloch Spaces 𝑏𝛼, 𝛼 ∈ ℝ, on the unit ball of ℝ𝑛. We define projections and characterize those

that are bounded from 𝐿𝛼∞ onto 𝑏𝛼 by using reproducing kernels of Bergman-Besov spaces.

The projections provide integral representations for the functions in these spaces. We solve

the Gleason problem and provide atomic decomposition for all 𝑏𝛼 , 𝛼 ∈ ℝ . Finally we give an

oscillatory characterization of 𝑏𝛼 when 𝛼 > −1. This is joint work with Adem Ersin Üreyen.

Keywords. Harmonic Bloch space, Bergman space, Reproducing kernel, Radial

fractional derivative, Bergman projection, Duality, Gleason problem, Atomic decomposition,

Oscillatory characterization.

AMS 2010. 31B05, 31B10, 26A33, 46E15.

References

[1] Axler, S., Bourdon, P., Ramey,W., Harmonic Function Theory, vol. 137, 2nd edn.

Springer, New York (2001). Grad. Texts in Math

[2] Choe, B.R., Koo, H., Yi, H., Derivatives of harmonic Bergman and Bloch functions on the

ball. J. Math. Anal. Appl. 260, 100–123, 2001.

[3] Choe, B.R., Lee, Y.J., Note on atomic decompositions of harmonic Bergman functions. In:

Complex Analysis and its Applications, OCAMI Studies, vol. 2, Osaka Munic. Univ. Press,

Osaka, pp. 11–24, 2007.

[4] Coifman, R.R., Rochberg, R., Representation theorems for holomorphic and harmonic

functions in 𝐿𝑝. Astérisque 77, 12–66, 1980.

[5] Gergün, S., Kaptanoğlu, H.T., Üreyen, A.E.: Reproducing kernels for harmonic Besov

spaces on the ball. C. R. Math. Acad. Sci. Paris 347, 735–738, 2009.

[6] Gergün, S., Kaptanoğlu, H. T., Üreyen, A. E., Harmonic Besov Spaces on the Ball,

Internat. J. Math., 27(9), 165007059, 2016.

[7] Holland, F., Walsh, D., Criteria for membership of Bloch space and its subspace, BMOA.

Math. Ann. 273, 317–335, 1986.

1 Namık Kemal University, Tekirdağ, TURKEY, [email protected]



79

[8] Hu, Z., Gleason’s problem for harmonic mixed norm and Bloch spaces in convex

domains. Math. Nachr. 279(1–2), 164–178, 2006.

[9] Jevtic, M., Pavlovic, M., Harmonic Bergman functions on the unit ball in ℝ𝑛. Acta Math.

Hung. 85, 81–96, 1999.

[10] Kaptanoğlu, H.T., Tülü, S., Weighted Bloch, Lipschitz, Zygmund, Bers, and growth

spaces of the ball: Bergman projections and characterizations. Taiwan. J. Math. 15(1), 101–

127, 2011.

[11] Ligocka, E., On the reproducing kernel for harmonic functions and the space of Bloch

harmonic functions on the unit ball in ℝ𝑛. Stud. Math. 87, 23–32, 1987.

[12] Miao, J., Reproducing kernels for harmonic Bergman spaces of the unit ball. Mon. Math.

125, 25–35, 1998.

[13] Ren, G., Kähler, U., Weighted harmonic Bloch spaces and Gleason’s problem. Complex

Var. Theory Appl. 48, 235–245, 2003.

[14] Ren, G., Kähler, U., Weighted Lipschitz continuity and harmonic Bloch and Besov spaces

in the unit real ball. Proc. Edinb. Math. Soc. 48, 743–755, 2005.

[15] Ren, G., Tu, C., Bloch spaces in the unit ball of ℂ𝑛. Proc. Am. Math. Soc. 133, 719–726,

2005.

[16] Stroethoff, K., The Bloch space and Besov spaces of analytic functions. Bull. Aust. Math.

Soc. 54, 211–219, 1996.

[17] Stroethoff, K., Harmonic Bergman Spaces, Holomorphic Spaces, vol. 33. Mathematical

Sciences Research Institute Publications, Cambridge University, Cambridge, 1998.

[18] Tanaka, K., Representation theorem for harmonic Bergman and Bloch functions. Osaka

J. Math. 50, 947–961, 2013.

[19] Zhao, R., A characterization of Bloch-type spaces on the unit ball of ℂ𝑛. J. Math. Anal.

Appl. 330, 291–297, 2007.

[20] Zhao, R., Zhu, K., Theory of Bergman Spaces in the Unit Ball of ℂ𝑛, Mém. Soc. Math.

Fr., vol. 115, pp. vi+103, 2008.


80

Majorization Problem on a Subclass of Analytic Functions

Öznur ÖZKAN KILIÇ1, Osman ALTINTAŞ2

Abstract. The main object of this paper is to investigate a majorization problem involving

a subclass H(α,A,B) of analytic functions in the unit disk U. Relevant connection of the result

presented here with those given by earlier workers on the subject are also indicated.

Keywords. Analytic function, Subordination, Majorization.

AMS 2010. 30C45.

References

[1] Altıntaş O., On the coefficients of functions majorized by univalent functions, Hacettepe

Bulletin of Natur. Sci. and Eng., 10, 23-30, 1981.

[2] Altıntaş O., On the coefficients of certain analytic functions, Hacettepe Bulletin of Natur.

Sci. and Eng., Vol. II, 147-156, 1982.

[3] Goodman A. W., Univalent functions, Vol. I, 1983.

[4] Kakeya S., On the limits of the roots of an algebra equation with positive coefficients,

Tohoku Math. J., 140-142, 1912.

[5] Kılıç Ö. Ö., Altıntaş O.,¨ Coefficients bounds for certain subclasses of analytic functions.

[6] MacGregor T.H. , Majorization by univalent functions, Duke Math. J. , 34, 95-102, 1967.

[7] Srivastava H. M. , Altintaş O. and Serenbay S. K. , Coefficient bounds for certain

subclasses of starlike functions of complex order, Appl. Math. Lett., 24, 1359-1363, 2011.


81

A Note on the Extended Type Riemann-Liouville Fractional Derivative Operator

Praveen AGARWAL1, İ. Onur KIYMAZ2, Shilpi JAIN3 and Ayşegül ÇETİNKAYA4

Abstract. In this paper, by using the extended type beta function, establish the new

extended definition for the Riemann-Liouville fractional derivative operator and discuses its

properties. Also, we establish the some relations to extended special functions of two and

three variables via generating functions.

Keywords. Beta function Hypergeometric function of two and three variables

Fractional derivative operator, Generating functions

AMS 2010. Primary 33C0533C1533C20; Secondary 33C6533C99.

References

[1] Chaudhry M. A., Qadir A., Raflque M., and Zubair S. M., Extension of Euler's Beta

function, J. Comput. Appl. Math. 78, 19-32, 1997.

[2]Chaudhry M. A., Qadir A., Srivastava H. M., and Paris R. B., Extended hypergeometric

and confluent hypergeometric functions, Appl. Math. Comput. 159, 589-602, 2004.

[3] Chaudhry M. A., Temme N. M., and Veling E. J. M., Asymptotic and closed form of a

generalized incomplete gamma function, J. Comput. Appl. Math. 67, 371-379, 1996.

[4] Chaudhry M. A. and Zubair S. M., Generalized incomplete gamma functions with

applications, J. Comput. Appl. Math. 55, 99-124, 1994.

[5] Chaudhry M. A. and Zubair S. M., On the decomposition of generalized incomplete

gamma functions with applications to Fourier transforms, J. Comput. Appl. Math. 59, 253-

284, 1995.

[6] Chaudhry M. A. and Zubair S. M., Extended incomplete gamma functions with

applications, J. Math. Anal. Appl. 274, 725-745, 2002.

1 Anand International Collage of Engineering, Jaipur, INDIA, [email protected] 2,4 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected] , [email protected] 3 Poornima Collage of Engineering, Jaipur, INDIA, [email protected]

Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number:

FEF.D1.16.001






82

[7] Chaudhry M. A. and Zubair S. M., On a Class of Incomplete Gamma Functions with

Applications, CRC Press (Chapman and Hall), Boca Raton, FL, 2002.

[8] J. Choi, A.K. Rathie and Rakesh K. Parmar, Extension of extended beta, hypergeometric

and confluent hypergeometric functions, Honam Math. J. 36(2), 339-367, 2004.

[9] Ozarslan M.A., Ozergin E., Some generating relations for extended hypergeometric

function via generalized fractional derivative operator, Math. Comput. Modelling 52, 1825—

1833, 2010.

[10] Emine Ozergin, Mehmet Ali Ozarslan, Abdullah Altin, Extension of gamma, beta and

hypergeometric function, Journal of Computational and Applied Mathematics, 235 (16),

4601-4610, 2011.


83

On the Harmonic Averages of Sequences of Fuzzy Numbers

Rahmet SAVAŞ1 and Sefa Anıl SEZER2

Abstract. In this study, we introduce summability methods of harmonic averages of

sequences fuzzy numbers and we state necessary and sufficient conditions which enable us to

obtain convergence of a sequence of fuzzy numbers from its summability by harmonic

methods.

Keywords. Harmonic averages, Abelian theorem, Tauberian theorem, slowly

oscillating sequences.

AMS 2010. 26E50, 40E05, 40C05.

References

[1] Bede, B., Mathematics of fuzzy sets and fuzzy logic, Springer, Berlin, 2013.

[2] Moricz, F., On the harmonic averages of numerical sequences, Arch. Math., 86, 375–384,

2006.

[3] Moricz, F., On the arithmetic and harmonic averages in nondiscrete setting, Math.

Inequal. Appl., 16, 633–643, 2013.

[4] Önder, Z., Sezer, S. A., Çanak, İ., A Tauberian theorem for the weighted mean method of

summability of sequences of fuzzy numbers, J. Intell. Fuzzy Systems, 28, 1403-1409, 2015.

[5] Yavuz, E., Çoşkun, H., On the logarithmic summability method for sequences of fuzzy

numbers, Soft Comput. doi:10.1007/s00500-016-2156-4, 2016.

1 Istanbul Medeniyet University, Istanbul, TURKEY, [email protected] 2 Istanbul Medeniyet University, Istanbul, TURKEY, [email protected]


84

Malliavin Calculus of Bismut Type for an Operator of Order Four on a Lie Group

Rémi LÉANDRE1

Abstract. I adapt Bismu’t way of the Malliavin Calculus to a four order generator on a

Lie group. In such a case, the semi-group does not preserve the positivity. We apply this

machinery to get Varadhan type estimates of the associated heat-kernel.

Keywords. Malliavin Calculus. Large deviation estimates.

AMS 2010. 60H07

References

[1] Léandre R, Malliavin Calculus of Bismut type for an operator of order four on a Lie

group, To appear in “Journal of pseudodifferential operators and its applications”.

[2] Léandre R.: Varadhan estimates for an operator of order four on a Lie group. In

“Control, decision and information technologies” IEEE (IEEE X-plore), 2016.

1 Université de Bourgogne-Franche-Comté, [email protected]


85

On Convergence Properties of Sequences of k-Positive Operators in the Subspace of

Analytic Functions

Seda KARATEKE1 and Tülin COŞKUN2

Abstract. In this presentation, we put in evidence that Korovkin type theorem is true

for the space 𝐴𝑔 that consists of analytic functions whose coefficients satisfy the inequality

|𝑓𝑘| ≤ 𝑀𝑓 𝑔(𝑘) such that 𝑔(𝑘) = 1 + 𝑘2𝑝. The theorem is proved by choosing the convenient

test functions. On the other hand we see that Korovkin type theorem is not valid in the space

of analytic functions by giving an example with graphics.

Keywords. Korovkin type theorem, k-positive linear operators, analytic functions.

References

[1] Gadjiev, A. D., Hacısalihoğlu, H. H., Lineer Pozitif Operatörler Dizilerinin

Yakınsaklığı, 1. Basım A.Ü.F.F Döner Sermaye İşletmesi Yayınları:31, Ankara, s. 78-100,

1995.

[2] Korovkin, P. P., Linear Operators and Approximation Theory, Hindustan Publishing,

1-100, 1960

[3] Evgrafov, M. A., Analytic Functions, W. B. Saunders Company, Philadelphia and

London, 24-46, 1966.

[4] Başkan, T., Kompleks Fonksiyonlar Teorisi, 5. Basım Nobel Yayınları, Ankara, 26-

200, 2003.

[5] Freitag, E., Busam, R., Complex Analysis, Springer-Verlag Berlin Heidelberg, New

York, 9-244, 2005.

[6] Evgrafov, M. A., The method of near systems in the space of analytic functions and its

application to interpolation, American Mathematical Society Translations, Series:2,

Volume:16, American Mathematical Society, USA, 89-201, 1960.

[7] Gadjiev, A. D., Ghorbanalizadeh, A. M., On an approximation process in the space of

analytic functions, Cent. Eur. J. Math. 8 (2), 389-398, 2010.

1 İstanbul Arel University,, İstanbul, TURKEY, [email protected] 2 Bülent Ecevit University, Zonguldak, TURKEY, [email protected]



86

[8] Gadjiev, A. D., Ghorbanalizadeh, A. M., Approximation of analytical functions by the

sequences of k-positive linear operators, Journal of Approximation Theory, (6), 1245-1255,

2010.

[9] Gadjiev, A. D., Simultaneous statistical approximation of analytic functions and their

derivatives by k-positive linear operators, Azerbaijan Jornal of Mathematics, (1), 57-66,

2011.

[10] Gadjiev, A. D., Duman, O., Ghorbanalizadeh, A. M., Ideal convergence of k-positive

linear operators, Journal of function spaces and applications, 2012.

[11] Ispir, N., Convergence of sequences of k-positive linear operators in subspaces of the

space of analytic functions, Hacettepe Bulletin of Natural Sciences and Engineering, (28), 47-

53, 1999.

[12] Kurt, S. B., Analitik fonksiyonlar uzayında lineer k-pozitif operatör dizilerinin

yakınsaklık koşulları, (YL), Bülent Ecevit Üniversitesi, Fen Bilimleri Enstitüsü (Danışman:

Doç. Dr. Tülin COŞKUN), Zonguldak, 1-73, 2010

[13] Gadjiev, A. D. and Ghorbanalizadeh, A M., On an approximation process in the space of

analytic functions, Cent. Eur. J. Math., 8 (2), 389-398, 2010.


87

Some Operator Inequalities Related to Means and Entropies

Shigeru FURUICHI1

Abstract. We give some new refinements and reverses Young inequalities for two positive

operators. Our results refine the ordering relations among the arithmetic mean, the geometric

mean and the harmonic mean for two positive invertible operators.

In addition, we give the tight bounds of Tsallis relative operator entropy by the use of

Hermite-Hadamard's inequality. We also give alternative tight bounds for the Tsallis relative

operator entropy.

Furthermore, we study the properties on monotonicity for the weight of operator means,

and for the parameter of relative operator entropies.

My talk will be composed by my recent results in [1,2,3,4,5,6,7,8].

Keywords. Operator inequality, positive operator, operator mean, Young inequality,

Hermite-Hadamard's inequality and relative operator entropy.

AMS 2010. 15A39, 15A45, 47A63, 46L05 and 47A60

References

[1] Furuichi, S., On refined Young inequalities and reverse inequalities, J. Math. Inequal.,

Vol.5(2011), 21-31.

[2] Furuichi, S., Refined Young inequalities with Specht's ratio, J. Egyptian Math. Soc.,

Vol.20(2012), 46-49.

[3] Furuichi, S., Operator inequalities among arithmetic mean, geometric mean and harmonic

mean, J. Math. Inequal., Vol.8, 669-672, 2014.

[4] Furuichi, S., M.B.Ghaemi, N.Gharakhanlu, Generalized reverse Young and Heinz

inequalities, Bull.Malays.Math.Sci.Soc., DOI 10.1007/s40840-017-0483-y

[5] Moradi, H. R., Furuichi, S., Minculete, N., Estimates for Tsallis relative operator entropy,

to appear in Math. Ineq. Appl.

[6] Furuichi, S. and Moradi, H.R., Operator inequalities among arithmetic mean, geometric

mean and harmonic mean. II, preprint.

[7] Furuichi, S. and Minculete, N., Alternative estimates for Tsallis relative operator entropy,

preprint.

[8] Furuichi, S. and Minculete, N., On monotonicity of the weight of operator means and

relative operator entropies, preprint.

1 Nihon University, Tokyo, Japan, e-mail:[email protected]


88

On a Subclass of Harmonic Univalent Functions Involving a Linear Operator

Sibel Yalçın TOKGÖZ1 and Şahsene ALTINKAYA2

Abstract. Making use of a multiplier transformation, we introduce a new class of

complex valued harmonic functions which are orientation preserving and univalent in the

open unit disk U . Relevant connections of the results presented here with various known

results are briefly indicated.

Keywords. Harmonic functions, univalent functions, Hadamard product, modified

generalized Sălăgean operator, multiplier transformation.

AMS 2010. 30C45, 30C80.

References

[1] Cho, N.E., Srivastava, H.M. Argument estimates of certain analytic functions defined by a

class of multiplier transformations, Math. Comput. Modelling, 37, 39-49, 2003.

[2] Cho, N.E., Kim, T.H., Multiplier transformations and strongly close-to-convex functions,

Bull. Korean Math. Soc. 40, 3, 399-410, 2003.

[3] Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I

Math. 9, 3-25, 1984.

[4] Dziok, J., Classes of harmonic functions defined by subordination, Abstr. Appl. Anal.

2015, Article ID 756928, 2015.

[5] Dziok, J, On Janowski harmonic functions, J. Appl. Anal. 21, 2, 99-107, 2015.

[6] Dziok, J., Jahangiri, J., Silverman, H., Harmonic functions with varying coefficients,

Journal of Inequalities and Applications, 139, 2016.

[7] Jahangiri, J.M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235,

470-477, 1999.

1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected]


89

[8] Jahangiri, J.M., Murugusundaramoorthy, G., Vijaya, K., Salagean-type harmonic

univalent functions, South J. Pure Appl. Math. 2, 77-82, 2002.

[9] Salagean, G.S., Subclasses of univalent functions, Lecture Notes in Math. Springer-

Verlag Heidelberg 1013, 362-372, 1983.

[10] Silverman, H., Harmonic univalent functions with negative coefficients, J. Math. Anal.

Appl. 220, 283-289, 1998.

[11] Silverman, H., Silvia, E.M., Subclasses of harmonic univalent functions, N. Z. J. Math.

28, 275-284, 1999.

[12] Uralegaddi, B.A. and Somanatha, C., Certain classes of univalent functions, Current

topics in analytic function theory, (Edited by H.M. Srivastava and S. Owa), 371-374, World

Sci. Publishing, Singapore, 1992.

[13] Yasar, E., Yalcin, S., Certain properties of a subclasses of harmonic functions, Appl.

Math. Inf. Sci. 7, 5, 1749-1753, 2013.


90

On a Coefficient Problem for a Subclass of Bi-Univalent Functions Defined by Using a

q-Derivative Operator

Sibel Yalçın TOKGÖZ1 and Şahsene ALTINKAYA 2

Abstract. We introduce a new class of bi-univalent functions defined by using a q-

derivative operator. Moreover, using the Faber polynomials, we obtain general coefficient

estimates for functions in this class.

Keywords. Bi-univalent functions, Faber polynomials, coefficient estimates, q-

derivative.

AMS 2010. 30C45, 05A30, 33D15.

References

[1] Airault, H., Symmetric sums associated to the factorization of Grunsky coefficients, in

Conference, Groups and Symmetries, Montreal, Canada, April, 2007.

[2] Airault, H., Bouali, H., Differential calculus on the Faber polynomials, Bulletin des

Sciences Mathematiques, 130, 179-222, 2006.

[3] Airault, H., Ren, J., An algebra of differential operators and generating functions on the

set of univalent functions, Bulletin des Sciences Mathematiques, 126, 343-367, 2002.

[4] Altınkaya, Ş., Yalçın, S., Faber polynomial coefficient bounds for a subclass of bi-

univalent functions, C. R. Acad. Sci. Paris, Ser. I, 353, 12, 1075-1080, 2015.

[5] Duren, P. L., Univalent functions, Grundlehren der Mathematischen Wissenschaften,

Springer, New York, USA, 259, 1983.

[6] Gasper, G., Rahman, M., Basic Hypergeometric Series (with a Foreword by Richard

Askey), Encyclopedia of Mathematics and Its Applications, Vol. 35, Cambridge University

Press, Cambridge, New York, Port Chester, Melbourne and Sydney, 1990; Second edition,



91

Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press,

Cambridge, London and New York, 2004.

[7] Hamidi, S. G., Jahangiri, J. M., Faber polynomial coefficient estimates for analytic bi-

close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I, 352, 17-20, 2014.

[8] Jackson, F.H., On q-functions and a certain difference operator, Transactions of the Royal

Society of Edinburgh, 46, 253-281, 1908.

[9] Srivastava, H.M., Univalent functions, fractional calculus, and associated generalized

hypergeometric functions, in Univalent Functions; Fractional Calculus; and Their

Applications (H. M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited,

Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.

[10] Zireh A., Hajiparvaneh, S., Bulut, S., Faber polynomial coefficient estimates for a

comprehensive subclass of analytic bi-univalent functions defined by subordination, Bull.

Belg. Math. Soc. Simon Stevin 23, 487-504, 2016.


92

On Asymptotically f-Statistical Equivalent Set Sequences

Sukran KONCA1 and Mehmet KUCUKASLAN2

Abstract. The aim of this paper is to introduce a generalization of statistical

convergence of asymptotically equivalent set sequences and examine some inclusion relations

related to a new concept of Wijsman asymptotically equivalent statistical convergence of

sequences of sets with respect to a modulus function f.

Keywords. Modulus function, asymptotically statistical equivalent set sequences,

Wijsman convergence.

AMS 2010. 46A45, 40G15, 40A05.

References

[1] Wijsman, R. A., Convergence of sequences of convex sets, cones and functions, Bull. Am.

Math. Soc. 70, 186-188, 1964.

[2] Aizpuru, A., Listan-Garcia, M. C., Rambla-Barreno, F., Density by moduli and statistical

convergence, Quaestiones Mathematicae. 37, 4, 525–530, 2014.

[3] Bhardwaj, V. K., Dhawan, S., Density by moduli and Wijsman lacunary statistical

convergence of sequences of sets, J. Ineq. Appl. 2017, 25, 2017. DOI 10.1186/s13660-017-

1294-2

[4] Seneta, E., Regularly varying functions, Lecture Notes in Mathematics, Springer-Verlag,

Berlin-Heidelberg-New York, 1976.

1 Bitlis Eren University, Bitlis, TURKEY, [email protected] 2 Mersin University, Mersin, TURKEY, [email protected]


93

Reverse Inequalities for the Berezin Numbers of Operators

U. YAMANCI1, M. GÜRDAL2 and M. T. GARAYEV3

Abstract. In this paper, we prove reverse inequalities for the so-called Berezin number

of some operators in reproducing kernel Hilbert space.

Acknowledgement: The second author is supported by TUBA through Young

Scientist Award Program (TUBA-GEBIP/2015).

Keywords. Positive operator, Berezin symbol, Berezin number.

AMS 2010. 47A63.

References

[1] Berezin, F.A., Covariant and contravariant symbols for operators, Math. USSR-Izv., 6,

1117-1151, 1972.

[2] Hardy, G., Littlewood, J.E., Polya, G., Inequalities, 2 nd ed. Cambridge University Press,

Cambridge, 1967.

[3] Dragomir, S.S., Reverse inequalities for the numerical radius of linear operators in

Hilbert spaces, Bull. Aust. Math. Soc., 73, 255-262, 2006.

[4] Saitoh, S., Sawano, Y., Theory of reproducing kernels and applications, Springer, 2016.

1 Suleyman Demirel University, Isparta, TURKEY, [email protected] 2 Suleyman Demirel University, Isparta, TURKEY, [email protected] 3 King Saud University, Riyadh, SAUDİ ARABİA, [email protected]


94

On the (p,q)-Analogue of Poly-Bernoulli Polynomials

Veli KURT 1

Abstract. One of the different generalization of the Bernoulli numbers and

polynomials are the (p,q)-analogue of Poly-Bernoulli polynomials. In this article, we

introduce and investigate a (p,q)-analogue of the poly-Bernoulli polynomials and numbers by

using the (p,q)-polylogarithm function. There new sequences are generalization of the poly-

Bernoulli numbers and polynomials. We give several combinatorial identities and properties

of these new polynomials.

Keywords. Poly-Bernoulli polynomials and numbers, Polylogarithm, Generating

Function.

AMS 2010. 11B68, 11B73, 11B83.

References

[1] Bayad, A, Hamahata, Y, Polylogarithms and poly-Bernoulli polynomials, Kyushu. J.

Math. 65, 15-24, 2011.

[2] Kim, D, Kim, T, A note on poly-bernoulli and higher order-poly-Bernoulli polynomials,

Russian J. of Math. Physics, 22, 1, 26-33, 2015.

[3] Komatsu, T, Ramirez, J, S, Sirvent, V.F, A (p,q)-Analogue of poly-Euler polynomials and

some related polynomials, Arxiv: 1604.3787 [Math.NT] 2016.

[4] Liu, H, Wang, W, Some identities on the Bernoulli,Euler and Genocchi polynomials via

power sums and alternate power sums, Discrete Math. 309, 3346-3363, 2009.

[5] Sandjang. P.N, On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor

formulas, Arxiv: 1309.3934 [math QA] 2013.

1 Antalya, TURKEY, [email protected]


95

APPLIED

MATHEMATICS


96

An Affine Scaling Method Using a Class of Differential Barrier Functions

Abdessamad BARBARA1

Abstract. In this paper we address a practical aspect of differential barrier penalty

functions in linear programming. In this respect we propose an affine scaling interior point

algorithm based on a large classes of differential barrier functions. The comparison of the

algorithm with a version of the classical affine scaling algorithm shows that the algorithm is

robust and efficient. We thus show that differential barrier functions open up new

perspectives in linear optimization.

1 Institut de Mathématiques de Bourgogne(IMB)- UMR 5584 CNRS, Université de Bourgogne, 9 avenue Alain Savary

BP 47870, 21078 Dijon cedex, FRANCE, [email protected]


97

Hyperbolic Tangent Solution to the Conformable Time Fractional Zakharov-Kuznetsov

Equation in 3D Space

Alper KORKMAZ1 and Ozlem Ersoy HEPSON2

Abstract. In the study, hyperbolic tangent (tanh) ansatz solution is investigated for the

conformable time fractional Zakharov-Kuznetsov Equation (fZKE) in 3D space. Transformation of the

fZKE to an ODE by the compatible wave transformation is the first step of the methodology. It is

assumed that there exists a solution of positive integer power of hyperbolic tangent form. Determining

the power of the predicted solution follows some algebra to find the relations among the other

parameters given in the solution. The final step is transforming the solution into original variables.

Keywords. Conformable fractional Zahharov-Kuznetsov Equation, exact solution, solitary

wave.

AMS 2010. 35R11, 35C07, 76B25.

References

[1] Zakharov, V., Kuznetsov, E., On three dimensional solitons, Zhurnal Eksp. Teoret. Fiz 66, 594–

597, 1974.

[2] Mace, R., Hellberg, M., The Korteweg-de Vries-Zakharov-Kuznetsov equation for electron-acoustic

waves, Physics of Plasmas 8, 2649–2656, 2001.

[3] Matebese, B., Adem, A., Khalique, C., Biswas, A., Solutions of Zakharov-Kuznetsov equation with

power law nonlinearity in (1+ 3) dimensions, Physics of Wave Phenomena 19, 148–154, 2011.

[4] Atangana, A., Baleanu, D., Alsaedi, A., New properties of conformable derivative, Open

Mathematics 13, 1–10, 2015.

1 Çankırı Karatekin University, Çankırı, TURKEY, [email protected] 2 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected]


98

Asymptotic Properties of the Lasota Equation in Various Functional Spaces

Anna POSKROBKO1 and Antoni Leon DAWIDOWICZ2

Abstract. We study asymptotic properties of the dynamical system generated by the Lasota

equation

).,()( xuFx

uxc

t

u

This equation is interesting in the context of its properties as well as its applications.

It describes the process of reproduction and differentiation of a population of red blood cells. We give

the conditions of its stability and chaos in the sense of Devaney in various functional spaces. A

examples illustrate the criteria when the semigroup generated by the equation has not asymptotic

behaviour.

Keywords. Lasota equation, chaos, stability.

AMS 2010. 35B10, 35B35, 37C75.

This work is supported by Bialystok University of Technology (Grant No. S/WI/1/2016) and founded

by the resources for research by Ministry of Science and Higher Education.

1 Faculty of Computer Science, Bialystok University of Technology, Białystok, POLAND, [email protected] 2 Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, POLAND,

[email protected]


99

Iterative Solutions to the Systems of Linear Differential Equations

Arzu GÜLEROĞLU1

Abstract. The solutions to the systems of three first-order linear differential equations with

variable coefficients are obtained by using the asymptotic iteration method. This method is applied to

some special linear systems.

Keywords. Asymptotic iteration method, linear systems.

AMS 2010. 35A24, 34G10.

References

[1] Polyanin, A. D., F. Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential

Equations, Chapman and Hall/CRC, America, 2003.

[2] Ciftci, H., Hall, R. L., Saad, N., Asymptotic iteration method for eigenvalue problems, J.

Phys. A: Math. Gen., 36, 47, 11807-11816, 2003.

[3] Ciftci, H., Hall, R. L., Saad, N, Iterative solutions to the Dirac equation, Phys. Rev. A, 72,

2, 022101, 2005.

[4] Robertson, K. M., Saad, N., Solvable systems of linear differential equations, J. Appl.

Math. Comput., 31, 475–494, 2009.

1Trakya University, Edirne, TURKEY, [email protected]


100

A Mathematical Model on Coral-Population Dynamics

Arzu ÜNAL

Abstract. Rapid increases in ocean temperatures are causing repeated thermal stresses to

populations of tropical corals, are causing differential coral mortality, and are consequently changing

the composition of coral-reef communities. Examining recovery rates of reef organisms after thermal

disturbances is a prerequisite to understanding population resilience. Here we use a nonlinear system of

differential equations to predict the dynamics of coral populations that are forced by the complexities of

multi-cyclic temperature cycles. We show that the intrinsic rates of recovery vary among species, from

0.4 to 2.3. Our forecasting simulations of the response of coral populations to three different climate

change scenarios predict that most population-recovery rates will dampen through time as thermal

stresses become more frequent.

Keywords. Nonlinear differential equation system, coral population, climate change.

AMS 2010. 92D25, 92D40, 60H10

References

[1] Crabbe, M. J. C., Global warming and coral reefs: modelling the effects of temperature on

Acropora palmata colony growth, Computational Biology and Chemistry, 31, 294–297, 2007.

[2] Crabbe, M. J. C., Climate change, global warming and coral reefs: modelling the effects of

temperature, Computational Biology and Chemistery, 32, 311–314, 2008.

[3] Hughes, T. P., Baird, A. H., Bellwood, D. R., Card, M., Connolly, S. R., Folke, C.,

Grosberg, R., Hoegh-Guldberg, O., Jackson, J. B. C., Kleypas, J., Lough, J. M., Marshall, P.,

Nystrom, M., Palumbi, S. R., Pandolfi, J. M., Rosen, B. and Roughtgarden, J., Climate change,

human impacts, and the resilience of coral reefs. Science, 301, 929-933, 2003.

[4] Van Woesik, R. and Koksal, S. , A coral population response (CPR) model for thermal

stress, Coral Reefs and Climate Change: Science and Management, 61, 129-144, 2006.

[5] Yee, S. H. , Santavy, D. L. and Barron, M. G., Assessing the effects of disease and

bleaching on Florida Keys corals by fitting population models to data. Ecological modelling,

222, 1323-1332, 2011.


101

The Class of 𝑳 ∩ 𝑫 and Its Application to Renewal Reward Process

Asli Bektaş KAMIŞLIK1, Tülay KESEMEN2, Tahir KHANIYEV3

Abstract. The class of 𝐿 ∩ 𝐷 is generated by intersection of two important subclasses of heavy

tailed distributions: The long tailed distributions and dominated varying distributions. This class itself

is also an important member of heavy tailed distributions and has some principal application areas

especially in renewal, renewal reward and random walk processes.

The aim of this study is to observe some well and less known results on renewal functions

generated by the class of 𝐿 ∩ 𝐷 and apply them into a special renewal reward process which is known in

the literature a semi Markovian inventory model of type (s,S). Especially we focused on Pareto

distribution which belongs to the 𝐿 ∩ 𝐷 subclass of heavy tailed distributions. As a first step we

obtained asymptotic results for renewal function generated by Pareto distribution from the class of 𝐿 ∩

𝐷 using some well-known results by Emrechts and Omey [3]. Then we applied the results we obtained

for Pareto distribution to renewal reward processes. As an application we investigate inventory model of

type (s,S) when demands have Pareto distribution from the class of 𝐿 ∩ 𝐷. We obtained asymptotic

expansion for ergodic distribution function and finally we reached asymptotic expansion for nth order

moments of distribution of this process.

Keywords. The class of 𝐿 ∩ 𝐷, Heavy tailed distributions, Renewal reward process, Asymptotic

expansion, Ergodic distribution, Moments for ergodic distribution.

AMS 2010. 60K05, 41A60.

Acknowledgements: The authors wish to thank to Scientific and Technological Research Council of

Turkey (TÜBİTAK), for the financial support. Project Number: 115F221

References

[1] Adler, R.J., Feldman, R.E. and Taqqu, M.S. A Practical Guide to Heavy Tails: Statistical

Techniques and Applications, Birkhäuser, Boston, 1998.

[2] Bekar, N.O., Aliyev,R. ve Khaniyev, T., Asymptotic expansions for a renewal-reward process with

Weibull distributed interference of chance, Contemporary Analysis and Applied Mathematics, 2, 1, 200-

211, 2013.

[3] Embrechts, P., Omey, E., A property of longtailed distributions, J. Appl. Prob., 21 80-87, 1984.

1 Recep Tayyip Erdoğan University, Rize, TURKEY, [email protected] 2 Karadeniz Technical University, Trabzon, TURKEY, [email protected] 3 TOBB University of Economics and Technology, Ankara, TURKEY, [email protected]




102

[4] Khaniyev, T.A., Kokangul, A. ve Aliyev, R.T., An asymptotic approach for a semi-Markovian

inventory model of type (s, S), Applied Stochastic Models in Business and Industry, 29, 5, 439 – 453,

2013.


103

Mathematical Modeling for Real Life Problems

Aysegul SAGLAM ARSLAN

Abstract. The aim of this study is to analyze the ability of students to solve real-life problems

requiring mathematical modeling. For this purpose, the researchers developed a data collection tool

aiming to evaluate students’ ability to create, apply and finalize models using experimental and

theoretical methods. A series of activities (such as Why does a spoon in a cup seem broken? What if an

object is threw down? Why do springs get jammed? How can I throw a ball into the basket pot? How is

a bicycle's energy calculated?) was organized with the participation of ten 9th grade students. Students'

answers were classified under five categories: prediction, observation, experience and measurement,

mathematical modeling, model testing. The problems, difficulties and mistakes that students

encountered in each of the mentioned categories have been identified.

The analysis of ongoing data has shown that students are more successful in the modeling

activities with the theoretical method than the modeling activities with the experimental method.

Accordingly, it has been determined that participant students can easily create a mathematical model of

a real-life problem they are working on by using an equation they have. On the other hand, it has been

observed that the participants experienced considerable difficulties while using the experimental method

in order to construct a mathematical model and to test the model by returning to the problem.

Keywords. Modelling, mathematical model, real life problems.


104

Higher Order Trigonometric B-Spline Algorithms to the Solution of Coupled Burgers’

Equation

Aysun Tok ONARCAN1 and Ozlem ERSOY HEPSON2

Abstract. Trigonometric B-spline functions of higher degrees have advantages over lower ones

since they can be used as approximate functions in the numerical methods if the differential equation

include higher order derivatives. Thus the collocation methods based on higher degree trigonometric B-

splines are set up to get numerical solutions to the Coupled Burgers’ equation. The unknowns of the

coupled Burgers equation are integrated in time by aid of the Crank-Nicolson method. Resulting time-

integrated coupled Burgers’ equation is discretized using the trigonometric B-spline collocation method.

Several initial and boundary value problems of the Coupled Burgers’ equation are studied by the

proposed technique.

Keywords. Trigonometric B-Spline, collocation method, Coupled Burgers’ Equation

AMS 2010. 41A15, 65M70.

References

[1] Esipov, S. E., Coupled Burgers Equations- A Model of Polydispersive sedimentation, James Franck

Institute and Department of Physics, University of Chicago, 1995.

[2] Mittal R. C., Arora, G., Numerical solution of the coupled viscous Burgers' equation, Commun

Nonlinear Sci Numer Simulat, 16, 1304-1313, 2011.

[3] Nikolis, A., Seimenis, I., Solving dynamical systems with cubic trigonometric splines, Applied

Mathematics E-notes, 5, 116-123, 2005

[4] Kutluay, S., Ucar, Y., Numerical solutions of the coupled Burgers' equation by the Galerkin

quadratic B-spline finite element method, Math. Meth. Appl. Sci., 36 2403-2415, 2013.

1 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected] 2 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected]


105

Numerical Solution of Nonlinear Parabolic Type Equation in the Class of Generalized

Functions with Degeneration

Bahaddin SINSOYSAL1, Hakan BAL2and Kubra YILMAZ3

Abstract. In this paper, a method for obtaining the numerical solution of the first type initial

boundary value problem for a degenerative second order parabolic equation is suggested. To this end, in

accordance with the smoothness of the solution of the investigated problem, we introduce two auxiliary

problems, as they have some advantages over the main problem, and are equivalent to the main problem

in a definite sense. Using the introduced auxiliary problems, we write more sensitive numerical scheme

whose solution express all of the physical properties of the problem accurately. Moreover, using the

auxiliary problems, we can write the higher order numerical schemes with respect to the time variable.

Some comparative results of the exact and numerical solutions are presented.

Keywords. Linear parabolic equation with degeneration, sensitive numerical algorithms

AMS 2010. 35K20, 35K65, 65M06.

References

[1] Antoncev, S. N., On the localization of solutions of non-linear degenerate elliptic and parabolic

equations, Soviet Mathematics. Doklady, 24, 420–424, 1981.

[2] Martinson, L. K. and Pavlov, K. B., On the problem of spatial localization of thermal perturbations

in the theory of non-linear heat conduction, USSR Computational Math and Math Physics, 12, 4, 261–

268, 1972.

[3] Rasulov, M. A., A numerical method of solving a parabolic equation with degeneration, Differential

Equations, 18, 8, 1418–1427, 1992.

[4] Sinsoysal, B., The Analytical and a Higher-Accuracy Numerical Solution of a Free

Boundary Problem in a Class of Discontinuous Functions, Mathematical Problems in

Engineering, Article ID 791026, 2012.

1 Beykent University, Department of Management Information Systems, Istanbul, TURKEY, [email protected] 2 Beykent University, Faculty of Economics and Administrative Sciences, Istanbul, TURKEY, [email protected] 3 Beykent University, Department of Mathematics and Computing, Istanbul, TURKEY, [email protected]


106

A Finite Differences Scheme for Solving the System of Differential Equations of Non-

Isentropic Flow in a Class of Discontinuous Functions

Bahaddin SINSOYSAL1, Hasan CARFI2 and Nazife KOC3

Abstract. In this paper for the Cauchy problem of system of equations

𝜌𝑡 + (𝜌𝑢)𝑥 = 0, (1)

(𝜌𝑢)𝑡 + (𝑝 + 𝜌𝑢2)𝑥 = 0, (2)

𝜖𝑡 + [𝜌𝑢𝜖 + 𝑝𝑢]𝑥 = 0, (3)

describing of non-isentropic flow in a class of discontinuous functions

𝜌(𝑥, 0) = 𝑓(𝑥), (4)

𝑢(𝑥, 0) = 𝑔(𝑥), (5)

𝜖(𝑥, 0) = ℎ(𝑥). (6)

Here 𝑓, 𝑔 and ℎ are known functions and 𝜖 = 𝜀 +𝑢2

2. The characteristic properties of the solution of

the system (1)-(3) have discontinuity points whose locations are unknown beforehand. These points do

not allow to apply classical numerical methods to the solution of problem (1)-(6). To this end, we

introduce the special auxiliary problem having some advantages over the main problem as follows:

𝑞𝑡 + 𝜌𝑢 = 0,

𝜋𝑡 + 𝑝 + 𝜌𝑢2 = 0,

𝜇𝑡 + 𝜌𝑢(𝜀 + 𝑝𝑢) = 0,

𝑞(𝑥, 0) = 𝑞(0)(𝑥),

𝜋(𝑥, 0) = 𝜋(0)(𝑥),

𝜇(𝑥, 0) = 𝜇(0)(𝑥),

where 𝑞(0)(𝑥), 𝜋(0)(𝑥) and 𝜇(0)(𝑥) are solutions of equations (𝑞(0))𝑥

= 𝑓(𝑥), (𝜋(0))𝑥

= 𝑔(𝑥), and

(𝜇(0))𝑥

= ℎ(𝑥), respectively.

Using the proposed auxiliary problem we develop a finite differences method to obtain

numerical solution of the problem (1)-(6) describing all of the physical properties.

Keywords. Non-isentropic flow, finite differences scheme in a class of discontinuous functions.

AMS 2010. 35L67, 65M06.

1 Beykent University, Department of Management Information Systems, Istanbul, TURKEY, [email protected] 2 Nisantasi University, Vocational SchoolUniversity, Istanbul, TURKEY, [email protected] 3 Beykent University, Institute of Sciences and Technology, Istanbul, TURKEY, [email protected]


107

A Finite Difference Method for Solving Generalized FitzHugh-Nagumo Equation

Bilge İNAN1

Abstract. In this paper, we propose Crank-Nicolson exponential finite difference method for

solving the generalized Fitzhugh-Nagumo equation. Numerical solutions obtained by this method when

compared with the exact solution are shown that the obtained solution produces high accurate results.

The comparisons showed that proposed method is reliable, precise and convenient alternative method

for solution of the generalized FitzHugh-Nagumo equation.

Keywords. Generalized Fitzhugh-Nagumo equation, Finite difference method, Exponential

finite difference method, Crank-Nicolson exponential finite difference method.

AMS 2010. 65N06, 35D99

References

[1] Bhrawy, A. H., A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-

Nagumo equation with time-dependent coefficients, Appl. Math. Comput., 222, 255-264, 2013.

[2] Ruiz-Ramírez, J., Macías-Díaz, J. E., A finite-difference scheme to approximate non-negative and

bounded solutions of a FitzHugh-Nagumo equation, Int. J. Comp. Math., 88, 3186-3201, 2011.

[3] Bhattacharya, M. C., An explicit conditionally stable finite difference equation for heat conduction

problems, Int. J. Num. Meth. Eng., 21, 239-265, 1985.

[4] Bahadır, A. R., Exponential finite-difference method applied to Korteweg-de Vries equation for

small times, Appl. Math. Comput., 160, 675-682, 2005.

[5] İnan, B., Bahadır, A. R., Numerical solution of the one-dimensional Burgers equation: Implicit and

fully implicit exponential finite difference methods, Pramana J. Phys., 81, 547-556, 2013.

1 Kilis 7 Aralık University, Kilis, TURKEY, [email protected]


108

Numerical Solutions of Boundary Value Problems for Nonlinear Delay Differential

Equations

Bülent YILMAZ1 and Volkan YAMAN2

Abstract. : In this paper we investigated solutions of boundary value problems for nonlinear

delay differential equations obtained by Adomian Decomposition Method, Differential Transformation

Method and Daftardar – Jafari Method. Comparison and error analysis of the solutions of 3 methods

have been given with three numerical examples.

Keywords. Delay Differential, Boundary value problems, Adomian Decomposition,

Differential Transformation, Daftardar-Jafari

AMS 2010. 34K28, 34L30, 34K06

References

[1] Adomian, G. and Rach, R., Inversion of nonlinear stochastic operators, Journal of Mathematical

Analysis and Applications, 91, 39–46, 1983.

[2] Adomian, G., Nonlinear Stochastic Operator Equations, Academic Press, New York, 1986.

[3] Adomian, G., Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic

Publishers, 1994.

[4] Erturk, V. S., The Numerical Solution of a Second-Order Differential Equation With Neumann

Boundary Conditions via Differential Transformation Method, Anadolu University Journal of Science

and Technology-B, Vol:1, 19-29, 2011

[5] Haziqah, C., Hussin, C., Kilicman, A., On the Solutions of Nonlinear Higher-Order Boundary Value

Problems by Using Differential Transformation Method and Adomian Decomposition Method,

Mathematical Problems In Engineering, vol: 2011, 1-19, 2011.

[6] Daftardar-Gejji, V. and Jafari, H., An Iterative Method For Solving Nonlinear Functional Equations,

Journal of Mathematical Analysis and Applications 316, 753-763, 2006.

[7] Ullah, I., Khan, H., Rahim, M.T., Numerical Solutions of Fifth and Sixth Order Nonlinear Boundary

Value Problems by Daftardar Jafari Method, Journal of Computational Engineering, Vol: 2014, Article

id: 286039, 2014

1Marmara University, Istanbul, TURKEY, [email protected] 2 Marmara University, Istanbul, TURKEY, [email protected]



109

The Analytical Solutions of the Space-Time Fractional Modified Kawahara Equation

Dilek Varol BAYRAM1, Sevil ÇULHA1, Ayşegül DAŞCIOĞLU1

Abstract. In this study, the analytical exact solutions of the space-time fractional modified

Kawahara equation are obtained for the first time in the literature. Here, the fractional derivative is

considered in conformable sense. Using Jacobi elliptic function expansion method, the solutions are

found in general form containing the hyperbolic, trigonometric, and rational functions. Also, the

complex valued solutions and soliton solutions are obtained.

Keywords. Jacobi elliptic function, fractional partial differential equation, modified Kawahara

equation.

AMS 2010. 35G20, 35L05, 35R11.

References

[1] Armitage, J. V., Eberlein, W. F., Elliptic Functions, New York: Cambridge University Press, 2006.

[2] Miller, K. S., An Introduction to Fractional Calculus and Fractional Differential Equations, J.

Wiley and Sons, New York, 1993.

[3] Kilbas, A., Srivastava, H., Trujillo, J., Theory and Applications of Fractional Differential

Equations in: Math. Studies., North-Holland, New York, 2006.

[4] Podlubny, I., Fractional Differential Equations, Academic Press, USA, 1999.

[5] Khalil, R., Horani, M. A., Yousef A., Sababheh, M., A new definition of fractional derivative,

Journal of Computational and Applied Mathematics, 264, 65–70, 2014.

[6] Abdeljawad, T., On conformable fractional calculus, Journal of Computational and Applied

Mathematics, 279, 57–66, 2015.

[7] Wazwaz, A. M., New solitary wave solutions to the modified Kawahara equation, Physics Letters

A, 360, 588–592, 2007.

[8] Fu, Z., Liu, S., Liu, S., Zhao, Q., The JEFE method and periodic solutions of two kinds of

nonlinear wave equations, Communications in Nonlinear Science and Numerical Simulation 8, 67–75,

2003.

1 Pamukkale University, Denizli, TURKEY, [email protected], [email protected], [email protected].





110

An Approximation Method for the Fractional Linear Fredholm Integrodifferential

Equations by Laguerre Polynomials

Dilek Varol Bayram1 and Ayşegül Daşcıoğlu1

Abstract. In this study, an approximation method based on the Laguerre polynomials is

presented to obtain the solutions of the fractional linear Fredholm integrodifferential equations. Here,

the fractional derivative is considered in Caputo sense. This method transforms the integrodifferential

equation to a system of linear algebraic equations by using the collocation points. Besides, some

examples are presented to illustrate the accuracy of the method and the results are discussed.

Keywords. Fredholm integrodifferential equations, Laguerre polynomials, Fractional

integrodifferential equations.

AMS 2010. 26A33, 45J05, 65N35.

References

[1] Khader, M. M., El Danaf, T. S., Hendy, A. S., Efficient spectral collocation method for solving

multi-term fractional differential equations based on the generalized Laguerre polynomials, Journal of

Fractional Calculus and Applications, 3 (13), 1-14, 2012.

[2] Miller, K.S., An Introduction to Fractional Calculus and Fractional Differential Equations, J.

Wiley and Sons, New York, 1993.

[3] Kilbas, A., Srivastava, H., Trujillo, J., Theory and Applications of Fractional Differential

Equations in: Math. Studies., North-Holland, New York, 2006.

[4] Podlubny, I., Fractional Differential Equations, Academic Press, USA, 1999.

[5] Mahdy, A. M. S., Shwayyea, R. T., Numerical solution of fractional integro-differential equations

by least squares method and shifted Laguerre polynomials pseudo-spectral method, International

Journal of Scientific & Engineering Research, 7(4), 1589-1596, 2016.

1 Pamukkale University, Denizli, Turkey, [email protected], [email protected].




111

Convergence of Backward Semi-Lagrangian Scheme with BDF2 for Two-Dimensional

Burgers’ Equation

Dojin KIM1, Soyoon BAK2, Philsu Kim3,

Abstract In this paper, we present a convergence analysis of a backward semi-Lagrangian

scheme for Burgers’ equation with Dirichlet boundary condition. We apply BDF2 for the total time

derivative and adopt the second-order central finite difference for the diffusion term together with the

local Lagrangian interpolation for the spatial discretization. For the temporal discretization, the

characteristic curve imposed by the solution of the Helmholtz equation is resolved by an error correction

method with the midpoint quadrature rule. The issues of the convergence analysis are resolved using the

discrete version of integration by parts combined with a symmetry property of the second-order central

finite difference and the Gronwall’s lemma to estimate the time evolution of the errors. Finally, we

show that the proposed scheme has the convergence order O(h2 + x2 + xp+1) in the sense of the

discrete l2-norm, where h and 4x are the step sizes of the time and the spatial grid, respectively, and p is

the degree of the interpolation polynomial. Numerical tests are presented to support the theoretical

analysis.

Keywords: Burgers’ equation, Semi-Lagrangian scheme, Non-linear advection-diffusion

equation, Convergence analysis

AMS subject classifications: 35K55, 65M06, 65M12, 65M25.

References

[1] Allievi, A., Bermejo, R., Finite element modified method of characteristics for the Navier-Stokes

equations, Int. J. Numer. Methods Fluids 32(4), 439-463, 2000.

[2] Bermúdez, A. , Nogueiras, M. R. , Vázquez, C., Numerical analysis of convection-diffusion-reaction

problems with higher order characteristics/finite elements I & III: time discretization, SIAM J. Numer.

Anal. 44, 2006.

[4] Falcone, M. , Ferretti, R., Convergence analysis for a class of high-order semi-Lagrangian

advection schemes, SIAM J. Numer. Anal. 35(3), 909-940, 1998.

[5] Kim, P., Piao, X., Kim, S. D., An error corrected Euler method for solving stiff problems based on

Chebyshev collocation, SIAM J. Numer. Anal. 49, 2211-2230, 2011.

1 Department of Mathematics, Kyungpook National University, Daegu, SOUTH KOREA, [email protected], 2 Department of Mathematics, Kyungpook National University, Daegu, SOUTH KOREA, [email protected], 3 Department of Mathematics, Kyungpook National University, Daegu, SOUTH KOREA, [email protected]


112

[6] Piao, X., Bu, S., Bak, S., Kim, P., An iteration free backward semi-Lagrangian scheme for solving

incompressible Navier-Stokes Equations, J. Comput. Phys. 283, 189-204, 2015.

[7] Piao, X. , Kim, S., Kim, P., An iteration free backward semi-Lagrangian scheme for Guiding center

problems, SIAM J. Numer. Anal. 53(1), 196-643, 2015.


113

Compressed Sensing with Cyclic-S Hadamard Matrix for TeraHertz Imaging

Applications

Esra Şengün ERMEYDAN1 and İlyas ÇANKAYA2

Abstract. In this study, for single pixel imaging applications, Compressed Sensing(CS) with

circular Hadamard matrix is proposed. In single pixel imaging scheme,𝑁2 samples should be taken for

√𝑁𝑥√𝑁 pixel image. CS is a popular technique claiming that the sparse signals can be reconstructed

with samples under Nyquist rate [1],[2]. CS precodes the analog signal, reduces acquisition time, power

consumption and hardware complexity since the image can be reconstructed using fewer samples.

Therefore to solve the slow data acquisition problem in Terahertz (THz) single pixel imaging, CS is a

good candidate [3]. In CS single pixel imaging, the beam is modulated via a number of √𝑁𝑥√𝑁 pixel

masks which is placed after the object and the imaged is reconstructed with generally Total variation

based algorithms using the acquired modulated signal and the information of the masks. The single pixel

cameras in optical band uses spatial light modulators(SLM) to realize the masks [4], however there is no

commercial SLM for THz. Therefore, the masks are printed on PCB to have ON/OFF switches, since

PCB material is considered as transparent for the THz beam and the copper material is considered

opaque. The mechanical PCB masks should be changed manually or mechanically with automatic

translation stage. Since each mask should be printed separately, most probably it will result more than

one PCB board depending on pixel size and resolution. The change of each mask or each PCB board for

each measurement is very disadvantageous especially for real-time or commercial real world THz

applications. To solve this problem, circular masks can be designed so that for each measurement only

one pixel column shift will be enough to change the mask [5]. In this study, the CS masks are designed

using cyclic-S matrixes based on Hadamard transform for 9x7 and 15x17 pixel images. The %50

compressed images are reconstructed using total variation based TVAL3 algorithm [6]. Matlab

simulations demonstrates that cyclic-S matrixes can be used for single pixel imaging based on CS. The

circular masks have the advantage to reduce the mechanical SLMs to a single sliding strip, whereas the

CS helps to reduce acquisition time and energy since it allows to reconstruct the image from fewer

samples.

Keywords. Compressed Sensing, Hadamard Transform, TeraHertz, single-pixel Imaging,

sparsity, total variation minimization.

AMS 2010. 53A40, 20M15.

References

1 Ankara Yıldırım Beyazıt University, Ankara, TURKEY, [email protected] 2 Ankara Yıldırım Beyazıt University, Ankara, TURKEY, [email protected]


114

[1] Donoho, D., Compressed Sensing, IEEE Transactions on Information Theory, 52(4), 1289–1306,

2006.

[2] Candes, E., Romberg J., Tao T., Robust Uncertainty Principles: Exact Signal Reconstruction from

Highly Incomplete Frequency Information, IEEE Transactions on Information Theory 52, 489–509,

2006.

[3] Chan, W. L., Charan, K., Takhar, D., Kelly, K. F., Baraniuk, R. G., Mittleman, D. M., A single-pixel

terahertz imaging system based on compressed sensing, Applied Physics Letters 93, 121105, 2008.

[4] Duarte, M. F., Davenport, M. A., Takbar, D., Laska, J. N., Sun, T., Kelly, K. F., Baraniuk, R. G.,

Single-pixel imaging via compressive sampling, IEEE Signal Processing Magazine 25, 83–91, 2008.

[5] Vasile, T., Damian, V., Coltuc, D., Petrovici, M., Single pixel sensing for THz laser beam profiler

based on Hadamard Transform, Optics & Laser Technology 79, 173–178, 2016.

[6] Li, C., Yin, W., Jiang, H., Zhang, Y., An efficient augmented lagrangian method with applications to

total variation minimization, Computational Optimization and Applications 56, 507–530, 2013.


115

Symbolic Regression: An Application to the Development of Models for the Surface

Tension

Eva T. López SANJUÁN1 and María Isabel Parra ARÉVALO2

Abstract. Genetic programming is an optimization technique based on stochastic, evolutionary

principles that is used to find global extreme of a given function [1]. In particular, symbolic regression

works with a set of possible expressions, represented as parse trees, and evolve them in order to

summarize the data response. It has emerged as one of the dominate modeling technology in recent

years, due to algorithmic and computer advances that have extended the scope of problems to which it

can be applied.

In this work, we study the symbolic regression algorithm and apply it to find a model for a

physical measure: the surface tension of fluids. We use a set of data, corresponding to the surface

tension of alcohols, that has been carefully selected and filtered in previous works [2], and perform

symbolic regression to develop mathematical expressions for the surface tension in terms of some

variables (temperature, critical temperature, critical pressure, molecular weight, and molar volume). We

analyze the validity of the models, and select the optimum ones.

Keywords. Symbolic Regression, Genetic Programming, Surface Tension.

AMS 2010. 62P35, 74A15, 97M10.

Acknowledgements This work was supported by GR15146 project from the Gobierno de Extremadura

and FEDER.

References

[1] Koza, J.R., Genetic Programming on the Programming of Computers by Means of Natural

Selection, MIT Press, 1998.

[2] Mulero, A., Cachadiña, I. and Sanjuán, E.L. Surface Tension of Alcohols. Data Selection and

Recommended Correlations, J. of Phys. and Chem. Ref. Data 44, 33104 2015.

1 University of Extremadura, Badajoz, SPAİN, [email protected] 2 University of Extremadura, Badajoz, SPAİN, [email protected]


116

On the Stability Analysis of Sturm-Liouville Operator with Jump Conditions

Etibar S. Panakhov1 and Ahu Ercan2

Abstract. In this paper we study stability of inverse problem for discontinuous Sturm-

Liouville equation using Ryabushko’s method. We have established certain stability of the

spectral functions of two spectral problems for discontinuous Sturm-Liouville operator. The

method which used was given firstly by Ryabushko for regular Sturm-Liouville problem in [2].

Keywords. Discontinuity, stability, spektral function.

AMS 2010. 34B09, 34D20.

References

[1] Marchenko, V. A., Maslov, K. V., Stability of the problem of recovering the Sturm-

Liouville operator from the spectral function, Mathematics of the USSR Sbornik, 81(123), 475-

502, 1970.

[2] Ryabushko T. I., Stability of the reconstruction of a Sturm–Liouville operator from two

spectra, II Teor. Funkts. Anal. Prilozhen, 18, 176–85, 1973 (in Russian).

[3] Hald, O. H., Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 37,

539-577, 1984.

[4] Panakhov, E. S., Sat, M., Reconstruction of potential function for Sturm-Liouville operator

with Coulomb potential, Boundary Value Problems, 49, 1-9, 2013.

1 Baku State University, Baku, Aizerbaijan, [email protected] 2 Firat University, Elazig, Turkey, [email protected]


117

Inverse Nodal Problem for p-Laplacian Diffusion Equation with Polynomially Dependent

Spectral Parameter

Etibar S. PANAKHOV1,2 Emrah YILMAZ3 and Tuba GULSEN4

Abstract. We consider the below p-Laplacian diffusion eigenvalue problem

( 1) 2 ( 1)( ) ( 1) ( ) 2 ( ) , 0 1,p p

m my p q x r x y x (1)

with the boundary conditions

(0) 0, (0) 1,y y (2)

(1, ) ( ) (1, ) 0,y f y (3)

where 1p is a constant,

2

1 2( ) ... , , 0, ,m

m i mf a a a a a m

is a spectral parameter; 2 1

20,1 , 0,1m mq x L r x W are real-valued functions defined

in the interval 0 1x for all m and

( 1)( 1) sgn ,ppy y y

is considered. To find the spectral datas as eigenvalues and nodal parameters of this problem,

we used a modified Prüfer substitution. Then, reconstruction formula of the potential function

is also given by using nodal lenghts. Furthermore, this method is similar to used in [6], our

results are more general.

Keywords. Inverse Nodal Problem, Prüfer Substitution, Diffusion Equation.

AMS 2010. 34A55, 34L05, 34L20.

References

1 Firat University, Elazig, Turkey, [email protected] 2 Baku State University, Baku, Azerbaijan, [email protected]

3 Firat University, Elazig, Turkey, [email protected] 4 Firat University, Elazig, Turkey, [email protected]




118

[1] Bairamov, E., Cakar O. and Celebi, A. O., Quadratic pencil of Schrödinger operators with

spectral singularities: discrete spectrum and principal functions, J. Math. Anal. Appl., 216(1),

303-320, 1997.

[2] Chen, H. Y., On generalized trigonometric functions, Master of Science, National Sun Yat-

sen University, Kaohsiung, Taiwan, 2009.

[3] Gulsen, T., Yilmaz, E., Inverse nodal problem for p_Laplacian diffusion equation with

polynomially dependent spectral parameter, Commun., Ser. A1; Math. Statist., 65(2), 23-36,

2016.

[4] Guseinov, G. S., On the spectral analysis of a quadratic pencil of Sturm-Liouville

operators, Sov. Math. Dokl., 32, 1292-1296, 1985.

[5] Jaulent, M. and Jean, C., The inverse s-wave scattering problem for a class of potentials

depending on energy, Commun. Math. Phys., 28(3), 177-220, 1972.

[6] Koyunbakan, H., Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville

equation, Bound. Value Probl., 2013:272, 2013. (Erratum: Inverse nodal problem for p-

Laplacian energy-dependent Sturm-Liouville equation, Bound. Value Probl., 2014: 222, 2014.

[7] Sat, M. and Panakhov, E. S., Spectral problem for diffusion operator, Appl. Anal., 93 (6),

1178-1186, 2014.

[8] Wang, W. C., Cheng, Y. H. and Lian, W. C., Inverse nodal problems for the p-Laplacian

with eigenparameter dependent boundary conditions, Math. Comput. Modelling, 54(11-12),

2718-2724, 2011.

[9] Yang, C. F. and Yang, X. P., Inverse nodal problem for the Sturm-Liouville equation with

polynomially dependent on the eigenparameter, Inverse Prob. Sci. Eng., 19(7), 951-961, 2011.


119

On the Stress Distribution in the Elastic Body with a Locally Curved Triple Walled

Carbon Nanotube

Fatma COBAN1and Reşat KOSKER2

Abstract. In the framework of the piecewise homogeneous body model with the use of the

three-dimensional exact equations of the theory of elasticity, the method developed for the problem

related to the determination of the stress distribution in the infinite elastic body containing a single

locally curved triple-walled carbon nanotube (TWCNT) is considered [1]. The TWCNT is modeled as

concentrically-nested three circular locally curved hollow cylinders between which there is free space

[2]. It is assumed that on the inner surface of the outer tube (cylinder) and on the outer surface of the

inner tube (cylinder) of the TWCNT full slipping conditions satisfy. In addition to this, it is assumed

that the difference between the radial displacements of the adjacent surfaces of the tubes resists with van

der Waals forces [3].

It is assumed that thicknesses and radii of tubes of the TWCNT are constants along the tubes. Under this

assumption, surface equations and the components of its normal are written. Expressions of the shear

stresses are obtained from that. The boundary value problem is obtained by using strain-displacement

relations, equilibrium equations with boundary conditions. Mentioned boundary value problem is solved

by boundary form perturbation method and shear stresses are calculated. Effect of the various

parameters on the shear stresses are investigated and interpreted.

Acknowledgement: This research has been supported by Yıldız Technical University Scientific

Research Projects Coordination Department. Project Number: 2013-07- 03-DOP01.

Keywords: Triple walled carbon nanotube, local curving, shear stress

References

[1] Akbarov, S. D., Stability Loss and Buckling Delamination: Three-Dimensional Linearized Approach

for Elastic and Viscoelastic Composites, Springer, 465 pp., 2012.

[2] Coban F., Yerel Eğrilikli İki ve Üç Duvarli Karbon Nanotüplerin Gerilme ve Stabilite Analizi,

Doktora Tezi, 2016.

[3] Akbarov, S. D., Microbuckling of a Double-Walled Carbon Nanotube Embedded in an

Elastic Matrix, International Journal of Solids and Structures, 50 (2013) 2584- 2596, 2013.

1 Yildiz Technical University, Istanbul, TURKEY, [email protected] 2 Yildiz Technical University, Istanbul, TURKEY, [email protected]


120

Construction of Exact Solutions to Partial Differential Equations via CRE Method

Filiz TAŞCAN1 and Arzu AKBULUT2

Abstract. In this paper, the consistent Riccati expansion (CRE) method [1], [2] is presented for

constructing new exact solutions of (1+1) dimensional nonlinear dispersive modified Benjamin Bona

Mahony (DMBBM) and mKdV-Burgers equations. The obtained exact solutions are consist of

hyperbolic and exponential functions [3], [4]. The results obtained confirm that the proposed method is

an efficient technique for analytic treatment of a wide variety of nonlinear partial differential equations.

Keywords. Partial differential equations, exact solution, the consistent Riccati expansion.

AMS 2010. 35R50, 83C15.

References

[1] Xiang-Li, J., Sen-Yue, L., CRE method for solving mKdV equation and new interactions between

solitons and cnoidal periodic waves, Commun. Theor. Phys., 63, 7-9, 2015.

[2] Chen, M., Hu H., Zhu H., Consistent Riccati expansion and exact solutions of the Kuramoto-

Sivashinsky equation, App. Math. Lett., 49, 147-151, 2015.

[3] Khan, K., Akbar, M. A., Raynaul Islam S. M., Exact solutions for (1 + 1)-dimensional nonlinear

dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations,

SpringerPlus, 3, 724, 2014.

[4] Wang, M., Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A., 213, 279-287,

1996.



121

Numerical Solution of an Inverse Problem for the Stokes Equations

Gullazata DAIRBAYEVA1

Abstract. In the domain 𝑄 = (𝑥, 𝑦) ∈ 𝑅2: −2𝜋 < 𝑥 < 2𝜋, 𝑐𝑜𝑠𝑥 + 1 < 𝑦 < 𝑐𝑜𝑠𝑥 + 3 we

consider the Cauchy problem for the Stokes system

∆𝑢 − ∇𝑝 = 0, (1)

𝑑𝑖𝑣 𝑢 = 0, (2)

𝑢 = 𝜑, (𝑥, 𝑦) ∈ 𝛾0, (3)

𝑝𝑛 −𝜕𝑢

𝜕𝑛 = 𝑓, (𝑥, 𝑦) ∈ 𝛾0, (4)

where 𝑢 = (𝑢1, 𝑢2), 𝜕𝑄 = 𝛾0 ∪ 𝛾1 is the boundary of the domain 𝑄, 𝛾1 = (2𝜋, 𝑦): 𝑐𝑜𝑠𝑥 + 1 ≤

𝑦 ≤ 𝑐𝑜𝑠𝑥 + 3, 𝑛 = (𝑛1, 𝑛2) is the outward unit normal to 𝜕𝑄, 𝜑 ∈ 𝐻00

12⁄

(𝛾0), 𝑓 ∈ (𝐻00

12⁄

(𝛾0))

∗

. This

spaces are considered in [1].

The value of the solution is not known on 𝛾1. The problem (1)-(4) is ill-posed. It can be

formulated as the inverse problem to some well-posed problem. Analogous approach for other problems

can be found in [2].

Let’s consider the problem (1)-(3) with the following condition

𝑝𝑛 −𝜕𝑢

𝜕𝑛 = 𝑞, (𝑥, 𝑦) ∈ 𝛾1, (5)

where the function 𝑞 = (𝑞1, 𝑞2) is given. The problem (1)-(3), (5) we shall call the direct problem. This

problem is well-posed. Then the initial problem (1)-(4) is reduced to the following inverse problem to

the direct problem: it is required to determine 𝑞 on 𝛾1 using the additional information (4) about the

solution of the (1)-(3), (5).

The inverse problem (1)-(5) is solved numerically on the bases combination of the finite

element method [3] end the optimization method. The estimate on convergence rate of the algorithm

with respect to the functional is obtained.

Keywords. Stokes Equations, Inverse Problem.

References

[1] Bastay, G., Johansson, T., Lesnik. D., Kozlov, K., An Alternating Method for the Stationary Stokes

System, ZAAM (Z. Angew.Math.Mech). 86, 268-280. 2006/

[2] Kabanikhin, Sr., Inverse and ill-posed problems, The Siberian scientific publishing house, 2009.

[3] Segerlind, Larry J., Applied finite element analysis, New York, 1984.

1 Al-Farabi Kazakh National University, Almaty, KAZAKHSTAN, [email protected]


122

Existence of Solution for the Problem with a Concentrated Source in a Subdiffusive

Medium

H. Terence Liu1

1 Tatung University, Taipei, TAIWAN, [email protected]


123

Numerical Solution of the Conformable Fractional Diffusion Equation

Handan Çerdik YASLAN1

Abstract. In this work, a numerical approach for solving space-time fractional diffusion

equation with variable coeffcients is proposed by using shifted Chebyshev polynomials. The fractional

derivatives are described in the conformable sense. The space-time fractional dffusion equation with

variable coeffcients is reduced to a system of ordinary differential equations by using the properties of

Chebyshev polynomials. The finite difference method is applied to solve this system of equations.

Numerical results are provided to verify the accuracy and effciency of the proposed approach.

Keywords. Space-time fractional diffusion equation, Conformable fractional derivative, Shifted

Chebyshev polynomials, Finite difference method.

AMS 2010. 26A33, 35K05, 65M70.

References

[1] Khader, M. M., Numerical treatment for solving fractional Riccati dif ferential equation, J. Egyptian

Math. Soc. 21, 32-37, 2013.

[2] Li, Y., Solving a nonlinear fractional di_erential equation using Chebyshev wavelets, Commun.

Nonlinear Sci. Numer. Simulat. 15, 2284-2292, 2010

[3] Balaji, S., Solution of nonlinear Riccati di_erential equation using Chebyshev wavelets, WSEAS

Trans. Math. 13, 441-451, 2014.

[4] Dalir, M., Bashour, M., Applications of fractional calculus, Appl. Math. Sci. 4, 1021-1032, 2010.

[5] Sweilam, N.H., Nagy, A.M., El-Sayed, A. A., Second kind shifted Chebyshev polynomials for

solving space fractional order diffusion equation. Chaos Soliton Fract 73, 141-147, 2015.

[6] Sweilam, N. H., Nagy, A. M., El-Sayed, A. A., On the numerical solution of space fractional order

di_usion equation via shifted Chebyshev polynomials of the third kind. J King Saud Univ Sci 28, 41-47,

2016.

1Pamukkale University, Sakarya, TURKEY, [email protected]


124

Oscillation Criteria Fractional Difference Equations with Mixed Nonlinearities

Hande BÜYÜKÇAVUŞOĞLU1, Mustafa Kemal YILDIZ2 and Tuğba YALÇIN UZUN3

Abstract. In this article, we consider higher order fractional nonlinear difference equation of the

type

1

1

1

1

)(

10,0),()()()()()()(

atx

qattvtxtxtgtxtptx

at

q

ta

m

i

i

q

ta

i

where Δ𝑞 is Riemann-Liouville like discrete fractional difference operator of order q. We

obtain some oscillation criteria for this equation.

Keywords. Difference Equations, Oscillation, Nonlinear, Fractional Order.

AMS 2010. 26A33, 39A11, 39A12.

References

[1] Shao, J., Zheng, Z. and Meng, F., Oscillation criteria for higher order fractional differential

equations with mixed nonlinearities, Advanced in Difference Equations, 2013:323, 2013.

[2] Chen, D. X., Qu, P. X. and Lan, Y. H., Forced oscillation of certain fractional differential

equations, Advanced in Difference Equations, 2013:125, 2013.

[3] Grace, S. R., Agarwal, R. P., Wong, P. J. Y. and Zafer, A., On the oscillation of fractional

differential equations, An International Journal for Theory and Applications, v:15, n:2, 2012.

[4] Diethelm, K., The Analysis of Fractional Differential Equations, Springer, Berlin 2010.

.

1 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected] 2 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected] 3 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected]


125

Free Vibration of a Tapered Euler-Bernoulli Beam with 3D Tip Mass by Multi-Step

Differential Transformation Method

Hilal Doğanay KATI1 and Hakan GÖKDAĞ2

Abstract. Free vibration of beam-rigid tip mass system has been studied by many investigators

[1-10]. The interest in the dynamic analysis of such a structure is partly because of its suitability for

providing successful design for structures such as robot arms and manipulators. In many studies the

vibration of uniform beam-tip mass system has been dealt with analytically. However, there are few

studies about non-uniform beam-tip mass systems. Besides, majority of these works assume the tip mass

as concentric and coincident with beam end point. Thus, in this work, coupled flexural and torsional free

vibration of a tapered beam with rigid 3D tip mass whose center of gravity is, in general, not at the beam

end point is considered with the assumption of the Euler-Bernoulli beam theory. First, a set of governing

differential equations and boundary conditions are obtained for the considered system by Hamilton’s

Principle. Since the exact solution of the equations of motion is not easy, a semi-analytical method

called Multi-Step Differential Transform Method (MDTM) is applied. Later, the effects of tip mass,

beam length and taper ratio on the natural frequencies are examined. In addition, the results are verified

by ANSYS, famous finite element software. It is proved that the results by MDTM and ANSYS are in

well agreement.

Keywords. Differential Transform Method, Multi-Step Differential Transform Method,

Bending Vibration, Torsional Vibration, Coupled Deformation

AMS 2010. 53A40, 20M15.

References

[1] Bhat, B. R., and Wagner, H., Natural frequencies of a uniform cantilever with a tip mass slender in

the axial direction, Journal of Sound and Vibration, 45, 2, 304-307, 1976.

[2] Auciello, N. M., Transverse Vibrations of a Linearly Tapered Cantilever Beam with Tip Mass of

Rotatory Inertia and Eccentricity, Journal of Sound and Vibration, 194, 1, 25-34, 1996.

[3] Hoa, S. V., Vibration of a Rotating Beam With Tip Mass, Journal of Sound and Vibration, 67, 3,

369-381, 1979.

[4] Oguamanam, D. C. D., Free Vibration of beams with Finite Mass Rigid Tip Load and flexural-

torsional coupling, International Journal of Mechanical Sciences, 45, 963-979, 2003.

1 Bursa Technical University, Bursa, TURKEY, [email protected] 2 Bursa Technical University, Bursa, TURKEY, [email protected]


126

[5] Gökdağ, H., Kopmaz, O., Coupled Bending and Torsional Vibration of a Beam with inspan and tip

attachments, Journal of Sound and Vibration, 287, 3, 591-610, 2005.

[6] Oguamanam, D.C.D., Arshad, M., On the natural frequencies of a flexible manipulator with a tip

load, Proceedings of the Institution of Mechanical Engineers, 219, 1199-1205, 2005.

[7] Salarieh, H., Ghorashi, M., Free Vibration of Timoshenko Beam with Finite Mass Rigid Tip Load

and Flexural Torsional Coupling, International Journal of Mechanical Science, 48, 763-779, 2006.

[8] Mabie, H. H., Rogers, C. B., Transverse vibrations of double-tapered cantilever beams with end

support and with end mass, Journal of the Acoustical Society of America, 55, 5, 986-991, 1974.

[9] Lee, T. W., Transverse Vibrations of a Tapered Beam Carrying a Concentrated Mass, Applied

Mechanics, 43, 2, 366-367, 1976.

[10] Kati H. D., Gokdag H., Free vibration of a Timoshenko beam carrying three dimensional tip mass:

Analytical solution and experimental modal testing, Material Testing, 59, 6, 591-597, 2017.

[11] Liu Z., Yin Y., Wang F., Zhao Y., Cai L., Study on modified differential transform method for free

vibration analysis of uniform Euler-Bernoulli beam, Structural Engineering and Mechanics, 48, 5, 697-

709, 2013.

[12] Özdemir Ö., Kaya M.O., Flapwise bending vibration analysis of a rotating tapered cantilever

Bernoulli-Euler Beam by differential transform method, Journal of Sound and Vibration, 289, 413-420,

2006.

[13] Torabi K., Afshari H., Zafari E., Transverse Vibration of Non-uniform Euler-Bernoulli Beam Using

Differential Transform Method (DTM), Applied Mechanics and Materials, 110-116, 2400-2405, 2012.

[14] Ertürk V. S, Odibat Z. M., Monami S., The Multi-Step Differential Transform Method and ıts

Application to Determine the Solutions of Non-Linear Oscillators, Adv. Appl. Math. Mech., 4, 4, 422-

438, 2012.


127

Image Segmentation with Fan-shaped Gradients

Hosook KIM1 and Hyoungseok KIM2

Abstract. Image segmentation is the process of partitioning a digital image into multiple

segments, which plays a fundamental role in computer vision [1]. The goal of segmentation is to find

out the boundaries of more meaningful regions in images. The pixels in a region are similar with respect

to some characteristic such as color, intensity, or texture. So the sudden changes occur at the pixels of

boundaries of the region.

The most important process of image segmentation is how to exactly and efficiently find out the

sudden changes. In a mathematical sense, a digital image is a two-dimensional one-valued function

f: N × N → R, where N is the set of natural numbers and R is the set of real numbers. Since the domain

is a discrete space, there are some difficulties in applying directly the concept of gradient of continuous

space to image segmentation in order to find out the sudden changes. The domain has only restricted

directions such as horizontal and vertical directions. The pixels of images may be in general grouped

because an image is acquired by taken a picture of real-world consisting of isolated objects. The

neighbor pixels have similar intensities and are influenced by each other.

In this research, we propose a new fan-shaped gradient of image which is one variety of

grouped gradients. To confirm pros and cons of our gradient, we apply the fan-shaped gradient to the

level-set algorithm proposed by Li et. al [2].

Keywords. Gradient, Image Segmentation.

AMS 2010. 68U10, 68R01.

References

[1] Szeliski, R., Computer Vision: Algorithms and Applications, Springer, New-York, 2011.

[2] Li, C., Xu, C., Cui, C., Fox, M., Distance Regularized Level Set Evolotion and Its Application to

Image Segmentation, IEEE Trans. On Image Processing, Vol. 19, No. 12, pp 3243-3254, 2010.

1 KSA of KAIST, Busan, REP. OF KOREA, [email protected] 2 Dong-Eui University, Busan, REP. OF KOREA, [email protected]


128

The Computational and Experimental Studies on (E)-4-Methyl-N'-(1-(Pyridin-2-Yl)

Ethylidene) Benzenesulfonohydrazide

Hümeyra BATI1, Erbil Murat AYDIN2, Murat ÇINARLI3, Nezihe ÇALIŞKAN4

Abstract. In this work, the single crystal structure, (E)-4-methyl-N'-(1-(pyridin-2-

yl)ethylidene)benzenesulfonohydrazide (I), has been synthesized and characterized by IR, UV spectra

and X-ray diffraction methods. The structure was solved by direct methods using SHELXS-97 and

refined by a full-matrix least-squares procedure using the SHELXL-97 program [1]. The crystal

structure of the compound (I) obtained in P 21/c space monoclinic space group, from the results of X –

ray diffraction. In addition, the optimized structure, the vibrational assignments, the molecular orbital

energies, molecular electrostatic potential maps have been investigated by using Density Functional

Theory with B3LYP/6-311G basis set. Gaussview molecular visualization program [2] and the Gaussian

03 program [3] were used to perform DFT calculations. UV–Visible spectrum of the compound was

recorded and the electronic properties HOMO and LUMO energies were measured by time-dependent

TD-DFT approach. The experimental results of the compound have been compared with theoretical

results and it is found that the experimental data show good agreement with calculated values.

Keywords. Hydrazone, DFT calculations, crystal structure analysis.

References

[1] Sheldrick, G. M., SHELXS-97 and SHELXL-97, Univ. Göttingen, Germany, 1997.

[2] Frish, A., Nielseni, A.B., Holder, A.J., Gaussview User Manuel, Gaussian Inc., Pittsburg, 2001.

[3] Frisch, M.J., Trucks, G.W., Schlegel, H.B., Scuseria, G.E., Robb, M.A., Cheeseman, J.R.,

Zakrzewski, V.G., Montgomery, J.A., Stratmann, R.E., Burant, J.C., Dapprich, S., Millam, J.M.,

Daniels, A.D., Kudin, K.N., Strain, M.C., Farkas, O., Tomasi, J., Barone, V., Cossi, M., Cammi, R.,

Mennucci, B., Pomelli, C., Adamo, C., Clifford, S., Ochterski, J., Petersson, G.A., Ayala, P.Y.,Cui, Q.,

Morokuma, K., Malick, D.K., Rabuck, A.D., Raghavachari, K., Foresman, J.B., Cioslowski, J., Ortiz,

J.V., Baboul, A.G., Stefanov, B.B., Liu, G., Liashenko, A., Piskorz, P., Komaromi, I., Gomperts, R.,

Martin, R.L., Fox, D.J., Keith, T., Al- Laham, M.A., Peng, C.Y., Nanayakkara, A., Gonzalez, C.,

Challacombe, M., Gill, P.M.W., Johnson, B., Chen, W., Wong, M.W., Andres, J.L., Gonzalez, C., Head

Gordon, M., Replogle, E.S., Pople, J.A., GAUSSIAN 03, Revision B.02, Wallingford, CT, 2004.

1 Ondokuz Mayis University, Samsun, Turkey, [email protected] 2 Gazi University, Ankara, Turkey, [email protected] 3 Ahi Evran University, Kırşehir, Turkey, [email protected] 4 Gazi University, Ankara, Turkey, [email protected]


129

An Application of Integration Interpolation Method to Cauchy Type Singular Integral

Equation System

Hüseyin OĞUZ1 and Elçin YUSUFOĞLU2

Abstract: In this study, a numerical solution method for the Cauchy type singular

integral equation system was developed by using the Gauss quadrature formula and Lagrange

interpolation polynomial.

The proposed algorithm is applied to a multi contact problem in order to show the

effectiveness of the algorithm.

Keywords: Singular Integral Equation, Plane Contact Problem, Cauchy kernel, Gauss–Jacobi

quadrature, A system of linear algebraic equation,

References

[1] Muskhelishvili, N. I, Singular İntegral Equations, Wolters-Noordhoff Publishing, Groningen, 1958.

[2] Erdoğan, F., Approximate Solutions of Systems of Singular Integral Equations, Society for Industrial

and Applied Mathematics, 17, 1041-1059, 1969.

[3] F. Erdogan and G.D. Gupta, Cook T.S, The numerical solution of singular integral equations, Q. J.

Appl. Math. 29, 525–534, 1973.

1 Dumlupınar University, Kütahya, TURKEY, [email protected] 2 Uşak University, Uşak, TURKEY, [email protected]




130

A New Analysis for Fractional Order Ebola Model with Non-Local and Non-Singular

Kernel

Ilknur KOCA

Abstract. The mathematical model portraying the spread of Ebola hemorrhagic fever is

considered here within the scope of fractional differentiation with non-local and non-singular kernel.

This fatal disease first occurred in Central Africa near tropical rainforests in 1976. Using classical

differentiation and fractional differentiation with power law kernel, the model was established. However

the model with classical differentiation does not take into account the memory effect as well as the

complexity of the dynamical system of the spread. On the other hand the model with fractional

differentiation based on power law kernel does not accurately take into account the full memory effect

due to the bad memory induced by the power law. Different from other works, we extend model by

replacing derivative with newly established derivative with fractional order called Atangana-Baleanu

derivative in Caputo sense. Detailed analysis of existence and uniqueness of exact solution is presented

using the Banach fixed point theorem

Keywords. fractional derivative, epidemic model, fixed point theorem.

AMS 2010. 34A34, 47H10, 65L07.

References

[1] http://www.who.int/mediacentre/factsheets/fs103/en/June 2017

[2] Odibat, Z. M., Momani, S., Application of variational iteration method to nonlinear di¤erential

equation of fractional order, International Journal of Nonlinear Sciences and Numerical Simulations, 7,

27-34, 2006.

[3] Atangana, A., Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel:

Theory and application to heat transfer model, Thermal Science, 20(2), 763-769, 2016.

[4] Atangana, A., Koca, I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with

fractional order, Chaos Solitons Fractals, 89, 447-454, 2016.

http://www.who.int/mediacentre/factsheets/fs103/en/June%202017


131

Linearization of the State Equation of a Nonlinearly Elastic Material of a Circular

Composite Shaft for Estimation of Natural Frequencies of Torsional Vibrations

I.A. TARASYUK1 and A.S. KRAVCHUK2

Abstract. The present work is devoted to generalization of the torsional vibrations equation of a

circular composite shaft [1, 2] to the case of nonlinearly elastic behavior of the material. The purpose of

the research is to obtain analytically approximate expression for natural frequencies depending on

deformation characteristics of material components which can be used in engineering and construction

designing.

The generalization has been carried out using new homogenization technique [3]. The method is

based on obtaining the effective characteristics of a composite, on average isotropic material from the

rule of a mixture for the state equations got by applying the Voigt’s [5] and the Reuss’ [6] hypotheses of

strains and stresses homogeneity, respectively. Nonlinearly elastic behavior of the material has been

modelled by using the bilinear Prandtl’s diagrams as the state equations of shaft components.

Relations for the effective secant and tangent shear moduli, and proportionality limits have been

obtained in the explicit form. Using the root-mean-square approximation of the effective bilinear

Prandtl’s diagram, estimated expressions for natural frequencies being functions of nonlinearly elastic

characteristics of the composite shaft have been determined in the closed form for the following

simplest cases:

1. the case of the smallness of nonlinear deformation region in comparison with elastic region;

2. the case of the small difference between the effective secant and tangent shear moduli.

The analysis of performed calculations has discovered that nonlinearity can be neglected in the

indicated simplest cases, since only pronounced nonlinearity of the effective state equation has a

significant effect on natural frequencies values.

Keywords. Torsional vibrations, composite material, nonlinear elasticity, effective

characteristics, homogenization.

AMS 2010. 74Q10, 74Q15, 74Q20.

References

1 Belarusian State University, Minsk, BELARUS, [email protected] 2 Belarusian State University, Minsk, BELARUS, [email protected]


132

[1] Kravchuk, A.S., Kravchuk, A.I., Tarasyuk, I.A., Equation of torsional vibrations of a circular

longitudinally fibrous, cross layered and structurally heterogeneous composite shaft (in Russian),

Vestnik VGU (Fiz. Mat. Ser.) 4, 53-62, 2015.

http://www.vestnik.vsu.ru/pdf/physmath/2015/04/2015-04-15.pdf

[2] Kravchuk, A.S., Kravchuk, A.I., Tarasyuk, I.A., Rheology of a circular structurally heterogeneous

composite shaft under torsion and torsional vibrations (in Russian), Zavodskaya Laboratoriya.

Diagnostika Materialov. 11, 147-159, 2015.

http://zldm.ru/content/article.php?ID=2070

[3] Tarasyuk, I.A., Kravchuk, A.S., Reducing the Voigt-Reuss range in the theory of elastic structurally

heterogeneous composite, on average isotropic solids without application of variational principles (in

Russian), APRIORI (Est. i Tehn. Nauki Ser.) 3 [online], 2014.

http://apriori-journal.ru/seria2/3-2014/Tarasyuk-Kravchuk.pdf

[4] Gorshkov, A.G., Starovoitov, E.I., Yarovaya, A.V., Mechanics of laminated viscoelastoplastic

structural elements (in Russian), Fizmatlit, Moscow, 2005.

[5] Voigt, W., Textbook of crystal physics (in German), Teubner, Berlin, 1928.

[6] Reuss, A., 1929 Calculation of the flow limit of mixed crystals due to the plasticity condition for

single crystals (in German), ZAMM 1, 49-58, 1929

http://dx.doi.org/10.1002/zamm.19290090104

[7] Peters, S.T. Handbook of Composites, Chapman & Hall, London, 1998.


133

Parameter Dependent Generalized Sylvester and T-Sylvester Equations

Ivana KUZMANOVIĆ1 and Ninoslav TRUHAR2

Abstract. In this talk Sherman-Morrison-Woodbury-type formulas for the solution of

generalized Sylvester equation of the form

(𝐴0 + 𝜇𝑈1𝑉1)𝑋𝐸1 + 𝐸2𝑋(𝐵0 + 𝜇𝑈2𝑉2) = 𝐸

as well as for generalized T-Sylvester equation of the form

(𝐴0 + 𝜇𝑈1𝑉1)𝑋𝐸1 + 𝐸2𝑋𝑇(𝐵0 + 𝜇𝑈2𝑉2) = 𝐸

will be presented, where 𝑈1, 𝑈2, 𝑉1 and 𝑉2 are low-rank matrices and 𝜇 ∈ 𝑅

is parameter.

Developed formulas can be used to obtain algorithms for efficient calculation of the solution of

structured generalized Sylvester and T-Sylvester equations for many values of parameter 𝜇 as well as

for optimization of the solution with respect to 𝜇.

One specific problem where optimization of parameter dependent Sylvester equation (or more

specific, Lyapunov equation) appears is a problem of damping optimization in mechanical systems (see

[1,2]).

Keywords. Matrix equations, Sylvester equation, parameter dependent problems.

AMS 2010. 65F99, 65Z05.

References

[1] Kuzmanović, I., Truhar, N., Optimization of the solution of the parameter-dependent Sylvester

equation and applications, Journal of Computational and Applied Mathematics, 237, 1, 136-144, 2013

[2] Kuzmanović, I., Truhar, N., Sherman-Morrison-Woodbury formula for Sylvester and T-Sylvester

equation with applications, International Journal of Computer Mathematics 90, 2,306-324, 2013

1 Department of Mathematics, University of Osijek, Osijek, CROATİA, [email protected] 2 Department of Mathematics, University of Osijek, Osijek, CROATİA, [email protected]


134

Generalization of the Linear Momentum Operator in Quantum Mechanics: A Position-

Dependent Effective Mass Approach

J. J. PEÑA1, J. MORALES2, J. GARCÍA-RAVELO3, Leonardo SALINAS4

Abstract. The problem of constant mass in quantum mechanics is well known, in this context,

the linear momentum operator p and the position operator x satisfy the conmutation relation [ x , p ]=iℏ,

likewise the uncertainty relation is 𝛥x𝛥p≥ℏ/2. Nevertheless, when the mass distribution m(x) depends on

the position, the linear momentum operator p no longer commute with m(x). For example, to the

particular case m(x)=(1+𝛾x)-2 [1], the modified linear momentum opertor becomes p𝛾=- iℏ(1+ 𝛾x)d/dx

which leads to [ x , p𝛾 ]=iℏ(1+𝛾x), a similar result is obtained for the uncertainty relation 𝛥x𝛥p𝛾

≥ℏ/2(1+𝛾‹x›). In this work, by using the commutation relation [d/dx , m(x)]=dm(x)/dx, we show that it is

possible to obtain a generalized structure for the linear momentum operator under the condition of the

hermiticity of the Hamiltonian [2] from a position-dependent effective mass approach.

Keywords. Position-dependent effective mass, modified linear momentum operator.

AMS 2010. 53B50, 35J10.

References

[1] Costa Filho, R. N. et.al., Displacement operator for quantum systems with position-dependent mass.

, Phys. Rev. A, 84, 050102(R), 2011.

[2] Levy-Leblond, J-M., Elementary quantum models with position-dependent mass, Eur. J. Phys. 215-

218, 1992.

1 Universidad Autonoma Metropolitana Azc, CDMX, MÉXİCO, [email protected] 2 Universidad Autonoma Metropolitana Azc, CDMX, MÉXİCO, [email protected] 3 Instituto Politécnico Nacional ESFM, CDMX, MÉXİCO, [email protected] 4 Instituto Politécnico Nacional ESFM, CDMX, MÉXİCO, [email protected]






135

On Trace Inequalities for Generalized Quasi-Metric Adjusted Skew Informations

Kenjiro YANAGI1

Abstract. In this talk we investigate some uncertainty relations for generalized quasi-metric

adjusted skew informations. We give generalized Heisenberg type or Schrodinger type uncertainty

relations. As applications we have new trace inequality related to fidelity and trace distance.

THEOREM(Extension of [3],[1]). For ,, ( )nA B M the following hold:

1 1

1 2 20 1

1( | | ) inf ( ) ( )

2A BTr A B L R I Tr A B Tr A B

2

21 11 1( ) ( ) (1 ) | | ,

2 2A BTr A B A B Tr A B Tr L R I

Where ( ) , ( )A AL X AX R X XA .

Key words and phrases: generalized quasi-metric adjusted skew information

2010 Mathematics Subject Classification: 15A45, 47A63, 91A17

The author was partially supported by JSPS KAKENHI Grant Number 26400119.

References

[1] Audenaert K. M. R., Calsamiglia J., Masancs L. I., Munnoz-Tapia R., Acin A., Bagan E. and

Verstraete F., The quantum Chernoff bound, Phys.Rev.Lett., 98, 160501, 2007.

[2] Audenaert K. M. R., Nussbaum M., Szkola A. and Verstraete F., Asymptotic error rates in quantum

hypothesis testing, Commun.Math.Phys., 279, 251-283, 2008.

[3] Powers R. T. and Stormer E., Free states of the canonical anticommutation relations, Commun.

Math. Phys., 16, 1-33, 1970.

[4] Yanagi K., Furuichi S. and Kuriyama, K., Uncertainty relations for generalized metric adjusted

skew information and generalized metric adjusted correlation measure, J.Uncertainty Analysis and

Applications, 1, 1-14, 2013.

1 Josai University, Sakado, Saitama, JAPAN, [email protected]



136

[5] Yanagi K. and Sekikawa, K., Non-hermitian extensions of Heisenberg type and Schrodinger type

uncertainty relations, J.Inequalities and Applications, 381, 1-9,2015.

[6] Yanagi, K., Generalized trace inequalities related to fidelity and trace distance, Linear and

Nonlinear Analysis, 2, 263-270, 2016.


137

The Solutions of Special Second Order Ordinary Differential Equations Using

Symmetries

Kısmet KASAPOĞLU1

Abstract. In this study, certain symmetries of second order ordinary differential equations 𝑦′′ +

𝑃(𝑥)𝑦′ + 𝑞(𝑥)𝑦 = 0 are found for 𝑝(𝑥) = 2

𝑥 and special cases of 𝑞(𝑥) and then these symmetries are

generalized. Also by using these symmetries, these equations are reduced to the first order ordinary

differential equations and analytic solutions of these equations are obtained and generalized.

Keywords. Symmetry, Ordinary differential equations.

AMS 2010. 76M60, 34B60, 34A05.

References

[1] Stephani, H., Differential Equations Their Solution Using Symmetries, Cambridge University Press,

New York, 2003.

[2] Polyanin, A. D., Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations,

Chapman & Hall/CRC. , New York, 2003.

[3] Starrett, J., Solving Differential Equations by Symmetry Groups, Mathematical Association of

America, 114, 9, 778-792, 2007.

[4] Oliveri, F., Lie Symmetries of Differential Equations: Classical Results and Recent Contributions,

Symmetry, 2, 8, 658-706, 2010.

[5] Robert, G., Lie Groups, Lie Algebras, and Some of Their Applications, John Wileyond. Sons, Inc.,

New York, 1974.

[6] Hydon, P. E., Symmetry Methods for Differential Equations, Cambridge University Prees, New

York, 2000.

[7] Zwillinger D., Handbook of Differential equations, Acedemic Pres, San Diego, 1989.

1Trakya University, Edirne, TURKEY, [email protected]


138

Schrödinger Equation with Variable Effective Mass: Linear and Staggered Mass

Distributions

Leonardo SALINAS1, H. LUNA-GARCÍA 2, J. GARCÍA-RAVELO3, J. J. PEÑA4

Abstract. The problem of position-dependent effective mass applied to the Schrödinger

equation has been studied under different formalisms on condition to the hermiticity of the Hamiltonian

[1]. Different approaches have been used in the treatment of this subject [2]. In this work, a point

canonical transformation method is used for transforming the Schrödinger equation with variable mass

in to a constant mass problem [3]. A combination of linear and staggered mass distribution for solving

the Schrödinger equation with position-dependent mass, has been used. Furthermore, the simplest case

of free particle leads to a new potential generated from the linearity properties of the mass distribution

Keywords. Schrödinger equation, position-dependent effective mass, point canonical

transformation.

AMS 2010. 53A40, 20M15.

References

[1] von Roos, O, Position-dependent effective mass in semiconductor theory, Phys. Rev. B, 27, 7547,

1983.

[2] Bagchi, B., Gorain, P., Quesne, C., Roychoudhury, R., A general scheme for the effective-mass

Schrödinger equation and the generation of the associated potentials, Mod Phys Lett A, 19, No. 37,7

2765-2775, 2004.

[3] Pacheco-García, C., García-Ravelo, J., Morales, J., Peña, J.J. Exactly solvable effective mass

Schrödinger equation with Coulomb-like potential, Int. J Quant. Chem. Vol 110, 2880–2885, 2010.

1 Instituto Politécnico Nacional, ESFM, CDMX, MÉXİCO, [email protected] 2 Universidad Autonoma Metropolitana Azc, CDMX, MÉXİCO, [email protected] 3 Instituto Politécnico Nacional, ESFM CDMX, MÉXİCO, [email protected] 4 Universidad Autonoma Metropolitana Azc, CDMX, MÉXİCO, [email protected]






139

Weak Solutions to Interdiffusion Models with Vegard Rule

Lucjan SAPA1, Bogusław BOŻEK2 and Marek DANIELEWSKI3

Abstract. We study the one and multidimensional models of interdiffusion. Let a region Ω ⊂

ℝ𝑛, 𝑛 ∈ ℕ with the smooth boundary 𝜕Ω, 𝑇 > 0 and 𝑟 ∈ ℕ ∖ 1 be fixed. Moreover, let diffusion

coefficients 𝐷𝑖, partial molar volumes Ω𝑖, initial concentrations 𝑐0𝑖 and flows on the boundaries 𝑗𝑖 of the

𝑖th components of a mixture, 𝑖 = 1, … , 𝑟 be given. The unknowns are concentrations 𝑐𝑖 and the potential

𝐹 of a drift velocity 𝑣𝑑.

The local mass conservation law for fluxes with the Darken drift term and the Vegard rule lead

to the parabolic-elliptic system of strongly coupled nonlinear differential equations

𝜕𝑡𝑐𝑖 = 𝑑𝑖𝑣(𝐷𝑖(𝑐1, … , 𝑐𝑟)∇𝑐𝑖 − 𝑐𝑖∇𝐹) 𝑜𝑛 [0, 𝑇] × Ω,

∆𝐹 = 𝑑𝑖𝑣(∑ Ω𝑘𝑟𝑘=1 𝐷𝑘(𝑐1, … , 𝑐𝑟)∇𝑐𝑘) 𝑜𝑛 [0, 𝑇] × Ω,

∫ 𝐹𝑑𝑥 = 0Ω

𝑜𝑛 [0, 𝑇]

(1)

with the initial condition and the coupled nonlinear boundary conditions

𝑐𝑖(0, 𝑥) = 𝑐0𝑖(𝑥) 𝑜𝑛 Ω, (2)

−𝐷𝑖(𝑐1, … , 𝑐𝑟)

𝜕𝑐𝑖

𝜕𝜈+ 𝑐𝑖

𝜕𝐹

𝜕𝜈= 𝑗𝑖(𝑡) 𝑜𝑛 [0, 𝑇] × 𝜕Ω,

𝜕𝐹

𝜕𝜈= ∑ Ω𝑘 (𝐷𝑘(𝑐1, … , 𝑐𝑟)

𝜕𝑐𝑘

𝜕𝜈+ 𝑗𝑘(𝑡)) 𝑜𝑛 [0, 𝑇] × 𝜕Ω𝑟

𝑘=1 , (3)

𝑖 = 1, … , 𝑟. System (1) in the one dimensional case can be reduced to the parabolic one

𝜕𝑡𝑐𝑖 = 𝜕𝑥 (𝐷𝑖(𝑐1, … , 𝑐𝑟)𝜕𝑥𝑐𝑖 − 𝑐𝑖(∑ Ω𝑘𝐷𝑘(𝑐1, … , 𝑐𝑟)𝜕𝑥𝑐𝑘𝑟𝑘=1 + 𝐾(𝑡))) 𝑜𝑛 [0, 𝑇] × Ω (4)

with the initial condition (2) and the boundary conditions

−𝐷𝑖(𝑐1, … , 𝑐𝑟)𝜕𝑐𝑖

𝜕𝜈+ 𝑐𝑖 (∑ Ω𝑘𝐷𝑘(𝑐1, … , 𝑐𝑟)

𝜕𝑐𝑘

𝜕𝜈𝑟𝑘=1 + 𝐾(𝑡)𝜈) = 𝑗𝑖(𝑡) 𝑜𝑛 [0, 𝑇] × 𝜕Ω, (5)

𝑖 = 1, … , 𝑟, where 𝐾 is a some calculated function.

The theorems about existence, uniqueness and properties of weak solutions in the Sobolev

spaces to the problems (4), (2), (5) and (1)-(3) are formulated and proved. The main tool used in the

proofs of the existence results is the Galerkin approximation method. The agreement between the

theoretical results, numerical simulations and experimental data is shown.

1 AGH University of Science and Technology, Faculty of Applied Mathematics, Cracow, POLAND, [email protected] 2 AGH University of Science and Technology, Faculty of Applied Mathematics, Cracow, POLAND, [email protected] 3 AGH University of Science and Technology, Faculty of Materials Science and Ceramics, Cracow, POLAND,

[email protected]


140

Keywords. Interdiffusion, Darken method, Vegard rule, parabolic-elliptic nonlinear system,

existence, uniqueness, properties of global weak solutions, Galerkin approximation.

AMS 2010. 35K61, 35K51, 35D30, 35Q70.


141

Self-Adjoint Extensions of Differential Operators with Potentials-Point Interactions

Manaf Dzh. MANAFOV1

Abstract. In this study, operators generated by the most general formally self-adjoint

differential expression of even order on the Hilbert space are studied and the main result consists in an

explicit construction of a boundary triplet for the associated symmetric minimal quasi-differential

operator. Notice that the quasi-differential operators were introduced first by Shin [2] and then

essentially developed by Zettl [3], see also the monograph [1] and references therein. After the

regularization of the formal differential expression using the quasi derivatives, the minimal and maximal

operators corresponding to potentials of this type on a finite interval are constructed. All self-adjoint

extensions of the minimal operator are described.

Keywords. Self-adjoint extensions of differential operators, Quasi-differential expressions,

Potentials-point interactions.

AMS 2010. 34B24, 34B05, 47N20.

References

[1] Naimark, M. A., Linear differential operators, Part II Linear differential operators, Frederick Ungar

Publ. Co., New York, 1969.

[2] Shin, D., Quasi-differential operators in Hilbert space, Math. Sb (N. S.) 13 (55), 1, 39-70, 1943 (in

Russian).

[3] Zettl, A., Formally self-adjoint quasi-differential operators, Rocky Mountain J. Math., 5, 3, 453-

474, 1975.

1 Adıyaman University, Adıyaman, TURKEY, [email protected]


142

Detecting a Hyperbolic Quadratic Eigenvalue Problem Using a Subspace Algorithm

Marija MILOLOŽA PANDUR1

Abstract. The quadratic eigenvalue problem (QEP) is to find scalars 𝛼 and nonzero vectors 𝑥

such that 𝑄(𝛼)𝑥 = 0 holds for a given quadratic matrix polynomial 𝑄(𝛼) = 𝛼2𝑀 + 𝛼𝐷 + 𝐾, where

𝑀, 𝐷, 𝐾 are complex matrices of order 𝑛. Such 𝛼 is called an eigenvalue, and 𝑥 the associated

eigenvector. A very important is hyperbolic QEP appearing for example in dynamical analysis of

structures, such as a damped mass-spring oscillator. QEP is hyperbolic if 𝑀, 𝐷, 𝐾 are Hermitian

matrices, 𝑀 is positive definite and there exists a real 𝜇 such that 𝑄(𝜇) is negative definite matrix. Any

such 𝜇 is called a definitizing shift. In this case, QEP has 2𝑛 real eigenvalues: 𝑛 neg-type eigenvalues

smaller than 𝜇 and 𝑛 pos-type eigenvalues greather than 𝜇. There is a number of algorithms that solves

the whole or partial hyperbolic QEP. Also, there is a number of algorithms that detect if a given QEP

with Hermitian matrices, is hyperbolic or not. In this talk we present a new algorithm for detecting a

hyperbolic QEP that is suited for large and sparse QEP. Our subspace algorithm is based on iterative

testing of small compressed QEPs formed using test-subspaces of small dimensions. The algorithm has

a monotonicity property based on Cauchy-type interlacing inequalities [1]. If a small nonhyperbolic

compressed QEP is find, the algorithm ends with the conclusion that the given large QEP is

nonhyperbolic. The algorithm finds a definitizing shift if the given large QEP is hyperbolic. In this case,

the definitizing shift can be used to form the preconditioner for the LOBPCG-type algorithm given in

[2].

Keywords. hyperbolic quadratic eigenvalue problem, definitizing shift, subspace algorithm.

AMS 2010. 65F30, 15A18

References

[1] K. Veselić, Note on interlacing for hyperbolic quadratic pencils, Recent Advances in Operator

Theory in Hilbert and Krein Spaces, Oper. Theory: Adv. Appl., Vol. 198, 305–307, Birkhäuser, Boston

2009.

[2] X. Liang, R.-C. Li, The hyperbolic quadratic eigenvalue problem, Forum Math. Sigma, 3, e13 (93

pages), 2015.

1University of Osijek, Osijek, CROATIA, [email protected]


143

A Generalization of Midpoint Type Inequality of m-Logarithmically Convex Functions

Mehmet Eyüp KİRİŞ 1

Abstract. In this study, we establish generalized new inequalities for differantiable

mappings that are connected with the celebrated Hermite-Hadamard integral inequality by

using m - log -convex function.The results presented here would provide extensions of those

given in earlier works.

Keywords. Hermite-Hadamard's inequalities, Riemann-Liouville fractional integral,

integral inequalities.

AMS 2010. 26B25, 35A23, 26D10, 52A41,

References

[1] Azpeitia, A.G., Convex functions and the Hadamard inequality, Rev. Colombiana Math.,

28, 7-12, 1994.

[2] Bakula M. K., and Pečarić, J., Note on some Hadamard-type inequalities, Journal of

Inequalities in Pure and Applied Mathematics, vol. 5, no. 3, article 74, 2004.

[3] Belarbi S., and Dahmani, Z., On some new fractional integral inequalities, J. Ineq. Pure

and Appl. Math., 10(3), Art. 86, 2009.

[4] Dahmani, Z., New inequalities in fractional integrals, International Journal of Nonlinear

Scinece, 9(4), 493-497, 2010.

[5] Dahmani, Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional

integration, Ann. Funct. Anal. 1(1), 51-58, 2010.

[6] Dahmani, Z., Tabharit, L., Taf, S., Some fractional integral inequalities, Nonl. Sci. Lett. A,

1(2), 155-160, 2010.

[7] Dahmani, Z., Tabharit, L., Taf, S., New generalizations of Gruss inequality usin Riemann-

Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3), 93-99, 2010.

1 Afyon Kocatepe University, Afyonkarahisar, Turkey, [email protected]


144

[8] Deng J., and Wang, J., Fractional Hermite-Hadamard inequalities for ( m, )-

logarithmically convex functions. J. Inequal.Appl. 2013, Article ID 364, 2013.

[9] Dragomir S. S. and Pearce, C. E. M., Selected Topics on Hermite-Hadamard Inequalities

and Applications, RGMIA Monographs, Victoria University, 2000.

[10] Dragomir S. S. and Agarwal, R.P., Two inequalities for differentiable mappings and

applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett.,

11(5), 91-95, 1998.

[11] Gorenflo, R., Mainardi, F., Fractional calculus: integral and differential equations of

fractional order, Springer Verlag, Wien, 223-276, 1976.

[12] Kilbas, A. A., Srivastava H. M., and Trujillo, J. J., Theory and Applications of Fractional

Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V.,

Amsterdam, 2006.

[13] Latif, M.A., Dragomir S.S., and Matouk, A.E., New inequalities of Ostrowski type for co-

ordinated convex functions via fractional integrals, J. Fract. Calc. Appl. 2, 1-15, 2012.

[14] Miller S. and Ross, B., An introduction to the Fractional Calculus and Fractional

Differential Equations, John Wiley & Sons, USA, 2, 1993.

[15] Pečarić, J.E., Proschan F., and Tong, Y.L., Convex Functions, Partial Orderings and

Statistical Applications, Academic Press, Boston, 1992.

[16] Podlubni, I., Fractional Differential Equations, Academic Press, San Diego, 1999.

[17] Sarikaya M.Z. and Ogunmez, H., On new inequalities via Riemann-Liouville fractional

integration, Abstract and Applied Analysis, Volume 2012, Article ID 428983, 10 pages, 2012.

[18] Sarikaya, M.Z., Set, E., Yaldiz H. and Basak, N., Hermite -Hadamard's inequalities for

fractional integrals and related fractional inequalities, Mathematical and Computer Modelling,

DOI:10.1016/j.mcm.2011.12.048, 57, 2403—2407, 2013.

[19] Sarikaya, M.Z., Filiz H. and Kiris, M.E., On some generalized integral inequalities for

Riemann-Liouville fractional integrals, Filomat, 29:6, 1307—1314, 2015.


145

[20] Sarikaya M.Z. and Yildirim, H., On Hermite-Hadamard type inequalities for Riemann-

Liouville fractional integrals,Miskolc Mathematical Notes, Vol. 17, No. 2, pp. 1049—1059,

2016.

[21] Zhang Y. and Wang, J., On some new Hermite-Hadamard inequalities involving Riemann-

Liouville fractional integrals, J. Inequal. Appl., Article ID 220, 2013.


146

Some Approximation Results for Lupaş-Kantorovich-Schurer Operators Based on (p, q )-

Calculus

Melek SOFYALIOĞLU1 and Kadir KANAT2

Abstract. The purpose of this presentation is to introduce Lupaş-Kantorovich-Schurer operators

based on (p,q)-integers. We investigate approximation properties for these operators according to

Korovkin type approximation theorem. We prove some auxiliary results which will be needed to

establish the main results. Moreover, we estimate the rate of convergence by using modulus of

continuity, with the help of functions of Lipschitz class and Peetre’s K-functionals.

Keywords. (p,q)-integers, Lupaş operators, rate of convergence.

AMS 2010. 41A10, 41M25, 41A36.

References

[1] Mursaleen, M., Nasiruzzaman, M., Nurgali, A., Some approximation results on Bernstein-Schurer

operators defined by (p,q)-integers, Journal of Inequal. Appl. (2015), 249, 2015.

[2] Khalid Khan, D.K. Loyibal, Bezier curves based on Lupaş (p,q)-analogue of Bernstein functions in

CAGD, Journal of Comp. and Applied Mathematics, Volume 317, 458-477, 2017.

[3] Mursaleen, M., Ansari, K.J., Khan, A., On(p,q)-analogue of Bernstein operators, Applied

Mathematics and Computation 266 (2015a) 874882.( Erratum to On (p,q)-analogue of Bernstein

Operators [Appl.Math.Comput.266 (2015) 874882], Applied Mathematics and Computation 278(2016)

7071.)

[4] Lupaş, A., A q-analogue of the Bernstein operator, in: Seminar on Numerical and Statistical

Calculus, No. 9, Univ. of Cluj-Napoca, 1987.

1 Gazi University, Ankara, TURKEY, [email protected] 2 Gazi University, Ankara, TURKEY, [email protected]


147

Conformable Fractional Klein-Gordon and Cahn-Allen Equations

Meryem ODABASI1

Abstract. Numerous types of physical models can be described by fractional order differential

equations. Because of the applications in chemistry, biology, elasticity, acoustics, fluid dynamics, solid

mechanics, optics, propagation of shallow water waves, quantum field theory and others, traveling wave

solutions of the fractional differential equations have become more important. Hence, a lot of methods

and different definitions of fractional derivatives have been used so far. Recently, a new definition

called conformable fractional derivative have been proposed. In the present study, conformable time-

fractional Klein-Gordon and Cahn-Allen equations are investigated in the sense of conformable

derivative. Using the modified trial equation method, some exact traveling wave solutions of these

equations that can allow us to understand the phenomena they describe have been obtained.

Keywords. Conformable fractional differential equations, traveling wave solutions.

AMS 2010. 35R11, 35C07.

References

[1] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and applications of fractional differential

equations, Elsevier: Amsterdam, 2006.

[2] Khalil, R., Horani, M.A., Yousef, A. and Sababheh, M., A new definition of fractional derivative,

Journal of Computational and Applied Mathematics, 264, 65–701, 2014.

[3] Hosseini, K., Mayeli P. and Ansari, R., Modified Kudryashov method for solving the conformable

time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities, Optik, 130, 737–742,

2017.

[4] Liu, C.S, A new trial equation method and its applications, Communications in Theoretical Physics,

45, 3, 395-397, 2006.

[5] Odabasi, M. and Misirli, E., On the solutions of the nonlinear fractional differential equations via

the modified trial equation method, Mathematical Methods in the Applied Sciences, Doi:

10.1002/mma.3533, 2015.

1 Ege University, Tire Kutsan Vocational School, İzmir, TURKEY, [email protected]


148

On the Finite Element Approximation in the Maximum Norm of Elliptic Quasi-

Variational Inequalities with Nonlinear Source Terms

Messaoud BOULBRACHENE1

Abstract. We are concerned with the standard finite element approximation of the semilinear

quasi-variational inequality (QVI) of obstacle type: Find u ∈ H¹(Ω) such that:

a(u,v-u) ≥ (f(u),v-u) ∀ v ∈ H¹(Ω)

u ≤ ψ(u), v ≤ ψ(u)

where, Ω is a bounded convex domain of RN , N ≥ 1, with sufficiently smooth boundary Γ, f is a positive

Lipschitz and nondecreasing nonlinearity with Lipschitz constant c, ψ(u) is an implicit obstacle, which

justifies the terminology quasi-variational inequality, (.,.) is the inner product in L²(Ω), a(.,.) is a

coercive bilinear form associated with linear second order elliptic differential operator, such that the

zero order term satisfies a₀(x)≧ β> 0 ∀ x ∈Ω.

In [1], by means of a fixed point approach, we derived error estimates, in the maximum norm,

between the exact solution and its finite element counterpart, under the assumption that c/β <1. In this

paper, we drop this assumption and derive error estimate in the same norm, under the sole Lipschitz

condition on the nonlinearty f(.), which enables us to encompass a larger class of semilinear QVIs. For

that, we introduce a new method that combines an iterative scheme of Bensoussan-Lions type [2] and

the concept of subsolutions for elliptic variational inequalities [3].

Keywords. Quasi-variational-inequalities, Iterative Scheme, Subsolutions, Finite Elements,

Error estimates.

AMS 2010. 65N15, 65N30.

References

[1] Boulbrachene, M., Pointwise error estimates for a class of elliptic quasi-variational inequalities

with nonlinear source terms. Applied Mathematics and Computation, 161, 129-138, 2005.

[2] Bensoussan, A., Lions, J. L., Impulse control and quasi-variational inequalities. Gauthiers-Villars,

Paris, 1984.

[3] Bensoussan, A., Lions, J.L., Applications des inequations variationnelles en controle stochastique.

North Holland, 1978.

1 Sultan Qaboos University, Muscat, OMAN, [email protected]; [email protected]




149

Exponentially Cusped Prismatic Shells in the 0N Approximation of I. Vekua’s

Hierarchical Models

Miranda GABELAIA1

Abstract. Bending problem of prismatic shell with the thickness as follows

)(0

22

21 xx

ehh

, 0),,(,0,0 210 xxconstconsth ,

is investigated. The solution of the posed boundary value problem is given in an explicit form.

Static problem of the prismatic shell with the following thickness

20

xehh

, 0),,(,0,0 210 xxconstconsth ,

is investigated as well.

Keywords. Cusped Prismatic Shells, Hierarchical Models, Partial Differential Equations.

AMS 2010. 74K20, 74K25, 35J47.

1 Iv. Javakhishvili Tbilisi State University, Tbilisi, GEORGİA, [email protected]



150

m m1

∈

∇

. Σ

Bounds for Blow up Time in Nonlinear Pseudo-Parabolic Equations

Müge MEYVACI1

Abstract. The point of this paper is to analyze the following initial boundary value problem:

ut − ∆ut − ∆u − u ux1 + g(t, x, u, ∇u) = |u| u, x ∈ Ω, t > 0, (1)

u(x, 0) = u0(x), x ∈ Ω, (2)

u (x, t) = 0, x ∈ ∂Ω, t ≥ 0, (3)

where m and m1 are given numbers, g(t, x, u, w) satisfies the property:

|g(t, x, u, w)| ≤ M0 |u|k + |w| , (x, t) ∈ Ω × R, u ∈ R, w ∈ RN , (4)

where M0, k are given numbers and Ω RN (N > 3) is a bounded domain with sufficiently smooth

boundary ∂Ω. During the last decades, the pseudo-parabolic equations have been contemplated by

numerous authors by different aspects. In recent years an increasing number of researchers focused in

the upper and lower bound for blow up time for parabolic and pseudo-parabolic equations. So we obtain

the sufficient conditions on initial data, m, m1 and the function g(t, x, u, u) for nonblow up case.

Moreover we decide the lower and upper bounds for the blow up time, if blow up happens.

Key words. Pseudo-parabolic equation, sobolev equation, lower bound, upper bound, blow-up,

global nonexistence.

AMS 2010. 35B40, 35B44.

References

[1] Korpusov, M. O. and Sveshnikov, A. G., Blow-Up of Solutions of Nonlinear Sobolev Type

Equations with Cubic Sources. Differential Equations, 42, 3, 431–443, 2006.

[2] Lu, Y., L. Fei, L., Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal

source. J. Ineq. Appl., 2016:229, 2016.

[3] Luo, P., Blow-up phenomena for a pseudo-parabolic equation. Math. Meth. Appl. Sci., 38, 12,

2636-2641, 2015.

1 Mimar Sinan Fine Art University, Istanbul, TURKEY, [email protected]


151

[4] Mei, M., Lq-Decay Rates of Solutions for Benjamin-Bona-Mahony-Burgers Equations Journal of

Differential Equations 158, 314–340, 1999.

[5] Peng, X., Shang, Y., Zheng, X., Blow-up phenomena for some nonlinear pseudo-parabolic

equations. Appl. Mat. Lett. 56, 17–22, 2016.

[6] Showalter, R. E, Ting, T. W., Pseudoparabolic Partial Differential Equations. SIAM J. Math. Anal.

1, 1–26, 1970.

[7] Showalter, R. E., Weak solutions of nonlinear evolution equations of Sobolev-Galpern type. Journal

of Differential Equations, 11 2, 252–265, 1972.

[8] Xu, R., Su, J., Global existence and finite time blow-up for a class of semilinear pseudo-parabolic

equations. Journal of Functional Analysis 264 12, 2732–2763, 2013.


152

Oscillation Criteria for Higher Order Fractional Difference Equations with Nonlinearities

Mustafa Kemal YILDIZ 1 , Ebru YILMAZ2 and Umut Mutlu ÖZKAN3

Abstract. This paper deals with some oscillation criteria of forced nonlinear fractional difference

equations of the form

Δ∗𝛼𝑥(𝑡) − 𝑔(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1)) + 𝑓1(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1)) = 𝜐(𝑡)

Δ∗𝛼𝑥(𝑡) − 𝑔(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1)) + 𝑓1(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1)) + 𝑓2(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1))

= 𝜐(𝑡)

where Δ∗𝛼 is a Caputo like discrete fractional difference operator, 𝑡 ∈ ℕ𝑎, 𝑚 − 1 < 𝛼 ≤ 𝑚,

𝑔, 𝑓𝑖: [0, +∞) × ℝ → ℝ, 𝑖 = 1,2 ve 𝜐: [0, +∞) → ℝ are continuous with respect 𝑡 and 𝑥 and 𝑁𝑎 =

𝑎, 𝑎 + 1, 𝑎 + 2, . . . .

Keywords. Difference Equations, Oscillation, Nonlinear, Fractional Order.

AMS 2010. 26A33, 39A11, 39A12.

References

[1] Marian, S. L., Sagayaraj, M. R., Maria Selvam, A. G., M. P. Loganathan Oscillation of caputo like

discrete fractional equations, International Journal of Pure and Applied Mathematics, v:89, n:5, 2013.

[2] Marian, S. L., Sagayaraj, M. R., Maria Selvam, A. G., M. P. Loganathan Oscillation of fractional

nonlinear difference equations, Mathematica Aeterna, v:2, n:9, 2012.

[3] Chen, D. X., Qu, P. X. and . Lan, Y. H., Forced oscillation of certain fractional differantial

equations, Advanced in Difference Equations, 125, 2013.

[4] Grace, S. R., Agarwal,R. P., Wong P. J. Y. and Zafer, A., On the oscillation of fractional differential

equations , An International Journal for Theory and Applications, v:15, n:2, 2012.

[5] Han, Z., Zhao, Y., Sun, Y., Zhang, C., Oscillation for a class of fractional differential equation,

Hindawi Publishing Corporation, DiscreteDynamics in Nature and Society, Volume 2013.

[6] Anastassiou, G. A., Discrete fractional calculus and inequalities, http://arxiv.org/abs/0911.3370 v:1.

[7] Chen, F., Luo, X., Zhou, Y., Existence results for nonlinear fractional difference equations,

Advances in Difference Equations,Volume, Article ID 713201, 12 pages, 2011.

[8] Atici, F. M., Eloe, P. W., Initial value problems in discrete fractional calculus, Proceedings of the

American Mathematical Society,v:137, n:3, pp:981-989, 2009.

1 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected] 2 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected] 3 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected]


153

[9] Alzabut, J. O., T. Abdeljawad, T., Sufficient Conditions For The Oscillation Of Nonlinear

Fractional Difference Equations, Journal of Fractional Calculus and Applications, v:5 pp:177-187,

2014.


154

Lyapunov Type Inequalities for nth-order Three point BVP Differential Equations with

Mixed Nonlinearities

Naceri MOSTEPHA1 and Abdullah OZBEKLER2

Abstract. In this paper, we present some new Lyapunov and Hartman type in equalities

for second order equations with mixed nonlinearities:

𝑥(𝑛)(𝑡) + 𝑝(𝑡)|𝑥(𝑡) | 𝛽−1𝑥(𝑡) + 𝑞(𝑡)|𝑥(𝑡) | 𝛾−1𝑥(𝑡) = 0,

where 𝑝(𝑡), 𝑞(𝑡) are real-valued functions and 0<𝛾<1<𝛽<2. No sign restrictions are imposed

on the potential functions 𝑝(𝑡)and 𝑞(𝑡). The inequalities obtained generalize and compliment

the existing results for the special cases of this equation in the literature.

Keywords. Lyapunov type inequality, Mixed nonlinear, Sub-linear.

AMS 2010. 34C10, 34B15, 34B40.

References

[1] A. M. Liapunov, Probleme général de la stabilité du mouvement, (French Translation of a

Russian paper dated 1893), Ann. Fac. Sci. Univ. Toulouse 2 (1907), 27-247, Reprinted as Ann.

Math. Studies, No. 17, Princeton, 1947.

[2] A. Wintner, On the nonexistence of conjugate points, Amer. J.73, 368—380, 1951.

[3] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964 and BirkhÃ¤user,

Boston 1982.

[4] A. Beurling, Un théorème sur les fonctions bornées et uniformément continues sur l’axe

réel. Acta Math.77, 127—136, 1945.

1 Oran school of economics, Oran, ALGERİA. [email protected] 2 Atilim University 06836, Incek, Ankara, TURKEY. [email protected]


155

3D FEM Analysis of Buckling of a Piezoelectric Rectangular Thick Plate

Nazmiye YAHNIOGLU1 and Fatih AYLIKCI2

Abstract. In this work a buckling problem of a sandwich PZT/Metal/PZT rectangular thick

plate within the scope of the 3D linearized theory of stability loss in the framework of the piecewise

homogeneous body model is studied. It is supposed that an ideal contact conditions are satisfied

between the layers and all the lateral surfaces of the plate is simply supported mechanically and also

grounded for the PZT layers' surface only. In addition, it is assumed that on the upper and lower surface

of the plate neither mechanically load nor electrically load are act. However, this plate is subjected to bi-

axial uniformly-distributed compressive mechanical forces acting on two edge surfaces.

In the analyzing procedure, before the plate is loaded (i.e. in the natural state), the free planes of

the plate have insignificant initial imperfections. Due to action of the aforementioned compressive

forces the evolution of the initial imperfections is investigated and as a result of this investigation the

values of the critical buckling forces for the considered sandwich plate are found from the criteria,

according to which, the initial imperfections grow indefinitely with the compressive forces [1-

3]. Mathematical modeling of the considered problem is formulated within the scope of the three

dimensional exact geometrically nonlinear equations of electro-elasticity in the framework of the

piecewise homogeneous body model. The corresponding boundary-value problems are solved

numerically by employing the 3D finite element method (3D-FEM).

Acknowledgement: This work was supported by Research Fund of the Yildiz Technical

University. Project Number: 2016-07-03-DOP03.

Keywords: Buckling, piezoelectric plate, 3D FEM

References

[1]. Akbarov, S. D., Stability Loss and Buckling Delamination: Three-Dimensional Linearized

Approach for Elastic and Viscoelastic Composites, Springer. Heidelberg, New York, 2013.

[2]. Akbarov, S. D., Yahnioglu, Y., Buckling delamination of a sandwich plate-strip with

piezoelectric face and elastic core layers, Appl. Math. Model. 37, 8029 – 8038, 2013.

[3]. Cafarova, F.I., Akbarov, S.D., Yahnioglu, N., Buckling delamination of the

PZT/Metal/PZT sandwich circular plate-disc with penny-shaped interface cracks, Smart

Structures and Systems, 19 (2) (2017) 163 – 179, 2017.

1 Yildiz Technical University, Istanbul, TURKEY, [email protected] 2 Yildiz Technical University, Istanbul, TURKEY, [email protected]


156

Identities for the double L22-integral transform

Neşe DERNEK1, Fatih AYLIKÇI2

Abstract. In this paper, the authors introduce the double Laplace-type integral transform L22

and its properties. Several simple theorems dealing with general properties of the L22-integral transform

are proved. The convolution, its properties and convolution theorem are given. The main focus of this

paper is to develop a method for the L22-integral transform to solve problems in applied mathematics

which involve integral equations.

Keywords: Laplace transform, Double Laplace transform, Convolution, Integral equations

2010 MSC: 44A10, 44A30, 44A35, 33B20, 45A05

References

[1] Aghili, A., Ansari, A., and Sedghi., A., An inversion technique for the L2-transform with

applications, Int. J. Contemp. Math. Sciences 2.28 1387-1394, 2007.

[2] Churchill, R., Operational mathematics 3rd edn., Mc Graw Hill, New York 1972.

[3] Debnath, L., The double Laplace transform and their properties with applications to functional,

integral and partial differential equations, Inter. J. App. Comput. Math. Vol.2 Issue 2 p.223-241, 2016.

[4] Debnath, L., Bhatta, D., Integral transforms and their applications 3rd edn., CRC Press, Chapman

& Hall, Boca Ratan, 2015.

[5] Dernek, N., Aylikci,F., Identities for the Ln-transform, the L2n-transform and the P2n-transform and

their applications, Journal of Inequalities and Special Functions Vol.5 Issue 4 1-16, 2014.

[6] Dernek, N., Aylikci,F., Kivrak, S., An alternative technique for solving ordinary differential

equations, Konuarlp Journal of Mathematics Vol.4 No.1 68-79, 2016.

[7] Erdelyi, A. et. al., Tables of integral transforms, Mc Graw Hill Book Company Vol.1 and Vol.2

(1954)

[8] Schiff, J.L., The Laplace transforms, Springer New York, 1999.

1 Marmara University, Istanbul, TURKEY, [email protected] 2 Yildiz Technical University, Istanbul, TURKEY, [email protected]


157

[9] Sneddon, T.N., Fourier transforms, Mc Graw Hill New York, 1951.

[10] Yurekli, O., New identities involving the Laplace and the L2-transform and their applications,

Applied Mathematics and Computation 99 141-151, 1999.

[11] Yurekli, O., Sayginsoy, O., A theorem on a Laplace-type integral transform and its applications,

Internat. J. Math. Ed. Sci. Tech. 29 561-567, 1998.

[12] Yurekli, O., Sadek, I., A Parseval-Goldstein type theorem on the Widder potential transform and

its applications, Internat. J. Math. Ed. Sci. Tech. 14 517-524, 1991.

[13] Yurekli, O., Scott, W., A new method of solving Bessel's di_erential equation using L2-transform,

Applied Mathematics and Computation 130 587-591, 2002.


158

Simplified Hirota Method for a Nonlinear Partial Differential Equation

Ömer ÜNSAL1, Ahmet BEKIR2, Murat KOPARAN3

Abstract. In the recent past, Ma introduced and obtained complexiton solutions for the

Korteweg-de Vries equation through its bilinear form [1], [2]. After finding of this novel class of

explicit exact solutions, some new exact solution methods have been developed and used to get

complexiton solutions of nonlinear partial differential equations. For instance, extended transformed

rational function method [3], multiple Riccati equations rational expansion method [4], generalized sub-

equations rational expansion method [5], generalized compound Riccati equations rational expansion

method [6].

In this paper, we apply the simplified Hirota method to a nonlinear partial differential equation

[7], [8]. Through the bilinear form, complexiton solutions and interaction solutions are obtained. Some

graphs associated with the solutions are illustrated. This approach can also be applied to other nonlinear

partial differential equations of which bilinear form is obtained.

Keywords. Simplified Hirota Method, Complexiton Solution, Nonlinear Partial Differential

Equation.

AMS 2010. 35C99, 35Q53.

Acknowledgements. This work was supported by Eskisehir Osmangazi University Scientific

Research Projects (Grant No. 2016-1079).

References

[1] Ma, W. X., Complexiton solutions to the Kortweg-de Vries equation, Phys. Lett. A, 301, 35-44,

2002.

[2] Ma, W. X., Complexiton solutions to integrable equations, Nonlinear Anal., 63 e2461-e2471, 2005.

[3] Zhang, H., Ma, W. X., Extended transformed rational function method and applications to

complexiton solutions, Applied Mathematics and Computation, 230, 509-515, 2014.

[4] Chen, Y., Wang, Q., Multiple Riccati equations rational expansion method and complexiton

solutions of the Whitham-Broer-Kaup equation, Physics Letters A, 347, 215-227, 2005.

1 Eskisehir Osmangazi University, Eskisehir, TURKEY, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, TURKEY, [email protected] 3 Anadolu University, Eskisehir, TURKEY, [email protected]


159

[5] Li, W., Zhang, H., A generalized sub-equatons rational expansion method for nonlinear evolution

equations, Commun. Nonlinear. Sci. Numer. Simulat., 15, 1454-1461, 2010.

[6] Li, W., Zhang, H., A new generalized compound Riccati equations rational expansion method to

construct many new exact complexiton solutions of nonlinear evolution equations with symbolic

computation, Chaos, Solitons and Fractals, 39, 2369-2377, 2009.

[7] Wazwaz, A. M., Zhaqilao, Nonsingular complexiton solutions for two higher-dimensional fifth order

nonlinear integrable equations, Physica Scripta, 88, 025001, 2013.

[8] Wazwaz, A.M., New solutions for two integrable cases of a generalized fifth-order nonlinear

equation, Modern Physics Letters B, 29, 1550065, 2015.


160

Extended Transformed Rational Function Method for Some (3+1) Dimensional Nonlinear

Partial Differential Equations

Ömer ÜNSAL1, Ahmet BEKIR2, Murat KOPARAN3

Abstract. Recently, some exact solution methods have been developed and employed to obtain

complexiton solutions of nonlinear partial differential equations [1], [2], [3], [4], [5], [6], [7]. In this

paper, we apply the extended transformed rational function method to two (3+1) dimensional nonlinear

partial differential equations [8]. Obtained solutions contain trigonometric and hyperbolic functions and

it is shown that the considered transform and method are very efficient and powerful in solving other

kinds of nonlinear partial differential equations.

Keywords. Extended Transformed Rational Function Method, Complexiton Solution,

Nonlinear Partial Differential Equation.

AMS 2010. 35C99, 35Q53.

Acknowledgements. This work was supported by Eskisehir Osmangazi University Scientific

Research Projects (Grant No. 2016-1079).

References

[1] Ma, W. X., Complexiton solutions to the Kortweg-de Vries equation, Phys. Lett. A, 301, 35-44,

2002.

[2] Ma, W. X., Complexiton solutions to integrable equations, Nonlinear Anal., 63 e2461-e2471, 2005.

[3] Chen, Y., Wang, Q., Multiple Riccati equations rational expansion method and complexiton

solutions of the Whitham-Broer-Kaup equation, Physics Letters A, 347, 215-227, 2005.

[4] Li, W., Zhang, H., A generalized sub-equatons rational expansion method for nonlinear evolution

equations, Commun. Nonlinear. Sci. Numer. Simulat., 15, 1454-1461, 2010.

[5] Li, W., Zhang, H., A new generalized compound Riccati equations rational expansion method to

construct many new exact complexiton solutions of nonlinear evolution equations with symbolic

computation, Chaos, Solitons and Fractals, 39, 2369-2377, 2009.

1 Eskisehir Osmangazi University, Eskisehir, TURKEY, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, TURKEY, [email protected] 3 Anadolu University, Eskisehir, TURKEY, [email protected]


161

[6] Wazwaz, A. M., Zhaqilao, Nonsingular complexiton solutions for two higher-dimensional fifth order

nonlinear integrable equations, Physica Scripta, 88, 025001, 2013.

[7] Wazwaz, A.M., New solutions for two integrable cases of a generalized fifth-order nonlinear

equation, Modern Physics Letters B, 29, 1550065, 2015.

[8] Zhang, H., Ma, W. X., Extended transformed rational function method and applications to

complexiton solutions, Applied Mathematics and Computation, 230, 509-515, 2014.


162

Some Results on Cooperative Grey Games

Osman PALANCI1, Rana S.M. BAKRI2, Marwah Imad Najm NAJM3, Sirma Zeynep

ALPARSLAN GOK4

Abstract. This contribution is located in the common area of Operations Research and

Economics, with a close relation and joint future potential with optimization and cooperative game

theory. We study some game theoretic solutions on the interesting class of cooperative games where the

coalitional values are interval grey numbers. These class of games are called cooperative grey games.

Further, we deal with an axiomatization of the grey Shapley value. We introduce the Banzhaf value and

the egalitarian rule by using the theory of cooperative grey games.

Keywords. Shapley value, Banzhaf value, egalitarian rule, grey uncertainty, fairness property,

economics.

1 Suleyman Demirel University, Isparta, TURKEY, [email protected] 2 Suleyman Demirel University, Isparta, TURKEY, [email protected] 3 Suleyman Demirel University, Isparta, TURKEY, [email protected] 4 Suleyman Demirel University, Isparta, TURKEY, [email protected]



163

A Cubic Subdomain Galerkin Method over the Geometrically Graded Mesh to the

Singularly Perturbed Problem

Ozlem Ersoy HEPSON1 and Idris DAG2

Abstract. In this paper, subdomain Galerkin method is set up to find solutions of singularly

perturbed boundary value problems which are used widely in many areas such as chemical reactor

theory, aerodynamics, quantum mechanics, reaction-diffusion process, optimal control, etc. A

combination of the cubic B-spline base functions as an approximation function is used to build up the

presented method over the geometrically graded mesh. Thus finer mesh can be established through the

end parts of the problem domain where steep solutions exist.

Keywords. Subdomain Galerkin, graded mesh, spline, singularly perturbed.

AMS 2010. 41A15, 65L60.

References

[1] Kumar, M., Singh, P., Mishra, H. K., A recent survey on computational techniques for solving

singularly perturbed, International Journal of Computer Mathematics, 84(10), 1439-1463, 2007.

[2] Dag, I., Sahin, A., Numerical solution of the Burgers’ equation overgeometrically graded mesh,

Kybernetes, 8, 2649-2656, 2001.

[3] Dag, I., Sahin, A., Numerical solution of singularly perturbed problems, Int. Journal of Nonlinear

Science, 8 (1), 32-39, 2009.

[4] Gupta, Y., Kumar, M., Anthology of spline based numerical tecniques for singularly perturbed

boundary value problem, International Journal of Pure and Applied Mathematics 74(4), 437-453, 2012.



164

Optimal Control of Distributed Systems in Problems of Quartz Optical Fiber Production

Pervadchuk V.P.1, Vladimirova D.B. 2, Gordeeva I.V.3

Abstract. The problems of optimal control of distributed systems arise in different fields of

science and technology [1]. Therefore, in the last decades the efforts of many researchers are aimed at

solving these problems. They are of special importance for high-precision processes and technologies

which include the production of special quartz optical fibers.

Fiber production involves two main stages: the first stage is quartz tube alloying (by Modified

Chemical Vapour Deposition (MCVD), as a rule), and the second one is fiber drawing from a preform.

In the MCVD process the temperature field distribution plays a decisive role. For the description of

temperature field distribution, two-dimensional and one-dimensional mathematical models of quartz

tube heating have been proposed in our research. The purpose of control was to minimize the deviation

of the actual (measured) temperature field on the outer surface of the tube from the programmed

(prescribed) temperature field. In this case, the heat flux from the moving burner was controlled. Two

problems of optimal control of distributed systems were formulated: a two-dimensional problem with

boundary control and a one-dimensional problem with distributed control. The optimality systems for

each of the problems were obtained and the algorithms for their realization were proposed [2].

A quasi-one-dimensional nonstationary model of the process was used to study fiber drawing.

The stationary mode of fiber drawing was considered as a programmed (given) movement. A

disturbance movement was the deviation of the real movement from the programmed one. The purpose

of the control was to minimize perturbations. The drawing speed (winding) of the fiber was the

controlled parameter. Thus, a problem with boundary control was formulated. The convex continuous

coercive functional of an integral form was considered as the objective functional. The optimality

system was obtained from the condition that the first variation of the considered functional was equal to

zero [3].

The optimality systems obtained are a set of partial differential equations. For some of them the

initial conditions are given at the zero point in time, and for the other equations at the final moment in

time. This feature makes it difficult to find a solution and requires special approaches to the numerical

implementation of such problems. However, the proposed method allows us to find the control functions

in an explicit form, depending on the optimality system solutions.

Keywords. Optimal control, distributed system, optimality system, moving heat source.

1 Perm national research polytechnic university, Perm, RUSSİA, [email protected] 2 Perm national research polytechnic university, Perm, RUSSİA, [email protected] 3 Perm national research polytechnic university, Perm, RUSSİA, [email protected]

http://www.multitran.ru/c/M.exe?t=4775776_2_1&s1=MCVD

http://www.multitran.ru/c/M.exe?t=4775776_2_1&s1=MCVD


165

AMS 2010. 49J20, 49K20, 93C20.

References

[1] Shumkova, D.B., Optimal control in problems with unknown boundaries and moving

sources [Text]: diss. ... PhD: 01.01.02: it is protected on December, 21st, 2006: approved

12.10.2007, 108-110, 2006.

[2] Pervadchuk, V. P., Shumkova, D. B., Optimal control of a moving heat source, St.

Petersburg State Polytechnical University Journal, 37-44, 2010.

[3] Pervadchuk, V. P., Shumkova, D. B., Gordeeva, I. V., Optimal stabilizing control in the

problems of optical fiber drawing process, Applied Mathematics and Control Sciences, 4, 125-

134, 2016.


166

On the Kinetics of the Dimer-Dimer Reactions over Supported Catalysts

Pranas KATAUSKIS1

Abstract. We consider a phenomenological model of the dimer-dimer reaction [1] proceeding

on the composite (supported) catalysts [2] which involves: the bulk diffusion of reactants from a

bounded vessel towards the adsorbent and the product bulk one from the catalyst surface into the same

vessel, adsorption and desorption reactants molecules, and surface diffusion of adsorbed particles. The

model is based on the Langmuir-Hinshelwood surface reaction mechanism. The bulk diffusion of both

reactants and product particles is described by the Fick law while the surface diffusion of the adsorbed

molecules is based on the particle jumping mechanism [3]. The kinetics of the reaction is described by a

coupled system of partial and ordinary differential equations. Simulations of the mean-field model were

performed using the finite difference technique. Two distinct arrangements of the adsorption sites are

used for numerical calculations: (i) the same total amount of active and inactive in the surface reaction

adsorption sites, (ii) the same concentrations of active and inactive sites. Two adsorption cases of both

reactants for each arrangement of adsorption sites are considered: (a) each reactant adsorbs on both

active and inactive sites, (ii) both reactants adsorb only on the support. The mathematical model where

concentrations of both reactants at the catalyst surface are given is also studied. The influence of the size

of the catalytic particle, surface diffusivity, adsorption rate constants, and particle jump rate constants

via the catalyst-support interface on the catalytic reactivity of the supported catalyst is studied.

Keywords. Heterogeneous reactions, spillover, surface diffusion.

AMS 2010. 00A69.

References

[1] Skakauskas, V., Katauskis, P., Modelling dimer-dimer reactions on supported catalysts, J. Math.

Chem., 53, 604-617, 2015.

[2 Zhdanov, V. P., Kasemo, B., Simulations of the reaction kinetics on nanometer supported catalyst

particles, Surf. Sci. Rep., 39, 25-104, 2000.

[3] Gorban, A. N., Sargsyan, H. P., Wahab, H. A., Quasichemical models of multicomponent nonlinear

diffusion, Math. Model. Nat. Phenom., 6, 184-262, 2011.

1 Vilnius University, Vilnius, LİTHUANİA, [email protected]


167

The Importance of Algebraic Geometry and Cusp Singularities for Phase Transition

Dynamics of Neural Populations in Cerebral Cortex

R Murat DEMİRER1 , Oya DEMİRER2

Abstract. Freeman revealed that background activity have pseudo-equlibrium states which

leads to a phase transitions at neuronal mass level. Background activity has often been modeled in

thermodynamic terms like Carnot Cycle. The distribution of phase patterns behave like from a gas-like,

disorganized phase to more organized liquid like phase at the analogical level that jumps stochastically

over time depending on its past state and environmental state. The measures were based on entropy,

gain and spectral power. Freeman showed singularity as null-spikes in theta dynamics for adaptation

from random to perceptıon. We clarify his information on thermodynamics structures in the importance

of algebraic geometry and stochastic flows. We will define null-spikes with hypersurface cusp

singularities as the germ of a holomorphic function near the origin: it is an isolated hypersurface

singularity. We show that phase transition of each amplitude modulation pattern carries distinguished

paths over electrodes on a smooth stochastic spatio-temporal surface.

Keywords. AM pattern, Hilbert transform, null spike, phase cone, phase transitions, singularity

References

[1]W. J. Freeman, “Origin, structure, and role of background EEG activity. Part 1. Analytic amplitude,”

Clin. Neurophysiol., vol. 115, no. 9, pp. 2077–2088, 2004.

[2]W. J. Freeman, “Origin, structure, and role of background EEG activity. Part 2. Analytic phase,”

Clin. Neurophysiol., vol. 115, no. 9, pp. 2089–2107, 2004.

[3] J. Bhattacharya, H. Petsche, U. Feldmann, and B. Rescher, “EEG gamma-band phase

synchronization between posterior and frontal cortex during mental rotation in humans,” Neurosci. Lett.,

vol. 311, no. 1, pp. 29–32, 2001.

[4] Ramon, C., and M. D. Holmes, M. D., “Stochastic behavior of phase synchronization index and

cross-frequency couplings in epileptogenic zones during interictal periods measured with scalp dEEG,”

Front. Neurol., vol. 4 MAY, 2013.

[5] A. Keating, A., Homological mirror symmetry for hypersurface cusp singularities, Selecta

Mathematica, New Series, pp. 1–42, 2017.

1Üsküdar University, Istanbul, TURKEY, [email protected] 2Arel University, Istanbul, TURKEY, [email protected]


168

Stress Distribution in the Infinite Elastic Body with Two Neighboring Periodical Curved

Carbon Nanotube Located on Different and Parallel Planes

Reşat KÖŞKER1 and İsmail GÜLTEN2

Abstract. Curvature of the fibers in the structure of the unidirectional fibrous composite

materials is due to design requirements or to technological processes [1], [2]. Usually the curvature

caused by the technological process is modeled as a local one, whereas the curvature caused by design

features is modeled as a periodical. It is known that, this curvature cause to arise self-balanced stresses

and the values of these stresses can pass over adhesion resistance values. The widely explanation and

interpretation of investigations carried out on this subject are given in [1], [2]. However in all

investigations detailed in the monographs [1], [2], it was assumed that the reinforcing elements of

composite materials are made of traditional materials. In the paper [3] the attempt is made for

development of the internal stability loss problems in the structure of the unidirectional fibrous

composites for the case where the reinforcing element in the composite is the double-walled carbon

nanotube (DWCNT). In [4], the case is considered where a single periodical curved carbon nanotube

(CNT) with an infinite length is contained by an infinite body with low concentration of fibers and

stress distribution in that is investigated. However investigation on how the stress distribution

mentioned above is effected by the reciprocal effect between fibers, as volume ratio of fibers get bigger

in composites, is very important. In this study, we investigate the stress distribution in an infinite elastic

body containing two neighboring fibers, which are located along two parallel lines. It is assumed that

the middle lines of the fibers are located on the different and parallel planes and the curving of these

lines is periodic and sin-phase. The stress distribution is studied when the body is loaded at infinity by

uniformly distributed normal forces with intensity p acting in the fibers direction.

We assume that on the inter-medium surfaces the completely cohesion conditions are satisfied.

For solution of this problem we use the boundary form perturbation method according to which the

unknown values are presented in series form in small parameter. So we obtain equations set for each

approximation and making some process as detailed in [1], [2], and we obtain contact conditions for

each approximation. The original of the sought values are determined numerically. In this study,

numerical results are obtained in the framework of the zeroth and the first approximations for the

normal stress and the self-equilibrium shear stresses on the contact surfaces between CNTs and matrix.

The numerical results related to the stress distribution in considered body and illustrated the influence of

the distance between the fibers to the self-equilibrium stress distributions are given. Moreover, the

influences of the geometrical and mechanical problem parameters to these distributions are also

analyzed.

1 Yildiz Technical University, İstanbul, TURKEY, [email protected] 2 Yildiz Technical University, İstanbul, TURKEY, [email protected]


169

Acknowledgement. This research has been supported by Yıldız Technical University Scientific

Research Projects Coordination Department. Project Number: 2014-07-03-DOP01.

Keywords. carbon nanotube, stress distribution, nanocomposite.

AMS 2010. 74A10, 74E30.

References

[1] Akbarov, S.D., Stability Loss and Buckling Delamination: Three-Dimensional Linearized Approach

for Elastic and Viscoelastic Composites, Springer, 2012.

[2] Akbarov, S. D. And Guz, A. N., Mechanics of Curved Composites, Kluwer Academic Publishers,

2000.

[3] Akbarov, S.D., Microbuckling of a Double-Walled Carbon Nanotube Embedded in an Elastic

Matrix, International Journal of Solids and Structures 50, 2584- 2596, 2013.

[4] Kosker R., Akbarov S.D., Gulten İ., Stress Distribution in the Infinite Elastic Body with a

Periodical Curved Carbon Nanotube, 3rd İnternational Eurasian Conference on Mathematical Sciences

and Applications (IECMSA-2014), Book of Abstract, pp:251, 25-28 August 2014.


170

Discrete Fractional Solutions of a Legendre Equation

Resat YILMAZER 1

Abstract. One of the most popular research interests of science and engineering is the fractional

calculus theory in recent times. Discrete fractional calculus has also an important position in fractional

calculus. In this work, we acquire new discrete fractional solutions of the homogeneous and

nonhomogeneous Legendre differential equation by using discrete fractional nabla operator.

Theorem. Let 𝜑 ∈ 𝜑: 0 ≠ |𝜑𝛼| < ∞; 𝛼 ∈ 𝑅. Then the homogeneous Legendre equation

𝜑2(1 − 𝑟2) − 2𝜑1𝑟 + 𝜑𝜀(𝜀 + 1) = 0 (1 − 𝑟2 ≠ 0)

has solution of the form;

𝜑 = 𝑘(1 − 𝑟2)−[𝜀(𝜀+1)+2]/2−[2+𝐸−1𝜀(𝜀+1)]/2

where 𝜀 > 0 are given constant.

Keywords. Discrete fractional calculus, Legendre equation, nabla operator.

AMS 2010. 26A33, 34A08.

References

[1] Sabatier, J., Agrawal, O.P., Tenreiro Machado,J.A., Advances in Fractional Calculus: Theoretical

Developments and Applications in Physics and Engineering, Springer, 2007.

[2] Yilmazer, R., N-fractional calculus operator N- method to a modified hydrogen atom equation,

Math. Commun., 15, 489-501, 2010.

[3] Atici, F.M., Eloe, P.W., A transform method in discrete fractional calculus, Int. J. Difference

Equations 2(2), 165-176, 2007.

[4] Acar, N., Atici, F.M., Exponential Functions of Discrete Fractional Calculus, Applicable Analysis

and Discrete Mathematics, Vol. 7 (2), 343—353, 2013.

[5] Yilmazer, R., Inc, M., Tchier, F., Baleanu, D. Particular Solutions of the Confluent Hypergeometric

Differential Equation by Using the Nabla Fractional Calculus Operator. Entropy, 18, 49, 2016.

1 Fırat University, Elazığ, TURKEY, [email protected]


171

Discrete Fractional Solutions of a Associated Laguerre Equation

Resat YILMAZER1 and Erdal BAS2

Abstract. In this article, we will give theorems for the discrete fractional solutions of the

homogeneous and nonhomogeneous Laguerre equation by using discrete fractional nabla operator.

Theorem. Let 𝑦 ∈ 𝑦: 0 ≠ |𝑦𝛼| < ∞; 𝛼 ∈ 𝑅. Then the homogeneous Laguerre equation

𝑦2𝑥 + 𝑦1(𝜇 + 1 − 𝑥) + 𝑦𝜆 = 0 (𝑥 ≠ 0)

has particular solution of the form;

𝑦 = ℎ𝑥−(𝜆+𝜇+1)−(1+𝑞−1𝜆)

where 𝜇, 𝜆 are given constant.

Keywords. Discrete fractional calculus, Laguerre equation, nabla operator.

AMS 2010. 26A33, 34A08.

References

[1] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.

[2] Kelley, W.G., Peterson, A.C., Difference Equations: An Introduction with Applications, Academic

Press, San Diego, 2001.

[3] Atici, F.M., Eloe, P.W., A transform method in discrete fractional calculus, Int. J. Difference

Equations 2(2), 165-176, 2007.

[4] Atıcı, F.M., Sengül, S., Modeling with fractional difference equations, J. Math. Anal. Appl. 369, 1-9,

2010.

[5] Yilmazer, R., Inc, M., Tchier, F., Baleanu, D. Particular Solutions of the Confluent Hypergeometric

Differential Equation by Using the Nabla Fractional Calculus Operator. Entropy, 18, 49, 2016.

1 Fırat University, Elazığ, TURKEY, [email protected] 2 Fırat University, Elazığ, TURKEY, [email protected]


172

Asymptotic Stability of an Evolutionary Nonlinear Boltzmann-Type Equation

Roksana Brodnicka1 and Henryk Gacki2

Abstract. In the presentation a sufficient condition for the asymptotic stability with respect to

total variation norm of semigroup generated by an abstract evolutionary non-linear Boltzmann-type

equation in the space of signed measures with the right-hand side being a collision operator is

presented. For this purpose a sufficient condition for the asymptotic stability of Markov semigroups

acting on the space of signed measures for any distance ([4]), adapted to the total variation norm,

joined with the maximum principle for this norm is used. The presentation generalizes the result in [4]

related to the same type of non-linear Boltzmann-type equation, where the asymptotic stability in the

weaker norm, Kantorovich-Wasserstein, was investigated.

Keywords. Asymptotic stability, Markov operators, maximum principle for the total variation

metric, nonlinear Boltzmann-type equation

AMS 2010. 82B40, 82D05, 35Q20.

References

[1] Barnsley, M. F. and H. Cornille, H., General solution of a Boltzmann equation and the formation

of Maxwellian tails, Proc. Roy. London A 374, 371–400, 1981.

[2] Brodnicka, R. and Gacki, H., Asymptotic stability of a linear Boltzmann-type equation,

Appl. Math., 41, 323–334, 2014.

[3] Crandall, M. G., Differential equations on convex sets, J. Math. Soc. Japan 22 (1970), 443–455,

1970.

[4] Gacki, H., Applications of the Kantorovich-Rubinstein maximum principle in the theory of Markov

semigroups, Dissertationes Math. 448, 1–59, 2007.

[5] Gacki, H. and Lasota, A., A nonlinear version of the Kantorovich-Rubinstein maximum principle,

Nonlinear Anal. 52, 117–125, 2003.

[6] Gacki, H., On the Kantorovich-Rubinstein maximum principle for the Fortet-Mourier norm, Ann.

Pol. Math. 86.2, 107–121, 2005.

[7] Lasota, A. and Mackey, M. C., Chaos, Fractals, and Noise, Springer-Verlag, Berlin 1994.

[8] Lasota, A., Invariant principle for discrete time dynamical systems, Univ. Jagellonicae Acta Math.,

111–127, 1994.

1 University of Silesia in Katowice, Poland, [email protected] 2 University of Silesia in Katowice, Poland, [email protected]


173

[9] Lasota, A., Asymptotic stability of some nonlinear Boltzmann-type equations, J. Math. Anal. Appl.

268, 291–309, 2002.

[10] Lasota, A. and J. Traple, J., An application of the Kantorovich-Rubinstein maximum principle in

the theory of the Tjon-Wu equation, J. Differential Equations 159, 578–596, 1999.

[11] Lasota, A. and J. Traple, J., Asymptotic stability of differential equations on convex sets, J.

Dynamics and Differential Equations 15, 335–355, 2003.

[12] Lasota, A. and J. Traple, J., Properties of stationary solutions of a generalized Tjon-Wu equation,

J. Math. Anal. Appl. 335 No. 1, 669–682, 2007.

[13] Rachev, S. T., Probability Metrics and the Stability of Stochastic Models, John Wiley and Sons,

New York 1991.

[14] Tjon, J. A. and WuT. T., Numerical aspects of the approach to a Maxwellian equation, Phys.

Rev. A. 19, 883–888, 1979.


174

Hybrid Interval Algorithms for the Simultaneous Inclusion of Polynomial Complex

Zeros: An Experimental Comparison

Roseleine Neves MACHADO1 and Luiz Guerreiro LOPES2

Abstract. The theoretical relevance and practical importance of the polynomial root

finding problem has motivated the development of many numerical floating-point and interval

algorithms for the simultaneous approximation or inclusion of polynomial complex zeros.

Although the iterative algorithms based on complex interval arithmetic [1] produce inclusion

intervals for the exact zeros, giving error bounds automatically, contrary to what happens with

the methods based on the usual floating-point arithmetic, the computational costs of the

complex interval operations are comparatively high. In order to attempt to combine the

advantages of both classes of simultaneous polynomial zero finding algorithms, several hybrid

interval methods, in the sense of [2], have also been developed (see, e.g., [3]). However,

relatively little has been done to evaluate and compare these different iterative root finding

methods in terms of their efficiency and quality of results. In this work, an extensive

computational experiment, using a large set of randomly generated polynomials, is carried out

to assess the efficiency and accuracy of different known hybrid interval algorithms for

simultaneous polynomial root finding. The obtained results provide information on the relative

performance of the hybrid algorithms and the sharpness of the interval results they produce.

Keywords. Polynomial zeros, Simultaneous iterative methods, Hybrid interval methods.

AMS 2010. 30C10, 65G20, 65H04, 65Y20.

References

[1] Petković, M. S., Petković, L. D., Complex interval arithmetic and its applications, Wiley-

VCH, Berlin, 1998.

[2] Caprani, O., Madsen, K., Iterative methods for interval inclusion of fixed points, BIT, 18,

42-51, 1978.

[3] Petković, M. S., On the efficiency of some combined methods for polynomial complex

zeros, J. Comput. Appl. Math., 30, 99-115, 1990.

1 Federal Institute of Rio Grande do Sul, IFRS, Bento Gonçalves, RS, BRAZİL, [email protected] 2 University of Madeira, Faculty of Exact Sciences and Engineering, Funchal, Madeira Is., PORTUGAL, [email protected]


175

An Investigation on the Free Vibration of a Sphere with Inhomogeneous Initial Stresses

S. D. AKBAROV3,4 , N. YAHNIOGLU5 and H. H. GULIYEV6

Abstract. In this study, an investigation of free vibration of a hollow sphere with non-

homogeneous initial stresses are made within the exact equations of three dimensional

linearized theory of elastic waves in initially stressed bodies[1]. It is assumed that the initial

stresses are caused by the uniform radial forces act on the inner or outer surfaces of the

sphere. The material of the sphere are also assumed homogeneous and isotropic. The solutions

of the corresponding eigenvalue problems for the initial stressed hollow sphere are made

using the discrete-analytical solution method proposed [2,3]. The influence of some

geometrical parameters as well as initial stresses on the spheroidal vibration of this sphere are

analyzed and discussed. Analyzing the numerical results reveal that initial stresses in the

sphere are significantly effect on the fundamental frequency of the hollow sphere.

Acknowledgement: This study was made according to the Project No. 5/3, 2015: “Complex

of theoretical and experimental investigations related to the study of the interdisciplinary

problems of the Geomechanics” of the National Academy of Sciences of Azerbaijan.

Keywords: Free vibration of a sphere, spheroidal vibration, initial stress

References

[1] A.N. Guz, Elastic Waves in Bodies with Initial (Residual) Stresses, “A.S.K.”, Kiev, 2004

[2] S.D. Akbarov, Dynamics of Pre-Strained Bi-Material Elastic Systems: Linearized Three-

Dimensional Approach, Springer – Heildelberg, New York, 2015.

[3] S.D. Akbarov, H.H. Guliyev and N. Yahnioglu, Natural vibration of the three-layered

solid sphere with middle layer made of FGM: three-dimensional approach, SEM:

Struct.Eng. Mech, 57 (2016) 239–263.

3 Yildiz Technical University, Istanbul, TURKEY, [email protected] 4 National Academy of Sciences of Azerbaijan, Baku, AZERBAIJAN, [email protected] 5 Yildiz Technical University, Istanbul, TURKEY, [email protected] 6 National Academy of Sciences of Azerbaijan, Baku, AZERBAIJAN, [email protected]


176

GDP and Efficiency of Russian Economy

Sergey M. BORODACHEV 1

Abstract. The goal is to study GDP (gross domestic product) as an unobservable

characteristic of the Russian national economy state on the basis of more reliable observed

data on gross output (system’s output) and final consumption (system’s control). To do this,

the dynamic Leontief model is presented in a system-like form and its parameters and GDP

dynamics are estimated by the Kalman filter (KF).

We consider that all previous year's investments affect the growth of the gross output

by the next year. The weights of these investments in the sum are equal to unity and decrease

in geometric progression. The estimation of the model parameters was carried out by the

maximum likelihood method. The original data on the gross output and final consumption in

the period from 1995 to 2015 where taken from the Rosstat website, where maximally

aggregated economy of Russia is reflected in the system of national accounts.The growth of

direct costs and capital expenditures at gross output increase has been discovered, which

indicates the extensive character of the development of the economy. Investments are being

absorbed 2 - 4 years; any change of them causes a surge of commissioned fixed assets

fluctuation with a period of 2 years.

Then these parameter values were used in the KF to estimate the states of the system.

The emerging tendency of the transition of GDP growth to its fall means that the rate of

growth of final consumption is higher than the rate of GDP growth. In general, the behavior

of the curve of Rosstat GDP obviously follows from the declared investments, whereas in the

present calculation it is closer to the behavior of final consumption. Estimated GDP and

investments that really increased it were significantly less after the crisis of 2008-2009 than

officially published data.

The results of the work mean that the traditional approaches to determining GDP are

too based on not always reliable declared information on investments. The proposed approach

to the model calculation of GDP uses more reliable data and provides an objective picture of

the country's economic development.

Keywords. Dynamic Leontief model, investments in production capital, Kalman filter.

AMS 2010. 62P20, 93E11.

1 Ural Federal University named after the first President of Russia Boris Yeltsin, Ekaterinburg, RUSSİA.

[email protected]


177

Stabilization of Discrete Systems by High Order Compensators

Şerife YILMAZ İRTEM1, Taner BÜYÜKKÖROĞLU2, Vakif DZHAFAROV3

Abstract. For a unstable discrete plant the problem of stabilization by affine

compensator is considered. We consider the case where the difference of the order of the

system and the dimension of the controller vector is two or three. The obtained results are

based on the investigation for a vanishing point of a vector-valued (two dimensional or three

dimensional) multilinear map defined on a box. The required assumptions allow to roughly

display of the image sets of this multilinear map. Number of illustrative examples are

provided.

Keywords. Stabilizing vector, stabilization, stability region.

AMS 2010. 93D05, 93D15.

References

[1] Barmish, B. R., New Tools for Robustness of Linear Systems, MacMillan, New York,

1994.

[2] Bhattacharya, S. P. , Chapellat, H., Keel, L., Robust Control: The Parametric Approach,

Prentice-Hall, New Jersey, 1995.

[3] Fam, A. T., Meditch J. S., A canonical parameter space for linear systems design. IEEE

Transactions on Automatic Control, 23, 3, 454–458, 1978.

[4] Nurges, Ü., Avanessov, S. Fixed-order stabilising controller design by a

mixedrandomized/deterministic method, Int. J. Control, 88, 2, 335–346, 2015.

[5] Polyak, B.T., Shcherbakov, P. S., Hard problems in linear control theory: possible

approaches to solution, Automation and Remote Control, 65, 5, 681–718, 2005.

1 Mehmet Akif Ersoy University, Burdur, TURKEY, [email protected] 2 Anadolu University, Eskişehir, TURKEY, [email protected] 3 Anadolu University, Eskişehir, TURKEY, [email protected]


178

[6] Waqar, A. M., Swaroop, D., Bhattacharyya, S. P. Synthesis of fixed structure controllers

for discrete time systems. Numerical Linear Algebra in Signals, Systems and Control. Volume

80 of the series Lecture Notes in Electrical Engineering, 367-385, 2011.


179

Numerical Investigation of a Steady Flow of an Incompressible Pseudoplastic Fluid in a

Lid Driven Cavity

Serpil ŞAHİN1 and Hüseyin DEMİR2

Abstract. In this paper, numerical investigation for 2-D steady-state, incompressible

pseudoplastic viscous flow is presented. Pseudo time derivative is used to solve the continuity

and momentum equations with suitable boundary conditions. Depending on high Reynolds

number, wall motions of flow are investigated with respect to nonlinear viscosity by using

Cross model. This study has been undertaken as a first step toward understanding in heat and

mass transport in solvent and polymer processing equipment. Solution to the vorticity

equation for moving top wall is obtained numerically and found to be stable and convergent

for high value of Reynolds numbers. In fact some new results, which are governed by inertia

and variable shear-rate, are obtained.

Keywords. Pseudoplastic viscous flow, Pseudo-time derivative, High Reynolds

number.

AMS 2010. 76A05, 76D05.

References

[1] Aydin, S.H., Nesliturk, A.I., Tezer-Sezgin, M., Two-level finite element method with a

stabilizing subgrid for the incompressible MHD equations, Int. J. Numer. Meth. Fluids, 62,

188-210, 2010.

[2] Benjamin, A.S., Denny, V.E., On the convergence of numerical solutions for 2-D flows in

a cavity at large Re, J. Comp. Physics, 33, 340-358, 1979.

[3] Demir, H., Sahin, S., Numerical investigation of a steady flow of an incompressible fluid

in a lid driven cavity, Turkish Journal of Mathematics and Computer Science, Article ID:

20130031, 2013.

[4] Erturk, E., Corke, T.C., Gökçöl, C., Numerical solutions of 2-D steady incompressible

driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids, 48, 747–774, 2005.

1 Amasya University, Amasya, TURKEY, [email protected] 2 Ondokuz Mayıs University, Samsun, TURKEY, [email protected]


180

Numerical Comparision of Newtonian and Dilatant Fluids in an Enclosed Cavity Region

Serpil ŞAHİN1 and Hüseyin DEMİR2

Abstract. In this study, numerical investigation for 2-D steady-state, incompressible

Newtonian and dilatant viscous flow is presented. Pseudo time derivative is used to solve the

continuity and momentum equations with suitable boundary conditions. Depending on high

Reynolds number, wall motions of flow are investigated with respect to nonlinear viscosity by

using Cross model. Therefore, the governing equations of fluid of vorticity-stream function

formulations are solved numerically using finite difference and Gauss Elimination method.

The stream function and vorticity results are obtained for Newtonian and dilatant fluids.

These results are presented both in tables and figures.

Keywords. Pseudo Time Parameter, Newtonian Fluid, Dilatant Fluid.

AMS 2010. 76A05, 76D05.

Acknowledgment

This study was supported financially by the Research Centre of Amasya University (Project

No: FMB‐BAP 16-0201).

References

[1] Barragy, E., Carey, G.F., Stream function-vorticity driven cavity solutions using p finite

elements, Computers and Fluids. 26, 453-468, 1997.

[2] Benjamin, A.S., Denny, V.E., On the convergence of numerical solutions for 2-D flows in

a cavity at large Re, J. Comp. Physics, 33, 340-358, 1979.

[3] Demir, H., Sahin, S., Numerical investigation of a steady flow of an incompressible fluid

in a lid driven cavity, Turkish Journal of Mathematics and Computer Science, Article ID:

20130031, 2013.

[4] Erturk, E., Corke, T.C., Gökçöl, C., Numerical solutions of 2-D steady incompressible

driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids, 48, 747–774, 2005.

1 Amasya University, Amasya, Turkey, [email protected] 2 Ondokuz Mayıs University, Samsun, Turkey, [email protected]


181

New Hybrid Conjugate Gradient Method as a Convex Combination of HS and FR

Conjugate Gradient Methods

Snezana S. DJORDJEVIC1

Abstract. In this paper we present a new hybrid conjugate gradient algorithm for

unconstrained optimization. This method is a convex combination of Hestenes-Stiefel

conjugate gradient method and Fletcher-Reeves conjugate gradient method. The convex

parameter is computed in such a way that the search direction satisfies the condition of the

Newton direction.

The strong Wolfe line search conditions are used. The global convergence of this new

method is proved.

Numerical comparisons show that the present hybrid conjugate gradient algorithm is

the efficient one.

1 Faculty of Technology, University of Nis, 16000 Leskovac, SERBIA, [email protected]

[email protected]


182

A New Formula for Higher Order Derivatives of Type xfdx

dx

n

k

and Its

Applications

Telhat ÖZDOĞAN, Melek ERASLAN, Metin ORBAY1

Abstract. We employed a new analytical formula for the higher derivatives of type

xfdx

dx

n

k

in terms of the derivatives of the function xf . Obtained formula for these

higher derivatives enable one to handle some special functions, such as spherical Bessel

functions and Hankel functions, easily [1-4]. As Its an application, the obtained formula has

been applied to spherical Bessel and Hankel functions and then analytical formulae have been

obtained for these special functions.

Keywords. Higher order derivatives, Spherical functions.

AMS 2010. 43A90.

References

[1] Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J., the siam 100-digit challenge. a

study in high-accuracy numerical computing, SIAM, Philadelphia, 2004.

[2] Arfken, G.B., Weber, H. J., mathematical methods for physicists, Elsevier Academic

Press, USA, 2005.

[3] Watson, N., Treatise A., on the theory of bessel functions, Cambridge Univ. Press,

Cambridge, 1966.

[4] Jeffrey A., and Hui Dai, H.,handbook of mathematical formulas and tables, Academic

Press, London, 2008.

1 Amasya University, Amasya, TURKEY, [email protected]



183

Asymptotic Behaviour for Functional Three-Dimensional Navier-Stokes-Voigt

Equations

T. CARABALLO 1 and Antonio M. Márquez DURÁN 2

Abstract. The Navier-Stokes-Voigt (NSV) model of the viscoelastic incompresible

fluid was introduced by Oskolkov in [1], giving an approximate description of the Kelvin-

Voigt fluid, and was recently proposed as a regularization of the 3D-Navier-Stokes equation

for the purpose of direct numerical simulation.

Our aim in this paper is consider the following version of this problem for a variety of

delay terms in a unified formulation, and to study the existence of pullback attractors, which

suppose a generalization of some recent works ([2], [3]):

2

0

. , , , ,

. 0, , ,

0, , ,

,

, , , ,0 , .

u uv u u u p g t u in

t t

u in

u on

u x u x in

u t x t x t h x

where τ 3 -dependent

delay term g(t, ut ) represents, for instance, the influence of an external force with some kind

of delay, memory or hereditary characteristic. Here, ut denotes a segment of solution, in other

words, given a function :u t

we can define the mapping ut

(s,x)=u(t+s,x), s (-∞,0], x .

Keywords. Navier-Stokes-Voigt model, delay terms, pullback attractors.

AMS 2010. Primary: 35R15, 35B41, 37B55; Secondary: 47J35.

References

[1] A. Oskolkov, The uniqueness and solvability in the large of boundary value problems for

the equations of motion of aqueous of polymers, Zap. Nauc. Semin. POMI 38, 98-116, 1973.

[2] L. Haiyan and Q. Yuming, Pullback attractors for three-dimensional Navier-Stokes-Voigt

equations with delays, Boundary value problems, 2013:191, 2013.

[3] JG. Luengo, P. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-

autonomous Navier-Stokes-Voigt equations. Nonlinearity 25, 905-930, 2012.

1Universidad de Sevilla, Sevilla, ESPAÑA, [email protected] 2Universidad Pablo de Olavide, Sevilla, ESPAÑA, [email protected]


184

Mixing Problems: Associated Matrices and Stability of Linear ODE Systems

Victor MARTINEZ-LUACES1

Abstract. In this paper, matrices related to chemical mixing problems (MP-matrices) are

characterized and studied. In previous works, mixing problems were analyzed [1], [2] and it is

possible to infer that the mathematical models are always linear ODE systems. The associated

matrices have different structures depending on whether or not there is recirculation of fluids.

In the last case, i.e., if there is not recirculation, the MP-matrix is an upper matrix; otherwise

the structure of the MP-matrix is not so simple. However, it is possible to give a necessary

condition to be satisfied by any MP-matrix. Applying the Gershgorin Circles Theorem [3] it

can be observed that all the matrix eigenvalues have non-positive real part. Even more, it can

be proved that if the real part is zero, it must be the null eigenvalue. Lastly, if the mixing

problem involves three or less components, all the eigenvalues have negative real part and so,

the corresponding ODE solutions are asymptotically stable.

Finally, a comparative analysis shows similarities and differences with other matrices

(FOCKM-matrices), usually involved in Chemical Kinetics problems [4], [5], [6]. There

exists some similarities; however the differences are important enough to obtain distinct

qualitative behaviors of the ODE solutions which are showed in this article through several

examples.

Keywords. Mixing problems, associated matrices, ODE stability.

AMS 2010. 49K15, 34L15, 97M60.

References

[1] Martinez Luaces, V., Inverse-modelling problems in chemical engineering courses, in

Vision and change for a new century. ISC-Delta, Montevideo, Uruguay, 2007.

[2] Martinez-Luaces, V., Modelling and inverse-modelling: experiences with O.D.E. linear

systems in engineering courses, International Journal of Mathematical Education in Science

and Technology 40(2), 259-268, 2009.

[3] Varga, R.S., Geršgorin and His Circles. Springer-Verlag, Berlin, 2004.

1 UdelaR, Montevideo, URUGUAY, [email protected]



185

[4] Martinez-Luaces, V., Chemical Kinetics and Inverse Modelling Problems, in Chemical

Kinetics, In Tech Open Science, Rijeka, Croatia, 2012.

[5] Martinez-Luaces, V., First Order Chemical Kinetics Matrices and Stability of O.D.E.

Systems, in Advances in Linear Algebra Research, Nova Publishers New York, 2015.

[6] Martinez-Luaces, V., Qualitative Behavior of Concentration Curves in First Order

Chemical Kinetics Mechanisms, in Advances in Chemistry Research, Nova Publishers, New

York, 2017.


186

Backward Semi-Lagrangian Scheme for Guiding Center Problems

Xiangfan PIAO1, Dojin KIM2 and Philsu KIM3

Abstract. In this talk, we disucss how to develop a numerical scheme based on

Backward semi-Lagrangian for solving guiding center models. A main problem in this time

discretization is to find a starting position of the characteristic curves arriving at each grid

point, which is a highly nonlinear and self-consistency problem imposed by the Poisson

equation. The proposed scheme is based on the error correction method recently developed by

the authors[1,2,3]. It is shown that the proposed scheme has a good performance in

computational cost together with a superior mass and total kinetic energy conservation

compared to the convectional second-order iteration shcme and recently developed higher

order method based on the Adams-Moulton rule[4].

Keywords. Backward semi-Lagrangian method, guiding center problem.

AMS 2010. 35Q83,65M12, 65M70.

Aknowlegement. This work was supported by Basic Science Research Program

through the National Research Foundation of Korea(NRF) funded by the Ministry of Science

ICT & Future Planning(grant number NRF-2017R1C1B1002370).

References

[1] Kim, P., Piao, X. and Kim, S.D., An error corrected Euler method for solving stiff

problems based on Chebyshev collocation, SIAM J. Numer. Anal., 49, 2211-2230, 2011.

[2] Kim, S.D., Piao, X. and Kim, P., Convergence on error correction methods for solving

initial value problems, J. Comp. App. Math., 236, 4448-4461, 2012.

[3] Piao, X., Kwon, J., Yi, D., Kim, S. and Kim, P., An iteration free backward semi-

Lagrangian scheme for guiding center problems, SIAM J. Numer. Anal. 53, 619-643, 2015.

[4] Filbet, F., Prouveur, C., High order time discretization for backward semi-Lagrangian

methods, J. Comp. Appl. Math., 303, 171-188, 2016.

1 Hannam University, Daejeon, KOREA, [email protected] 2 KyungPook National University, DAEGU, KOREA, [email protected] 3 KyungPook National University, DAEGU, KOREA, [email protected]


187

A Comparison of Multi-step and Multi-stage Method

Yonghyeon JEON1 and Sunyoung BU2, Philsu KIM3

Abstract. We study a multi-stage method compared with a multi-step method for

solving a stiff initial value problem. Due to expensive computational costs of the multi-stage

methods for solving a massive linear system induced from the linearization of a highly stiff

system, stiff problems are usually solved by the multi-step method, rather than the multi-stage

method. In this work, we investigate properties of both the multi-step and the multi-stage

methods and discuss the difference between the two methods through numerical tests in

several examples. Furthermore, the advantages of multi-stage methods can be heuristically

proved even for stiff systems by the comparison of two methods with several numerical tests.

Keywords. Ordinary differential equation, Multi-step method, Multi-stage method

References

[1] Prothero, A., Robinson, A., On the stability and accuracy of one step methods for solving

stiff systems of ordinary differential equations , Math. Comput. 28 (125)145-162, 1974.

[2] Fredebeul, C., A-BDF: a generalization of the backward differentiation formulae, SIAM J.

Numer. Anal. 35 (1998) 1917–1938.

[3] Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations,

Prentice–Hall, 1971.

[4] Hairer, E., Nørsett, S.P., Wanner, G., Solving Ordinary Differential Equations.I Nontstiff

Problems, Springer Ser. Comput. Math., Springer, 1993.

1 Kyungpook National University, REPUBLCİ OF KOREA, [email protected] 2 Hongik University, REPUBLCİ OF KOREA, [email protected] 3 Kyungpook National University, REPUBLCİ OF KOREA, [email protected]


188

Green's Functions For Problems Simulating Potential Fields in Thin-Walled Assemblies

of Irregular Configuration

Yu. A. MELNIKOV1 and V. N. BORODIN2

Abstract. Green's functions are targeted for a specific class of boundary value problems that

simulate potential fields generated in assemblies of thin-walled fragments of irregular configuration.

Computational efficiency is explored of an approach based on a Green's function version of the

classical [1] method of functional equations. It is shown that the proposed approach appears workable

for a broad range of problem settings. Representing the key component in the approach, matrices of

Green's type are analytically constructed to a number of boundary-contact value problems for

compound regions of regular configuration.

Construction of Green's functions for applied partial differential equations and incorporation of

them into numerical schemes have been a notable part in the classical boundary integral equation

method and its various implementations. An extensive database, accumulated so far in this segment of

research, convincingly illustrates the computational potential of already developed Green's function-

based numerical algorithms. The latter provide high accuracy level achieved at a low computational

cost in solving different classes of problems for partial differential equations applicable in various

areas of engineering and science.

Prior to the actual numerical work, we obtain either closed analytical forms or computer-friendly

series containing representations of required matrices of Green's type for sets of elliptic partial

differential equations. Each of such equations is written in geographical coordinates associated with a

single fragment of the given assembly. Construction of such matrices is not a trivial exercise.

Predetermining the non-triviality, is a factor related to problems set up for assemblies of elements,

where each element hosts a single governing equation. This yields a specific type of boundary-contact

value problems for sets of equations each posed in an individual region, while some contact

conditions, imposed on the interface lines, convert the problem statement into a system format. The

classical Green's function formalism appeared inapplicable to this type of problems, requiring a special

modification, which was described in detail in [2].

Keywords. Potential fields on surfaces, Green's function method

AMS 2010. 35J15

References

1 Middle Tennessee State University, Murfreesboro, USA, [email protected] 2 Tennessee Technological University, Cookeville, USA, [email protected]


189

[1] Kupradze, V. D. and Aleksidze, M. A., The method of functional equations for approximate

solution of certain boundary value problems, USSR Computational Mathematics and Mathematical

Physics, 1964, 4, 82-126.

[2] Yuri A. Melnikov and Max Y. Melnikov, Green's Functions. Construction and Applications, De

Gruyter, Berlin-Boston, 2012.


190

The Hurwitz’s Formula and Boundary Value Problems for Second-Order Equations

Yu. E. ANIKONOV1, İsmet GÖLGELEYEN2 and Mustafa YILDIZ3

Abstract. In this work, we consider an ill-posed boundary value problem for

multidimensional second-order evolution equations with variable coefficients. We reduce the

boundary value problem to a functional equation by using the given data, and then we obtain

the solution by means of the Hurwitz formula.

Keywords. Ill-posed problem, functional equation, second-order differential equation.

AMS 2010. 65N20, 35R10.

References

[1] Anikonov, Yu. E., Neshchadim, M. V., Representations for the Solutions and Coefficients

of Second-Order Differential Equations, Journal of Applied and Industrial Mathematics, 7

(1):15-21, 2013.

[2] Markushevich, A. I., Introduction to the classical theory of Abelian functions,

Translations of Mathematical Monographs, 96, American Mathematical Society, Providence,

R. I., 1992.

1 Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Science, Acad. Koptyug prosp., 4, Novosibirsk,

630090, RUSSİA. [email protected] 2,3 Department of Mathematics, Faculty of Arts and Sciences, Bülent Ecevit University, Zonguldak, 67100, TURKEY.

[email protected], [email protected]


191

DISCRETE

MATHEMATICS


192

Secret Sharing Schemes Based on Extension Fields

Selda ÇALKAVUR1

Abstract. A ,(t )n secret sharing scheme is a method of distribution of information

among n participants such that 1t can reconstruct the secret but 1t cannot. There is

numerous research about secret sharing schemes. However there is little research on secret

sharing schemes based on extension fields. In this paper, we study some secret sharing

schemes based on extension fields over finite fields. We use two methods to recover the

secret. We define the access structure and the accessibility degree for these secret sharing

schemes. We also describe our theorems, definitions and a corollary.

Keywords. Secret sharing scheme, extension fields, trace and norm functions.

AMS 2010. 94A62, 12E20.

References

[1] Blakley, G.R., Safeguarding cryptographic keys, American Federation of Information

Processing Societies, National Computer Conference, pp. 313-317, 1979.

[2] Carreras, F., Magana, A. and Munuera, C., The accessibility of an access structure,

RAIRO- Theoretical Informatics and Applications, 40.04, pp. 559-567. 2006.

[3] Ding, C., Kohel, D. R. and Ling, S., Secret-sharing with a class of ternary codes,

Theoretical Computer Science, 246(1), pp.285-298, 2000.

[4] Dougherty, S. T., Mesnager, S., Solé, P., Secret sharing schemes based on self-dual codes,

Information Theory Workshop (2008), ITW'08, IEEE, 2008.

[5] Kim. J.L., Lee, N., Secret sharing schemes based on additive codes over )4(GF ,

Applicable Algebra in Engineering, Communication and Computing, pp. 1-19, 2016.

[6] R. Lidl, H. Niederreiter, Finite Fields, vol. 20, Cambridge.

[7] Li, Zhihui, Xue, Ting Xue and Lai, Hong, Secret sharing schemes from binary linear

codes, Information Sciences 180(22), pp. 4412-4419, 2010.



193

[8] Massey, J. L., Minimal codewords and secret sharing, in Proc. 6th Joint Swedish-Russian

Workshop on Information Theory, Mölle, Sweden, pp. 276-279, 1993.

[9] McEliece R, J. and Sarwate D. V., On sharing secrets and Reed-Solomon codes, Common.

Assoc. Comp. Mach., vol. 24, pp. 583-584, 1981.

[10] Piepryzk, J. and Zhang, X. M., Ideal threshold schemes from MDS codes, Proceedings of

Information Security and Cryptology ICISC 2002, Lecture Notes in Computer Science,

vol. 2587, Springer-Verlag, Berlin, pp. 269-279, 2003.

[11] Shamir, A., How to share a secret, Comm. of the ACM 22 pp. 612-613, 1979.


194

GEOMETRY


195

Pseudo Spherical Indicatrix Elastic Curves

Ahmet YÜCESAN1 Gözde ÖZKAN TÜKEL2 and Tunahan TURHAN3

Abstract. The tangent, normal and binormal indicatrix of a curve in Minkowski 3-

space may be positioned in De Sitter 2-space, hyperbolic 2-space and 2- dimensional lightlike

cone in terms of causal character of the curve. We separately derive Euler- Lagrange

equations of pseudo spherical indicatrix elastic curves with regard to structure of the

Minkowski 3-space. Then we discuss the solution of the equations.

Keywords. Elastic curve, Euler-Lagrange equation, pseudo-spherical image

AMS 2010. 53A35, 53B30, 74B05.

References

[1] Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. Int.

Electron. J. Geom. 7, no. 1, 44-107, 2014.

[2] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Pres.,

New York, 1993.

[3] Singer, D.A., Lectures on Elastic Curves and Rods, AIP Conf. Proc., 1002, Amer. Inst.

Phys., Melville, NY, 2008.

[4] Tukel, G.O., Elastic Strips in Minkowski 3-space, Ph.D. Thesis, Suleyman Demirel

University, Isparta, 2010.

[5] Tükel, G. O., Yücesan, A., Elastic Curves in a Two-dimensional Lightlike Cone. Int.

Electron. J. Geom. 8 , no. 2, 1–8, 2015.

[6] Weinstock, R., Calculus of Variations with Application to Physics and Engineering,

Dover Publications, Inc., New York, 1974.

[7] Yücesan, A., Oral, M., Elastica on 2-dimensional de Sitter Space, V. International

Meeting on Lorentzian Geometry, July 8-11 2009, Martina Franca (Taranto), Italy.

[8] Yücesan, A., Oral, M., Elastica on 2-dimensional Anti-de Sitter Space. Int. J. Geom.

Methods Mod. Phys. 8 , no. 1, 107–113, 2011.

1 Suleyman Demirel University, Isparta, TURKEY, [email protected] 2 Suleyman Demirel University, Isparta, TURKEY, [email protected] 3 Suleyman Demirel University, Isparta, TURKEY, [email protected]

http://www.ams.org/mathscinet/search/journaldoc.html?id=6464


http://www.ams.org/mathscinet/search/series.html?id=1099

http://www.ams.org/mathscinet/search/author.html?mrauthid=951603




http://www.ams.org/mathscinet/search/publications.html?pg1=ISSI&s1=336902&sort=oldest












196

Problems of g- lifts

Arif SALIMOV1 and Rabia CAKAN2

Abstract. The main purpose of this paper is to transfer complete lifts from the tangent

bundle to the cotangent bundle by using a musical isomorphism between these bundles. The

g-lifts of tensor fields are described in this study newly. We study some problems of g-lifts

that are constituted by transferring the tensor fields via musical isomorphism. The results

obtained give new remarks about g-lifts on the cotangent bundle.

Keywords. Tangent bundle, cotangent bundle, complete lift, musical isomorphism, g-

lifts.

AMS 2010. 55R10, 53C15.

References

[1] Cakan, R., Akbulut, K., Salimov, A., Musical isomorphisms and problems of lifts, Chin.

Ann. Math. Ser. B., 37, 3, 323-330, 2016.

[2] Aslancı, S., Cakan, R., On a Cotangent Bundle with Deformed Riemannian Extension,

Mediterr. J. Math., 11, 4, 1251-1260, 2014,

[3] Salimov, A.A., On operators associated with tensor fields, J. Geom., 99, 1-2, 107-145,

2010.

[4] Yano, A., Ako, M, On certain operators associated with tensor fields, Kodai Math. Sem.

Rep., 20, 414-436, 1968.

[5] Yano, K., Ishihara, S., Tangent and cotangent bundles, Pure and Applied Mathematics,

Marcel Dekker, Inc., New York, 1973.

1 Atatürk University, Erzurum, TURKEY, [email protected] 2 Kafkas University, Kars, TURKEY, [email protected]


197

Position Vectors of Curves with Respect to Darboux Frame in the Galilen 3-Space

Buket CEYLAN DİRİŞEN1 and Tevfik ŞAHİN2

Abstract. This article investigates the position vector of a curve on the surface in the Galilean

3-space G3. Firstly, the position vector of a curve with respect to the Darboux frame is

determined. Secondly, we obtain the standard representation of the position vector of a curve

with respect to Darboux frame in terms of the geodesic, normal curvature and geodesic

torsion. As a result of this, we define the position vectors of geodesic, asymptotic and normal

line along with some special curves with respect to Darboux frame. Finally, we elaborate on

some examples and provide their graphs.

Keywords. : Position vector, Darboux frame, Geodesic, Galilean 3-space.

AMS 2010. 53B30, 53A35.

Acknowledgment

This study was supported financially by the Research Centre of Amasya University (Project

No: FMB‐BAP 16-0213).

References

[1] Ali A. T., Position vectors of curves in the Galilean space G3, Matematicki Vesnik,

64(3), 200-210, 2012.

[2] Roschel O., Die Geometrie des Galileischen Raumes, Habilitation Schrift, Leoben, 1984.

[3] Yaglom I. M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag,

New York, 1979.

[4] Dirişen B.C., Şahin T., Position vector of a curve with respect to the Darboux frame in the

Galilean space G3, arXiv:1707. 03930v1, 2017.

1 Amasya University, Amasya, Turkey, [email protected] 2 Amasya University, Amasya, Turkey, [email protected], [email protected]





198

Polynomial Helices in 𝑬𝒏

Bülent ALTUNKAYA1 and Levent KULA2

Abstract. In En (𝑛 ≥ 4), there are only a few examples of general helices being valid

in special conditions. In this work, we introduce general helices by using polynomial curves

in En (𝑛 ≥ 4) and we present new results, respectively.

Keywords. General Helix, Polynomial Curve.

AMS 2010. 53A04, 53A05.

References

[1] Ali, Ahmad T., Turgut, M., Some Characterizations of Slant Helices in the Euclidean

Space 𝐸𝑛 , Hacettepe Journal of Mathematics and Statistics, 39, 327-336, 2010.

[2] Camcı, C., İlarslan, K., Kula, L. and Hacısalihoğlu, H. H., Harmonic curvatures and

generalized helices in 𝐸𝑛, Chaos, Solitons and Fractals, 40, 2590–2596, 2009.

[3] Hacısalihoğlu, H. H., Diferansiyel Geometri I, Ankara, 1980.

1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected]


199

On Normal Magnetic Curves in 3-Dimensional Heisenberg Group H3

Cihan ÖZGÜR1

Abstract. We study normal magnetic curves in 3-dimensional Heisenberg Group H3

and we give the classification of this type curves. We obtain the parametric equations of all

normal magnetic curves whose trajectories are described by the equation ' ' ( ') q in

H3. As an application, we consider normal magnetic curves in 3-dimensional Nil space Nil3.

Keywords. Magnetic curve, Legendre curve, slant curve, Heisenberg group.

AMS 2010. 53C25, 53C40, 53A05.

References

[1] Adachi T., Curvature bound and trajectories for magnetic fields on a Hadamard surface,

Tsukuba J. Math., 20, no. 1, 225-230, 1996.

[2] Cabrerizo J. L., Fernández M., Gómez J. S., On the existence of almost contact structure

and the contact magnetic field, Acta Math. Hungar., 125, no. 1-2, 191-199, 2009.

[3] Calvaruso G., Munteanu M., Perrone A., Killing magnetic curves in three-dimensional

almost paracontact manifolds, J. Math. Anal. Appl., 426, no. 1, 423-439, 2015.

[4] Druţă-Romaniuc S., Inoguchi J., Munteanu M., Nistor A., Magnetic curves in Sasakian

manifolds, J. Nonlinear Math. Phys., 22, no. 3, 428-447, 2015.

1 Balıkesir University, Balıkesir, TURKEY, [email protected]


200

On Chen Invariants for Submanifolds of Riemannian Product Manifolds

Erol KILIÇ1, Mehmet GÜLBAHAR2 and Sadık KELEŞ3

Abstract. The Chen invariant for submanifolds of a Riemannian product manifold is

studied. Some relations dealing the Chen invariant are presented. With the help of these

relations, some characterizations for invariant and anti-invariant submanifolds of a

Riemannian product manifold are obtained.

Keywords. Chen invariant, submanifold, Riemannian product manifold.

AMS 2010. 53C15, 53C40, 53C42.

References

[1] Atçeken, M., Slant submanifolds of a Riemannian product manifold, Acta Math. Sci.Ser.

B. Eng. Ed. 30(1), 215-224, 2010.

[2] Chen, B.-Y., Pseudo-Riemannian geometry, invariants and applications, World

Scientific Publishing, Hackensack, NJ 2011.

[3] Kılıç, E., Tripathi, M. M., Gülbahar, M., Chen-Ricci inequalities for submanifolds of

Riemannian and Kaehlerian product manifolds, Ann. Polon. Math. 116(1), 27-56, 2016.

[4] Sahin, B., Slant submanifolds of an almost product Riemannian manifold, J. Korean Math.

Soc. 43, 717-732, 2016.

[5] Yano, K., Structures on manifolds, World Sci. Singapore, 1984.

1 İnönü University, Malatya, TURKEY, [email protected] 2 Siirt University, Siirt, TURKEY, [email protected] 3 İnönü University, Malatya, TURKEY, [email protected]


201

Special Curves in Finsler Space

Fatma ATEŞ1, Zehra ÖZDEMİR2, F. Nejat EKMEKCİ3 and Mustafa ÇALIŞKAN4

Abstract. In this study, firstly we give the necessary notions and results in Finsler

geometry. Then, we investigate some new characterizations for special Finslerian curves.

Also, we obtain some results for these curves with using the Finslerian Metrics.

Keywords. Finsler space, special curves.

AMS 2010. 53B40, 14H45.

References

[1] Bao, D., Chern, S.S., Shen, Z., An Introduction to Riemann-Finsler Geometry (Graduate

Texts in Mathematics; 200), Springer, 2000.

[2] Bucataru, R. M., Finsler-Lagrange Geometry - Applications to Dynamical Systems, Ed.

Acad. Romane, Bucharest, 2007.

[3] Mo, X., An Introduction to Finsler Geometry, World Scientific, 2006.

1 Ankara University, Ankara, TURKEY, [email protected] 2 Ankara University, Ankara, TURKEY, [email protected] 3 Ankara University, Ankara, TURKEY, [email protected] 4 Gazi University, Ankara, TURKEY, [email protected]



202

A Differential Calculus on Rh(1|2)

Fatma BULUT1

Abstract. Quantum superspaces are introduced in [1] to represent quantum

supergroups. In this presentation, we set up a differential calculus on the superspace Rh(1|2)

[2]. It contains functions on Rh(1|2), their differentials and differential forms [3].

Keywords. h-deformed, superspace, differential calculus.

AMS 2010. 17B37, 81R60.

References

[1] Manin, Y. I.: Multiparametric quantum deformation of the general linear supergroups.

Commun. Math. Phys. 123, 163–175, 1989.

[2] Aizawa, N., Chakrabarti, R.: Noncommutative geometry of super-jordanian OSph(2/1)

covariant quantum space. J. Math. Phys. 45, 1623–1638, 2004.

[3] Celik, S., Bicovariant differential calculus on the superspace Rq(1|2), J. Alg. Appl., 15,

1650172 (2016) [17 pages], doi:10.1142/S0219498816501723.

1 Bitlis Eren Üniversitesi, Bitlis, TURKEY, [email protected]


203

Application Example of the Processing of the Cubes in the Geometry Lesson in

Elementary Schools

Gönül TÜRKAN DEMİR1, Keziban ORBAY2 and Emine ALTUNAY ŞAM3

Abstract. Mathematics has had an important place in Turkish educational history for

long. Philosophy, mathematics, geometry, astronomy etc. collectively known as the rational

sciences had been taught in the madrasahs, which were the most important institutions for the

educational system of Classical Ottoman Period, beginning from the reign of Mehmed the

Conqueror, as well as the traditional/religious subjects. However, with madrasahs getting

corrupted during the mid-17th century, besides other rational sciences, geometry was also

abandoned.

19th century is when the Ottoman Empire had started a reformation and transition,

implementing western models in military, politics and administration. This is also the period,

during which the most significant improvements in education and science were observed.

Transferred from the western civilization, course books on fundamental sciences that were

taught only in higher at the beginning, were also published and used in primary and secondary

education as well, in the following years.

Educational Statute 1869, involved geometry in educational system. Following this

Statute, whereas in the elementary schools, only calculus was taught, geometry took its place

in the curriculums of junior high schools (rüştiye), high schools (idadi) and higher education

(sultani). In those elementary schools, called “Mekatib-iptidâiye” and today known as

“secondary school”, formed in 1870, there were no geometry courses in the beginning,

however in the following years, geometry was attached to the curriculum. This research

studies the articles “An Example Geometry Course on Cubes for 1st Grades” published on the

Volume 4 / Issue 47 of the Journal of Education in 1919. This articles define the stages of

course planning in geometry and the equipment required and present an example course.

This is a descriptive research, conducted using screening model in order to introduce

textual contents of example geometry courses in 1919. The research uses document review

method.

The text reviewed exhibits preparation phases, review of the previous courses,

introduction of new concepts, and association of those concepts with real-life and exercises

for students. All stages presented as teacher-student dialog, it allows identifying the teacher-

1 Amasya University, Amasya, TURKEY, [email protected] 2 Amasya University, Amasya, TURKEY, [email protected] 3 Amasya University, Amasya, TURKEY, [email protected]


204

student interaction of that period. This research, raising awareness on the historical

development of geometry education in terms of how the courses were conducted, is expected

to contribute to the studies of educators of this field.

Key Words: Geometry course, History of Mathematic, Secondary school, Planning in

geometry.

References

[1] Abdullatif, Hendese dersi numunesi ‘devre-i aliyede’ (mika’blar hakkında). Tedrisat

Mecmuası 4, 47, 236-240, 1919.

[2] Aslan E. ve Olkun S., Elementary school mathematics in the first curricula of Turkish

Republic. Elementary Education Online, 10, 3, 991-1009, http://ilkogretim-online.org.tr.

2011.

[3] Baki, A., Matematik tarihi ve felsefesi, Pegem Akademi, Ankara, 2014.

[4] Uzunçarşılı, İ., Osmanlı Devletinin ilmiye teşkilâtı, Ankara, 1988.

[5] Çınar S., Eskişehir eğitim tarihi (1876–2004). (Yüksek Lisans Tezi). Eskişehir Osmangazi

Üniversitesi Sosyal Bilimler Enstitüsü, Eskişehir, 2005.

[6] Göker, L., Matematik tarihi ve Türk-İslâm matematikçilerinin yeri, Ankara, Gazi

Üniversitesi, 1981.

[7] İlk Mektepler Müfredat Programı, İstanbul, Matba-ı Amire, 1924.

[8] İlk Mektepler Müfredat Programı, İstanbul, Matba-ı Amire, 1927.

[9] Süveysi, M., Hendese. Diyanet İslâm Ansiklopedisi. 17, 196-199.

http://www.diyanetislamansiklopedisi.com/hendese/.

[10] Ülger, A., Matematiğin kısa bir tarihi – İkinci dönem: Eski Yunan matematiği,

Matematik Dünyası, Yaz, 49-53, 2003.

[11] Yıldırım A. ve Şimşek H., Sosyal bilimlerde nitel araştırma yöntemleri. Ankara, Seçkin.

2013.

http://www.diyanetislamansiklopedisi.com/hendese/


205

A study on Elastic Strips

Gözde ÖZKAN TÜKEL 1 and Ahmet YÜCESAN2

Abstract. We obtain the solution of the variational problem which gives elastic strips

with spacelike directrix in Minkowski 3-space. By changing the variation, we derive two

conservation laws of elastic strips with spacelike directrix, and so we define two new types of

elastic strips. Then, we establish some connections between elastic curves on De Sitter 3-

space and hyperbolic 2-space and elastic strips with spacelike directrix. Finally, we define P-

functional by means of the tangent vector of a spacelike curve defining force-free strip.

Keywords. Elastic strip, elastic curve, Euler-Lagrange equation, conservation laws.

AMS 2010. 53A35, 53B30, 35A15.

References

[1] Chubelaschwili, D., Pinkall, U., Elastic Strips, Manuscripta Math. 133 , no. 3-4, 307–326,

2010.

[2] Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. Int.

Electron. J. Geom. 7, no. 1, 44-107, 2014.

[3] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Pres.,

New York, 1993.

[4] Tukel, G.O., Yucesan, A., Elastic Strips with Timelike Directrix, Mathematical Reports, In

Press.

[5] Tukel, G.O., Elastic Strips in Minkowski 3-space, Ph.D. Thesis, Suleyman Demirel

University, Isparta, 2010.

[6] Weinstock, R., Calculus of Variations with Application to Physics and Engineering,

Dover Publications, Inc., New York, 1974.

1 Süleyman Demirel University, Isparta, TURKEY, [email protected] 2 Süleyman Demirel University, Isparta, TURKEY, [email protected]










206

Involutes of Order k of a Space-like Curve in 4

1IE

Günay ÖZTÜRK1, İlim KİŞİ2, Sezgin BÜYÜKKÜTÜK3 and Kadri ARSLAN4

Abstract. The orthogonal trajectories of the first tangents of a curve x are called the

involutes of x. In this study, we give a characterization of involutes of order k of a space-like

curve x with time-like principal normal in Minkowski 4-space 4

1IE .

Keywords. Involute, space-like curve, W-curve, helix.

AMS 2010. 53A04, 53A05.

References

[1] Blazenka, D., Zeljka M. S., 1999. Involutes and evolutes in n-dimensional simply isotropic

space, Journal of Information and Organizational Sciences, 2(3), 71-79, 1999.

[2] Kılıc, B., Arslan, K., Öztürk, G., Tangentially cubic curves in Euclidean spaces,

Differential Geometry-Dynamical Systems, 10, 186-196, 2008.

[3] O'Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press,

New York, 1983.

[4] Öztürk, G., Arslan, K., Hacisalihoglu, H. H., A characterization of ccr-curves in mIR .

Proccedings of Estonian Academy Science, 57(4), 217-224, 2008.

[4] Turgut, M., Ali, A.T., Lopez-Bonilla, J. L., Time-like involutes of a space-like helix in

Minkowski space-time, Apeiron, 17(1), 28-41, 2010.

[5] Walfare, J., Curves and surfaces in Minkowski space, PhD thesis, K.U. Leuven, Faculty of

Science, Leuven, 1995.

[6] Yılmaz, S. Turgut, M., On the differential geometry of the curves in Minkowski space-time

I, International Journal of Contemporary Mathematical Science, 3(27), 1343-1349, 2008.

1 Kocaeli University, Kocaeli, TURKEY, [email protected] 2 Kocaeli University, Kocaeli, TURKEY, [email protected] 3 Kocaeli University, Kocaeli, TURKEY, [email protected] 4 Uludag University, Bursa, TURKEY, [email protected]






207

Similarity Geometry of Null Cartan Curves

Hakan ŞİMŞEK1

Abstract. In this paper, the differential geometric properties of null Cartan curves

according to the similarity map of the Minkowski spacetime are investigated. The shape

Cartan curvatures and the pseudo-de Sitter parameter of a null Cartan curve are introduced

and the fundamental theorems (uniqueness and existence) under the similarity map in

Minkowski spacetime are given. Also, self-similar null Cartan curves parameterized by the

pseudo-de Sitter parameter are obtained.

Keywords. pseudo-de Sitter parameter, null Cartan curve, p-similarity.

AMS 2010. 14H50, 14H81, 53A35, 53A55, 53B30.

References

[1] Bejancu A., Lightlike curves in Lorentz manifolds, Publ. Math. Debrecen, 44, 145--155,

1994.

[2] Berger M. Geometry I. Springer, New York 1998.

[3] Bonnor W. B., Null curves in a Minkowski space-time, Tensor N.S. 20, 229--242, 1969.

[4] Cöken A. C, Ciftci Ü., On the Cartan curvatures of a null curve in Minkowski space-time,

Geom. Dedicata, 114, 71-78, 2005.

[5] Duggal K. L., Bejancu A. Lightlike Submanifolds of Semi-Riemannian Manifolds and

Applications, volume 364 of Mathematics and its Aplications. Kluwer Academic Publishers

Group, Dordrecht, The Netherlands, 1996.

[6] Encheva R., Georgiev G., Similar Frenet curves. Results in Mathematics, vol. 55, no. 3-4,

359--372, 2009.

[7] Falconer K., Fractal Geometry: Mathematical Foundations and Applications, Second

Edition, John Wiley & Sons, Ltd., 2003.

[8] Ferrandez A., Gimenez A, Lucas P., Null helices in Lorentzian space forms, Int. J. Mod.

Phys. A 16, 4845-4863, 2001.

1Antalya International University, Antalya, TURKEY, [email protected]


208

[9] Simsek H., Özdemir M., Similar and Self-similar Curves in Minkowski n-space, Bull. of

Korean Math. Soc., 52 (6), 2071-2093, 2015.


209

Semisimilarity and Consemisimilarity for matrices over a Commutative Quaternion

Hidayet Hüda KÖSAL1 , Mahmut AKYİĞİT2 and Murat TOSUN3

Abstract. In this paper, we investigate some equivalence relations and results over the

commutative quaternions and their matrices. In this sense, semisimilarity, and

consemisimilarity over the commutative quaternions and their matrices are established. Also,

Equalities of these equivalence relations are explicitly determined.

Keywords. Commutative quaternions, Consemisimilarity, Semisimilarity.

AMS 2010. 53A40, 20M15.

References

[1] Hamilton, W. R., Lectures on quaternions, Hodges and Smith, Dublin: 1853.

[2] Segre, C., The real representations of complex elements and extension to bicomplex,

Systems, Math. Ann., 40, 413, 1892.

[3] Huang L., Consimilarity of quaternion matrices and complex matrices, Linear Algebra

Appl. 331, 21-30, 2001.

[4] Hartwig, R. E., Putcha, M. S. Semisimilarity for matrices over a division ring. Linear

Algebra and its Applications, 39, 125-132, 1981.

[5] Bevis, J. H., Hall, F. J. Pseudo-consimilarity and semi-consimilarity of complex matrices.

Linear Algebra and its Applications, 90, 73-80, 1987.

1 Sakarya University, Sakarya, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected] 3 Sakarya University, Sakarya, TURKEY, [email protected]


210

On an Almost Contact Metric Manifold with a Type of Semi-symmetric Non-metric

Connection

Hulya Bagdatli YILMAZ1

Abstract. In this paper, we consider an almost contact metric manifold endowed with

a type of semi-symmetric non-metric connection. We find the expression for the curvature

tensor of an almost contact metric manifold that admits a type of semi-symmetric non-metric

connection. Furthermore, we study the properties of the curvature tensor and the projective

curvature tensor.

Keywords. Almost contact metric manifold, Semi-symmetric non-metric connection,

Curvature tensor.

AMS 2010. 53B15.

References

[1] Agashe, N. S. and Chafle, M. R., A semi-symmetric non-metric connection on a

Riemannian manifold, Indian J. Pure Appl. Math , 23,399-409, 1992.

[2] Chaubey, S. K.,On semi symmetric non-metric connection, Prog. of Math., 41-42, 11-20,

2007.

[3] Chaubey, S. K.,and Ojha, R. H., On semi symmetric non-metric connection and quarter

symmetric connections, Tensor N. S., 70 No:2, 202-213, 2008.

[4] Chaubey, S. K.,and Ojha, R. H., On semi symmetric non-metric connection, Filomat, 25:4,

19-27, 2011.

[5] Blair, D. E.,The theory of quasi-Sasakian structures, J. Diff. Geom., 1, 331-345, 1967.

[6] Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in math.509,

Springer-Verlag, Berlin 1976.

[7] De, U. C. and Biswas, S.C., On a type of semi symmetric non-metric connection on a

Riemannian manifold, Pub. De L'institut Math. Nouvelle serie tome 61, 90-96, 1997.

1 Marmara University, İstanbul, TURKEY, [email protected]


211

[8] Friedmann, A. and Schouten, J. A., Über die Geometric der holbsymmetrischen

Übertragurgen, Math. Z., 21, 211-233, 1924.

[9] Gray,J., Some global properties of contact structures, Ann. of Math.,69, 421-450, 1959.

[10] Hayden, H. A., Subspaces of space with torsion, Proc. London Math. Soc., 34, 27- 50,

1932.

[11] Janssens, D. and Vanhecke, L., Almost contact structures and curvature tensors, Kodai

Math. J., 4, 1-27, 1981.

[12] Oubina, J., New classes of almost contact metric structures, Publicationes Mathematicae,

32, 187-193, 1985.

[13] Sasaki, S., On diff. manifolds with certain structures which are closely related to almost

contact structure I, Tohoku Math. J., 12, 459-476, 1960.

[14] Sengupta, J., De, U. C. and Binh, T. Q., On a type of semi symmetric non-metric

connection on a Riemannian manifold, Indian J. Pure Appl. Math.,31(12),1659-1670, 2000.

[15] Yano, K., On semi-symmetric metric connections, Revue Roumania De Math. Pures

ppl., 15, 1579-1586, 1970.

[16] Yılmaz, H. B., On weakly symmetric manifolds with a type of semi-symmetric non-metric

connection, Ann. Polonici Math., 102.3, 301-308, 2011.


212

On Invariants of a System of Vectors in Minkowski Spacetime

İdris ÖREN1

Abstract. Let M be Minkowski spacetime (or the 4- dimensional pseudo-Euclidean

space of index 1), O(3, 1) be the group of all pseudo-orthogonal transformations of M and m-

uple x1, x2, … , xm of vectors in M. According to the group O(3, 1), invariants of an m-uple

x1, x2, … , xm are investigated. We define the new O(3,1) -invariant function Lxk , which is

the k-linearly dependently ratio of an m-uple x1, x2, … , xm in M. The invariants can be

used to judge whether two m-uples x1, x2, … , xm and y1, y2, … , ym in M are equivalent.

Methods and results of this study will be useful in the geometry. Since invariants obtained in

the present paper are computable, they are useful in problems of Bézier curves, Bézier

surfaces and Bézier vector fields.

Keywords. Invariant, Minkowski spacetime.

AMS 2010. 13A50, 51B20, 53B30.

References

[1] Misiak , A., Stasiak, E., Equivariant maps between certain G-spaces with G=O(n-1,1),

Math Bohem., 126, 555-560, 2001.

[2] O'Neill, B., Semi-Riemannian geometry: with applications to relativity, Academic

Press.,1983.

[3] Stasiak, E., Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of

index 1, Publ.Math.Debrecen, 57, 55-69, 2001.

[4] Gohberg, I.,Lancaster, P and Rodman, L., Invariant subspaces of matrices with

application, Siam, 2006.

[5] Bognar, J., Indefinite inner product spaces, Springer-Verlag, 1974.

[6] Höfer , R., m-point invariants of real geometries, Beitrage Algebra Geom.,40, 261-266,

1999.

[7] Greub, W, Linear Algebra, Springer-Verlag, 1967.

1 Karadeniz Technical University, Trabzon, TURKEY, [email protected]


213

[8] Ören, İ., Complete system of invariants of subspaces of Lorentzian space,

Iran.J.Sci.Technol.Trans. A Sci., 2014.(in pres)


214

Determining by Control Points of Null Bézier Curves with Degree 2 and 3 Curves in 3-

Dimensional Minkowski Spacetime

İdris ÖREN

Abstract. Let E13 be 3-dimensional Minkowski spacetime. In this study, the following

problems are investigated:

1) Does there exists a null Bézier curve with degree 2 and 3 in E13?

2) Determination by control points of a null Bézier curve with degree 2 and 3 in E13.

Therefore, some examples to null Bézier curves with degree 3 and 4 in E13 are given.

Keywords. Bézier curve, null curve, Minkowski spacetime.

AMS 2010. 65D17, 53B30, 51B20

References

[1] Georgiev, G. H., Space-like Bézier curves in the three-dimensional Minkowski space, AIP

Conf. Proc., 1067, Amer. Inst. Phys., Melville, NY, 2008.

[2] Pekşen,Ö,Khadjiev,D., On invariants of null curves in the pseudo-Euclidean

geometry,Differential Geom. Appl., 29, 183-187, 2013.

[3] Wang, W., Zhang, H., Liu, X., Paul, J. C., Conditions for coincidence of two cubic Bézier

curves, J. Comput. Appl. Math., 235, no. 17, 5198–5202, 2011.

[4] Chen, X., Ma, W., Deng, C., Conditions for the coincidence of two quartic Bézier curves,

Appl. Math. Comput. 225, 731–736, 2013.

[5] Ören, İ., Complete system of invariants of subspaces of Lorentzian space, Iran J Sci

Technol Trans Sci. 41,401–408,2017.

[6] Ören, İ., Equivalence conditions of two Bézier Curves in the Euclidean geometry, Iran J

Sci Technol Trans Sci.(article in pres) DOI 10.1007/s40995-016-0129-1.

[7] Pokorna, B. Quadratic space-like Bézier curves in three dimensional Minkowski space,

2012, project of Dissertation Thesis.

Karadeniz Technical University, Trabzon, Turkey, [email protected]



215

The Elliptic Matrices Associated with Eliptic Biquaternions and De-Moivre’s Formula

for These Matrices

Murat Tosun1 and Kahraman Esen Özen2

Abstract. In this study, with the aid of the Hamilton operators, we obtain the 4 4 elliptic

matrix representations of elliptic biquaternions and call them as right matrix representation

and left matrix representation by analogous Hamilton. Also, we have showed that the matrix

algebra which includes the left representations is isomorphic to the algebra of elliptic

biquaternions. Because of this isomorphism, we have called especially the left matrix

representations as the elliptic matrices associated with elliptic biquaternions. Afterwards, we

have expressed the De-Moivre’s and Euler Formulas for the elliptic matrices associated with

elliptic biquaternions by writing them in the polar form.

Keywords. De-moivre’s formula, Hamilton operator, complex quaternions.

AMS 2010. 11R52, 15A99.

References

[1] van der Waerden, B. L.. Hamilton’s discovery of quaternions, Math. Magazine, 49, 227-

234, 1976.

[2] Zhang, F.. Quaternions and Matrices of Quaternions, Linear Algebra and its Applications,

251, 21-57, 1997.

[3] Grob, J.. Trenkler, G.. Troschke, S.-O, Quaternions: further contributions to a matrix

oriented approach, Linear Algebra and its Applications 326, 205-213, 2001.

[4] Farebrother, R. W.. Grob, J.. Troschke, S-O, Matrix representation of quaternions. Linear

Algebra and its Applications 362, 251-255, 2003.

[5] Jafari, M.. Mortazaasl, H.. Yaylı, Y, De Moivre’s Formula for Matrices of Quaternions,

JP Journal of Algebra, Number Theory and Applications 21, 57-67, 2011.

1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]


216

New Results for General Helix in Euclidean 3-Space

Kazım İLARSLAN1

Abstract. Finding parametric equation of a space curve is not easy task when its

curvature functions are given. It is well known that this problem is known as fundamental

theorem of a space curve. If the curvature functions are functions of arc-length, the solution of

this problem is usually impossible. In this talk, we discuss the answer of the problem when

the curve is a general helix in Euclidean 3-space. By using slope axis of general helix, we get

parametric equations of a general helix. Also, we give some examples and their figures

Keywords. General helix, Euclidean 3-space, slope axis,.

AMS 2010. 53A04

References

[1] Ali, Ahmad T., Position vectors of general helices in Euclidean 3-space. Bull. Math.

Anal. Appl. 3, no. 2, 198–205, 2011.

[2] Barros, M., General helices and a theorem of Lancret. Proc. Amer. Math. Soc. 125, no. 5,

1503–1509, 1997.

[3] İlarslan, Kazım, Characterizations of spacelike general helices in Lorentzian manifolds.

Kragujevac J. Math. 25, 209–218, 2003.

[4] Uçum, A., Camcı, Ç. and İlarslan, K., General helices with timelike slope axis in

Minkowski 3-space. Adv. Appl. Clifford Algebr. 26, no. 2, 793–807, 2016.

1 Kırıkkale University, Kırıkkale, TURKEY, [email protected]


217

On Cardan Position for the Lorentzian Plane Motion of a Rigid Body

Kemal EREN1, Soley ERSOY2 and Mahmut ERGÜT3

Abstract. Cardan motion is defined as being generated by a circle rolling within

another circle of twice its size. A moving plane is said to be at a Cardan position if there is a

Cardan motion such that instantaneously the trajectories of the two motions hyper-osculate

[1]. In this paper, we study the instantaneous geometric properties of motion of rigid bodies in

the Lorentzian plane. For this purpose, we define Lorentzian form of Bottema’s instantaneous

invariants. We obtain the necessary and sufficient condition of a Lorentzian plane to be at

Cardan position with respect to these invariants.

Keywords. Cardan position, instantaneous invariants, Lorentzian plane

AMS 2010. 53A17, 53A35.

References

[1] Bottema, O. and Roth B., Theoretical Kinematics, North-Holland, Amsterdam, 1979.

[2] Dörrie, H., 100 Great problems of elementary mathematics: their history and solutions,

New York: Dover, 1965.

[3] Rauh, K., Marks, H., Bündgens, M., Otto, K. (1938): Kardanbewegung und

Koppelbewegung (Cardan motion and coupler motion), Schriftenreihe Praktische

Getriebetechnik, Heft 2, Berlin: VDI-Verlag, 1938.

[4] Bottema, O., On Cardan Positions for the Plane Motion of a Rigid Body, Indagationes

Mathematicae, V. XI, fasc. 3, 1949.

[5] Freudenstein, F., The Cardan Positions of a Plane, Trans. Sixth Conf. Mechanisms,

Purdue University, West Lafayette, Ind., pp. 129–133, October 1960.

1 Fatsa Science High School, Ordu, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected] 3 Namık Kemal University, Tekirdağ, TURKEY, [email protected]


218

[6] Ergüt, M., Aydin, A.,P. and Bildik N., The Geometry of the Canonical Relative System and

One-Parameter Motions in 2-Lorentzian Space, The Journal of Fırat University, 3(1) 113-

122, 1988.

[7] Ergin. A.A., On the one-parameter Lorentzian motion, Communications, Faculty of

Science, University of Ankara, Series A 40, 59–66, 1991.

[8] Tutar, A., Kuruoğlu, N. and Duldul, M., On the moving coordinate system and pole points

on the Lorentzian plane, International Journal of Applied Mathematics, 7(4), 439-445, 2001.

[9] Güngör, M. A., Pirdal A.Z. and M. Tosun, Euler-Savary Formula for the Lorentzian

Planar Homothetic Motions, Int. J. Math. Comb. 2, 102–111, 2010.


219

Some Characterizations for Screen Isotropic Leaves on Lightlike Hypersurfaces

Mehmet GÜLBAHAR1, Sadık KELEŞ2 and Erol KILIÇ3

Abstract. Isotropic leaves in the integrable screen distribution for lightlike

hypersurfaces of a Lorentzian manifold are investigated. Some examples of such leaves are

presented. Furthermore, some relations on these leaves are obtained.

Keywords. Isotropic leaf, distribution, lightlike hypersurface, Lorentzian manifold.

AMS 2010. 53C42, 53C50.

References

[1] ] Bejan C. L., Duggal K. L., Global lightlike manifolds and harmonicity, Kodai Math. J.

28, 131-145, 2005.

[2] Duggal, K. L., Sahin, B., Differential geometry of lightlike submanifolds, Birkhauser,

Basel, 2010.

[3] Duggal, K. L., Jin D. H., Null curves and hypersurfaces of semi-Riemannian manifolds,

World Scientific Publishing, Hackensack, NJ, 2007.

[4] O’Neill, B., Isotropic and Kaehler immersions, Canad. J. Math. 17, 907-915, 1965.

1 Siirt University, Siirt, TURKEY, [email protected] 2 İnönü University, Malatya, TURKEY, [email protected] 3 İnönü University, Malatya, TURKEY, [email protected]


220

Special Smarandache Curves with Respect to Darboux Frame in the Galilean 3-Space

Merve OKUR1 and Tevfik ŞAHİN2

Abstract. This article investigates special Smarandache curves of arbitrary curves respect to

Darboux frame in Galilean space. As a result of this, we define the special Smarandache

curves of geodesic, asymptotic and line of curvature respect to Darboux frame in Galilean

space.

Keywords. : Smarandache curves, Galilean space, Geodesic.

AMS 2010. 51B20, 53A35.

References

[1] Abdel-Aziz H.S., Saad M.K., Smarandache curves of some special curves in the Galilean

3-space, Honam Mathematical J., 253-264, 2015.

[2] Ali A. T., Special Smarandache Curves in the Euclidean Space, International Journal of

Mathematical Combinatorics, 2, 30-36, 2012.

[3] Bektaş Ö, Yüce S., Special Smarandache Curves According to Darboux Frame in

Euclidean3- Space, Romanian Journal of Mathematics and Computer Science, 3, 48-59, 2013.

[4] Çetin M, Tunçer Y., Karacan M. K., Smarandache Curves According to Bishop Frame in

Euclidean 3- Space, General Mathematics Notes, 20 (2) , 50-66, 2014.

[5] Dirişen B.C., Şahin T., Position vector of a curve with respect to the Darboux frame in

the Galilean space G3, arXiv:1707. 03930v1, 2017.

[6] Okur M., Şahin T., Special Smarandache Curves with Respect to Darboux Frame in the

Galilean 3-Space, arXiv: 1707.03935v1, 2017.

1 Amasya University, Amasya, Turkey, [email protected] 2 Amasya University, Amasya, Turkey, [email protected], [email protected]




221

Elliptical Motion on a Given Ellipsoid

Mustafa ÖZDEMİR1

Abstract. The aim of this study is to explain the motion on the ellipsoid

((x²)/(a²))+((y²)/(b²))+((z²)/(c²))=1,

as a rotation, using the proper inner product, vector product and elliptical orthogonal matrices.

In this method, the elliptical inner product, the vector product and the angles are compatible

with the parameters θ and β of the parametrization ϕ(θ,β)=(acosθcosv, bcosθsinβ, csinθ). We

use the classical methods to generate elliptical rotation matrices such as Rodrigues, Cayley

and Hauseholder transformations.

Keywords. Elliptical Motion, Rotation Matrix, Elliptical Inner and Vector Product,

Rodrigues Formula, Cayley Formula.

AMS 2010. 15A63, 15A66.

References

[1] Özdemir, M., An Alternative Approach to Elliptical Motion, Adv. Appl. Clifford Algebras

26, 279–304, 2016.

[2] Rodríguez-Andrade, M. A., Aragón-González, G., Aragón, J. L., Verde-Star, L., An

algorithm for the Cartan-Dieudonné theorem on generalized scalar product spaces, Linear

Algebra and Its Applications, Vol. 434, Issue 5, 1238-1254, 2011.

[3] Mackey, D. S., Mackey, N., Tisseur, F., G-reflectors : Analogues of Householder

transformations in scalar product spaces, Linear Algebra and its Applications Vol. 385,

187—213, 2004.

[4] Özdemir, M., Ergin, A.A., Rotations with unit timelike quaternions in Minkowski 3-space

Journal of Geometry and Physics 56, 322-336, 2006.

[6] Şimşek, H, Özdemir M., Generating hyperbolical rotation matrix for a given hyperboloid,

Linear Algebra and Its Applications, Vol, 496, 221-245, 2016.

1 Akdeniz University, Antalya, TURKEY, [email protected], [email protected]




222

[7] Şimşek H., Özdemir M., Rotations on a lightcone in Minkowski 3-space, Adv. Appl.

Clifford Algebras, 2017.

[8] M. Özdemir, M. Erdoğdu, On the Rotation Matrix in Minkowski Space-time. Reports on

Mathematical Physics 74, 27-38, 2014.

[9] M. Erdoğdu, M. Özdemir, Cayley Formula on Matrix in Minkowski Space-time, Int. J. of

Geometric Methods in Modern Physics, Vol 12, 2015.

[10] Aragón-González, G., Aragón, J. L., Rodríguez-Andrade, M. A., The decomposition of

an orthogonal transformation as a product of reflections, J. Math. Phys. 47, 2006.

[11] Bükçü, B., On the Rotation Matrices in Semi-Euclidean Space, Commun. Fac. Sci. Univ.

Ank. Series A1. 55, 7-13, 2006.

[12] Özkaldı, S., Gündoğan H., Cayley Formula, Euler Parameters and Rotations in 3-

Dimensional Lorentzian Space, Adv. in Applied Clifford Algebras 20 367-377, 2010.


223

Ruled Surfaces in Contact Geometry

Nural YÜKSEL1, Murat Kemal KARACAN2, Hasibe İKİZ3 and Çağlar Zeki ODABAŞI4

Abstract. In this paper, we study ruled surface in 3-dimensional almost contact metric

manifolds by using surface theory in contact geometry .We obtain the distribution parameters

of the ruled surface and then present some results and theorems with special cases. Moreover,

some relationships among asymptotic curve and striction line of the base curve of the ruled

surface were obtained.

Keywords. Sasakian manifold; contact manifold; ruled surface

AMS 2010. 53C15, 53C25.

References

[1] Baikoussis C. and Blair, D. E., Finite type integral submanifold of the contact manifold,

Bull. Math. Acad. Sinica 19, 327–350, 1991.

[2] Camcı, C., Extended cross product in a 3-dimensional almost contact metric manifold

with applications to curve theory, Turk. J. Math. 35, 1–14, 2011.

[3] Blair, D. E., Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics,

Vol. 509 (Springer-Verlag, 1976).

[4] Gök, İ., Surfaces theory in contact geometry, PhD thesis, 2010.

1 Erciyes University, Kayseri, TURKEY, yukseln @erciyes.edu.tr 2 Uşak University, Uşak, TURKEY, [email protected] 3 Erciyes University, Kayseri, TURKEY, [email protected] 4 Erciyes University, Kayseri, TURKEY, [email protected]


224

A Note on Spherical Orthotomic of Lorentzian Spherical Spacelike Curves

Önder Gökmen YILDIZ1

Abstract. In this paper the spherical orthotomic of spacelike curve on Lorentzian

sphere is defined. Then, its local diffeomorphic image is investigated.

Keywords. Orhotomic, spherical curve, Lorentzian sphere.

AMS 2010. 53C50, 53C40.

References

[1] Bruce, J. W., Giblin, P. J., Curves and singularities: a geometrical introduction to

singularity theory, Second Edition, Cambridge, University Press, 1992.

[2] O’neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press,

1983.

[3] Pekmen, U., Paşalı, S., Some characterization of Lorentzian spherical space-like curves,

Mathematica Moravica, 3, 33-37, 1999.

[4] Petrovic-Torgasev, M Sucurovic, E., Some characterizations of Lorentzian spherical

spacelike curves with the timelike and null principal normal, Mathematica Moravica, 4, 83-

92, 2000.

[5] Xiong, J. F., Spherical orthotomic and spherical antiorthotomic, Acta Mathematica

Sinica, 23, 9, 1673-1682, 2007.

1 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected]


225

Local Diffeomorphic Image of Spherical Antiorthotomic of Lorentzian Spherical

Timelike Curves

Önder Gökmen YILDIZ1

Abstract. In this paper, spherical antiorthotomic of Lorentzian spherical curve on the

Lorentzian sphere is defined. Then, spherical antiorthotomic is obtained as an envelope of the

family of great circles, lying on the planes that perpendicularly bisect the chords. Finally, by

using the technique in [1] its local diffeorporhic type is determined.

Keywords. Orhotomic, antiorthotomic, spherical curve, Lorentzian sphere.

AMS 2010. 53C50, 53C40.

References

[1] Bruce, J. W., Giblin, P. J., Curves and singularities: a geometrical introduction to

singularity theory, Second Edition, Cambridge, University Press, 1992.

[2] O’neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press,

1983.

[3] Pekmen, U., Paşalı, S., Some characterization of Lorentzian spherical space-like curves,

Mathematica Moravica, 3, 33-37, 1999.

[4] Petrovic-Torgasev, M Sucurovic, E., Some characterizations of Lorentzian spherical

spacelike curves with the timelike and null principal normal, Mathematica Moravica, 4, 83-

92, 2000.

[4] Petrovic-Torgasev, M Sucurovic, E., Some characterizations of the Lorentzian spherical

timelike and null curves, Mathematicki Vesnik, 53.(1-2), 21-27, 2001.

[5] Xiong, J. F., Spherical orthotomic and spherical antiorthotomic, Acta Mathematica

Sinica, 23, 9, 1673-1682, 2007.

1 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected]


226

The Kinetic Energy Formula for the Closed Planar Homothetic Motions in Complex

Plane

Önder ŞENER1, Ayhan TUTAR2 and Serdar SOYLU3

Abstract. In this paper, the kinetic energy formula was expressed during one-

parameter closed planar homothetic motions in complex plane. Then the relation between the

kinetic energy formula and the Steiner formula was given. As an example the sagittal motion

of a telescopic crane was considered. This motion was described by a double hinge consisting

of the fixed control panel of telescopic crane and the moving arm of telescopic crane. The

results were applied to experimentally measured motion.

Keywords. kinetic energy, Steiner Formula, homothetic motions.

AMS 2010. 53A17, 70B10.

References

[1] Dathe, H., Gezzi, R., Characteristic directions of closed planar motions, Zeitschrift für

Angewandte Mathematik und Mechanik, 92, 2-13, 2012.

[2] Dathe, H., Gezzi, R., Addenda and Erratum to: Characteristic directions of closed planar

motions, Zeitschrift für Angewandte Mathematik und Mechanik, 94, 551–554, 2014.

[3] Müller, H. R., Verallgemeinerung einer Formel von Steiner, Abh. Braunschweig. Wiss.

Ges., 29, 107-113, 1978.

[4] Steiner, J., Von dem Krümmungs-Schwerpuncte ebener Curven, Journal für die reine und

angewandte Mathematik, 21, 33-63, 1840.

[5] Tutar, A., Kuruoğlu, N., The Steiner formula and the Holditch theorem for the homothetic

motions on the planar kinematics, Mechanism and Machine Theory, 34, 1-6, 1999.

[6] Tutar, A., Sener, O., The Steiner formula and the Polar moment of inertia for the closed

planar homothetic motions in complex plane, Advances In Mathematical Physics, 2015, 1-5,

2015.

1 Ondokuz Mayis University, Samsun, TURKEY, [email protected] 2 Present address: Kyrgyz-Turk Manas University, Bishkek, KYRGYZSTAN

Permanent address: Ondokuz Mayis University, Samsun, TURKEY, [email protected] 3 Giresun University, Giresun, TURKEY, [email protected]



227

New Developments in Lightlike Submanifolds Theory

Sadık KELEŞ1, Erol KILIÇ2 and Mehmet GÜLBAHAR3

Abstract. Sectional curvature map on r lightlike submanifolds of a semi-

Riemannian manifold is investigated. Some special lightlike submanifolds are mentioned.

Some inequalities for lightlike submanifolds obtained. Using these inequalities, some

characterizations are obtained for special lightlike submanifolds.

Keywords. Curvature, lightlike submanifold, semi-Riemannian manifold.

AMS 2010. 53C40, 53C42, 53C50.

References

[1] Bejan C. L., Duggal K. L., Global lightlike manifolds and harmonicity, Kodai Math. J. 28,

131-145, 2005.

[2] Chen, B.-Y., Pseudo-Riemannian geometry, invariants and applications, World

Scientific Publishing, Hackensack, NJ 2011.

[3] Duggal, K. L., Sahin, B., Differential geometry of lightlike submanifolds, Birkhauser,

Basel, 2010.

[4] Kulkarni, R. S., The values of sectional curvature in indefinite metric, Comment Math.

Helv. Phys. 54, 173-176, 1979.

1 İnönü University, Malatya, TURKEY, [email protected] 2 İnönü University, Malatya, TURKEY, [email protected] 3 Siirt University, Siirt, TURKEY, [email protected]


228

Holditch Type Theorem for Kinetic Energy of Projective Curve Under the 1-Parameter

Spatial Motion

Serdar SOYLU 1, Ayhan TUTAR2 and Önder ŞENER3

Abstract. Firstly, we obtain the kinetic energy of the projective curve for the closed

spatial motion as following,

2𝑆 = 2𝑆0 + 𝜌 ∑ 𝑥𝑖2 −

3

𝑖=1

∑ 𝑏𝑖𝑗𝑥𝑖𝑥𝑗 +

3

𝑖,𝑗=1

∑ 𝑐𝑖𝑥𝑖

3

𝑖=1

Then, we show the Holditch type theorem for kinetic energy of the projective curve

and we get some results related with that theorem.

Keywords. Kinetic, holditch, motion.

References

[1] Dathe H. and Gezzi R. Characteristic directions of closed planar motions, Zeitschrift für

Angewandte Mathematik und Mechanik, 2-13, 2012.

[2] Dathe H. and Gezzi R., Addenda and Erratum to: Characteristic Directions of Closed

Planar Motions, Zeitschrift für Angewandte Mathematik und Mechanik, 92(9), 731-748,

2014.

[3] Dathe H., Gezzi R., Kubein-Meesenburg D. and Nagerl H., Characteristic Point and

Cycles in Planar Kinematiks with Applications to Human Gait. Acta Bioengineering and

Biomechanics. Vol 17, No.1, 2015.

[4] Düldül M., Kuruoğlu N. and Tutar A., Generalization of Steiner Formula for the

homothetic motions on the planar kinematics, Applied Mathematics and Mechanics-English

Edition, 24(8), 945-949, 2003.

[5] Düldül M., Homotetik Uzay Hareketleri ve Holditch Teoremi. Doktora Tezi, Ondokuz

Mayıs Üniversitesi Fen Bilimleri Enstitüsü, Samsun, 2004.

1 Giresun University, Giresun, TURKEY, [email protected] 2 Present address: Kyrgyz-Turk Manas University, Bishkek, KYRGYZSTAN

Permanent address: Ondokuz Mayis University, Samsun, TURKEY, [email protected] 3 Ondokuz Mayis University, Samsun, TURKEY, [email protected]



229

[6] Düldül M., Kuruoğlu N. and Yüce S., The polar moment of inertia of the enveloping

curve, Novi Sad Journal of Mathematics, 38(2), 1-4, 2008.


230

On the Casual Characteristics of Higher Order Bertand Mate, Involute Curve,

Mannheim of a Non-Null Curve in IL³

Şeyda KILIÇOĞLU1 and Süleyman ŞENYURT2

Abstract. We have already defined and worked on the second order Bertand mate,

involute curve, Mannheim partner of a unit-speed curve in IE 3. In this paper, first we have

defined nth order Bertand mate, involute curve, Mannheim partner of a non-null curve in IL 3.

Also their casual characteristics are examined. Further, the casual characteristics of the nth

order Bertand mate, involute curve, Mannheim partner of a non-null curve is given in a table

with the number of different type of casual characteristics. The number of the different forms

of the casual characteristics of nth order involute and Mannheim mate of a space-like curve

can be given using Fibonacci sequence. A time-like Bertrand curve has 2ⁿ different forms,

also the nth order Bertrand mate of a space-like Bertrand curve with time-like principal normal

is always space-like Bertrand curve with time-like principal normal.

Keywords. Lorentz metric, casual characteristic, involute, Bertand mate, Mannheim

partner.

AMS 2010. 53A04, 53B30

References

[1]. Bilici M.., Çalışkan, M., On the Involutes of the Spacelike Curve with a Timelike

Binormal in Minkowski 3-Space. International Mathematical Forum, 4, no. 31, 1497 – 1509,

2009.

[2]. Bukcu, B, Karacan M. K., On the involute and evolute curves of the spacelike curve with

a spacelike binormal in Minkowski 3-space. Int. Journal of Contemp. Math. Sciences, Vol.

2,no. 5-8, 221-232, 2007.

[3]. Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd

ed. Boca Raton, FL: CRC Press, pp. 205, 1997.

1 Baskent University, Ankara, TURKEY, [email protected] 2 Ordu University, Ordu, TURKEY, [email protected]

http://www.m-hikari.com/ijcms-password2007/5-8-2007/karacanIJCMS5-8-2007.pdf



231

[4]. Hacısalihoğlu, H.H., Differential Geometry (in Turkish), Academic Press Inc. Ankara,

1994.

[5]. Kılıçoğlu, Ş., Senyurt, S., On the second order involute curves in IE³. Commun. Fac. Sci.

Univ. Ank. Series A1, 66(2), 332-339., Doi: 10.1501/Commua1_0000000823 (Yayın No:

3472826), 2017.

[6]. Kılıçoğlu, S., Senyurt, S., On the Second Order Mannheim Partner Curve in IE³.

International J.Math. Combin., 1, 71-77, 2017 (Yayın No: 3472825) , 2017.

[7]. Lipschutz, M.M., Differential Geometry, Schaum's Outlines, 1969 . Liu H. and Wang

F.,Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No 1-2, 120-

126(7),2008.

[8]. Senyurt, S., Kılıçoğlu S., On the Differential Geometric Elements of Bertrandian

Darboux ruled surface in E3., Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi. Sayı 3,

Cilt 21, 2017.

[9]. Schief, W.K., On the integrability of Bertrand curves and Razzaboni surfaces. Journal of

Geometry and Physics, Volume 45, Issues 1--2, Pages 130--150, February, 2003.

[10]. O’Neil, B., Semi-Riemannian geometry with applications to relativitiy, Academic

Press, Inc. USA 1983.

[11]. Orbay K., Kasap E., On mannheim partner curves, International Journal of Physical

Sciences, Vol. 4 (5), 261-264, 2009.


232

On the Second Order Involute of a Space-like Curve with Time-like Binormal in IL³

Şeyda KILIÇOĞLU1 and Süleyman ŞENYURT2

Abstract. We have already defined and worked on the second order involute curve of

a unit-speed curve in IE 3. In this paper, we consider the second order involute of a space-like

curve with time-like binormal in IL³. All Frenet apparatus of them are examined in terms of

Frenet apparatus of the space-like curve with time-like binormal α.

Keywords. Lorentz metric, casual characteristic, involute, Bertand mate, Mannheim

partner.

AMS 2010. 53A04, 53B30

References

[1]. Bilici M.., Çalışkan, M., On the Involutes of the Spacelike Curve with a Timelike

Binormal in Minkowski 3-Space. International Mathematical Forum, 4, no. 31, 1497 – 1509,

2009.

[2]. Bukcu, B, Karacan M. K., On the involute and evolute curves of the spacelike curve with

a spacelike binormal in Minkowski 3-space. Int. Journal of Contemp. Math. Sciences, Vol.

2,no. 5-8, 221-232, 2007.

[3]. Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd

ed. Boca Raton, FL: CRC Press, pp. 205, 1997.

[4]. Hacısalihoğlu, H. H., Differential Geometry (in Turkish), Academic Press Inc. Ankara,

1994.

[5]. Kılıçoğlu, Ş., Senyurt, S., On the second order involute curves in IE³. Commun. Fac. Sci.

Univ. Ank. Series A1, 66(2), 332-339., Doi: 10.1501/Commua1_0000000823 (Yayın No:

3472826), 2017.

[6]. Kılıçoğlu, S., Senyurt, S., On the Second Order Mannheim Partner Curve in IE³.

International J.Math. Combin., 1, 71-77, 2017 (No: 3472825) , 2017.

[7]. Lipschutz, M.M., Differential Geometry, Schaum's Outlines, 1969. Liu H. and Wang F.,

Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No 1-2, 120-126(7), 2008.

1 Baskent University, Ankara, TURKEY, [email protected] 2 Ordu University, Ordu, TURKEY, [email protected]




233

On Biharmonic Hypersurfaces in Semi-Euclidean Spaces

Sibel SEVİNÇ1, Gülşah AYDIN ŞEKERCİ2 and A. Ceylan ÇÖKEN3

Abstract. In this paper we study biharmonic hypersurfaces of semi-Euclidean spaces.

We consider the Chen conjecture which states that the only biharmonic submanifolds of

Euclidean spaces are the minimal ones. In particular, we obtain some conditions for

biharmonic hypersurfaces in semi-Euclidean spaces. Finally, we give some examples for

different conditions.

Keywords. Biharmonic hypersurface, Chen conjecture, semi-Euclidean space.

AMS 2010. 53C40, 53C42.

References

[1] Chen, B-Y., Ishikawa, S., Biharmonic pseudo-Riemannian submanifolds in pseudo-

Euclidean spaces, Kyushu J. Math., 52, 1, 167-185, 1998.

[2] Chen, B-Y., Ishikawa, S., Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac.

Sci. Kyushu Univ. Ser. A, 45, 2, 323-347, 1991.

[3] Chen, B-Y., Munteanu, M. I., Biharmonic ideal hypersurfaces in Euclidean spaces,

Differential Geometry and its Applications, 31, 1-16, 2013.

[4] Turgay, N. C., A classification of biharmonic hypersurfaces in the Minkowski spaces of

arbitrary dimension, Hacettepe J. Math. and Statistics, 45, 4, 1125-1134, 2016.

[5] Gupta, R. S., Biharmonic hypersurfaces in E⁵, An. Ştiint. Univ. Al. I. Cuza Iaşi Mat. (N.

S.), 2, 2, 585-593, 2016.

[6] Gupta, R. S., Biharmonic hypersurfaces in E6 with constant scalar curvature, International

J. Geo., 5, 2, 39-50, 2016.

1 Cumhuriyet University, Sivas, TURKEY, [email protected] 2 Süleyman Demirel University, Isparta, TURKEY, [email protected] 3 Akdeniz University, Antalya, TURKEY, [email protected]


234

Timelike and Spacelike Split Quaternion Ruled Surfaces in Minkowski 3-space

Kıvanç KARAKAŞ1 , Mesut ALTINOK2 and Levent KULA3

Abstract. Quaternion rational surface which generated from quaternion product of two

rational space curves is defined by Wang and Goldman in [1]. They also defined quaternion

rational ruled surface which is special quaternion rational surface. In this work, we investigate

the new surface in semi Euclidean space which generated from the split quaternion product of

a line and a space curve. We give a timelike and spacelike split quaternion ruled surface

examples by Mathematica 10. Moreover, we describe of syzygy, mu-basis for a timelike and

spacelike split quaternion ruled surfaces and give its implicit equations.

Keywords. Syzygy, mu-basis, split quaternion rational ruled surface, spacelike

quaternion ruled surface, timelike quaternion ruled surface.

This work is supported by Ahi Evran University Scientific Research Project Coordination

Unit. Project number: PYO- FEF.A3.17.002.

AMS 2010. 53A04, 53C50.

References.

[1] Wang, X. ; Goldman, Quaternion rational surfaces: Rational surfaces generated from the

quaternion product of two rational space curves, Journal of Graphical Models, 2015, no.81,

18-32.

[2] Chen, F. ; Zheng, J. ; Sederberg, T. µ-basis of a rational ruled surface, Journal of

Computer Aided Geometric Design, 2001, no.18, 61-72.

[3] Kula, L. Bölünmüş Kuaterniyonlar ve Geometrik Uygulamalari, Ph.D. Thesis, Ankara

University Institute of Science, Ankara, 2003.

1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Mersin University, Mersin, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected]


235

MATHEMATICS

EDUCATION


236

Parameterized Examination in Econometrics

Anna MALINOVA1, Vesselin KYURKCHIEV2 and Georgi SPASOV3

Abstract. This paper presents a parameterization of basic types of exam questions in

Econometrics to automate and facilitate the process of examination, assessment and self-

preparation of a large number of students. The proposed parameterization of testing questions

reduces the time required to author tests and course assignments. It enables tutors to generate

a large number of different but equivalent dynamic questions (with dynamic answers) on a

certain topic, which are automatically assessed. The presented methods are implemented in

DisPeL (Distributed Platform for e-Learning) [1], [5], [6], [7], [8] and provide questions on

the areas of filtering and smoothing time-series data, forecasting, building and analysis of

single-equation econometric models. Questions also cover elasticity, average and marginal

characteristics, product and cost functions, measurement of monopoly power, supply, demand

and equilibrium price, consumer and product surplus, etc. Several approaches are used to

enable the required numerical computations in DisPeL – integration of third-party

mathematical libraries, developing our own procedures from scratch, and wrapping our legacy

math codes in order to modernize and reuse them [2], [3], [4].

Keywords. Econometrics, parameterization, dynamic questions, automatic test

generation, e-testing,

AMS 2010. 91G70, 97D40, 97R20, 97U50.

References

[1] Arnaudova, V., T. Terzieva, A. Rahnev, A Methodological Approach for Implementation

of Adaptive e-Learning, CBU International Conference on Innovations in Science and

Education, March 23-25, 2016, Prague, Czech Republic, CBU International Conference

Proceedings, Vol. 4, 910-917, 2016.

[2] Malinova, A., Approaches and Techniques for Legacy Software Modernization, Scientific

Works, Plovdiv University, Vol. 37, Book 3, 2010 - Mathematics, 77-85, 2010.

1 University of Plovdiv Paisii Hilendarski, Plovdiv, BULGARİA, [email protected] 2 University of Plovdiv Paisii Hilendarski, Plovdiv, BULGARIA, [email protected] 3 University of Plovdiv Paisii Hilendarski, Plovdiv, BULGARIA, [email protected]


237

[3] Malinova, A., Modernizing Legacy Physics Applications for Reuse in Web and SOA,

Serdica Journal of Computing, Vol. 5, No. 1, 79-100, 2011.

[4] Malinova, A., Ivanova, V., & Rahnev, A., Computer algebra aided generation of English

language tests, The 11th Annual International Conference on Computer Science and

Education in Computer Science CSECS 2015, June 04-07 2015, Boston, MA, USA,

Computer Science Education & Computer Science Research Journal, 11, 66-74, 2015.

[5] Rahnev, А., Pavlov, N., Kyurkchiev, V., Distributed Platform for e-Learning – DisPeL,

European International Journal of Science and Technology (EIJST), 3(1), 95-109, 2014.

[6] Rahnev, A., Pavlov, N., Golev, A., Stieger, M., Gardjeva, T., New Electronic Education

Services Using the Distributed E-Learning Platform (DisPeL), International Electronic

Journal of Pure and Applied Mathematics (IEJPAM), 7(2), 63-71, 2014.

[7] Rahnev A., Rahneva O., Testing and assessment in accounting at the Distributed e-

Testing Cluster – DeTC, Scientific Works, ECEM, Book 4, 182-189, 2008.

[8] Rahneva O., Golev, A., Pavlov, N., Dynamic Generation of Testing Question in SQL in

DeTC, Cybernetics and Information Technologies, 8(1), 73-81, 2008.


238

The Effect of Differentiated Instruction on Mathematical Attitudes of Students

Ayten Pinar BAL1 and Onur EKİNCİ2

Abstract. Differentiated teaching approach that accounts personal differences within a

class, considers different students’ different skills and learning needs, and draws on students’

strengths gains importance [1], [2]. In this study has been carried out in order to define the

effects of differentiated instruction method on the mathematical attitudes of primary school

third grade learners. It is designed for quantitative models. The quantitative data has been

designed on semi-experimental model with pretest and posttest control group. The

experimental group of the study has been formed by semi-randomly selected 20 students who

got similar points from tests and had similar learning styles. During the application, the topic

of “Fractions” was taught by Differentiated Instruction Method in the experimental group,

while it was taught by Traditional Method in the control group according to the course book

of mathematics. In the experimental group, the courses were taught by the researcher and in

the control group it was taught by the teacher of the class. The data has been collected by

using “Mathematics Attitude Scale”. “Mathematics Attitude Scale” has been applied to both

experimental and control groups as pretest, posttest and permanence test. One factor Ancova

has been used in order to find out whether the Differentiated Instruction Method has any

statistical effects on the points of “Mathematics Attitude Test” before and after the

experiment. According to the data obtained at the end of the research, there is not a significant

difference in the post test results of attitude test in favor of the experimental group.

Keywords. Mathematics education, mathemetics attitude, differentiated instruction

References

[1] Gregory, G. H., Chapman, C., Differentiated instructional strategies: One size doesn’t fit

all. Thousand Oaks, CA: Corwin Press, 2002.

[2] Sondergeld, T. A., Schultz, R. A., Science, standards, and differentiation: It really can be

fun!, Gifted Child Today, 31(1), 34-40, 2008.

1 Çukurova University, Adana, TURKEY, [email protected] 2 Çukurova University, Adana, TURKEY, [email protected]


239

Investigation of the Test Anxieties of 8th Grade Students*

Bülent ALTUNKAYA1, Kamile ŞANLI KULA2 and Cahit AYTEKİN3

Abstract. Today, the guidance and psychological counseling services in our schools

spend most of their time trying to reduce students' test concerns [1]. Students who experience

high level of test anxiety cannot use their knowledge and skills effectively during the exams

and their success decreases as a result. In a study in Turkey, it is stated that the level of

anxiety for the university exam is considerably higher than that of patients waiting to be

operated [2]. For this reason, the research that will be done in order to decrease the students'

test concerns will guide the actions to be taken. If the conditions affecting students' test

concerns are well recognized, then successive approaches in all courses, especially

mathematics, can be put into practice. In this study, it is aimed to examine the test anxieties of

the 8th grade students according to some variables. A total of 952 students from 14 different

primary schools participated in the research. 468 of these students were females, while 484

were males.

According to the research findings, it was found that female students' test concerns

were significantly higher than male students. When the scores from the burnout scale are

examined, it is seen that boys burnout levels are significantly higher than girls. Children from

low income levels were found to have higher test concerns in the sub-dimension of delusions.

There is no significant difference in the test anxiety according to the mother's education level.

However, it has been determined that burnout levels of students who have mothers of

university graduates are significantly higher than those of mothers of primary and junior high

school graduates. A similar result was found for father's educational status. Burnout levels of

students whose mothers working are higher than those of mothers who do not work. It was

determined that the test anxiety of students who took support courses for the high school

placement exam were lower than the students who did not take. Burnout levels of students

who do not attend the support course are also significantly higher than those who took support

courses. It was found that there is no significant difference in the test anxiety and burnout

levels according to the situation of receiving private lessons for the high school placement

exam. It has been determined that the students who declared that they work 0-1 hours daily

have higher delusional test anxiety than the students who declare that they are working for 4-5

hours daily. A similar result is determined about burnout levels. It is seen that the students

who declared that they never worked or that they worked 0-1 hours were found to have

significantly higher burnout levels than the other students. Students who do leisure time

activities such as painting, music and sports had significantly higher test anxiety than those

who do not. There are significant differences in the test anxieties and burnout levels according

* This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project

Number: FEF.A4.17.003 1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected]





240

to the perceived level of students about relation life and lesson in school. Students who say

that their school is not related to their life at all have significantly higher test anxiety and

burnout than others. The students who stated that the teacher defined himself/herself as

unsuccessful were found to have significantly higher delusional test anxiety and burnout than

other students. A similar finding was found for students who stated that their family identified

themselves as unsuccessful. The students who say that they work only on interesting topics

have higher test anxiety and burnout level than those who work for not to take a poor grade in

school. A similar result was found in burnout levels. Finally, it was determined that the

burnout levels of the students who were waiting for their teachers to teach the information to

be useful in the daily life rather than the exam had significantly higher burnout levels than the

other students.

Keywords. Test anxiety, burnout levels, high school placement exam

References

[1] Kayapınar, E. Ortaöğretim Kurumları Öğrenci Seçme Ve Yerleştirme Sınavı (Oks)’na

Hazırlanan İlköğretim 8. Sınıf Öğrencilerinin Kaygı Düzeylerinin İncelenmesi

(Afyonkarahisar İli Örneği). Yayınlanmamış Yüksek Lisans Tezi. Afyon Kocatepe

Üniversitesi Sosyal Bilimler Enstitüsü, Afyon, 2006

[2] Baltaş A., Baltaş Z. Başarılı ve Sağlıklı Olmak İçin Stresle Başa Çıkmanın Yolları, Remzi

Kitabevi Yayınları, İstanbul, 1998.


241

Examining Pre-service Teachers’ Perceptions and Evaluations about Student-Invented

Strategies

Büşra KARTAL1 and Yasemin KIYMAZ2

Abstract. Student-invented strategies are referred to any strategies which are different

from traditional algorithm and not including manipulatives [1]. Students who invent or adopt

strategies actively engage in making sense of mathematics [2]. Evaluating and responding to

students’ thinking and strategies that they invent are considered as a major goal of effective

mathematics teaching [3]. It was aimed to examine, describe and interpret pre-service

teachers’ views about student-invented strategies in subtraction. Within the context of this

purpose, case study was utilized. 85 pre-service teachers from elementary education and

mathematics education participated in the study. A semi-conducted interview form was used

to collect data. Pre-service teachers were introduced to three different student solutions to the

problem 173-46. Two of the solutions were student-invented strategies and the last one was

traditional algorithm. Pre-service teachers were asked (i) which of the strategies they choose

for their future students, (ii) how to respond to each of these students, and (iii) how to

introduce traditional algorithm to students that invent strategies. Descriptive and content

analyses will be used in analyzing preferred strategies and the reasons of preference; pre-

service teachers’ evaluation, feedback and comparison of students’ solutions, and approaches

that pre-service teachers state they will use in introducing traditional algorithm.

Keywords. Mathematics education, teacher education, student-invented strategies.

AMS 2010. 97D99, 97B50.

Acknowledgement.This work is supported by Ahi Evran University PYO with project

number EGT.E2.17.043.

References

[1] Van de Walle, J. A., Karp, K. S., & Williams, J. M. B., Elementary and middle school

mathematics: Teaching developmentally (7th Ed.). New York: Longman, 2010.

[2] Carpenter, T. P., Fennema, E., & Franke, M. L., Cognitively guided instruction: Building

the primary mathematics curriculum on children’s informal mathematical knowledge. A

paper presented at the annual meeting of the American Educational Research Association,

1992.

[3] National Council of Teachers of Mathematics, Principles and standards for school

mathematics. Reston, VA: Author, 2000.

1 Ahi Evran University, Kırşehir, TURKEY, [email protected]



242

Investigation of Parent Expectations of Middle School Students from Mathematics

Education*

Cahit AYTEKİN1, Serdal BALTACI2, Bülent ALTUNKAYA3,

Avni YILDIZ4, Yasemin KIYMAZ5

Abstract.

It was stated that the academic achievement and the emotional development of the

students whose family supports and actively followed their education process develop faster

than the other students [1]. Similarly, it is known that the academic achievement of students

has significantly increased in a teaching environment actively supported by the family [3]. In

one study, the students whose parents actively participated in education were examined and

found that these students were more likely to have upper education, higher grades and more

social developments [2].

In this study, middle school student parents' expectations of "conceptual understanding

and active student participation", "positive attitude and behavior", and "authority and rule-

oriented teaching" were examined. In the research, a valid and reliable three-factor instrument

developed by the researchers was used. A total of 749 parents participated in the research. As

a result of the research, it was determined that the expectation of the parents did not show any

significant difference according to the parents' type (mother, father and other) and parental

age level. It has been found out that the university graduate student parents have less authority

and rule-oriented teaching expectation than others. It has also been determined that

individuals with moderate monthly income have a lower expectation of "authority and rule-

based mathematics instruction" than those with low-gestational age groups. Moreover, it was

found that parents who love mathematics have high "conceptual understanding and active

student participation" expectation. On the other hand, it is revealed that parents who do not

like mathematics have low level "authority and rule-oriented teaching" expectation. It was

also found that parent who have a high degree of support for their child have high expectancy

of "conceptual understanding and active student participation" and have low expectancy of

"authority and rule oriented teaching". It was also found that the parents of successful in

* This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number:

EGT.A3.16.013

1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected] 4 Bülent Ecevit University, Zonguldak, TURKEY, [email protected]



243

mathematics have higher "conceptual understanding and active student participation"

expectations. Based on these findings, it can be said that parents' expectations have an

important role in education.

Keywords. Parents Expectation, Mathematics Education

References

[1] Booth, A.,& Dunn, J. F., Family school links: How do they affect educational outcomes?

Hillsdale, NJ: Erlbaum, 1996.

[2] Henderson, A. T. & Berla, N., A New Generation of evidence: The family is critical to

student achievement. Washington DC: National Committee for Citizens in Education, 2004.

[3] Nyabuto, A. N.,& Njoroge P. M., Parental involvement on pupils' performance in

mathematics in public primary schools in Kenya. Journal of Educational and Social Research,

4(1), 19-26, 2014.


244

The Impact of the Probability Explorer Software on Preparing Probability Teaching

Plans of Elementary Mathematics Teacher Candidates*

Cahit AYTEKİN1, Serdal BALTACI2, Yasemin KIYMAZ3, Avni Yıldız4

Abstract

Studies on teaching probability in Turkey draw attention finding the most effective

teaching method to effect achievements of students [2], [3], [4], [5], [6]. Many studies show

that students have difficulty in making sense of the concepts of probability [7]. Considering

the difficulties in the teaching of probability and misconceptions, studies focused on teaching

probabiliry are very important [1]. In this study, it is aimed to develop teacher candidates

about teaching probability who will start teaching as a mathematics teacher in the near future.

This research aims to develop elementary school mathematics teachers’ skills to

prepare teaching plan about of probability. For this purpose, "Probability Explorer" software

has been taught to prospective teachers. Probability Explorer" software was developed for

probabilistic teaching. The instruction plans prepared by Teacher candidates before and after

this instruction were compared. The data of the study were analyzed by qualitative research

methods.

Thirteen different themes were found in the second plan of teacher candidates, which

were not found in their initial plans. These situations are thought to be very effective in the

teaching of probability. It was seen that from these themes "observing and interpreting

experimental results", "conducting a large number of experiments on an event", "approaching

the theoretical probability as the number of experiments increases", "estimating the

experimental results", "encouraging the student to do his own probabilistic experiments",

"comparison of the theoretical probability values" were more frequent.

When the findings of the research are examined, it has been seen that simulation

softwares are very important in teaching probability. These softwares make it possible to use

different methods in teaching probability by making difficult experiments very fast. It has

been observed that there are considerable differences in teaching plans based on using a

* This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number:

EGT.A3.17.009

1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected] 4 Bülent Ecevit University, Zonguldak, TURKEY, [email protected]


245

simulation program for elementary school mathematics teacher candidates compared to

teaching plans prepared in the traditional way.

Keywords. Teaching Probability, Preparing Teaching Plan, Simulation Softwares.

References

[1] Bulut, S., Yetkin, E.İ ve Kazak, S., Matematik Öğretmen Adaylarının Olasılık Başarısı,

Olasılık ve Matematiğe Yönelik Tutumlarının Cinsiyete Göre İncelenmesi. Hacettepe

Üniversitesi Eğitim Fakültesi Dergisi, 22: 21-28, 2002.

[2] Çubuk, Ş., Matematik Öğretiminde “Permütasyon ve Olasılık” Konusunun Bilgisayar

Destekli Öğretim Materyalleri İle Öğretilmesinin Öğrenci Başarısına Etkisi. Marmara

Üniversitesi Eğitim Bilimleri Enstitüsü Yüksek Lisans Tezi, 2005.

[3] Ercan, Ö., Çoklu Zeka Kuramına Dayalı Öğretim Etkinliklerinin 8.Sınıf Öğrencilerinin

Matematik Dersi “Permütasyon ve Olasılık” Ünitesindeki Akademik Başarıları Etkisi. Gazi

Üniversitesi Eğitim Bilimleri Enstitüsü Yüksek Lisans Tezi, 2008.

[4] Erkin Kavasoğlu, B., İlköğretim 6,7 ve 8 Matematik Dersinde Olasılık Konusunun Oyuna

Dayalı Öğretiminin Öğrenci Başarısına Etkisi. Gazi üniversitesi eğitim bilimleri enstitüsü,

2010.

[5] Esen, B., Matematik Eğitiminde İlköğretim 6.Sınıflarda Olasılık Konusunun Öğretiminde

Bilgisayar Destekli Eğitimin Rolü. Selçuk Üniversitesi Fen Bilimleri Enstitüsü Yüksek Lisans

Tezi, 2009.

[6] Gürbüz, R., Olasılık Kavramları İle İlgili Geliştirilen Öğretim Materyallerinin

Öğrencilerin Kavramsal Gelişimine Etkisi. Dokuz eylül üniversitesi buca eğitim fakültesi

dergisi, 2006.

[7] Sev Lekesiz, E.Ç., Dördüncü ve Beşinci Sınıf Öğrencilerinin Olasılık Konusunun

Öğretiminde Karşılaştıkları Zorluklar. Eskişehir Osmangazi Üniversitesi Eğitim Bilimleri

Enstitüsü yüksek lisans tezi, 2011.


246

Proof Evaluation: Mathematicians and Prospective Mathematics Teachers’ Perspectives

Esra Selcen YAKICI-TOPBAŞ1, Elçin EMRE-AKDOĞAN2, Fatma Nur AKTAŞ3, Gönül

YAZGAN-SAĞ4 and Ziya ARGÜN5

Abstract. Abstract. Proof plays an important role in mathematics to verify whether a

statement is true or not, to explain why a statement is true, and to communicate mathematical

knowledge ([1], [5]). “Determining whether a proof is correct and valid and also how good it

is regarding a wide range of features such as clarity, logical correctness, insight,

convincingness, fluency” are fundamental features for proof evaluation [4]. Therefore, proof

evaluation provides us with an understanding of mathematicians, mathematics teachers, and

prospective mathematics teachers’ perspectives on how they construct proofs and thus it helps

to obtain an understanding of what constitutes a proof. Also, mathematical proof is critical to

constitute the undergraduate mathematics curriculum for mathematics teachers and

prospective mathematics teachers [3]. Researchers have argued that mathematicians want to

know why a statement is correct rather than whether a statement is correct or not, that is,

proof evaluation is wider than judgment and validation [2]. In this context, the current study

aims to investigate the differences and similarities between mathematicians and prospective

teachers’ perspectives while evaluating a basic abstract mathematics theorem’s proof

examples, which include standard proof methods: direct, contradiction, and contraposition.

This study is a multiple-case study that has a qualitative approach [6]. The cases of this study

are mathematicians and prospective mathematics teachers. Two mathematicians (who are

professors, who have completed their doctorate in mathematics) and six senior prospective

mathematics teachers are participants of the study. The participants were selected through

purposeful and convenience sampling. The data were collected through individual semi-

structured interviews which were recorded with a video camera. In the interviews, two tasks

were completed by participants. We gave a proposition for proving in the first task. Then in

the second task we gave propositions and its nine different proofs including constituted direct,

contradiction, contraposition standard basic proof methods and a visual proof. We asked

participants to examine nine proofs given in the second task. We deeply examined the

participants' evaluation on the proofs. We analyzed the data through content analysis. We

shared the findings in the context of a comparison of mathematicians’ and prospective

mathematics teachers’ perspectives on the evaluation and investigation of the given proof

examples. We explored that mathematicians do not accept visual proof as a mathematical

proof, contrary to the fact that prospective teachers evaluate visual proof as a mathematical

proof that is essential for their students. On the process of evaluating proofs, all participants

pay attention to the mathematical language and the beginning of the proof.

Keywords. Proof evaluation, prospective mathematics teacher, mathematics

education.

AMS 2010. 03F03, 97-XX

1 Gazi University, Ankara, Turkey, selcenyakici@gmailcom 2 Gazi University, Ankara, Turkey, elcı[email protected]

3 Gazi University, Ankara, Turkey,[email protected]

4 Gazi University, Ankara, Turkey, [email protected]

5 Gazi University, Ankara, Turkey, [email protected]


247

References

[1] Hanna, G., Some pedagogical aspects of proof, Interchange, 21(1), 6–13, 1990.

[2] Hersh, R., Proving is convincing and explaining, Educational Studies in Mathematics, 24,

389–399,1993.

[3] Moore, R. C., Mathematics Professors’ Evaluation of Students’ Proofs: A Complex

Teaching Practice, International Journal of Research in Undergraduate Mathematics

Education, 2(2), 246-278, 2016.

[4] Pfeiffer, K., Features and purposes of mathematical proofs in the view of novice students:

Observations from proof validation and evaluation performances (Doctoral dissertation,

2010).

[5] Schoenfeld, A., What do we know about mathematics curricula? Journal of Mathematical

Behavior, 13(1), 55–80, 1994.

[6] Yin, R. K., Case study research: design and methods, Applied social research methods

series. Thousand Oaks, CA: Sage Publications, Inc. Afacan, 2003.


248

New Method of Grading Exams based on Computational Statistics

Evgeny GERSHİKOV1

Abstract. The common way to grade academic exams is by dividing the maximal

number of grade points among the exam questions and their paragraphs (if relevant), later

trying to estimate the amount of points that an individual student has scored in a each

question. The problem with this process is that sometimes a question level is not assessed

correctly and the question turns out to be too hard or too easy for the examinees, which brings

forward a question mark about the correctness of the points allocation to this particular

question. In this work, we use computational statistics to provide a different method to

allocate grade points to questions and we test our method on simulated exam results and real-

life experiments. Our tests were done using mainly multiple choice questions due to the

following reasons. Multiple choice questions have a long history [1] and are very common in

academic institutions around the world as well as in psychometric evaluations. Also, we have

designed and implemented an automatic grading system for this kind of questions and a great

benefit of this system is automatic collection of the student scores in all questions [2]. Thus,

the examiner does not have to enter the scores manually into the computerized system [3].

There are many approaches available to analyze multiple choice test data [4], however none

target question efficiency explicitly. In this work we propose statistical efficiency measures

for each question. The group of examinees is divided into three subgroups: the “strong”

students, the “weak” students and the “average” ones. The initial division is according to the

regular grades. Now a question is considered efficient if most “strong” students succeed in it

while most “weak” ones fail. We provide a numeric formula to calculate the efficiencies.

Then we weight the question scores by these efficiencies to calculate weighted grades that

represent, in our opinion, a better way of student evaluation than the regular non-weighted

grades. Optionally, we discard the totally inefficient questions, so that they do not affect the

weighted grades. The process is repeated in an iterative fashion until convergence, that is, we

repeat the step of dividing the students into three subgroups, now using the weighted grades

for the division, we calculate the new efficiencies and new weighted grades and so on. Results

and simulations are provided in our work. We conclude that this new way of grading is

beneficial to evaluate the efficiency of individual exam questions as well as the whole exam.

1 Braude Academic College, Carmiel, ISRAEL, [email protected]


249

Keywords. Efficiency measures, new grading method, computational statistical

measures.

References

[1] Wood R., Multiple choice: A state of the art report, Evaluation in Education, International

Progress, 1, 3, 191-280, 1977.

[2] Gershikov E., Kosolapov S., On Image Based Fast Feedback Systems for Academic

Evaluation, International Journal of Signal Processing Systems, 3, 1, 19-24, 2014.

[3] Gershikov E., Kosolapov S., Camera-Based Instant Feedback Systems, International

Journal of Advanced Computing, 47, 1, 1463-1473, 2014.

[4] Ding L., Beichner R., Approaches to data analysis of multiple-choice questions, Physical

Review Special Topics - Physics Education Research, 5, 1-17, 2009.


250

Difference Equations in Economics

Jelena STANOJEVIC1 and Katarina KUKIC2

Abstract. In this paper we give review of some well-known models in economy

described by difference equations. We presented application of cobweb analysis on one

supply-demand model with naive and also with rational expectations for dynamical system

described by difference equations. For same examples we briefly analyze their stability

properties and through this part we emphasize the importance of studding difference

equations in basic mathematics courses for economics. That motivated us to make a little

study with two groups of students at the University of Belgrade concerning the students'

ability and will to learn about difference equations. In the second part of our paper we present

the results of that study.

1 Faculty of Economics, University of Belgrade 2 Faculty of Traffic and Transport Engineering, University of Belgrade


251

Factors Affecting Scoring Open-Ended Questions

Kübra Akkaya1 and Selahattin Arslan2

Abstract. Although there are different tools available for assessment, written

examinations are often used in our educational system. Written examinations outclass other

assessment tool such as multiple choice questions, but differences in scoring are their main

disadvantages. There are many factors which engender the differences in scoring that occur

when the written exams are evaluated. By using the case study method in this research, it was

aimed to determine the factors that give rise to the scoring differences that arise when the

secondary school mathematics teachers evaluate written exams. For this purpose, firstly semi-

structured interviews were made with 11 randomly selected mathematics teachers and

secondly clinical interviews were made with them with the help of by purpose-selected

student examination papers. The result of the study showed that teachers gave different scores

to the same examination paper. Further results of the study are that some teachers have

focused on the solution process, and yet others were satisfied with the results. Some factors

such paper regularity, student achievement, also affect the score given.

Keywords. Measurement and evaluation, open-ended examination, grading.

1 Milli Eğitim Müdürlüğü, Trabzon, Türkiye, [email protected] 2 Karadeniz Technical University, Trabzon, Türkiye, [email protected]


252

Development Process of In-Service Training Intended for Teachers to Perform Teaching

of Mathematics with Computer Algebra Systems

Mehmet Alper ARDIÇ1 and Tevfik İŞLEYEN2

Abstract. In this study, it was dealt with the development process of in-service

training activities designed in order for mathematics teachers of secondary education to

realize teaching of mathematics, utilizing computer algebra systems. In addition, the results

obtained from the researches carried out during and after the in-service training were

summarized. In the last part of the study, suggestions were made to teachers and researchers

who want to carry out activities aimed at using computer algebra systems in teaching

environments.

Keywords. Computer-assisted mathematics instruction, computer algebra system, in-

service training.

AMS 2010. 97A99, 97D40, 97U50

1 Adıyaman University, Adıyaman, TURKEY, [email protected] 2 Atatürk University, Erzurum. TURKEY, [email protected]


253

Examination of the Activities about Enciphering Done with Fifth Grade Students and

Their Views about the Activities

Muhammet ŞAHAL 1 and Ahmet Şükrü ÖZDEMİR 2

Abstract. Mathematics is defined as intangible form of real life. It is assumed as a

difficult course by students due to its discrete structure and other different factors. In order to

deal with such considerations, it is believed that to utilize activities in the course, which are

attractive for students and enable them to be active through learning process and motivate

them. Middle school students are eager to learn with games, physical activities and dealing

with problems about daily life context. One of such kind of activities are enciphering ones.

Throughout history, human have been wondering about hidden things and hiding information

from others is always seemed as interesting. It is known that enciphering was used for

military purposes in the time of the Spartans and the Roman Empire. Colossus, which was

invented by Turing, helped to decipher the codes developed by Nazi armies and gave rise to

finish the Second World War in shorter time. That’s why, the victory was dedicated to

mathematicians. Enciphering was not only crucial in the past, but also it is even more

important in recent time. In this study, it was investigated that whether activities about

enciphering is appropriate for fifth grade students or not. In addition, revealing the students’

views about the activities was another goal in the study. In order to reach the goals, activities

developed by Mathematical Enhancement Programme (1995) were used after translating into

Turkish. Then students’ views about the activities were gathered and it was asked them to

produce similar activities that they had done. Case study, which is type of qualitative research

methodology, was used in the study. Participants of the study are 30 fifth grade students. It

was reached that majority of the students were accomplished the activities successfully but

they couldn’t do deciphered ones. They also produced similar enciphering activities. Besides

it was also found that enciphering activities attracted students’ attentions, increased their

motivations and interests to the course and made the course enjoyable for them.

Keywords: enciphering; deciphered mathematics; mathematics education; motivation.

1 Yıldız Technical University, Istanbul, TURKEY, [email protected] 2 Marmara University, Istanbul, TURKEY, [email protected]


254

The Concept of Zero: A Comparison of Students with Learning Disability vs Non-

Learning Disability

N. Dilşad GÜVEN1 and Esra Selcen YAKICI TOPBAŞ2 and Ziya ARGÜN3

Abstract. Labeling to non-existed object is a challenging process in history ([1], [5]).

It could be difficult for students to regard zero as a number because of labeling a non-existing

object with a number. Understanding zero and its using is different than understanding other

numbers due to its abstract nature. The numbers that are used to state multitude constitute a

symbolic system. Students with learning disability (LD) could not make connection with

multitude and symbolic system [3]. On the other side, studies in the literature remark that

students have weak knowledge about zero ([2], [4], [7]). Since students with LD having

difficulty with symbolic system and students with non-LD having difficulty about zero, zero

is a problematic concept for all students. It is issue of concern how students with LD and non-

LD conceptualize zero, what are the similarities and differences. Accordingly, the current

study aims to investigate similarities and differences between zero conception of students

with LD and non-LD.

It is a multiple-case study having qualitative approach [6]. The cases of study are

students with LD and non-LD. 3 students with LD and 3 students with non-LD are

participants of the study. The data were collected through clinical interviews. The conception

of zero with different usage (as a number, as a “nothing”, as a symbol, as a cardinality of a set

etc.) is examined deeply in the interviews. The data were analyzed through content analysis

method.

The results suggest that all students are aware of the symbol using of zero, especially

zero as a place holder. While students with non-LD consider that zero is a number, students

with LD doesn’t. However, students with non-LD attach meaning of nothing to zero similar to

students with LD. Thus, all of the students have misconceptions and mistakes about the

operations involving zero, especially multiplication and division. Students' explanations about

meaning of zero in multiplication are limited by words such as “eating, swallowing, or

absorbing”. Students could not make sense of absorbing role of zero in multiplication.

Students (especially students with LD) use algorithms for operations involving zero

1 Gazi University, Ankara, TURKEY, [email protected] /[email protected] 2 Gazi University, Ankara, TURKEY, selcenyakici@gmailcom/ [email protected] 3 Gazi University, Ankara, TURKEY, [email protected]/ [email protected]



255

developed by themselves. Thus, this could be a result or a reason of the lack of understanding

about zero concept. Conversely, students' conception about operations affect their

understanding of operations involving zero.

Keywords. Zero concept, Learning Disability, Mathematics education.

AMS 2010. 97-XX

References

[1] Gullberg, J., Mathematics: From the birth of numbers. New York: Norton, 1997.

[2] Reys, R. E., & Grouws, D. A., Division involving zero: Some revealing thoughts from

interviewing children. School science and mathematics, 75(7), 593-605, 1975.

[3] Rousselle, L., & Noël, M. P., Basic numerical skills in children with mathematics learning

disabilities: A comparison of symbolic vs non-symbolic number magnitude

processing. Cognition, 102(3),361-395, 2007.

[4] Tsamir, P., Sheffer, R., & Tirosh, D., Intuitions and Undefined Operations: The Cases of

Division by Zero. Focus on Learning Problems in Mathematics, 22(1), 1-16, 2000.

[5] Seife, C., Zero: The biography of a dangerous idea. Penguin, 2000.

[6] Yin, R. K., Case study research: design and methods, Applied social research methods

series. Thousand Oaks, CA: Sage Publications, Inc. Afacan, 2003

[7] Quinn, J. R., Lamberg, T. D., & Perrin, J. R., Teacher perceptions of division by zero. The

Clearing House, 81(3), 101-104, 2008.


256

Preservice Mathematics Teachers’ Mathematizing Skills in The Process of Solving a

Real Life Problem

Nuray Çalışkan DEDEOĞLU1

Abstract. Trends in mathematics curricula focus on concepts like mathematical

literacy, modelling and STEM which enable to work on real life applications of mathematics.

These approaches occur within the frame of problem solving which is an essential part of the

school mathematics. Beyond providing application opportunity traditionally, problem solving

improves mathematical thinking and it also help create mathematical knowledge [1-3].

According to Freudenthal who is pioneer of Realistic Mathematics Education, during the

process of problem solving which is called mathematizing, it is required to reach formal

mathematical knowledge from informal mathematical knowledge which consists of real life

situations [4].

It is important to have mathematizing skills by mathematics teachers and preservice

mathematics teachers to have an alternative approach to create knowledge. The purpose of the

study is examining the process of creating mathematical knowledge within the frame of a real

life problem of preservice elementary mathematics teachers. The solutions of a real life

problem which is individually solved by preservice teachers generate the data of the study

which is designed as a case study that is a qualitative method. Data were analyzed by content

analysis technique. Findings show that preservice teachers could not present some important

skills such as using generalization method which is necessary to create mathematical

knowledge properly and supporting the solution of problem with mathematical models and

explanations.

Keywords. Real life problem, realistic mathematics education, mathematizing,

modelling, preservice mathematics teachers.

AMS 2010. 97-XX, 68T20, 97F90 97B50.

References

[1] Polya, G., How to solve it (2nd Ed.), Princeton University Press, 1973.

1 Sakarya University, Sakarya, TURKEY, [email protected]



257

[2] Schoenfeld, A. H., Learning to think mathematically: Problem solving, metacognition,

and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on

Mathematics Teaching and Learning, MacMillan, New-York, 334-370, 1992.

[3] Tall, D., A theory of mathematical growth through embodiment, symbolism and

proof. IREM de Strasbourg, Annales de Didactique et de Sciences Cognitives, 11, 195–215,

2006.

[4] Gravemeijer, K., RME theory and mathematics teacher education, In D. Tirosh and T.

Wood (Eds.), The International Handbook of Mathematics Teacher Education: Tools and

Processes in Mathematics Teacher Education, Sense Publishers, 283–302, 2008.


258

Learning Implementations about Cooperative Learning Method: A Case Study in

Türkiye

Perihan DINC ARTUT1 and Ayten Pınar BAL2

Abstract. Cooperative learning method is used prevalently in many subject areas and

class levels. One of the reasons of this is that it has positive effects on academic success [1],

[2]. Although the studies about this learning method were focused on elementary and

secondary education, it is expressed in discussions about the benefits of this method that it can

also be implemented in higher education classes [3], [4]. Cooperative learning has many

different forms, such as Jigsaw, Cooperative Integrated Reading and Composition (CIRC),

Teams-Games-Tournament (TGT), Learning Together (LT), Student Teams-Achievement

Divisions (STAD). In addition to many other fields, STAD can be used in mathematics. By

the help of STAD, students interact mutually, become responsible for each other’s learning

and they develop their other skills. In the practice of STAD, students start working in their

teams and attempt to complete the task given to them on their own after the teacher’s

presentation of the topic. In line with the explanations above, it was aimed to implement

Student Teams-Achievement Divisions (STAD), which is a technique in cooperative learning

method, in the teaching of limits and derivatives topic in the course of calculus of the students

who were attending the department of science and technology teaching and to evaluate the

opinions of the students and instructors about this implementation. In this research, an

interview form was used so as to determine the opinions of students about working in groups

based on cooperation while learning mathematics. The findings of this research revealed that

the implementations of STAD technique in lessons created a positive effect, provided active

participation to the lessons and made the students understand more easily by collaborating

with their friends on their learning of limits and derivatives according to the opinions of the

students.

Keywords. Calculus, case study, cooperative learning, student teams-achievement divisions

References



259

[1] Slavin, R. E.; Madden, N.A.; Leavey, M., Effects of team-assisted individualisation on

the mathematics achievement of academically handicapped and nonhandicapped students.

Journal of Educational Psychology, 76(5), 813-819, 1984.

[2] Häsel-Weide, U. and Nührenbörger, M., Replacing counting strategies: Children’s

constructs working on number sequences. Proceedings of the Eighth Congress of the

European Society for Research in Mathematics Education. Ankara/ Turkey: Middle East

Technical University, 2013

[3] Artut, P. D., Effect of Cooperative Learning Method on Prospective Teachers’ Non-

routine Problem-Solving Skills and Their Views About the Method. US-China Education

Review, 6(4), 244-254, 2016.

[4] Johnson, D. W., Johnson, R. T., & Smith, K. A., Cooperative learning: Improving

university instruction by basing practice on validated theory, Journal on Excellence in

University Teaching, 2013.


260

Investigation of Number Sense Strategies Used By the 8th Grade Students in Turkey

Perihan DINC ARTUT1 and Zübeyde ER2

Abstract. The concept of number sense is a significant one in mathematics education.

According to the standards of National Council of Teachers of Mathematics (NCTM) (2000,

p.32) [1], it is stated that students understand the numbers, the ways of representing the

numbers, the relationships between the numbers and number systems, the meanings of

operations, the relationships of the numbers with each other and they can do fluent

computations and appropriate predictions and it is emphasized that the concept of number

sense should have been developed based on these standards. There are various definitions

about the concept of number sense in the literature. McIntosh, Reys and Reys (1992) [2]

describe number sense as: “a person’s general understanding of number and operations along

with the ability and inclination to use this understanding in flexible ways to make

mathematical judgments and to develop useful strategies for handling numbers and

operations” (cited Morais and Serrazina, 2013). In this research aimed to investigate the

strategies which the students use while solving number sense problems. This study was

conducted in the scope of case study design which is one of the qualitative research designs.

The research was conducted on 28 students, 17 of which were girls and 11 of which were

boys, who were attending the 8th grade in three secondary school having similar

characteristics in a city centre which is located in the south of Turkey. As the data collection

tool, a number sense test (NST) which consisted of some questions in the number sense scale

of Singh (2009) [3], which was originally adapted from McIntosh, Reys, Reys Bana, and

Farrell, (1997) was used. In the test, there are 5 components about understanding the concept

of number, using the multiple expressions of numbers, understanding the effect of numbers,

using equivalent expressions and using computing and counting strategies. NST consists of 25

items about the topics of whole numbers, decimal numbers, fractions and percentages. In this

research was seen that the students used the strategies based on rules in their solutions more

when all of the solutions obtained from the students in this research were considered. When

the strategies that the students used according to the components of number sense were

analyzed in this study, it was concluded that the component of understanding the concept of

number was the one which the students used the strategies based on number sense the least.

1 Çukurova University, Adana, TURKEY, [email protected] 2 MEB, Adana, TURKEY, [email protected]

https://www.google.com.tr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwid78LA0Y3UAhXIaFAKHQuGBKoQFggnMAA&url=http%3A%2F%2Fwww.nctm.org%2F&usg=AFQjCNG1lDpx7I59-l0huTGwIS9FUwFE7g&sig2=1MqZfjzwkJ_DzHy4HhJcmQ


261

Keywords. Number sense, number sense components, number sense strategies

References

[1] National Council of Teachers of Mathematics (NCTM)., The principles and standards for

school mathematics. Reston, VA: Author, 2000.

[2] McIntosh, A., Reys, B. J., and Reys, R. E., A proposed framework for examining basic

number sense. For the learning of mathematics, 12(3), 2-44, 1992.

[3] Singh, P., An Assessment of Number Sense among Secondary School Students,

International Journal for Mathematics Teaching and Learning, 2009.


262

Examining Teacher Candidates’ Thinking Skills

Perihan DİNÇ ARTUT1and Ayten Pınar BAL2

Abstract

Mathematics is one of the most effective fields of thinking. This aspect of mathematics

is also being emphasized in the educational process. In this context, it is observed tha

mathematics teaching programme in our country has given more importance to basic skills in

mathematical thinking (Ministry of NationalEducation [MEB], 2013).

The goals of mathematics curriculum to develop mathematical thinking skills have led

to focus on questions about mathematical thinking and its key components. Mathematical

thinking is defined as a set of complex processes such as "guessing, induction, deduction,

description, generalization, sampling, formal and informaL reasoning, verification, and so on"

(Liu Po-Hung, 2003).

Considering that one of the most important basic components effective in mathematics

education is mathematics teachers, it is thought that teacher training process and in this

context candidate teachers should have mathematical thinking skills. In this direction, it was

aimed to determine themathematical thinking skills of the class and mathematics teachers in

this research.

This research is a descriptive study done within the technique of survey model and

conducted on mathematics and classroom teacher candidates. As a data collection tool

“Mathematical Thinking Scale” whic was developed by Ersoy and Başer (2003) was used. As

data analyses, descriptive statistic was used. The findings revaled that both classroom and

mathematic teachers’ thinking skills are similar to each other.

Keywords.Thinking mathematicaly, mathematics, pre service teacher.

References

[1] Milli Eğitim Bakanlığı [MEB], İlköğretim matematik (1-5. Sınıflar) dersi öğretim

programı. Ankara: Talim Terbiye Başkanlığı Yayınları, 2013.

1CukurovaUniversity, Adana, Turkey, [email protected] 2CukurovaUniversity, Adana, Turkey, [email protected]


263

[2] Po-Hung, L., Do Teachers need to incorporate the history of mathematics in their

teaching?. The Mathematics Teacher, 96(6), 416, 2003.


264

Difficulties of High School Students in Complex Numbers

Selahattin ARSLAN1

Abstract. Complex numbers field is considered as difficult to understand by students

and many countries high school mathematics programmes do not accord place to this field. In

our country, this concept has been removed from high school mathematics programs with the

revision made in 2017. In this study, it was aimed to reveal the difficulties in complex

numbers of high school students and to compare students in different programmes. For this

purpose, a test of 29 questions, which is available in the literature, has been applied to 70 high

school second year students (35 studying at Linguistic/Verbal programs and 35 at language

Mathematical/Numerical program. Analyses realised has been revealed that both groups’

students had difficulties with complex numbers, but numerical learners were more successful.

Keywords. Complex numbers, difficulty, high school students.

1 Karadeniz Technical University, Trabzon, TURKEY, [email protected]


265

Investigation of Mathematical Viewpoints of Primary School Teacher Program Students

Tuğba BARAN KAYA1 and Ahmet IŞIK2

Abstract. In a learning environment; The teachers have important roles such as

making their lessons attractive, ensuring students’ learning, and making the relevant lesson

feel important [1]. Particularly, teacher candidates’ perceptions towards mathematics, who are

soon to be responsible for the teaching of basic mathematical concepts in the first stage of

primary education can give deep clues about this role [2]. Various researches can be done to

mention in depth about such a perception. This research is aimed to reveal the point of view

of Primary School Teaching Program students' mathematics and mathematics relations with

other sciences. In this research using the case study method of qualitative research methods,

the data were collected with the help of test items consisting of two open ended questions

directed to the students. The data obtained by this data collection tool were analyzed by

content analysis tecniques and then the frequency of the resulting themes was found. As a

result of the research, it was revealed that students perceive mathematics mostly in the context

of components of mathematics (numbers, symbols, operations). In addition, although students

in the survey wanted to explain the connection between mathematics and other sciences, the

vast majority of students avoided doing this, but went to sort out the sciences and other fields

they thought were related to mathematics.

Keywords. Mathematics, perception, primary school teaching program students.

AMS 2010. 53A40, 20M15.

References

[1] Şahin, B., Öğretmen adaylarının “matematik öğretmeni”, “matematik” ve “matematik

dersi” kavramlarına ilişkin sahip oldukları metaforik algılar, Mersin University Journal of

the Faculty of Education, 9(1), 313-321, 2013.

[2] Güveli, E., İpek, A. S., Atasoy, E., Güveli, H., Prospective primary teachers’

metaphorical perceptions towards mathematics, Turkish Journal of Computer and

Mathematics Education, 2(2), 140-159, 2011.

1 Kırıkkale University, Kırıkkale, TURKEY, [email protected] 2 Kırıkkale University, Kırıkkale, TURKEY, [email protected]


266

A Framework Suggestion for the Analyze of the Solving Process of a Geometrical

Construction Problem

Tuğçe KOZAKLI ÜLGER1, Işıl BOZKURT2 and Murat ALTUN3

Abstract. Many researchers on the problem-solving process seem to refer to the

problem-solving phases of Polya and interpret these phases for the required field like

cognitive and metacognitive processes. It brings the question to the mind; could it be more

useful to provide specific problem-solving process steps for subject areas (algebra, geometry

etc.) that are at least separated from each other by this outline? In this study, the analysis of

the solving process of any geometrical construction problem using the compass- straightedge

is discussed.

Geometrical construction refers to the standard procedures required to construct

geometric structures by using a compass-straightedge and is thought to play an important role

in the development of geometric thinking. Geometrical construction activities are important

because they provide a deep insight into the properties of the geometric structure created.

There are phases (analysis, construction, evidence, discussion) suggested by Smart regarding

the solving process of a geometrical construction problem. In this study, it is aimed to

elaborate using the steps proposed by Smart (1998), in order to provide a detailed study of

mental processes. The solutions made by the participants in the study group are used as data

to detail these mental processes.

This research used a qualitative case study approach in order to enable the in-depth

analysis of problem-solving processes. In line with the aim of the research, the study group

consists of four participants who have experience on the geometrical construction studies with

the help of a compass- straightedge and know the possibilities and limitations provided by

these tools. The data collection tools of the study were video recordings taken during drawing,

documents (reports and drafts which the solutions are detailed) and unstructured interviews

after problem solving. As a result of the study, a new model has been proposed with the

addition of sub-steps and refinement to the framework of Smart, which is included in the

literature on geometrical construction problems in the solving process.

Keywords. Geometrical construction, problem solving, using compass-straightedge

1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected] 3 Uludag University, Bursa, TURKEY, [email protected]





267

AMS 2010. 53A40, 20M15.

References

[1] Polya, G., How to solve it. Princeton, NJ: Princeton University Press, 1957.

[2] Smart, J. R., Modern geometries (5th ed.). CA: Brooks/Cole Publishing Company, Pacific

Grove.


268

Active Learning in Flipped College Algebra Class

Violeta VASILEVSKA1

Abstract. The recent popularity of flipped classrooms in undergraduate classes and

research showing the effectiveness of it on student learning [1, 2] prompted the presenter’s

teaching endeavor in that direction. Namely, in the last few years, the presenter has been

teaching a hybrid College Algebra class in a flipped classroom. In this presentation, the

specifically designed class structure will be discussed.

For the on-line learning activities students use MyMathLab (Pearson’s on-line system that

provides online homework, tutorials, and assessment tools). Students are asked to view the

video lectures on-line and solve the on-line homework before they come to class.

The in-class activities are divided in two parts. The first part is a question/answer segment.

During this part, the instructor (presenter) does not lecture, but rather just helps answer

questions about concepts/topics that caused difficulty to students at home. The second part is

a problem-solving segment. Students are given a group quiz, which is a unique way to involve

students in collaborative problem-solving active learning. They work as teams of two on the

quiz, apply the knowledge and practice the new concepts, discuss the questions, and learn

from each other.

This structure allows students to go over/review the new concepts several times, to encourage

them to ask questions, to lead them to answer their own questions, and to spark

communication between them.

During the presentation, it will be demonstrated how the question/answer part of the class is

conducted. In addition, some survey results about the structure and effectiveness of this

course will be shared. Moreover, it will be discussed how student feedback has been

incorporated in the structure of the class, and how that change has impacted the student

learning.

Keywords. Flipped classroom, hybrid class, active learning.

1 Utah Valley University, Orem, UT, USA, [email protected]


269

AMS 2010. 97D40, 97U50.

References

[1] Herreid, C. F., Schiller, N. A., Case studies and the Flipped Classroom, Journal of

College Science Teaching 42 (5), 62-66, 2013.

[2] Love, B., Hodge, A., Ernst, D. C., Inquiry-based learning and the Flipped Classroom

Model, PRIMUS (Problem, Rosources, and Issues in Undergraduate Mathematics Studies) 25

(5), 745-762, 2015.


270

STATISTICS


271

New Method for Obtaining of Weighted Distributions

Božidar V. POPOVİĆ1

Abstract. İn the real life there are many situations where symmetric distributions are

not appropriate for fitting. In that way, Azzalini’s method [1] is widely used to get

asymmetric or skewed distribution. Shortly this technique can be described as follows: a

random variable Z possesses the skew normal distribution with parameter α if its probability

density function (pdf) is 𝑓(𝑧; 𝛼) = 2𝜙(𝛼𝑧)𝜑(𝑧), where 𝜙(∙) and 𝜑(∙) are cumulative

density function (cdf) and pdf, respectively, of the standard normal random variable. In terms

of the conditional distribution Azzalini’s method consists of the following: let 𝑋𝑖 𝑖 = 1,2 are

independent random variables with pdfs 𝑓𝑖(𝑥𝑖) and cdfs 𝐹𝑖(𝑥𝑖). Then the conditional pdf of

𝑋 = 𝑋1 given 𝛼𝑋1 > 𝑋2 is

𝑓𝑋(𝑥) =𝑓1(𝑥)𝐹2(𝛼𝑥)

𝑃𝛼𝑋1 > 𝑋2, (1)

where 𝑃𝛼𝑋1 > 𝑋2 = ∫ [∫ 𝑓2(𝑥2)𝑑𝑥2𝛼𝑥1

0]𝑓1(𝑥1)𝑑𝑥1

∞

0= 𝐸𝑋1

(𝐹2(𝛼𝑥)). The Eq.(1) can be

interpreted as weighted distribution with weight 𝜔 = 𝐹2(𝛼𝑥). Described method has been

used to construct new skewed distributions from a given symmetric distribution, for example,

skew-t [2], skew-Cauchy [3], skew-Laplace [4] and skew-logistic [5].

The aim of this note is to generalize Azzalini’s method such that it is applicable in

case when 𝑋𝑖 𝑖 = 1,2 are dependent random variables. Dependence can be modelled using

copula function. In case of dependence of the random variables 𝑋𝑖 𝑖 = 1,2 the equation (1)

becomes

𝑓𝑋(𝑥) =𝑓1(𝑥) ∫ 𝑐(𝐹1(𝑥1), 𝐹2(𝑥2); 𝜃)

𝛼𝑥

−∞𝑓2(𝑥2)𝑑𝑥2

𝑃𝛼𝑋1 > 𝑋2,

where 𝑃𝛼𝑋1 > 𝑋2 =𝐸𝑋1[∫ 𝑐(𝐹1(𝑋1), 𝐹2(𝑋2); 𝜃)𝑓2(𝑥2)

𝛼𝑋1

−∞] 𝑑𝑥2 and 𝑐(∙) is the copula

function. Some statistical properties will be derived together with some particular cases.

Keywords. Weighted distributions, generalization of Azzalini’s method.

AMS 2010. 60E05, 62P10, 62G30, 62F10.

1 University of Montenegro, Faculty of Science and Mathematics, Podgorica, Montenegro, [email protected]


272

References

[1] Azzalini, A., A class of distributions which includes normal ones, Scand. J. Stat., 12, 171-

178, 1985.

[2] Gupta, A.K., Chang, F.C. and Haung, W.J., Some skew-symmetric models, Random Oper.

Stoch. Equ. 10, 133–140, 2002.

[3] Arnold, B.C., Beaver, R.J., The skew-Cauchy distribution, Statist. Probab. Lett. 49, 285–

290, 2000.

[4] Aryal, G., Nadarajah, S., On the skew-Laplace distribution, J. Inf. Optim. Sci. 26, 205–

217, 2005.

[5] Nadarajah, S., On the skew-logistic distribution, AStA Adv. Stat. Anal. 93, 187–203,

2009.


273

Transmuted Exponential Power Distribution and its Distributional Properties

Buğra SARAÇOĞLU1

Abstract. In this study, a new statistical distribution called “Transmuted Exponential

Power distribution” through the quadratic rank transmutation map studied by Shaw et al. [6] is

introduced. The various statistical properties of this distribution are derived. The method of

maximum likelihood estimation has been used to estimate the parameters of this distribution.

Keywords. Transmuted exponential power distribution, maximum likelihood

estimation, keyword three.

AMS 2010. 62F10, 62G30.

References

[1] Shaw, W. T., Buckley, I. R., The alchemy of probability distributions: beyond Gram-

Charlier expansions, and a skewkurtotic-normal distribution from a rank transmutation map,

arXiv preprint arXiv:0901.0434, 2009.

[2] Aryal, G. R., Tsokos, C. P., On the transmuted extreme value distribution with

application. Nonlinear Analysis: Theory, Methods & Applications, 71(12), 1401-1407, 2009.

[3] Mahmoud, M. R., & Mandouh, R. M. (2013). On the transmuted Fréchet

distribution. Journal of Applied Sciences Research, 9(10), 5553-5561, 2013.

[4] Al-Babtain, A., Fattah, A. A., Ahmed, A. H. N., & Merovci, F., The Kumaraswamy-

transmuted exponentiated modified Weibull distribution. Communications in Statistics-

Simulation and Computation, 46(5), 3812-3832, 2017.

1 Selcuk University, Konya, TURKEY, [email protected]


274

A New Bivariate Archimedean Copula Based on a Transformed Generator Function

Çiğdem TOPÇU GÜLÖKSÜZ1, Nuri ÇELİK2

Abstract. Copula functions are commonly used to model the relationships among

multivariate outcomes. A special class that is called Archimedean has many important

properties that make it very popular in applications. Archimedean copulas can be generated

by using generator functions. In literature, there are many examples of generator functions

and related Archimedean copulas. In this study, we apply a transformation to the generator

function of Gumbel copula that is one of well-known Archimedean copulas and obtain a new

generator function. Since different generator functions generate different copulas, by using

this function we construct a new bivariate Archimedean copula. Properties of these new

functions are examined. We consider some selected parameter values of Gumbel generator

function and estimate the parameter of the new copula function.

In the application part of the study, two real data set example are used to present that the new

copula function has a potential to model some dependence structures.

Keywords. Copula, Archimedean Copulas, Gumbel Copula, dependence

AMS 2010. 53A40, 20M15.

References

[1] Belgorodski, N. Selecting pair copula families for regular vines with application to the

multivariate analysis of European stock market indices, Master Thesis, Technische

Universität München, 2010.

[2] Genest, C., and Rivest, L.P., Statistical inference procedures for bivariate archimedean

copulas, American Statistician, 88(423),pp. 1034-1043, 1993.

[3] Gumbel, E.J., Bivariate exponential distributions, Journal of the American Statistical

Association., 55, pp. 698-707, 1960.

[4] Nelsen, R., An Introduction to Copulas, Springer, NewYork, 2006.

1 Bartin University, Bartin, TURKEY, [email protected] 2 Bartin University, Bartin, TURKEY, [email protected]


275

The Comparison of Population Means under Unequal Variances

Neriman AKDAM1, M. Fedai KAYA2 and Buğra SARAÇOĞLU3

Abstract. In this study, The usage of F test is not proper for testing the equality of

means of k normal distributed populations when the assumption of variances’ homogeneity is

violated. Accordingly, a test is proposed for testing the equality of means of k normal

distributed populatons under unequal variances. As regards this test, a critical values table is

constituted for certain significance levels. A simulation study based on The power values and

Type-I errors of the test is performed. In addition, The performances of this test are compared

with those of some tests proposed for comparison of population means under unequal

variances for different population sizes.

Keywords. Critical valuess, Power value, Simulation Study, Type-I error.

AMS 2010. 62F03, 62H10.

References

[1] David, H., Order Statistics. John Wiley, New York, 1970.

[2] Ekiz, M. and Gamgam, H., On The Comparison Of The Welch Test And The Single-Stage

Test: A Simulation Study ,Commun. Fac. Sci. Uni. Ank. Series, 51-61, 2007.

[3] Welch, B.L., On the comparison of several mean values: An alternative approach,

Biometrica, 38, 330-336, 1951.

[4] Chen, S. Y., & Chen, H. J. (1998). Single-stage analysis of variance under

heteroscedasticity. Communications in Statistics-Simulation and Computation, 27(3), 641-

666.

1 Selcuk University, Konya, TURKEY, [email protected]


276

A New Lifetime Distribution

Nuri CELİK1 and Cigdem TOPCU GULOKSUZ2

Abstract. Some probability distributions have been proposed to fit real life data with

both increasing and decreasing failure rates. In this study, we propose a new life time

distribution based on the generalization of exponential distribution. The distribution has both

increasing and decreasing failure rate functions. Additionally, we also define survival

function, the hazard function and the mean time to failure of this new distribution. Type II

censoring procedure is also considered for this distribution. Additionally, stress-strength

reliability and the maximum likelihood estimators (MLE) of the unknown parameters are

obtained. As an application, a real data set is used to show that the proposed distribution gives

best fit than the alternative ones.

Keywords. Exponential Distribution, Failure Rate, Type II Censoring.

AMS 2010. 62N01, 62N05.

References

[1] Gomez Y M, Bolfarine H, Gomez H W. A New Extension of Exponential Distribution.

Revista Colombiana de Estadistica 37: 25-34, 2014.

[2] Gupta R D, Kundu K. Generalized Exponential Distributions. Australian and New Zeland

Journal of Statistics 41, 2, 173-188, 1999.

[3] Kus C. A New Lifetime Distribution. Computational Statistics and Data Analysis 51, 4497-

4509, 2007.

[4] Raqab M Z, Kundu D. Comparison of different estimators of P(Y < X) for a scaled Burr

type X distribution. Communication in Statistics Simulation and Computation 34, 2, 465-483,

2005.

1 Bartin University, Bartin, TURKEY, [email protected] 2 Bartin University, Bartin, TURKEY, [email protected]


277

TOPOLOGY


278

A Topological Study of Harmony and Counterpoint in Music Using Quotient Orbifolds

Aditya Sivakumar1 and Dmitri Tymoczko2

Abstract: Western music is based on two concepts – harmony, which is the simultaneous

sounding of multiple notes (chords), and voice leading, which is the motion of these notes

(voices) through time. The incorporation of these two concepts is called counterpoint.

Geometry and topology are powerful tools for studying counterpoint in music. A musical

chord can be defined as an n-tuple in an n-dimensional non-Euclidean geometric quotient

space called an orbifold. The location and characteristics of a chord in the orbifold is defined

by symmetry as well as the consonance or dissonance of the chord. The voice leading from

one chord to another within the orbifold is defined by vectors in the space. In this study, a

new technique using quotient orbifolds has been developed to identify good sounding music

and the limits of playability as a guide for efficiently composing music. Over 300,000 chord

transitions were analyzed with a Python program and a random sample was graphed onto the

orbifold. Some extremely interesting geometric properties of voice leadings were revealed

visually for the first time and this innovative visual technique easily identified the ones that

violated counterpoint rules or were unplayable. In conclusion, the study describes the

development of a powerful new technique using orbifolds and assisted by computational

methods to efficiently compose new music, and this should prove extremely useful to

musicians and composers.

Keywords: Geometry, Topology, Voice Leading, Music

1 Beaverton High School, Beaverton Oregon USA. [email protected] 2 Princeton University, Princeton, New Jersey, USA.


279

New Functors from Fuzzy Normed Spaces Category

Deniz Pınar SUNAOĞLU 1 and Erdal GÜNER 2

Abstract. For each mathematical discipline we define at first objects and then

admissible maps for describing the objects. This procedure is formalized by the concept

‘category’. As well-known mathematical objects may be described by means of maps. There

is an analogous description of categories via so called functors. A functor is a morphism of

categories. Functors were first explicitly recognized in algebraic topology, where they arise

naturally when geometric properties are described by means of algebraic invariants. Category

theory and functors provide a tool by which many parallel techniques used in several branches

of mathematics can be linked and treated in a unified manner.

The aim of this study is to give some information about fuzzy normed spaces, then try

to find some new functors from fuzzy normed spaces category to anothers.

Keywords. Category, functor, fuzzy normed spaces.

AMS 2010. 18Axx, o3E72.

References

[1] Bag, T., Samanta, S.K.: A Comparative Study of Fuzzy Norms on a Linear Spaces. Fuzzy

Sets and Systems, 159, 673-684, 2008.

[2] Bag, T., Samanta, S.K.: Operator’s Fuzzy Norm and Some Properties. Fuzzy Information

and Engineering, 7, 151-164, 2015.

[3] Cheng, S.C., Mordeson, J.N.: Fuzzy Linear Operator and Fuzzy Normed Linear Spaces.

Bull. Calcutta Math. Soc., 86, 429-436, 1994.

[4] Nadaban,S.: Fuzzy Continuous Mappings in Fuzzy Normed Linear Spaces. İnternational

Journal Of Computers Communications and Control, ISSN 1841-9836, 10(6):834-843,

December, 2015.

[5] Preuss, G.: Theory of Topological Structures- An Approach to Categorical Topology.

D.Reidel Publishing Company, 1944.

[6] Zadeh, L. A. ,"Fuzzy Sets" , Inform. Contr., 8 : 338-353, 1965.

1 University of Ankara, Ankara, TURKEY, [email protected] 2 University of Ankara, Ankara, TURKEY, [email protected]


280

Soft Metric Spaces

Ebru Aydoğdu1, Abdülkadir Aygünoğlu2 and Halis Aygün3

Abstract. In this work, we firstly introduce the concept of soft metric and we give

relationships between soft metrics and fuzzy metrics by the sense of George, Veeramani [5].

Later we discuss the relationships between soft metrics and clasical metrics with examples.

Keywords. Soft set, Soft metric spaces

AMS 2010. 06D72, 03B52

References

[1] Atanassov, K. T., Intuitionistic fuzzy sets, Fuzzy sets and Systems 20.1, 87-96, 1986.

[2] Aygünoğlu, A., and Halis A., Some notes on soft topological spaces, Neural Computing

and Applications 21.1, 113-119, 2012.

[3] Çağman, N., and Enginoğlu S., Soft set theory and uni-int decision making, European

Journal of Operational Research 207.2, 848-855, 2010.

[4] Das S., Samanta S.K., Soft real sets,soft real numbers and their properties, J.Fuzzy Math.

20(3), 551-576,2012.

[5] George, A., P. Veeramani. On some results in fuzzy metric spaces. Fuzzy sets and

systems 64.3, 395-399, 1994.

[6] Maji, P.K., Biswas R., Roy A.R., Soft set theory, Computers & Mathematics with

Applications 45, 555-562, 2003.

[7] Maji P.K., Roy A.R, and Biswas R., An application of soft sets in a decision making

problem, Computers & Mathematics with Applications 44.8, 1077-1083, 2002.

[8] Shabir M., Naz M., On soft topological spaces., Comput Math Appl, 61,1786-1799, 2011.

1 Kocaeli University, Kocaeli, Turkey, [email protected] 2 Kocaeli University, Kocaeli, Turkey, [email protected] 3 Kocaeli University, Kocaeli, Turkey, [email protected]


281

A New Property between Compactness and Completeness in Generalized Metric Spaces

Emrah Evren KARA1 and Merve İLKHAN 2

Abstract. The main purpose of this presentation is to study on a new concept stronger

than completeness but weaker than compactness in a generalized metric space. For this aim,

we carry some basic concepts to generalized metric spaces. We define three new types of

Cauchy sequences belonging to a larger class than the class of all G-Cauchy sequences. Also,

we examine the relations between these new G-Cauchy sequences with corresponding ones in

the metric generated by G-metric. By defining three new types of G-completeness, we prove

some attracting theorems related to G-compactness, G-relative compactness, G-uniformly

locally compactness and some other concepts.

Keywords. Cauchy sequences, compactness, completeness.

AMS 2010. 54E35, 54E50, 54A20.

References

[1] Mustafa, Z. and Sims, B., A new approach to generalized metric spaces, J. Nonlinear

Convex Anal. 7(2), 289-297, 2006.

[2] Kaewcharoen, A., Common fixed point theorems for contractive mappings satisfying 𝛷-

maps in G-metric spaces, Banach J. Math. Anal. 6(1), 101-111, 2012.

[3] Beer, G., Between compactness and completeness, Topology Appl. 155(6), 503-514, 2008.

[4] Garrido, M. I. and Merono, A. S., New types of completeness in metric spaces, Ann. Acad.

Sci. Fenn. Math. 39, 733-758, 2014.

[5] Mustafa, Z., Khandagji, M. and Shatanawi, W., Fixed point results on complete G-metric

spaces, Studia Sci. Math. Hungar. 48(3), 304-319, 2011.

1 Düzce University, Düzce, TURKEY, [email protected] 2 Düzce University, Düzce, TURKEY, [email protected]


282

Extensions of T_0-quasi-metrics

Filiz YILDIZ1 and Hans-Peter KÜNZİ2

Abstract. Let ( , , )X m be a partially ordered metric space, that is, a metric space

(X,m) equipped with a partial order on X.

We say that a T_0-quasi-metric d on X is m-splitting provided that 1d d m .

Furthermore d is said to be ( , , )X m producing provided that d is m-splitting and the

specialization order of d is equal to .

It is known and easy to see that if ( , , )X m is a partially ordered metric space that is

produced by a T_0-quasi-metric and is a total order, then there exists exactly one producing

T_0-quasi-metric on X.

We first shall give an example that shows that a partially ordered metric space can be

uniquely produced by a T_0-quasi-metric although is not total.

Then we present solutions to the following two problems: Let ( , , )X m be a partially

ordered metric space and A a subset of X.

(1) Suppose that d is a T_0-quasi-metric on A which is | ( )m A A splitting . When

can d be extended to an m-splitting T_0-quasi-metric d on X?

(2) Suppose that d is a T_0-quasi-metric on A which is ( , | ( ), | ( ))A m A A A A

producing. When can d be extended to a T_0-quasi-metric d on X that produces ( , , )X m ?

References

[1] Gaba, Y.U. and Kunzi, H.-P.A., Splitting metrics by T_0-quasi-metrics, Topology

Applications 193, 84-96, 2015.

[2] Gaba, Y.U. and Kunzi, H.-P.A., Partially ordered metric spaces produced by T_0-quasi-

metrics, Topology Applications 202, 366-383, 2016.

1 Hacettepe University, Ankara, TURKEY, [email protected] 2 University of Cape Town, Rondebosch , SOUTH AFRICA, [email protected]


283

Some Categorical Approaches to the Relations between Inverse Systems of Bitopological

and Ditopological Spaces

Filiz YILDIZ1

Abstract. Followed by the constructing in [4] a detailed analysis of the theory of

inverse (projective) systems [2] and limits insofar as the subcategories of textures are

concerned, the relations between bitopological and ditopological inverse systems are

investigated and studied in a categorical context.

Especially, many interesting categorical [1] and functoral results on the

relationships between the inverse systems in a subcategory of Bitop consisting of the

bitopological spaces and the inverse systems in a corresponding subcategory of ifDitop

consisting of ditopological spaces are obtained in a natural way.

Finally, by introducing the suitable functor isomorphisms [3], we actually proved that

a natural transformation described between those particular subcategories is identity.

Keywords. Bitopology, Ditopology, Inverse System, Isomorphism Functor

AMS 2010. 46M40, 54B30, 18A22, 54E55

References

[1] Adámek J., Herrlich H, Strecker G. E., Abstract and Concrete Categories, John Wiley

& Sons, Inc., 1990.

[2] Eilenberg S, Steenrod N., Foundations of Algebraic Topology, Princeton University

Press, Princeton, New Jersey, 1952.

[3] Yıldız F., Connections between real compactifications in various categories, Quaestiones

Mathematicae, 38 (3) , 431-455, 2015.

[4] Yıldız F., Inverse Systems and Inverse Limits in the Category of Plain Textures, Topology

and Its Applications, (201), 217-234, 2016.

1 Hacettepe University, Ankara, TURKEY, [email protected]


284

Normal Product Adjacency for Simplicial Homology Groups of Digital Images

Gülseli BURAK1

Abstract: We study to compute a simplicial homology groups for Cartesian product of digital

images based on the normal product adjacency. We give some properties of the normal

product adjacency for finite for Cartesian product of digital images and give some examples

concerning homology groups of digital images.

Keywords: Digital topology, digital simplicial homology group, normal product adjacency

References

[1] Boxer, L., Generalized normal product adjacency in digital topology, arXiv.org,

arXiv:1608.03204v2, 2017.

[2] Karaca, I.,Boxer, L., Öztel, A., Topological invariants in digital images, Jour. Of

Mathematical Sciences: Advances and Applications, 11, No:2,109-140, 2011.

[3]Karaca, I., Demir, E., Simplicial homology groups of certain surfaces, Hacettepe Journal of

Mathematics and Statistic, 44(5), 1011-1022, 2015.



285

Bracket Polynomials of Torus Links as Fibonacci Polynomials

Kemal TAŞKÖPRÜ1, İsmet ALTINTAŞ2 and Merve BEYAZTAŞ3

Abstract. In this paper we work the bracket polynomial of (2,n)-torus link as a

Fibonacci polynomial. We show that the bracket polynomial of (2,n)-torus link provides a

recurrence relation as similar to Fibonacci polynomial and give its some fundamental

properties. We also prove important identities, which are similar to the Fibonacci identities,

for the bracket polynomial of (2,n)-torus link and prove Fibonacci-like identities of the Jones

polynomial of (2,n)-torus link as a result of the bracket polynomial. Finally, we observe that

the bracket polynomial of (2,n)-torus link and therefore its Jones polynomial can be derived

from its Alexander-Conway polynomial or classical Fibonacci polynomial.

Keywords. Bracket polynomial, torus link, Fibonacci polynomial, Fibonacci

identities, Jones polynomial.

AMS 2010. 57M25, 11B39, 11C08.

References

[1] Jones, V. F. R., A polynomial invariant for knots via von Neumann algebras, Bull. Amer.

Math. Soc., 12, 103-111, 1985.

[2] Kauffman, L. H., State models and the Jones polynomial, Topology, 26, 395-407, 1987.

[3] Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc., 318, 417-471,

1990.

[4] Koshy, T., Fibonacci and Lucas numbers with applications, Pure and Applied

Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley, USA, 2011.

[5] Nalli, A., Haukkanen, P., On generalized Fibonacci and Lucas polynomials, Chaos,

Solitons and Fractals, 42, 3179-3186, 2009.

[6] Taşköprü, K., Altıntaş, İ., HOMFLY polynomials of torus links as generalized Fibonacci

polynomials, Electron. J. Combin., 22, 4.8, 2015.

1 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected] 3 Sakarya University, Sakarya, TURKEY, [email protected]


286

Controllability of Affine Control Systems on Solvable Lie Groups

Memet KULE1

Abstract. In this paper we study controllability by taking in consideration the

eigenvalues of associated derivations. As soon as the state space is a solvable connected Lie

group, an affine control system is controllable if the induced invariant control system is

controllable and any eigenvalues of the derivations have zero real parts.

Keywords. Affine control systems; solvable Lie groups; controllability.

AMS 2010. 37N35, 22E25, 93C10.

References

[1] Ayala V, Tirao J., Linear control systems on Lie groups and Controllability, Proceedings

of Symposia in Pure Mathematics, 64, 47-64, 1999.

[2] Da Silva A.J., Controllability of linear systems on solvable Lie groups, SIAM J. Control

Opt., 54 No. 1, 372-390, 2016.

[3] Jouan, Ph., Equivalence of control systems with linear systems on Lie groups and

homogeneous spaces, ESAIM: Control Optim Calc Var. , 16, 956-973, 2010.

[4] Jurdjevic, V., Sallet, G., Controllability Properties of Affine Systems, SIAM J.Control

Opt., 22, 501-518, 1984.

[5] Kara, A., Kule, M., Controllability of Affine Control Systems on Carnot Groups, Int. J. of

Contemp. Math. Sci., 5, 2167-2172, 2010.

[6] Kara, A., San Martin. L., Controllability of Affine Control System for the Generalized

Heisenberg Lie Groups, Int. J. Pure Appl. Math., 29, 1-6, 2006.

[7] Kule, M., Controllability of Affine Control Systems on Lie Groups, Mediterr. J. Math.

3(2), 873-882, 2016.

1 1 Kilis 7 Aralık University, Kilis, TURKEY, [email protected]


287

[8] Kule, M., Controllability of affine control systems on graded Lie groups, Kuwait J. Sci.

43(1), 61-68, 2016.


288

Some Effective Results Related to the New Statistical Cauchy Sequence

Merve İLKHAN1 and Emrah Evren KARA2

Abstract. In this presentation, we introduce the statistical Bourbaki Cauchy sequence

as a new concept in a metric space. Whereas a Bourbaki Cauchy sequence or a statistical

Cauchy sequence is statistical Bourbaki Cauchy, we show with examples that the reverse

statements are false. Further, we give a new condition by using a statistical Bourbaki Cauchy

sequence and prove that this is equivalent to Bourbaki completeness. Moreover, we

characterize Bourbaki completeness and Bourbaki boundedness of a metric space by using

functions preserving these generalized sequences.

Keywords. Statistical Cauchy sequence, completeness, Bourbaki completeness,

Bourbaki boundedness.

AMS 2010. 54E35, 54E50, 54A20, 40A05, 11B05.

References

[1] Atsuji, M., Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8,

11-16, 1958.

[2] Beer, G., Metric spaces on which continuous functions are uniformly continuous and

Hausdorff distance, Proc. Amer. Math. Soc. 95, 653-658, 1985.

[3] Beer, G., More about metric spaces on which continuous functions are uniformly

continuous, Bull. Aust. Math. Soc. 33, 397-406, 1986.

[4] Garrido M. I. and Merono, A. S., New types of completeness in metric spaces, Ann. Acad.

Sci. Fenn. Math. 39, 733-758, 2014.

[5] Schoenberg, I. J., The integrability of certain functions and related summability methods,


1 Düzce University, Düzce, TURKEY, [email protected] 2 Düzce University, Düzce, TURKEY, [email protected]


289

Products of Hopfian Manifolds and Their Shape Fibrators' Properties

Violeta VASILEVSKA1

Abstract. Approximate fibrations were introduced and studied by Coram and Duvall

in the late 1970s [1, 2], as a generalization of the concept of Hurewicz fibrations. In the late

1980s, Daverman introduced the concept of codimension-k fibrator [3] and later the concept

of PL fibrator [4]. PL fibrators, by definition, are PL manifolds that provide detection of PL

approximate fibrations. Following this concept of Daverman, the presenter introduced a

slightly changed (‘special’) PL setting [5, 6]. In this talk, we present to what extent such

results can be obtained in the new setting. We introduce manifolds that can ‘detect’

approximate fibrations in the ‘special’ PL setting, called shape msimpl fibrators. In addition, the

shape msimpl fibrators properties of Hopfian manifolds and their products will be discussed.

Additionally, a particular type of Hopfian groups will be introduced and their properties

discussed. It will be shown how important these special groups are as fundamental groups of

the discussed shape msimpl fibrators.

Keywords. Approximate fibration, Hopfian manifolds, Hopfian groups.

AMS 2010. 57N15, 57M07.

References

[1] Coram, D. S., Duvall, P. F., Approximate fibrations, Rocky Mountain J. Math. 7, 275-288,

1970.

[2] Coram, D. S., Duvall, P.F., Approximate fibrations and movability conditions for a map,

Pacific J. Math. 72, 41-56, 1977.

[3] Daverman, R. J., Submanifold decomposition that induce approximate fibrations,

Topology Appl. 33, 173-184, 1989.

[4] Daverman, R. J., PL manifolds with manifolds fibers, J. London Math. Soc. (2) 45, 180-

192, 1992.

[5] Vasilevska, V., Homology n-spheres as codimension-(n+1) shape msimpl-fibrators,

Topology Appl. 155, 1140-1148, 2008.

1 Utah Valley University, Orem, UT, USA, [email protected]


290

[6] Vasilevska, V., Special manifolds and shape fibrators properties, Topology Appl. 153,

2765-2781, 2006.


291

Soft AB-Sets and Soft αAB-Sets in Soft Topological Spaces

Zehra Güzel ERGÜL1 and Naime TOZLU2

Abstract. The present study aims to give some new concepts in soft topological

spaces such as soft semi-regular sets, soft AB-sets and soft αAB-sets. We investigate many

basic properties of these concepts with the help of some counterexamples. We discuss their

relationships with different types of subsets of soft topological spaces. In addition, we

introduce soft AB-continuous and soft αAB-continuous functions and we obtain the new

decompositions of soft continuity. We introduce these concepts which are defined over an

initial universe with a fixed set of parameters.

Keywords. Soft set, soft topological space.

AMS 2010. 06D72, 54A40.

Acknowledgements: This work is supported by Ahi Evran University Scientific Research

Projects Coordination Unit. Project Number: FEF. A3.16.020.

References

[1] Akdağ, M., Özkan, A., Soft α-open sets and soft α-continuous functions, Abstr. Appl.

Anal., http://dx.doi.org/10.1155/2014/891341, 2014.

[2] Arockiarani, I., Arokialancy, A., Generalized soft gβ-closed sets and soft gsβ-closed sets

in soft topological spaces, International Journal of Mathematical Archive, 4, 2, 1-7, 2013.

[3] Aygünoğlu, A., Aygün, H., Some notes on soft topological spaces, Neural Computing and

Applications, 21, 1, 113-119, 2012.

[4] Chen, B., Soft semi-open sets and related properties in soft topological spaces, Appl.

Math. Inf. Sci., 7, 1, 287-294, 2013.

[5] Feng, F., Jun, Y. B., Zhao, X. Z., Soft semirings, Comput. Math. Appl., 56, 2621-2628,

2008.

1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ömer Halisdemir University, Niğde, TURKEY, [email protected]


292

[6] Kandil, A., Tantawy, O. A. E., El-Sheikh, S. A., Abd El-Latif, A. M., γ-operation and

decompositions of some forms of soft continuity in soft topological spaces, Ann. Fuzzy Math.

Inform., 7, 2, 181-196, 2014.

[7] Kharal, A., Ahmad, B., Mappings on soft classes, New Math. Nat. Comput., 7, 3, 471-481,

2011.

[8] Mahanta, J., Das, P. K., On soft topological space via semi-open and semi-closed soft sets,

Cornell University Library, 2012.

[9] Maji, P. K., Biswas, R., Roy, A. R., Soft set theory, Comput. Math. Appl., 45, 4-5, 555-

562, 2003.

[10] Molodtsov, D., Soft set theory-first results, Comput. Math. Appl., 37, 4-5, 19-31, 1999.

[11] Shabir, M., Naz, M., On soft topological spaces, Comput. Math. Appl., 61, 7, 1786-1799,

2011.

[12] Tozlu, N., Yüksel, Ş., Soft A-sets and soft B-sets in soft topological spaces, Accepted by

Mathematical Sciences and Applications E-Notes, 2017.

[13] Yüksel, Ş., Tozlu, N., Güzel Ergül, Z., Soft regular generalized closed sets in soft

topological spaces, International Journal of Mathematical Analysis, 8, 8, 355-367, 2014.

[14] Zorlutuna, I., Akdağ, M., Min, W. K., Atmaca, S., Remarks on soft topological spaces,

Ann. Fuzzy Math. Inform., 3, 2, 171-185, 2012.


293

THE OTHER

AREAS


294

Approximation by Certain Linear Positive Operators of Three Variables

Afşin Kürşat GAZANFER1

Abstract. We introduce positive linear operators which are combined with the

Chlodowsky and Szasz type operators and study some approximation properties of these

operators in the space of continuous functions of three variables on a compact set. The

convergence rate of these operators are obtained by means of the modulus of continuity. And

we also obtain weighted approximation properties for these positive linear operators in a

weighted space of functions of three variables and find the convergence rate for these

operators by using the weighted modulus of continuity.

Keywords. Chlodowsky-Szasz operators, linear positive operators, weighted modulus

of continuity.

References

[1] Gazanfer A. K., Büyükyazıcı İ., Approximation by certain linear positive operators of two

variables, Abstract and Applied Analysis, DOI 10.1155, 2014.

[2] Agrawal P. N., İspir N., Degree of approximation for bivariate Chlodowsky-Szasz-

Charlier type operators, Results in Mathematics, DOI 10.1007/s00025-015-0495-6, 2015.

[3] İspir N., Quantitative estimates for GBS operators of Chlodowsky-Szasz type, Faculty of

Sciences and Mathematics, University of Nis, Serbia, 31:5, 1175-1184, 2017.

[4] İspir N., Büyükyazıcı, İ., Quantitative estimates for a certain bivariate Chlodowsky-Szasz-

Kantorovich type operators, Mathematical Communications, 21(1):31-44, 2016.

[5] İbikli E., On approximation for functions of two variables on a triangular domain by

Bernstein-Chlodowsky polynomials, Rocky Mountain Journal of Mathematics, 35(5):1523-

1531, 2005.

1 Bülent Ecevit University, Zonguldak, TURKEY, [email protected]


295

Extension of Silver-Meal Algorithm with Variable Demand and Delivery Time to Cope

with Bullwhip Effect in Multi-Echelon Supply Chains

Halil İbrahim CEBECİ1 and Doğan ÜNAL2

Abstract. In multi-echelon supply chains, inventory management is very important to

avoid the adverse effect known by the bullwhip effect. Therefore, dynamic stock control

policies are preferred in the ordering process. The Dynamic Inventory Control Policy (DSKP)

approach based on demand forecasting, a proactive approach, is a dynamic inventory control

method that requires recalculating (r, Q) continuous replenishment inventory control

parameters at each forecast interval. The Silver-Meal algorithm, which is generally used in

Inventory quantity calculation, works under constant delivery time and demand conditions. In

our work, our purpose to design a stochastic extension of the silver-meal algorithm for use in

dynamic inventory control policies in multi-echelon supply chains. A simulation study has

been carried out for the validity of the relevant model.

Keywords. Bullwhip Effect, Dynamic Inventory Control Policy, Silver-Meal

Alghoritm.

AMS 2010. 68W01,60G15.

References

[1] Babai, M.Z. and Dallery, Y., “Dynamic versus static control policies in single stage

production-inventory systems”, International Journal of Production Research, 47(2), pp. 415-

433, 2009.

[2] Babai, M.Z. and Dallery, Y., “Inventory management: forecast based approach vs.

standard approach”, International Conference on Industrial Engineering and Systems

Management, pp. 57-67, 2005.

[3] Bregman, R.L. and Silver, E.A., “A Modification of silver meal heuristic to handle MRP

purchase discount situations”, Journal of Operational Research Society, 44(7), pp. 717-723,

1993.

[4] Burbidge, J.L., “Period batch control (PBC) with GT – the way forward from MRP”,

BPCIS Annual Conference, 1991.

[5] Forrester J., “Industrial Dynamics”, MIT Press, 1961.

[6] Houlihan J.B., “International supply chain management”, International Journal of

Physical Distribution and Materials Management, 17(2), pp.51-66, 198.

[7] Lee, H.L., Padmanabhan, P. and Whang, S., “Information distortion in a s supply chain:

the bullwhip effect”, Management Science, 43(4), 543-558, 1997a.

[8] Lee, H.L., Padmanabhan, P. and Whang, S., “The Bullwhip effect in supply chains”, Sloan

Management Review, 38(3), pp. 93-102, 1997b.



296

[9] Omar, M. and Smith, D.K., “An optimal batch size for a production system under linearly

increasing time-varying demand process”, Computers and Industrial Engineering, 42(1), pp.

35-42, 2002.

[10] Silver, E., Pyke, F.D. and Peterson, R., “Inventory management and production planning

and scheduling”, John Wiley & Sons, New York, 1998.

[11] Şenyiğit, E., “New heuristics to stochastic dynamic lot sizing problem”, Gazi University

Journal of Science, 22(2), pp. 97-106, 2009.


297

Consumer and Producer Surpluses of the Quadratic Demand and Supply Functions by

Using Trapezoidal Fuzzy Numbers and Signed Distance

İsmail ÖZCAN1 and Salih AYTAR2

Abstract. In the present study, we calculate the consumer surplus of the quadratic

demand function 𝑃𝐷(𝑥) = − 𝑥 − 𝑥2 and the producer surplus of the quadratic supply

function 𝑃𝑆(𝑥) = + 𝑥 + 𝑥2, where 𝑥 is a crisp quantity and , , , , and are

trapezoidal fuzzy numbers.

Keywords. Trapezoidal Fuzzy Number, Signed Distance Defuzzification Method,

Consumer Surplus, Producer Surplus.

AMS 2010. 03E72, 91B42, 91B02

References

[1] Wu K., Consumer surplus and producer surplus in fuzzy sense, Fuzzy Sets and Systems

103, 405-419, 1999.

[2] Kaufmann, A., Gupta, M.M, Introduction to Fuzzy Arithmetic: Theory and Application,

Van Nostrand Reinhold, New York, 1992.



298

On Creating Tessellations with Using Transformational Geometry

Kübra ÖZLÜ DEĞER1 and Ali Hikmet DEĞER2

Abstract. Translation, rotation and reflection are the bases of transformational

geometry. Using these transformations we can create beautiful tessellating shapes and cover

the plane with them. These two-dimensional patterns is decorative works in art and design.

Maurits Cornelis Escher (1898-1972) was a famous graphic artist using these types of

tessellations in his works.

In the present study we use transformational geometry to design some decorative

patterns of application to different surfaces. We also offer suggestions on the use of

transformational geometry for the mathematical education of interior design students.

Keywords. Transformational geometry, Tessellation, Decorative patterns.

AMS 2010. 97B10, 97D10, 97D40, 97G40, 97G50.

References

[1] Deger, K.O., Deger, A.H., An application of mathematical tessellation method in interior

designing, Procedia-Social and Behavioral Sciences, 51, 249-256, 2012.

[2] Hilden, H.M., Montesinos, J.M., Tejada, D.M., Toro, M.M., Artifacts for stamping

symmetric designs, The American Mathematical Monthly, 118, 4, 327-343, 2015.

[3] Feijs, L., Bartneck, C., Teaching geometric principles to design students, Digital Culture

& Education, 1,2, 104-115, 2009.

[4] Toth, L.F., Tessellation of the Plane with Convex Polygons Having a Constant Number of

Neighbors, The American Mathematical Monthly, 82, 3, 273–276, 1975.

[5] Dobson, J., Gage, J., Using art to teach maths, using maths to create art, Bridges

Donostia: Mathematics, Music, Art, Architecture, Culture, 445-452, 2007.

[6] Hart, G., Heathfield, E., Making math visible, Proceedings of Bridges 2017: Mathematics,

Art, Music, Architecture, Education, Culture, 63-70, 2017.



299

Relationship between Electricity Consumption and Economic Growth in some

Developing Countries: MS-VECM Analysis

Melike BİLDİRİCİ1 , Fazıl KAYIKÇI2

Abstract. This study aims at analyzing the relationship between electricity

consumption and economic growth in Some Developing Countries with variety of

econometric techniques: Markov Switching – Vector Error Correction Model. Empirical

results reveal that electricity consumption and economic growth are cointegrated for these

countries. Furthermore, there is a causality between electricity consumption and economic

growth such that policymakers in these countries should consider this reality when designing

energy and growth policies.

Keywords. Electricity Consumption, Economic Growth, MS-VECM

References

[1] Lee, C.C., Chang, C.P., Energy consumption and GDP revisited: a panel analysis of

developed and developing countries, Energy Economics, 29, 1206-1223, 2007.

[2] Levin, A., Lin, C.F., Chu, C.S.J., Unit Root Tests in Panel Data: Asymptotic and Finite-

Sample properties, Journal of Econometrics, 108, 1-24, 2002.

[3] Pedroni, P., Critical Values for Cointegration Tests in Heterogeneous Panels with

Multiple Regressors, Oxford Bulletin of Economics and Statistics, 61, 653-670, 1999.

[4] Pesaran, M.H., Shin, Y., Smith, R.J., Pooled mean group estimation of dynamic

heterogeneous panels, Journal of the American Statistical Association, 94, 621-634, 1999.

1 Yıldız Technical University, İstanbul, TURKEY, [email protected] 2 Yıldız Technical University, İstanbul, TURKEY, [email protected]

http://ideas.repec.org/p/oxf/wpaper/9958.html





300

Test of validity of Baltic Dry index with MS-ARMA and SET-ARMA models

Melike BİLDİRİCİ1 , Fazıl KAYIKÇI2 and Işıl Şahin ONAT3

Abstract. The Baltic Dry Index has become one of the foremost indicators on the cost

of shipping and an important barometer on the volume of worldwide trade and manufacturing

activity since its establishment. Global factors also play important role in supply and demand

of BDI index. Average of price of 23 different shipping routes around the world compiles

daily to form the Baltic Dry Index. Economic indicators such as unemployment rate, inflation

and oil prices that can be manipulated or influenced by governments and speculators,

however, Baltic Dry Index is difficult to manipulate because it is driven by clear forces of

supply and demand. In this paper, some econometic models such as Markov Switching Auto

Regressive Moving Average and SET- Auto Regressive Moving Average models were used

to analyze the validity of the BDI index for some developed and developing countries in the

world.

Keywords. BDI index, MS-ARMA, SET-ARMA

References

[1] Clements, M.C., Krolzig, H.-M., Modelling Business CycleFeatures Using

SwitchingRegimeModels, ," Economics Series WorkingPapers 9958, 1-17, 2002.

[2] Psaradakis, Z., Morten, O.R. and Sola M., Markov-Switching causality and the Money-

Output Relationship, Journal of Applied Econometrics, 665-83, 2005.

[3] Geman, H. and Smith, W Shipping Markets and Freight Rates: An Analysis of the Baltic

Dry Index, The Journal of Alternative Investments, 1, 98-109, 2012.

[4] Hamilton, J.D., A new approach to the economic analysis of nonstationary time series and

the business cycle, Econometrica, 57, 357-384, 1989.

1 Yıldız Technical University, İstanbul, TURKEY, [email protected] 2 Yıldız Technical University, İstanbul, TURKEY, [email protected] 3 Yıldız Technical University, İstanbul, TURKEY, [email protected]



http://ideas.repec.org/s/oxf/wpaper.html





301

Chaotic Examination of Turkish Financial Market

Melike E. BİLDİRİCİ1 , Fulya ÖZAKSOY 2 and Bahri SONÜSTÜN3

Abstract. Although chaos theory aspired great interest in physics by Lorenz attractor

in 1960s, its application to many branches of social sciences such as economics, mathematics,

psychology, behavioral sciences, neurosciences dates back to 1980s. In the regards of this

‘new economy’, ‘new concepts’ such as complexity, determinism, quantum mechanics,

relativity, multiple equilibrium, complexity, (continuously) instability, nonlinearity,

heterogeneous agents, irregularity were widely questioned in economics. It is noticed that

linear models are insufficient for analyzing unpredictable, irregular and noncyclical

oscillations of economies, and for predicting bubbles, financial crisis and business cycles in

financial markets. Therefore, economists gave great consequence to use appropriate tools for

modelling non-linear dynamical structures and chaotic behaviors of economies especially in

macro and financial economy. In this paper, we aim to model chaotic structure of exchange

rates (USD-TL and EUR-TL) in Turkish financial market. To determine non-linear pattern of

the selected financial time series, daily returns of the exchange rates were tested by BDS

during the period from January 01, 2002 to May 11, 2017 which covers after the era of 2001

financial crisis. After specifying the non-linear structure of the selected financial time series,

it was aimed to examine the chaotic characteristic in Turkish financial market for the selected

time period by Lyapunov Exponents. The findings verify the existence of the chaotic structure

of the exchange rate returns in the analyzed time period.

Keywords: Chaos, Non-linearity, BDS test, Exchange rate returns, Lyapunov

Exponent

1 Yıldız Technical University, İstanbul, TURKEY, [email protected] 2 Dogus University, İstanbul, TURKEY, [email protected] 3 Yıldız Technical University, İstanbul, TURKEY, [email protected]



302

Chaotic Structure of Oil Price

Melike E. BİLDİRİCİ1 , Fulya ÖZAKSOY 2

Abstract. The fluctuations in oil prices are very complicated and therefore, it is unable

to predict the dynamics in economies. However, there is a main constraints of linear economic

models is that they are not sufficient for modelling price movements of complex patterns of

economies such as oil prices. Thus, in recent years, economists attached great attention to

non-linear structure of oil prices. For analyzing this relationship, GARCH models are mostly

used in the papers. Distinctively from other papers, in this paper, we aim to analyze chaotic

pattern of oil prices. Thus, firstly to determine chaotic behavior of oil prices, BDS model was

used. Following, this structure was modelled by Lyapunov Exponents for the selected time

period and forecasting achievement of the models were compared with each others. The

findings confirm the existence of the non-linear and chaotic framework of oil prices.

Keywords: Oil price, BDS test, Lyapunov Exponent, Chaos, Non-linearity

1 Yıldız Technical University, İstanbul, TURKEY, [email protected] 2 Dogus University, İstanbul, TURKEY, [email protected]



303

Optimization of MANET Routing Tables with Plant Propagation Algorithm

Mevlüt ERSOY1 and Tuncay YİĞİT2

Abstract. Mobile Ad-hoc Network (MANET) is a network in which many mobile

nodes communicate with each other without any central device [1]. In these networks, the

communication between the source and the target node is achieved by routing through the

nodes in the network. The routing process of the packet sent from the source node to the

destination node is provided by the request of the nodes in the network or by the tables

maintained in their memory [2]. In MANET, routing roots can change dynamically due to

newly added nodes, battery life of the nodes, etc. Finding the optimal path in terms of the

energy efficiency and delay time of the package sent from the source to the destination is an

important and complex problem. For this reason, natural - inspired heuristic methods can be

used in MANET routing problem. Especially, there are many studies with GA to solve the

MANET routing problem [3] [4]. In recent years, many problems have been solved with the

Plant Propagation Algorithm (PPA) [5] [6]. In this work, we used PPA for finding the optimal

path of the package sent from the source to the destination. The algorithm has been

implemented by taking battery life of the nodes and the new nodes added to the network into

account. In this study, the performance of the routing tables formed with PPA and GA has

been analyzed in terms of delay times and energy costs. The results have been obtained from a

MANET consisting of 10 nodes. It has been observed that PPA is better in terms of energy

costs.

Keywords. MANET, routing optimization, naturel inspired algorithms

References

[1] Bayılmış C., Erturk İ., Çeken C., Bandırmalı N. DSR ve AODV MANET Yönlendirme

Protokollerinin Başarım Değerlendirmesi, Elektrik Mühendisleri Odası, 2–5, 2005.

[2] Doğru İA., Şimşek M., Akcayol MA. Hareketli Ad-Hoc Ağlarda Bir Hareketlilik Yönetimi

Protokolü, Politeknik Dergisi, 11, 4, 313–318, 2008.



304

[3] Lu T., Zhu J. Genetic algorithm for energy-efficient QoS multicast routing, IEEE

Communications Letters, 17, 1, 31–34, 2013.

[4] Bari A., Wazed S., Jaekel A., Bandyopadhyay S. A genetic algorithm based approach for

energy efficient routing in two-tiered sensor networks, Ad Hoc Networks, 7, 4, 665–676,

2009.

[5] Selamoglu B.I., Salhi A., Alsoufi G. Solving Container Terminal Scheduling Problems

with the Plant Propagation Algorithm, 5th Internatıonal Eurasıan Conference On

Mathematıcal Scıences And Applıcatıons, Begrad,Serbia, 82–83, 2016.

[6] Selamoğlu Bİ., Salhi A. The Plant Propagation Algorithm for Discrete Optimisation: The

Case of the Travelling Salesman Problem, Nature-Inspired Computation in Engineering,

Cham, 43–61, 2016.


305

The Integrability and the Zero-Hopf Bifurcation of the Three-Dimensional Lotka-

Volterra Systems

Rizgar H. SALIH1

Abstract: This paper is devoted to study the zero-Hopf bifurcation of the three-

dimensional Lotka-Volterra systems. The explicit conditions for the existence of two first

integrals for the system and a line of singularities with zero eigenvalue are given. We

characteristic the parameters for which a zero-Hopf equilibrium point takes place at any

points on the line. We prove that there are three 3-parameter families exhibiting such

equilibria. The averaging theory of the first order is also applied but any information about the

possible periodic orbits bifurcating from the zero-Hopf equilibria is not provided by this

theorem.

1 University of Raparin-College of Science, Kurdistan Region-IRAQ. [email protected]


306

POSTERS


307

Estimates of Initial Coefficients of a New Analytic and Bi-Univalent Function Class

Defined by Integral Operator Involving Polylogarithm Function

Arzu AKGÜL1

Abstract. In this study, the author aims at introducing a new subclass of the function

class of bi-univalent functions defined by integral operator involving polylogarithm function.

Furthermore, it is established that bounds for the coefficients for this subclass and several

related classes are also considered and connections to earlier known results are made.

Keywords: Bi-univalent functions, bi-starlike functions, coefficient estimates,

polylogarithm function, integral operator.

Mathematics Subject Classification: 2010: 30C45, 30C50.

References

[1] Akgul, A., Identi.cation of Taylor-Maclaurin coefficients for generalizes subclass of bi-

univalent functions, Sahand Communications in Mathematical Analysis, 2017 (Accepted).

[2] Altınkaya, Ş., Yalçın, S., Second Hankel determinant for a general subclass of bi-

univalent functions, TWMS J.Pure Appl. Math. V.7, N.1, 98-104, 2016.

[3] Altınkaya, Ş., Yalçın, S., Coefficient Estimates for Two New Subclasses of Bi-univalent

Functions, Acta Universitatis Apulensis, 43, 53-63, 2015.

[4] Bateman, H., Higher transendental functions, vol. 1 of Edited by: A. Erdely, W. Mangnus,

F. Oberhettinger, F. G. Tricomi, Mc Graw-Hill, New York, NY, USA, 1953

[5] Brannan, D.A., Taha, T.S., On some classes of bi-univalent functions, in: S.M. Mazhar, A.

Hamoui, N.S. Faour (Eds.), Math. Anal. and Appl.,Kuwait; February 18.21, 1985, in: KFAS

Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp.

53.60. see also Studia Univ. Babe¸s-Bolyai Math. 31 (2)70.77, 1986.



308

[6] Brannan D. A. and Clunie, J. G., Aspects of comtemporary complex analysis, (Proceedings

of the NATO Advanced Study Instute Held at University of Durham: July 1-20, 1979). New

York: Academic Press, 1980.

[7] Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften,

Springer, New York, NY, USA, 259, 1983.

[8] Frasin, B. A. and Aouf, M. K., New subclasses of bi-univalent functions, Applied

Mathematics Letters, 24, 1569-1573, 2011.

[9] Hamidi S. G. and Jahangiri, J. M., Faber polynomial coefficient estimates for analytic bi-

close-to-convex functions, C. R. Acad. Sci. Paris, Ser.I 352 (1), 17.20, 2014.

[10] Jahangiri, J. M. and Hamidi, S. G., Coefficient estimates for certain classes of bi-

univalent functions, Int. J. Math. Math. Sci., ArticleID 190560, 4 pp, 2013.

[11] Lewin, M., On a coe¢ cient problem for bi-univalent functions, Proceeding of the

American Mathematical Society, 18, 63-68, 1967.


second coe¢ cient of a univalent function in jzj < 1; Archive for Rational Mechanics and

Analysis, 32, 100-112, 1969.

[13] AL-Shaqsi, K., Strong differential subordinations obtained with new integral operator

defined by polylogarithm function, International Journal of Mathematics and Mathematical

Sciences, Vol2014,Art ID 260198, 6 pages.

[14] Srivastava, H. M., Mishra, A. K. and Gochhayat, P., Certain subclasses of analytic and

bi-univalent functions, Applied Mathematics Letters, 23, 10, 1188-1192, 2010.

[15] Xu, Q. H., Gui, Y. C., and Srivastava, H. M., Coefficient estimates for a certain subclass

of analytic and bi-univalent functions, Applied Mathematics Letters, 25, 990-994, 2012.

[16] Srivastava, H. M. and Attiya, A. A., An integral operator Associated with Hurwitz-

Lerch Zeta function and differential subordination, Integral Transforms and Special functions,

18, 3,207-216, 2007.


309

[18] Xu, Q. H., Xiao, H. G. and Srivastava, H. M., A certain general subclass of analytic and

bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput.,218

2012.

[19] Srivastava, H. M., Joshi, S. B., Joshi, S. S., Pawar, H., Coefficient estimates for certain

subclasses of meromorphically bi-univalent functions, Palest. J. Math., 5 (Special Issue: 1),

250-258, 2016.

[20] Taha, T. S., Topics in univalent function theory, Ph.D. Thesis, University of London,

1981.


310

Finding Initial Coefficients for a Class of Bi-Univalent Functions Given by q-Derivative

Operators

Arzu AKGÜL1

Abstract. In the present study, the author _nds coincident bounds for functions in a

subclasses of analytic and bi-univalent functions introduced by using q-derivative operator.

Some corollaries and consequences of the main results are also considered. The results are

presented in this paper generalize the recent work of Srivastava et all. [21].

Keywords: Analytic and univalent functions; bi-univalent functions; bi-starlike and

bi-convex functions; coefficient bounds; q-derivative operator.

References

[1] Ali, R. M., Lee, S. K., Ravichandran, V. and Supramanian, S., Coefficient estimates for bi-

univalent Ma-Minda starlike and convex functions, App. Math. Lett., 25(3), 344-351, 2012.

[2] Altınkaya, S. and Yalçın, S., Coefficient bounds for a subclass of bi-univalent functions,

TWMS J. Pure Appl. Math., 6(2), 180-185, 2015.

[3] Aral,A.,Gupta, V. and Agarwal, R. P., Applications of q-calculus in operator theory,

Springer, New York, USA, 2013.

[4] Brannan, D. A., Taha, T. S., On some classes of bi-univalent functions, Stad. Univ. Babes-

Bolyai, Math. 31(2), 70-77, 1986.

[5] Caglar, M., Orhan, H., Yagmur, N., Coe_cient bounds for new subclass of bi-univalent

functions, Filomat, 27, 1165-1171, 2013.

[6] Crisan,O., Coefficient estimates of certain subclass of bi-univalent functions, Gen. Math.

Notes, 16(2), 93-102, 2013.

[7] Duren, P. L., Grundlehren der Mathematischen Wissenchaften, Springer, New York,NY,

USA, 1983.



311

[8] Frasin, B. A. and Aouf, M.K., New subclass of bi-univalent functions, Appl. Math.

Lett.,24, 1569-1573, 2011.

[9] Jackson, F. H., On q-functions and a certain di_erence operator, Trans. R. Soc. Edinb.,46,

253-281, 1908.

[10] Lewin, M., On a coe_cient problem of bi-univalent functions, Proc. Amer. Math. Soc.,

18, 63-68, 1967.

[11] Li, X. F. and Wang, A. P., Two new subclasses of bi-univalent functions, International

Mathematical Forum, 7, 1495-1504, 2012.

[12] Magesh, N. and Yamini, J., Coe_cient bounds for a certain subclass of bi-univalent

functions, International Mathematical Forum 8(22), 1337-1344, 2013.


second coefficient of a univalent function in mid z j< 1 , Archive for Rational Mechanics and

Analysis,32, 100-112, 1969.

[14] Pommerenke, C. H., Univalent functions, Vandenhoeck and Rupercht, Göttingen, 1975.

[15] Ponnusamy, S., Inclusion theorems for convolution product of second order

polylogariyhms and functions with the derivative in a half plane, Rocky Montain J. Math.,

28(2), 695-733, 1998.

[16] Ponnusamy, S. and Sabapathy, S., Polylogarithms in the theory of univalent functions,

Result in Mathematics, 30,136-150, 1996.

[17] Porwal, S. and Darus, M., On a new subclass of bi-univalent functions, J. Egypt. Math.

Soc.,21(13),190-193, 2013

[18] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49,109-

115, 1975.

[19] Sâlâgean, G. S., Subclasses of univalent functions, Lecture Notes in Math., Springer

Verlag, 1013,362-372, 1983.

[20] Al, K. Shaqsi and Darus, M., An oparator de_ned by convolution involving the

polylogarithms functions, Journal of Mathematics and Statics, 4(1), 46-50, 2008.


312

[21] Srivastava, H. M., Mishra, A. K. and Gochhayat, P., Certain subclass of analytic and bi-

univalent functions, Appl. Math. Lett., 23,1188- 1192, 2010.

[22] Srutha Koerthi, B. and Raja, B., Coeffcient inequality for certain new subclass of analytic

bi-univalnt functions, Theoratical Mathematics and Applications, 3(1), 1-10, 2013.

[23] Xu, Q. H., Gui, Y. C. and Srivastava, H. M., Coe_cient estimates for a certain subclass

of analytic and bi-univalent functions, Appl. Math. Lett., 23,990-994, 2012.

[24] Xu, Q. H., Xiao, H. G. and Srivastava, H. M., A certain general subclass of analytic and

bi-univalent functions and associated coe_cient estimate problems, Appl. Math. Comput.,218

(2012), 11461-11465.See also Studia Univ. BabeÅ- Bolyai Math. 31(2), 70-77, 1986.

[25] Xu, Q. H. Srivastava, H. M. and Li, Z. A certain subclass of analytic and close-to-convex

functions, Appl. Math. Lett.,24, 396-401, 2011.


313

Dual-Hyperbolic Fibonacci and Lucas Numbers

Arzu CİHAN1, Ayşe Zeynep AZAK2 and Mehmet Ali GÜNGÖR3

Abstract: In this study, we initially defined the dual hyperbolic Fibonacci and dual

hyperbolic Lucas numbers. Then, the fundamental identities were proved for these numbers.

Additionally, we gave the identities with respect to negadual hyperbolic Fibonacci and

negadual hyperbolic Lucas numbers. At last, Binet formulas, D’cogne’s, Catalan and Cassini

identities were obtained for dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers.

Keywords: Dual hyperbolic numbers, dual hyperbolic Fibonacci numbers, dual

hyperbolic Lucas numbers

Referrences

[1] Alfsmann, D., On families of 2N dimensional Hypercomplex Algebras suitable for digital

signal processing. In: Proceedings of EURASIP 14th European Signal Processing

Conference (EUSIPCO 2006), Florence, Italy 2006.

[2] Akyiğit M., Kösal H.H., Tosun M., Split Fibonacci Quaternions, Adv. Appl. Clifford

Algebras, 23, 535-545, 2013.

[3] Catoni, F., Cannata, R., Catoni, V., Zampetti, P., Hyperbolic trigonometry in two-

dimensional space-time geometry. N. Cim. B 118, 475–491, 2003.

[4] Clifford W.K., Preliminary Sketch of Biquaternions, Proc. London Mathematical Society,

4, 64, 381-395, 1873.

[5] Dunlap R.A., The Golden Ratio and Fibonacci Numbers, World Scientific, 1997.

[6] Fjelstad, P. Extending special relativity via the perplex numbers. Am. J. Phys. 54(5), 416–

422 1986.

[7] Halıcı S., On Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 22, 321-327, 2012.

1 Sakarya University, Sakarya, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected] 3 Sakarya University, Sakarya, TURKEY, [email protected]


314

[8] Halıcı S., On Complex Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 23, 105-

112, 2013.

[9] Horadam A. F., Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math.

Monthly, 70, 289-291, 1963.

[10] Iyer M.R., Some Results on Fibonacci Quaternions, The Fibonacci Quarterly, 7, 2, 201-

210, 1969.

[11] Kantor I. L., Solodownikow, V. N. , Hyperkomplexe Zahlen, Teubner, Leipzig 1978.

[12] Knuth D., Negafibonacci Numbers and Hyperbolic Plane, Annual Meeting of the Math.

Association of America, 2013.

[13] Koshy T., Fibonacci and Lucas Numbers with Applications, A Wiley-Intersience

Publication, USA, 2001.

[14] Majernik V., Multicompenent Number Systems, Acta Physica Polonica A, 90, 491-498,

1996.

[15] Nurkan S.K., Güven I.A., A Note on Bicomplex Fibonacci and Lucas Numbers,

https://arxiv.org/abs/1508.03972v1, 2015.

[16] Swamy M.N., On Generalized Fibonacci Quaternions, The Fibonacci Quarterly, 5, 547-

550, 1973.

[17] Vajda S., Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Books

Math. Appl., Ellis Horwood Ltd./Halsted Press; Chichester/New York etc., 1989.

[18] Verner E., Hoggatt Jr., Fibonacci and Lucas Numbers, The Fibonacci Association, 1969.

[19] Yüce S., Aydın F.T., A New Aspect of Dual Fibonacci Quaternions, Adv. Appl. Clifford

Algebras, 26, 873-884, 2016.

[20] Güngör M.A., Azak A.Z. Investigation of Dual-Complex Fibonacci,Dual-Complex Lucas

Numbers and Their Properties, submitted.

https://arxiv.org/abs/1508.03972v1


315

Toeplitz Operators with Symbols in Some Function Spaces

Ayşe SANDIKÇI 1

Abstract. Toeplitz operators are introduced as a class of pseudo differential operators

which depend on a symbol and two different window functions. We investigate new

conditions for the boundedness properties on Lorentz type modulation spaces using symbols

in some function spaces. Some key references are given below.

Keywords. Modulation space, Lorentz type modulation space, Toeplitz operator.

AMS 2010. 42B10, 47B38.

References

[1] Cordero E. and Tabacco A., Localization Operators via Time-Frequency Analysis, In:

Ashino R., Boggiatto P., Wong M.W. (Eds) Advances in Pseudo-Differential Operators.

Operator Theory: Advances and Applications, vol 155. Birkhäuser, Basel, 2004.

[2] Cordero, E. and Gröchenig, K., Time-frequency analysis of localization operators, J.

Funct. Anal., 205(1) (2003), 107-131.

[3] Gröchenig,K., Foundation of Time-Frequency Analysis. Birkhäuser, Boston 2001, ISBN

0-8176-4022-3.

[4] O'Neil, R., Integral Transforms and Tensor Products on Orlicz Spaces and L(p,q) spaces,

J. d'Analyse Math., 21, 1-276, 1968.

[5] Sandıkçı, A., On Lorentz mixed normed modulation spaces, J. Pseudo-Differ. Oper. Appl.,

3 (2012), 263-281.

[6] Sandıkçı, A., Boundedness of localization operators on Lorentz mixed normed modulation

spaces, Journal of Inequalities and Applications, 2014, 2014: 430.



316

A Further Generalization of Gamma, Beta and Hypergeometric Functions

Ayşegül ÇETINKAYA1, İ. Onur KIYMAZ1, Praveen AGARWAL2 and Shilpi JAIN3

Abstract. In this work, we present further generalization of gamma and beta functions

by using a confluent hypergeometric function with six parameters. Then we introduce new

generalization of Gauss and confluent hypergeometric functions by using this generalization

of beta function and systematically investigate their properties.

Keywords. Gamma function, beta function, hypergeometric functions.

AMS 2010. 33B15, 33C05, 33C15.

References

[1] Chaudhry M. A., Qadir A., Rafique M., Zubair S.M., Extension of Euler's beta function, J.

Comput. Appl. Math., 78, 19-32, 1997.

[2] Chaudhry M.A., Qadir A., Srivastava H.M., Paris R.B., Extended hypergeometric and

confluent hypergeometric functions, Appl. Math. Comput., 159 (2), 589-602, 2004.

[3] Choi J., Rathie A.K., Parmar R.K., Extension of extended beta, hypergeometric and

confluent hypergeometric functions, Honam Mathematical J., 36 (2), 357-385, 2014.

[4] Luo Min-Jie, Milovanovic G.V., Agarwal P., Some results on the extended beta and

extended hypergeometric functions, Appl. Math. Comput., 248, 631-651, 2014.

[5] Özergin E., Özarslan M.A., Altın A., Extension of gamma, beta and hypergeometric

functions, J. Comput. Appl. Math., 235, 4601-4610, 2011.

[6] Parmar R.K., A new generalization of Gamma, Beta, hypergeometric and confluent

hypergeometric functions, Matematiche (Catania) 69, 33-52, 2013.

[7] Rainville E.D., Special Functions, Macmillan Company, New York, 1960; Reprinted by

1 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected] , [email protected] 2 Anand International Collage of Engineering, Jaipur, INDİA, [email protected] 3 Poornima Collage of Engineering, Jaipur, INDIA, [email protected] Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number: FEF.D1.16.001






317

Chelsea Publishing Company, Bronx, New York, 1971.

[8] Srivastava H.M., Agarwal P., Jain S., Generating functions for the generalized Gauss

hypergeometric functions, Appl. Math. Comput., 247, 348-352, 2014.

[9] Srivastava H.M., Jain R., Bansal M.K., A Study of the S-Generalized Gauss

Hypergeometric Function and Its Associated Integral Transforms, Turkish Journal of

Analysis and Number Theory, 3 (5), 116-119, 2015.

[10] Srivastava H.M., Karlsson P.W., Multiple Gaussian hypergeometric series, Ellis

Horwood Limited, 1985.

[11] Srivastava H.M., Manocha H.L., A treatise on generating functions, Ellis Horwood

Limited, Chichester, 1984.


318

Teachers’ Perceptions About Written Examination Preparation Process

Ayten Pınar BAL1 and Fatma SADIK2

Abstract. The purpose of this study is to figure out teachers’ perceptions about written

examination preparation process. The study carried out with twenty secondary school

mathematics teachers working at different public secondary schools in Adana. The data was

collected through document analysis. At the end of the research it was found out that teachers

generally use multiple choice and classic tests and ask 20 questions on average. Moreover, the

other preffered question types are True-False and Fill in the blanks. In terms of the difficulty

levels of the questions; It was seen that there are comprehension and application types of

questions on all of the exam papers. In addition, it was also found out that the teachers in the

study are tend to give equal points for the questions that they ask.

NOTE: This article was supported by Cukurova University Scientific Research Unit. Project

No: EF2013BAP11

Keywords. Assessment and evaluation, mathematics teacher, success test, written

examination.



319

The Analysis of Mistakes and Solution Strategies Applıed in Algebraic Verbal Problems

Ayten Pınar BAL1 and Ahmet KARACAOĞLU2

Abstract. This study is a scanning model research in order to obtain algebraic verbal

problem solution strategies of 6th, 7th and 8th class students and their mistakes during this

period. Totally 1017 students from 6th, 7th and 8th grade students educating at medium socio

economic level of central districts of Adana province, formed working group of this study. As

data collection tool “Identification Test of Algebraic Verbal Problems Solution Strategies and

Mistakes of Students” developed by researchers was used in the study. As a result of research

it was found out that secondary school students especially use systematic distribution, inverse

operation, ordering after division, empirical and equation strategies in solving of algebraic

verbal problems and as error type they frequent make logic and calculation errors.

Keywords. Algebraic verbal problem, error type, problem solving strategy

1 Cukurova University, Adana, TURKEY, [email protected] 2 Cukurova University, Adana, TURKEY, [email protected]


320

Examination of Burnout and Life Satisfaction of Nurses

Emrah GÜRLEK1 and Kamile ŞANLI KULA2 and Mehmet YETİŞ3 and Aysu YETİŞ4

Abstract. In this study, burnout and life satisfaction of nurses working at Ahi Evran

University Training and Research Hospital will be examined in terms of various variables.

The study will be conducted with the nurses working in Ahi Evran University Education and

Research Hospital, volunteers who agree to participate in the research. For this purpose,

Personal Information form developed by researchers as a data collection tool, Burnout Scale

and Life Satisfaction Scale will be used.

Keywords. Nurse, Burnout, Life Satisfaction.

MSC 2010. 62P10.

This work was supported by the Scientific Research Projects Council of Ahi Evran

University, Kırşehir, Turkey under Grant TIP.A3.17.005.

References

[1] Neugarten, B., Havighurst, R. and Tobin, S., The Measure of Life Satisfaction. Journal of

Gerontology, 16, 134-143, 1961.

[2] Çelebi, B., Workers' burnout and job satisfaction: Alanya State Hospital Nurses sample.

Unpublished Master's Thesis, Beykent University Social Sciences Institute. İstanbul, 2014.

[3] Durmuş, S. ve Günay, O., Factors affecting job satisfaction and anxiety levels in the

nurses. Erciyes Medical Journal. 29(2), 139-146, 2007.

[4] Maslach, C., Schaufeli, W. B., ve Leiter, M. P., Job Burnout. Annual Reviews of

Psychology, 52, 397-422, 2001.

[5] Kaplan, H., The relatıonshıp between job satısfactıon and lıfe satısfactıon: Denızlı

Servergazı State Hospıtal examples of nurse and mıdwıves. Unpublished Master's Thesis.

Beykent University Social Sciences Institute, Istanbul, 2014.

1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected] 4 Ahi Evran University Education and Research Hospital, Kırşehir, TURKEY, [email protected]






321

[6] Şenyüz, Z., Burnout ın nurses and midwives working ın hospitals. Unpublished Master's

Thesis. Beykent University Social Sciences Institute, İstanbul, 2015.


322

Ruled Surface Pair Generated by Darboux Vectors of a Curve and Its Natural Lift in

𝐈𝐑𝟏𝟑

Evren ERGÜN1 , Mustafa ÇALIŞKAN2 and Keziban ORBAY3

Abstract. In this study, firstly, the darboux vector W of the natural lift 𝛼 of a

curve 𝛼 are calculated in terms of those of 𝛼 in IR13. Secondly, we obtained striction lines and

distribution parameters of ruled surface pair generated by Darboux Vectors of the curve 𝛼 and

its natural lift 𝛼 . Finally, for 𝛼 and 𝛼 those notions are compared with each other.

Keywords. Natural Lift, Ruled Surface, Striction Line, Distribution Parameter.

AMS 2010. 51B20, 53A15, 53A04.

References

[1] Çalışkan, M., Sivridağ, A.İ., Hacısalihoğlu, H.H, Some Characterizations for the natural

lift curves and the geodesic spray, Communications, Fac. Sci.Univ. Ankara Ser. A Math. 33,

Num. 28,235-242, 1984

[2] Çalışkan, M., Ergün, E.,On The -Integral Curves and -Geodesic Sprays In Minkowski

3-Space International Journal of Contemp. Math. Sciences,Vol. 6, no. 39, 1935-1939, 2011.

[3] Ergün, E., Çalışkan, M., On Geodesic Sprays in Minkowski 3-Space, International Journal

of Contemp. Math. Sciences, Vol. 6, no. 39, 1929-1933, 2011.

[4] Sivridağ A.İ. Çalışkan M. On the -Integral Curves and -Geodesic Sprays Erc.Uni. Fen

Bil. Derg. 7, 2, 1283-1287, 1991

[5] Thorpe, J.A., Elementary Topics In Differential Geometry,Springer-Verlag,New York,

Heidelberg-Berlin, 1979.

[6] Walrave, J., Curves and Surfaces in Minkowski Space K. U. Leuven Faculteit,Der

Wetenschappen, 1995.

1 Ondokuz Mayıs University, Samsun, TURKEY, [email protected] 2 Gazi University, Ankara, TURKEY, [email protected] 3 Amasya University, Amasya, TURKEY, [email protected]



323

Caratheodory’s Theorem in B-1-convexity

Gabil ADILOV1 and Ilknur YESILCE2

Abstract. B-1-convexity is an abstract convexity type [1, 2]. B-1-convex sets are

introduced in [3]. Separation in B-1-convexity is investigated in [4].

In this work, for continuing studies regarding B-1-convexity, we give some important

theorems like Carathedory’s Theorem for B-1-convex sets.

Keywords. Abstract convexity, B-1-convex sets, Carathedory’s Theorem.

AMS 2010. 52A20, 52A35.

References

[1] Rubinov, A., Abstract Convexity and Global Optimization, Kluwer Academic Publishers,

Boston-Dordrecht-London, 2000.

[2] Singer, I., Abstract Convex Analysis, John Wiley & Sons., New York, 1997.

[3] Adilov, G. and Yesilce, I., B-1-convex Sets and B-1-measurable Maps, Numerical

Functional Analysis and Optimization, 33, 2, 131-141, 2012.

[4] Tinaztepe, G., Yesilce, I. and Adilov, G., Separation of B-1-convex Sets by B-1-measurable

Maps, Journal of Convex Analysis 21, 2, 571-580, 2014.

1 Akdeniz University, Antalya, TURKEY, [email protected] 2 Mersin University, Mersin, TURKEY, [email protected]


324

The Simulation of the Order Parameter Probability Distribution for the Four

Dimensional Ising Model on the Creutz Cellular Automaton with the Linear Dimensions

L=24, 26 and 28

Ganimet MÜLAZIMOĞLU KIZILIRMAK1

Abstract. The aim of this study is study of the finite-size scaling relation for the order-

parameter probability distribution of the four dimensional İsing model with Creutz Cellular

Automaton. The four-dimensional İsing model is simulated on the Creutz Cellular Automaton

with the linear dimensions L=24, 26 and 28. The order parameter probability distribution are

calculated at the critical temperature values of infinitive lattice. The finite size scaling relation

for the order parameter probability distribution is tested and verified numerically by the

Creutz Cellular Automaton simulation. The constants of the analytical function are estimated

by fitting it to probability function obtained numerically at the finite size critical point [1-4].

Keywords. Ising Model, cellular automaton, critical temperature, order parameter,

magnetic susceptibility,

Acknowledgement. This work was supported by the Ahi Evran University Scientific

Research Projects Coordination Unit. Project Number: FEF.A3.17.004

References

[1] Aktekin, N., Dört Boyutlu Isıng Modelinde Düzen Parametresi İhtimaliyet Dağılımı İçin

Sonlu Örgü Ölçekleme Bağıntısı, G.Ü. Fen Bilimleri Dergisi 17(3), 59-70, 2004.

[2] Binder, K., Finite-size scaling analysis of Ising model block distribution functions, Z.

Phys. B, 43, 119- 140, 1981.

[3] Binder, K., Critical properties from Monte Carlo coarse graining and renormalization,

Phys. Rev. Lett., 47, 693-696, 1981.

[4] Brezin, E. and Zinn-Justin, J., Finite-size effects in phase transitions, Nucl. Phys. B, 257,

867- 893, 1985.

1Ahi Evran University, Kırşehir, Turkey, [email protected]


325

Some Numerical Experiments for a Sharper Version of Hölder Inequality

Gültekin TINAZTEPE1

Abstract. In this work, the sharper version of Hölder Inequality which is given [1], [2]

and its derivation is presented. Also, to show the efficiency of given sharper version, some

new numerical experiments carried out are presented.

Keywords. Hölder inequality, convexity.

AMS 2010. 26D07.

References

[1] Tınaztepe, G., The Sharpening Hölder Inequality via Abstract Convexity, Turk. J. Math.,

40, 438-444, 2016.

[2] Tınaztepe, G., Tınaztepe, R. and Kemali, S., On Some Inequalities and Their Refinements,

4th International Eurasian Conference on Mathematical Sciences and Applications, Greece,

2015.

[3] Beckenbach, E. and Bellman, R., Inequalities, Springer-Verlag, 1961.

1 Akdeniz University, Antalya, TURKEY, [email protected]


326

Conformable variational iteration method for solving the time-fractional Fornberg-

Whitham equation

Handan Çerdik YASLAN

Abstract. In this work, conformable variational iteration method (CVIM) is applied to

the time fractional Fornberg-Whitham equation in the sense of the conformable derivative.

The exact and numerical solutions obtained by the conformable variational iteration method

are compared. Numerical results show that the approach performs extremely well in terms of

efficiency and simplicity

Keywords. Time fractional FornbergWhitham equation, Conformable derivative,

Variational iteration method.

AMS 2010. 35R11, 65K15.

References

[1] He, J. H., Variational iteration method for delay differential equations,

Commun.nonlinear Sci. Numer. Simul. 2 (4), 235236, 1997.

[2] He, J. H., Variational iteration method a kind of non-linear analytical technique: some

examples, Int. J. Nonlinear Mech. 34, 699708, 1999.

[3] Prakash, A., Kumar, M., He's Variational Iteration Method for the Solution of Nonlinear

Newell-Whitehead-Segel Equation. JAAC 6, 738-748, 2016.

[4] He, J. H., Approximate analytical solution for seepage ow with fractional derivatives in

porous media, Comput. Methods Appl. Mech. Engrg. 167 110, 5768, 1998.

[5] Odibat, Z., Momani, S., The variational iteration method: an effcient scheme for handling

fractional partial di_erential equations in fluid mechanics, Comput. Math. Appl. 58,

21992208, 2009.

[6] Yulita Molliq, R., Noorani, M.S.M., Hashim, I., Ahmad, R.R., Approximate solutions of

fractional ZakharovKuznetsov equations by VIM, J. Comput. Appl. Math. 233 (2)103108,

2009.


327

[7] Inc, M., The approximate and exact solutions of the space- and timefractional burgers

equations with initial conditions by variational iteration method, J. Math. Anal. Appl. 345 (1)

476484, 2008.


328

A Further Generalization of Fractional Operators

İ. Onur KIYMAZ1, Ayşegül ÇETİNKAYA1, Praveen AGARWAL2 and Shilpi JAIN3

Abstract. The purpose of this work is to present a systematic study of a further

generalization of fractional operators by using a confluent hypergeometric function with six

parameters. We investigate some of their properties such as integral representations, Mellin

transforms, generating functions, etc.

Keywords. Generalized beta function, fractional operators, Mellin transform.

AMS 2010. 26A33, 33B15, 33C15.

References

[1] Agarwal P., Choi J., Paris R.B., Extended Riemann-Liouville fractional derivative

operator and its applications, J. Nonlinear Sci. Appl., 8, 451-466, 2015.

[2] Baleanu D., Parmar R.K., Agarwal P., Salahshourd S., Extension of the fractional

derivative operator of the Riemann-Liouville, J. Nonlinear Sci. Appl., 10, 2914–2924, 2017.

[3] Chaudhry M.A., Qadir A., Rafique M., Zubair S.M., Extension of Euler's beta function, J.

Comput. Appl. Math., 78, 19-32, 1997.

[4] Chaudhry M.A., Qadir A., Srivastava H.M., Paris R.B., Extended hypergeometric and

confluent hypergeometric functions, Appl. Math. Comput., 159 (2), 589-602, 2004.

[5] Choi, J., Parmar, R.K., Extension of extended beta, hypergeometric and confluent

hypergeometric functions, Honam Mathematical J., 36 (2), 357-385, 2014.

[6] Kıymaz İ.O., Çetinkaya A., Agarwal P., An extension of Caputo fractional derivative

operator and its applications, J. Nonlinear Sci. Appl., 9, 3611-3621, 2016.

1 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected], [email protected] 2 Anand International Collage of Engineering, Jaipur, INDIA, [email protected] 3 Poornima Collage of Engineering, Jaipur, INDİA, [email protected] Acknowladgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number: FEF.D1.16.001






329

[7] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional

Differential Equations, Elsevier, Amsterdam etc., 2006.

[8] Özarslan M.A., Özergin E., Some generating relations for extended hypergeometric

functions via generalized fractional derivative operator, Math. Comp. Mod., 52, 1825-1833,

2011.

[9] Podlubny, I. Fractional differential equations, Academic Pres, New York, 1999.

[10] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives Theory

and Applications, Gordon and Breach, Longhorne, 1993.

[11] Srivastava H.M., Karlsson P.W., Multiple Gaussian hypergeometric series, Ellis

Horwood Limited, 1985.

[12] Srivastava H. M., Manocha H.L., A treatise on generating functions, Ellis Horwood

Limited, Chichester, 1984.


330

Higher-Order Multi-Point Fractional Boundary Value Problems

İsmail YASLAN

Abstract. In this paper, we concerned with existence of positive solutions for higher-

order multi-point fractional boundary value problem. We establish the criteria for the

existence of at least one and three positive solutions for higher order nonlinear multi-point

fractional boundary value problem by using the Krasnosel'skii fixed point theorem and the

Legget-Williams fixed point theorem, respectively.

Keywords. Boundary value problems, cone, fixed point theorems, positive solutions,

Riemann-Liouville fractional derivative.

AMS 2010. 34B07, 34D05, 34L20, 34K37.

References

[1] Fen, F.T., Karaca, I.Y., Ozen, O.B., Positive Solutions of Boundary Value Problems for p-

Laplacian Fractional Differential Equations, Filomat, 31, 1265-1277, 2017.

[2] Günendi, M., Yaslan, I., Positive solutions of higher-order nonlinear multi-point

fractional equations with integral boundary conditions, Fract. Calc. Appl. Anal., 19, 989-

1009, 2016.

[3] Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Academic Press,

San Diego, 1988.

[4] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional

Differential Equations, Elsevier, Amsterdam, 2006.

[5] Legget, R.W., Williams, L.R., Multiple positive fixed points of nonlinear operators on

ordered Banach space, Indiana Univ. Math. J., 28, 673-688, 1979.

[6] Sabatier, O.P., Agrawal, J.A., Machado, T., Advances in Fractional Calculus, Springer,

Dordrecht, The Netherlands, 2007.

[7] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integral and Derivatives: Theory

and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.


331

Elliptic Biquaternion Algebra

Kahraman Esen Özen1 and Murat Tosun2

Abstract. In this study, we define the elliptic biquaternion algebra and give basic properties

of elliptic biquaternions. An elliptic biquaternion is in the form 0 1 2 3A A A A i j k which is a

linear combination of 1, , ,i j k where the four components 0 1 2, ,A A A and 3A are elliptic

numbers and 1, , ,i j k is the quaternion basis which are exactly the same in the quaternion

algebra and complexified quaternion algebra and satisfies the same multiplication rules. Also,

we express the De-Moivre’s and Euler Formulas for elliptic biquaternions by writing elliptic

biquaternions in the polar form. Furthermore, with the aid of these formulas, we find powers

and roots of any elliptic biquaternions.

Keywords.Complex quaternion, De-Moivre’s formula, Elliptical numbers.

AMS 2010. 15A06, 11R52.

References

[1] van der Waerden, B. L.. Hamilton’s discovery of quaternions, Math. Magazine, 49, 227-

234, 1976.

[2] Clifford, W. K.. Mathematical Papers, edited by R. Tucker (MacMillan Co., London),

1882.

[3] Muses C.. Applied hypernumbers: Computational concepts, Appl. Math. Comput. 3, 211-

216, 1976.

[4] Muses C.. Hypernumbers and quantum field theory with a summary of physically

applicable hypernumber aritmetics and their geometries, Appl. Math. Comput. 6, 63-94,

1980.

[5] Carmody K.. Circular and hyperbolic quaternions, octanions, and sedenions, Appl. Math.

Comput. 28, 47-72, 1988.



332

Beliefs about Mathematical Problem Solving of University Students

Kamile ŞANLI KULA1 and Ezgi ÇAĞATAY İN2

Abstract. In this study, Ahi Evran University students of different faculties and colleges

to solve mathematical problems related to the beliefs examine in terms of their gender, class,

the faculties/colleges, the felt level in mathematics, and weekly schedule. For this purpose,

"Personal Information Form" developed by the researcher and developed by Kloosterman and

Stage (1992), Hacıömeroğlu (2011) by Turkish adapted to the scale of the "Belief in Relation

to Mathematical Problems Solving" will be used.

Keywords. Mathematics, Problem Solving, University, Student.

MSC 2010. 62P25, 97C20.

This work was supported by the Scientific Research Projects Council of Ahi Evran

University, Kırşehir, Turkey under Grant FEF.A3.16.036.

References

[1] Duatepe Paksu, A., Comparing Teachers’ Beliefs about Mathematics in term of their

Branches and Gender, Hacettepe University Journal of Education, 35, 87-97, 2008.

[2] Hacıömeroğlu, G, Turkish Adaptation of Beliefs about Mathematical Problem Solving

Instrument, Dicle University Journal of Ziya Gökalp Faculty of Education, 17, 119-132, 2011.

[3] Kayan, F. ve Çakıroğlu, E., Preservice Elementary Mathematics Teachers’ Mathematical

Problem Solving Beliefs, Hacettepe University Journal of Education, 35, 218-226, 2008.

[4] Kloosterman, P. and Stage, F. K., Measuring beliefs about mathematical problem solving.

School Science and Mathematics, 92(3), 109-115, 1992.

1 Ahi Evran University, Kırşehir, TÜRKİYE, [email protected] 2 Ahi Evran University, Kırşehir, TÜRKİYE, [email protected]


333

Investigation of Structural Phase Transformation in Iridium Dioxide (IrO2) under High

Pressure

Köksal KIZILIRMAK1, Ganimet M. KIZILIRMAK2 and Hülya ÖZTÜRK3

Abstract. The structural phase transformation in iridium dioxide (IrO2) has been

investigated using first-principles calculations by the density functional theory (DFT) with the

ab initio program SIESTA under the hydrostatic pressure. We used the generalized gradient

approximation (GGA) for the exchange-correlation energy. The rutile type structure of IrO2

with space group P42/mnm transformed to CaCl2 type structure with space group Pnnm. This

phase transformation is also analyzed from the total energy and enthalpy calculations.

Keywords. Structural Phase Transformation, Density Functional Theory, Metal

Dioxides

Acknowledgement. This work was supported by the Ahi Evran University Scientific

Research Projects Coordination Unit. Project Number: FEF.A4.17.015

References

[1] Hohenberg, P.; Kohn, W., Phys. Rev B 1964, 136, 864.

[2] Parr, R. G.; Yang, W. J. Am. Chem. Soc. 1980, 106, 4049-4050.

[3] Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1133.

[4] Ordejon, P.; Artacho, E.; Soler, J. M. Phys. Rev. B 1996, 53, (R10) 411-415.

[5] Ceperley, D. M.; Alder, M. J. Phys. Rev. Lett. 1980, 45, 566-569.

[6] Perdew, J. P.; Wang, Y. Phys. Rev. B 1992, 45, 13244-13249.

[7] Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868.

1Ahi Evran University, Kırşehir, TURKEY, [email protected] 2Ahi Evran University, Kırşehir, TURKEY, [email protected] 3Ahi Evran University, Kırşehir, TURKEY, [email protected]



334

[8] Troullier, N.; Martins, J. L. Phys. Rev. B 1991, 43, 1993-2006.

[9] Parrinello, M.; Rahman, A. Phys. Rev. Lett. 1980, 45, 1196.

[10] Birch, F. Phys. Rev. 1947, 71, 809.

[11] Murnaghan, F. D. Proc. Natl. Acad. Sci. U.S.A. 1944, 30, 244.

[12] Kürkçü, C. MgF2'de yüksek basınç etkisiyle yapısal faz dönüşümleri ve dönüşüm

mekanizmasının incelenmesi, Yüksek Lisans Tezi, Ahi Evran Üniversitesi, Kırşehir, 73s,

2014.

[13] Bolzan, A. A., Fong, C., Kennedy, B. J. and Howard, C. J., Acta Cryst, B53, 373-380,

1997.

[14] Szpunar, B., Szpunar, J., Journal of Physics and Chemistry of Solids, 74, 1632–1639,

2013.


335

Helixes on Clifford Surfaces in a Hyperbolic Space of Positive Curvature

Lyudmila ROMAKINA1

Abstract. Analogs of a Clifford surface in a hyperbolic space 3 of positive curvature

[1] are investigated. It is proved that the Clifford surfaces of the space 3 are proejectively

equivalents of an oval surface and form two types [2]. A Clifford surface has two rotation

axes which are mutually polar with respect to the absolute oval surface of the space 3 and

belong to the different types. A hyperbolic (or elliptic) Clifford surface of the space 3 can be

formed by one of two rotations. These rotations are as follows: (a) the rotation of a hyperbolic

(or, respectively, elliptic) cycle of the hyperbolic plane of positive curvature around its base;

(b) the rotation of a hypercycle of the hyperbolic plane of positive curvature (or, respectively,

a circle of the elliptic plane) around its base. The Clifford surfaces of the space 3 can be

used for the construction of orthogonal coordinate systems at the volumes calculation [3]. The

volume formulae for the orthogonal layers of the Clifford surfaces are obtained. A Clifford

surface of the space 3 contains helixes which can be represented by spirals with two poles

on rotation ellipsoids in Euclidean space. All helixes on a hyperbolic Clifford surface are

elliptic. Any helix on an elliptic Clifford surface belongs to one of three types. Depending on

the type of tangents it can be an elliptic, hyperbolic or parabolic curve.

Keywords. Hyperbolic space of positive curvature, Clifford surface, helix.

AMS 2010. 51F10, 51N25, 51N30.

References

[1] Romakina, L., Dihedrons of a hyperbolic three-space of positive curvature, International

Electronic Journal of Geometry, 9, 2, 50-58, 2016.

[2] Romakina, L., Clifford surfaces in a hyperbolic space of positive curvature, Sci. articles

collection of the 5th Int. Sci. Conf. of Eurasian Sci. Association, 5(27), 30-33, 2017.

[3] Romakina, L., The volume of a monopolar tetrahedron in a hyperbolic space of positive

curvature, Sci. articles collection of the 5th Int. Sci. Conf. of Eurasian Sci. Association, 5(27),

27-30, 2017.

1 Saratov State University, Saratov, RUSSİA, [email protected]


336

A New Generalization of Whittaker Function and its Properties

M. Baki YAĞBASAN1, Ayşegül ÇETİNKAYA1 and İ. Onur KIYMAZ1

Abstract. We introduce a new generalization of the extended Whittaker function by

using a generalization of confluent hypergeometric function of the first kind and investigate

its properties such as integral representations, integral transforms, differential formula and

recurrence relations.

Keywords. Whittaker function, generalized beta function, generalized confluent

hypergeometric function, generalized Gauss hypergeometric function.

AMS 2010. 33B15, 33C05, 33C15

References

[1] Chaudhry, M. A., Qadir, A., Rafique, M., and Zubair, S. M., Extension of Euler's beta

function, J. Comput. Appl. Math., 78, no. 1, 19-32, 1997.

[2] Chaudhry, M. A., Qadir, A., Srivastava, H. M. and Paris, R. B., Extended

hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 no. 2,

589-602, 2004.

[3] Choi, J., Ghayasuddin, M., Khan, N., Generalized Extended Whittaker Function and

Its Properties, Applied Mathematical Sciences, Vol. 9, 2015, no. 131, 6529-6541.

[4] Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Table of Integral

Transforms, Vol. 1, MaGraw-Hill, New York, 1954.

[5] Khan, N., Ghayasuddin, M., A Note on generalized extended Whittaker function,

Honam Mathematical J. 38, No. 2, pp. 325-335, 2016.

[6] Nagar, D. K., Vasquez, R. A. M. and Gupta, A. K., Properties of the extended

Whittaker function, Progr. Appl. Math., 6, no. 2, 70-80, 2013.

[7] Özergin, E., Özarslan, M. A. and Altın, A., Extension of gamma, beta and

hypergeometric functions, J. Comput. Appl. Math., 235, 4601-4610, 2011.

1 Ahi EvranUniversity, Kırşehir, Turkey, [email protected] Acknowladgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number:

FEF.A4.17.002



337

[8] Parmar, R. K., A new generalization of gamma, beta hypergeometric and confluent

hypergeometric functions, Matematiche (Catania), LXVIII, 33-52, 2013.

[9] Rainville, E. D., Special Functions, Macmillan Company, New York, 1960,

[10] Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press,

Cambridge, London and New York, 1960.

[11] Srivastava, H. M. and Manocha, H. L., A Treatise on Generating Functions, Halsted

Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester,

Brisbane, and Toronto, 1984.


338

Classification of Homothetical Hypersurfaces and Its Applications to Production

Functions in Economics

Mahmut ERGUT 1 and Muhittin Evren AYDIN 2

Abstract. In this study, we completely classify the homothetical hypersurfaces

having null Gauss- Kronocker curvature in a Euclidean 1n space. Several applications

to the production functions in economics areal so given.

Keywords. Homothetical hypersurface, Gauss- Kronocker curvature, production

function

AMS 2010. 53B25, 91B38.

References

[1] Aydin, M. E., Ergut, M., Hessian determinants of composite functions with applications

or production functions in economics, Kragujevac J. Math. 38(2), 259–268, 2014.

[2]nAlodan, H.,Chen, B.-Y.,Deshmukh, S., Vilcu, G.E.,On some geometric properties

of quasi- produc tproduction models, arXiv:1512.05190v1.

[3] Van de Woestyne, I., Minimal homothetical hypersurfaces of a semi-Euclidian

space, Results in Mathematics 27, 333-342, 1995.

[4] Lopez, R., Moruz, M., Translation and homothetical surfaces in Euclidean space with

constant curvature, J. Korean Math. Soc. 52(3), 523---535, 2015.

[5] Saglam, D., Sabuncuoglu, A., Minimal homothetical lightlike (degenerate)

hypersurfaces of semi- Euclidean spaces, Kuwait J. Sci. Eng. 38 (1A), 1-14, 2011.

1 Namik Kemal University, Tekirdağ, TURKEY, [email protected] 2 Firat University, Elazığ, TURKEY, [email protected]


339

Generation of Pseudo-Random Numbers from Given Probabilistic Distribution with the

Use of Chaotic Maps

Marcin LAWNİK1

Abstract. The scope of the paper is the presentation of a new method of generating

numbers from a given distribution. The method uses the inverse cumulative distribution

function and a method of flattening of probabilistic distributions [1]. On the grounds of these,

a new constructions of chaotic maps were derived:

𝑥𝑘+1 = 𝐹−1(𝑈𝑛(|𝑎𝑥𝑘|)), (1)

𝑥𝑘+1 = 𝑇(𝑈𝑛(|𝑎𝑥𝑘|)), (2)

where 𝐹−1 is a inverse cumulative distribution function, 𝑇 is a given transformation, 𝑈𝑛 is the

𝑛-th iteration of the chaotic map with a uniform distribution, and 𝑎 is a normative coefficient.

The analysis of (1) and (2) was conducted on the example of maps constructed on the

basis of Box-Muller transformation and the quantile function of the exponential distribution.

The obtained results certify, that the proposed method may be successively applicable for the

construction of generators of pseudo-random numbers.

Keywords. Pseudo-random numbers, chaotic maps, standard normal distribution.

AMS 2010. 65C10, 37M25.

References

[1] Lawnik, M., Generation of numbers with the distribution close to uniform with the use of

chaotic maps, Proceedings of the 4th International Conference on Simulation and Modeling

Methodologies, Technologies and Applications, Scitepress, 451–455, 2014.

1 Silesian University of Technology, Faculty of Applied Mathematics, Gliwice, POLAND, [email protected]


340

A New Approach on Slant Curves in three Dimensional Lie Groups

Osman Zeki OKUYUCU1, Caner DEĞIRMEN2 and İsmail GÖK3

Abstract. In this paper, the alternative moving frame ℕ, ℂ, 𝕎 is obtained for a unit

speed curve given in the 3-dimensional Lie group 𝐺 and ℂ -slant helices are defined according

to this frame. In addition, some characterizations of these curves are investigated.

Keywords. General helices, Slant helices, ℂ -slant helices, Lie groups.

AMS 2010. 53A04, 22E15.

References

[1] Crouch, P., Silva, F. L., The Dynamic Interpolation Problem: On Riemannian Manifolds

Lie Groups and Symmetric Spaces, Journal of Dyn. Control Syst., 1(2), 177-202, 1995.

[2] Çiftçi, Ü., A Generalization of Lancret’s Theorem, Journal of Geometry Physics, 59,

1597-1603, 2009.

[3] Izumiya, S., Takeuchi, N., New Special Curves and Developable Surfaces, Turkish Journal

of Mathematics, 28, 153-163, 2004.

[4] Lancret, M. A., Mémoire sur les courbes à double courbure, Mémoires présentés à

I’Institut 1, 416-454, 1806.

[5] Okuyucu, O. Z., Gök, İ., Yaylı, Y., Ekmekci, N., Slant Helices in three Dimensional Lie

Groups, Appl. Math. and Comp., 221, 672-683, 2013.

[6] Uzunoğlu, B., Gök, İ., Yaylı, Y., “A New Approach on Curves of Constant Precession”,

Applied Mathematics and Computation, 275, 317-323, 2016.

1 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected] 2 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected] 3 Ankara University, Ankara, Turkey, [email protected]


341

Analysis of Antimicrobial Resistance among Clinical Isolates from Denizli

Selma KIRAÇ1, Dilek KESKIN2 and Muradiye YARAR3

Abstract. The main causes of many diseases are bacterial infections. Antibiotics are

used for the treatment of these infections. Antibiotic resistance to bacterial infections is an

important health problem. The aim of this study was to analyze the resistance status of gram

negative bacterial strains isolated from clinical specimens in Denizli.

Multiple antibiotic resistance was calculated using the mar index value = number of

antibiotics in which the strains were resistant / number of antibiotics tested. If the result is

greater than 0.2, it is evaluated as having multiple antibiotic resistance.

Multiple antibiotic resistance was detected in 88% of Proteus mirabilis strains and 59%

of Pseudomonas aeroginosa strains and %100 Acinetobacter baumanii strains. This study is

important in order to guide the correct treatment of infections caused by Gram negative

bacteria in Denizli.

Keywords. Analysis Resistance, Antibiotic, Gram negative bacteria, infection control.

References

[1] Zhao, B., Zhang, X., Mathematical analysis of multi-antibiotic resistance. International

Journal of Cardiology 219, 33–37, 2016.

[2] Chen, C.,Chen, Y., Lu, P.L., Lin, W.R., Chen,T.C., Lin,C.Y., Proteus mirabilis urinary

tract infection and bacteremia: Risk factors, clinical presentation, and outcomes. Journal of

Microbiology, Immunology and Infection 45, 228-236, 2012.

[3] M. Haber, B.R. Levin, P. Kramarz, Antibiotic control of antibiotic resistance in

hospitals:a simulation study, BMC Infect. Dis. 10 1–10, 2010.

[4] Singer, B. Mathematical Models of infectious diseases:seeking new tools for planning and

evaluating control programs. Supplement to Popul. Dev. Rev., 10:347–365, 1984.

1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Adnan Menderes University Cine-Aydin, TURKEY 3 Denizli State Hospital, Denizli, TURKEY

http://aac.asm.org/content/45/6/1721.short


342

[5] Altunsoy, A., Aypak, C., Azap, A., Ergonul, O., Balik, I., The impact of a nationwide

antibiotic restriction program on antibiotic usage and resistance against nosocomial

pathogens in Turkey. Int J Med Sci 8(4):339–344, 2011.

[6] Somily A. M., Absar M. M., Arshad M.Z., Antimicrobial susceptibility patterns of

multidrug resistant Pseudomonas aeruginosa and Acinetobacter baumannii against

carbapenems, colistin, and tigecycline. Saudi Med J.; 33(7): 750-5, 2012.


343

An Approach to Volterra and Fredholm Integral Equations by Homotopy Perturbation

Method

Serpil ŞAHİN 1

Abstract. In this study, an application of the homotopy perturbation method for

Volterra and Fredholm integral equations is presented. The proposed method is based on the

homotopy perturbation method, which consists in constructing the series whose sum is the

solution of the problem considered. In this method, a homotopy with an imbedding parameter

pÎ 0,1[ ] is constructed. Furthermore, these integral equations are solved by Adomian

Decomposition Method, Variational Iteration Method and Successive Approximations

Method. The effectiveness and practicality of the homotopy perturbation method is evaluated

according to analytical and numerical results.

Keywords. Integral equations, Homotopy perturbation method.

AMS 2010. 33E30

References

[1] Abbasbandy, S., Iterated He’s homotopy perturbation method for quadratic Riccati

differential equation, Applied Mathematics and Computation, 175, 581-589, 2006.

[2] Arikoğlu, A., Özkol, I., Solutions of integral and integro-differential equation systems by

using differential transform method, Computers and Mathematics with Applications, 56,

2411-2417, 2008.

1 Amasya University, Amasya, Turkey, [email protected]



344

Composition Formulae Associated Fractional Integral Operator with the Multi-Index

Mittag-Leffler Functions

Shilpi JAIN1, Praveen AGARWAL2, Ayşegül ÇETINKAYA3 and İ. Onur KIYMAZ4

Abstract. In this paper, we aim at establishing new composition formulae for the

Marichev-Saigo-Maeda (M-S-M) fractional integral operator and the multi-index Mittag-

Leffler functions. Here, we record four such new and interesting special cases of our main

results. The main results of this paper generalizes the results obtained by Choi and Agarwal

[2]. Further, we obtain Laplace transforms of these composition formulae.

Keywords. Marichev Saigo Maeda Fractional Integral Opearators, generalized multi-

index Mittag-Leffler function, generalized Wright function

AMS 2010. 26A33, 33E12, Secondary 33C60,33E20.

References

[1] Agarwal, P. and Choi, J., Fractional Calculus Operators and Their Images Formulas, J.

Korean Math. Soc., 53(5), 1183-1210, 2016.

[2] Choi, J. and Agarwal, P., A note on Fractional Integral Operator Associated with

Multiindex Mittag-Leffler Functions, Filomat, 30:7, 1931-1939, 2016.

[3] Debnath, L. and Bhatta, D., Integral Transforms and Their Applications, Chapman &

Hall/CRC, Boca Raton, London, New York, 2007.

[4] Haubold, H. J., Mathai, A. M. and . Saxena, R. K, Mittag-Leffler functions and their

applications, J. Appl. Math., 2011.

[5] Kiryakova, V., Multiple (multiindex) Mittag-Leffler functions and relations to generalized

fractional calculus, J. Comput. Appl. Math., 118, 241-259, 2000.

1 Poornima Collage of Engineering, Jaipur, INDIA, [email protected] 2 Anand International Collage of Engineering, Jaipur, INDİA, [email protected] 3 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected] 4 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected] Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number: FEF.D1.16.001



345

[6] Mathai, A. M. and Saxena, R. K., The H-function with Applications in Statistics and Other

Disciplines,Halsted Press [John Wiley & Sons], New York, London, Sydney, 1978

[7] Saxena, R. K. and Nishimoto, K., N-fractional calculus of generalized Mittag- Leer

functions, J. Fract.Calc., 37, 43-52, 2010.

[8] Saxena, R. K., Pogany, T. K., Ram, J. and Daiya, J., Dirichlet averages of generalized

multi-index Mittag-Leffler functions, Armen. J. Math., 3(4), 174-187, 2010.

[9] Srivastava, H. M., Harjule, P. and Jain, R., A General Fractional Di

erential Equation Associated With an Integral Operator With the H-Function in the Kernel,

Russian Journal of Mathematical Physics, 22(1), 112-126, 2015

[10] Wright, E. M., The asymptotic expansion of the generalized hypergeometric function, J.

London Math. Soc., 10, 286-293, 1935.


346

On a General Subclass of Univalent Functions Based on the q-Derivative Operator


Abstract. In our present investigation, we aim at investigating a general subclass of

univalent functions based on the q-derivative operator in the open unit disc

1and: zzzU C . Making use of Sigmoid function, we find estimates on the

coefficients 2a and 3a . Furthermore, upper bounds are obtained for 223 aa , where

C .

Keywords. Coefficient estimates, Sigmoid function, q-derivative operator, Fekete-

Szegö inequalities..

AMS 2010. 30C45, 33D15.

References

[1] Aydoğan, M., Kahramaner, Y., Polatoğlu, Y., Close-to-convex functions defined by

fractional operator, Applied Mathematical Sciences, 7, 56, 2769-2775, 2013.

[2] Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften,


[3] Jackson, F. H., On q-functions and a certain difference operator, Transactions of the

Royal Society of Edinburgh, 46, 253-281, 1908.

[4] Fadipe-Joseph, O. A., Oladipo, A.T., Ezeafulukwe, U. A., Modified Sigmoid function in

univalent function theory, International J. of Math. Sci. & Engg. Appls. 7, 313-317, 2013.

[5] Kanas, S., Raducanu, D., Some class of analytic functions related to conic domain,

Mathematica Slovaca, 64 (5)(2014) 1183-1196.

[6] Y. Polatoğlu, Growth and distortion theorems for generalized q-starlike functions,

Advances in Mathematics: Scientific Journal, 5, 1, 7-12, 2016.



347

[7] Purohit, S. D., Raina, R. K., Certain subclass of analytic functions associated with

fractional q-calculus operators, Math. Scand. 109, 55-70, 2011.

[8] Robertson, M. S., On the theory of univalent functions, Ann. Math., 37, 374-408, 1936.

[9] Srivastava, H. M. Univalent functions, fractional calculus, and associated generalized

hypergeometric functions. in Univalent Functions, Fractional Calculus, and Their

Applications, (H. M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited,

Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.

[10] Seoudy, T. M., Aouf, M. K., Coefficient estimates of new classes of q-starlike and q-

convex functions of complex order, 10, 1, 135-145, 2016.


348

Estimates of Coefficients for a Subclass of Bi-Univalent Functions by Making Use of

Faber Polynomial Expansions


Abstract. In the present investigation, we give a new subclass of bi-univalent

functions which is defined by subordinations in the open unit disk

1and: zzzU C .

Making use of Faber polynomial expansions, we determine the general coefficient bounds

na of functions belonging to this new analytic bi-univalent function class.

Keywords. Coefficient estimates, Faber polynomials, subordination.

AMS 2010. 30C45.

References

[1] Airault, H., Symmetric sums associated to the factorization of Grunsky coefficients, in

Conference, Groups and Symmetries, Montreal, Canada, April, 2007.

[2] Airault, H., Bouali, H., Differential calculus on the Faber polynomials, Bulletin des

Sciences Mathematiques, 130, 179-222, 2006.

[3] Airault, H., Ren, J., An algebra of differential operators and generating functions on the

set of univalent functions, Bulletin des Sciences Mathematiques, 126, 343-367, 2002.

[4] Altınkaya, Ş., Yalçın, S., Coefficient estimates for a subclass of analytic and bi-univalent

functions, Acta Universitatis Apulensis, 40, 347-354, 2014.

[5] Altınkaya, Ş., Yalçın, S., Faber polynomial coefficient bounds for a subclass of bi-

univalent functions, C. R. Acad. Sci. Paris, Ser. I, 353, 12, 1075-1080, 2015.

[6] Duren, P. L., Univalent functions, Grundlehren der Mathematischen Wissenschaften,




349

[7] Hamidi, G. S., Halim, S. A., Jahangiri, J. M., Faber polynomial coefficient estimates for

meromorphic bi-starlike functions, International Journal of Mathematics and Mathematical

Sciences, Article ID 498159, 4 pages, 2013.

[8] Hamidi, S. G., Jahangiri, J. M., Faber polynomial coefficient estimates for analytic bi-

close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I, 352, 17-20, 2014.

[9] Lewin, M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc.,

18, 63-68, 1967.

[10] Netanyahu, E., The minimal distance of the image boundary from the origin and the

second coefficient of a univalent function in 1z , Archive for Rational Mechanics and

Analysis, 32, 100-112, 1969.

[11] Salagean, G. S., Subclasses of univalent functions, in: Complex Analysis- Fifth

Romanian Finish Seminar, Bucharest, 1, 362-372, 1983.

[12] Srivastava, H. M., Mishra, A. K., Gochhayat, P., Certain subclasses of analytic and bi-

univalent functions, Applied Mathematics Letters, 23, 1188-1192, 2010.


350

On a Subclass of Univalent Functions Defined by Chebyshev Polynomials


Abstract. Making use of Chebyshev polynomials, we get upper bound estimates for

the initial coefficients 2a , 3a and the Fekete-Szegö functional 223 aa of univalent

functions.

Keywords. Coefficient estimates, Chebyshev polynomials, Fekete-Szegö functional,

subordination.

AMS 2010. 30C45, 30C50.

References

[1] Altınkaya, Ş., Yalçın, S., On the Chebyshev polynomial bounds for classes of univalent

functions, Khayyam J. Math., 2, 1-5, 2016.

[2] Doha, E. H., The first and second kind Chebyshev coefficients of the moments of the

general-order derivative of an infinitely differentiable function, Intern. J. Comput. Math., 51,

21-35, 1994.

[3] Fekete, M., Szegö, G. Eine bemerkung über ungerade schlichte funktionen, Journal of the

London Mathematical Society, 2, 85-89, 1933.

[4] Mason, J. C., Chebyshev polynomials approximations for the L-membrane eigenvalue

problem, SIAM J. Appl. Math., 15, 172-186, 1967.



351

Coefficient Estimates for Analytic Bi-Bazilevi cFunctions of Order and Type


Abstract. A function is said to be bi-univalent in the open unit disk

1and: zzzU C if both the function and its inverse are univalent in U . By the same

token, a function is said to be bi-Bazilevi c in U if both the function and its inverse are

Bazilevi c there. We find estimates on the Taylor-Maclaurin coefficients 2a , 3a and by

using the Faber polynomials, we obtain general coefficient bound na for functions in the

class ),( B . Furthermore, we get the upper bound for the functional 23422 )1( aaaH .

Keywords. Coefficient estimates, Hankel determinant, bi-Bazilevi c functions, Faber

polynomials.

AMS 2010. 30C45.

References

[1] Airault, H., Remarks on Faber polynomials, Int. Math. Forum, 3, 449-456 , 2008.

[2] Airault, H. and Ren, J., An algebra of differential operators and generating functions on

the set of univalent functions, Bulletin des Sciences Mathematiques, 126, 343-367, 2002.

[3] Altınkaya Ş., Yalçın S., Upper Bound of Second Hankel Determinant for Bi-Bazilevic

Functions, Mediterranean Journal of Mathematics (MJOM), 13, 4081–4090, 2016.

[4] Altınkaya Ş., Yalçın S., On the Faber polynomial coefficient bounds of bi-Bazilevic

functions, Communications Faculty of Sciences University of Ankara Series A1: Mathematics

and Statistics, 66, 289-296, 2017.

[5] Bazilevic, I.E., On a case of integrability in quadratures of the Loewner-Kufarev

equation, Matematicheskii Sbornik, 37, 471–476, 1955.



352

[6] Duren, P.L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften,

Springer 259, New York, 1983.

[7] Noonan, J.W., Thomas, D.K., On the second Hankel determinant of areally mean p-valent

functions, Trans. Am. Math. Soc. 223, 337–346, 1976.

[8] Pommerenke, C., Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.


353

Singularities of the Darboux Ruled Surface of a Space Curve in the Pseudo-Galilean

Space

Tevfik ŞAHİN1

Abstract. In this article, we establish the singularity theory in a pseudo-Galilean space

G

1

3, a special case of Cayley- Klein spaces. We consider the cases where the Darboux ruled

surface in G

1

3 is diffeomorphic to some surfaces in the neighbourhood of a singular point. In

addition, we investigate the relationship between singularities of discriminant, bifurcation sets

of the function, and geometric invariants of curves in G

1

3.

Keywords. Height function, singularities, Darboux ruled surface, pseudo-Galilean

space.

AMS 2010. 58K05, 53A35.

References

[1] Mond, D., Singularities of the tangent developable surface of a space curve, The

Quarterly Journal of Mathematics, 40, 79-91, 1989.

[2] Sahin, T., Yılmaz, M., On singularities of the Galilean spherical darboux ruled surface of

a space curve in G

3, Ukranian Mathematical Journal, 62(10), 1597- 1610, 2010.

[3] Sahin, T., Yılmaz, M., The rectifying developable and the tangent indicatrix of a curve in

Galilean 3-space, Acta Mathematica Hungarica, 132(1-2), 154-167, 2011.

[4] Izumiya, S., Katsumi, H., Yamasaki, T., The Rectifying developable and the spherical

darboux image of a space curve, Geometry and Topology of Caustics '98 - Banach Center

Publications., 50, 137-149,1999.

1 Amasya University, Amasya, Turkey, [email protected], [email protected]




354

On the Hyperbolic Spinors and Split Quaternions

Tülay ERİŞİR1 and Mehmet Ali GÜNGÖR2

Abstract. In [1], the relationships between quaternions and spinors with complex

components and the kinematics of quaternion and spinor were given by J. Kronsbein. In

addition, Vivarelli offered a new approach to quaternions and spinors in the Euclidean 3-

space deriving from the vector formulation of the Euler’s theorem on the general

displacement of a rigid body with a fixed point in [2].

In this study, considering the studies mentioned above, firstly, we have introduced

hyperbolic spinors with two hyperbolic components and split quaternions. Then, we have

given the hyperbolic spinor representation of the rotations can be expressed with split

quaternions in Minkowski space. Finally, we have showed the hyperbolic spinor model of the

some characterizations of the rotations with the aid of split quaternions.

Keywords. Split quaternions, Hyperbolic spinors.

AMS 2010. 15A66, 51B20.

References

[1] Vivarelli, M. D., Development of spinors descriptions of rotational mechanics from

Euler’s rigid body displacement theorem, Celes. Mech. 32, 193-207, 1984

[2] Hacısalihoğlu, H. H., Hareket geometrisi ve kuaterniyonlar teorisi, Gazi Üniversitesi,

Fen-Edebiyat Fakultesi Yayinlari, Ankara, 2, 1983.

[3] Cartan, É., The theory of Spinors, The M.I.T. Press, Cambridge, 1966.

[4] Erisir, T., Gungor, M. A. and Tosun, M., Geometry of Hyperbolic Spinors Corresponding

to Alternative Frame, Adv. in Appl. Clifford Algebr., 25, 4, 799-810, 2015.

1 Erzincan University, Erzincan, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected]

http://link.springer.com/journal/6


355

An Introduction to Fibonaci, Lucas and Generalized Fibonacci Commutative

Quaternions

Ömer Tetik1 and The Mahmut Akyiğit2

Abstract. In this article, we will introduce the commutative Fibonacci quaternion, the

commutative Lucas quaternion and the commutative generalized Fibonacci quaternion. We

procure the relations between the commutative Fibonacci quaternion, the commutative Lucas

quaternion and the commutative generalized Fibonacci quaternion. To find these relations we

used the well-known identities related to the Fibonacci and Lucas numbers. Furthermore we

give Binet formulas and Cassini identities for commutative quaternions.

Keywords. Keyword one, keyword two, keyword three.

AMS 2010. 53A40, 20M15.

References

[1] Horadam, A. F., A Generalized Fibonacci Sequence, Amer. Math. Monthly 68, 455-459,

1961.

[2] Catoni, F., Cannata, R. and Zampetti, P., An Introduction to Commutative Quaternions,

Adv. Appl.Clifford Algebras 16, 1-28, 2006.

[3] Kosal, H. H., Akyigit, M. and Tosun, M., Consimilarity of Commutative Quaternion

Matrices, Miskolc Math. Notes 16 (2015), 965-977.

[4] Akyiğit, M., Kosal, H. H. and Tosun, M., Fibonacci Generalized Quaternions. Adv.

Appl. Clifford Algebras 24, 631–641, 2014.

[5] Dunlap, R. A., The Golden Ratio and Fibonacci Numbers, World Scientific, 1997.

[6] Vajda, S., Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Limited

Publ., England, 1989.

1 Sakarya university, Sakarya, Turkey, [email protected] 2 Sakarya university, Sakarya, Turkey, [email protected]


356

PARTICIPANTS


357

List of Participants of IECMSA-2017 Title Name-Surname University

Prof. Dr. Ahmet Yucesan Suleyman Demirel University

Prof. Dr. A. Ranjbari University of Tabriz,

Prof. Dr. Ayhan Tutar Ondokuz Mayis University

Prof. Dr. Ayse Nese Dernek Marmara University

Prof. Dr. Cihan Ozgur Balikesir University

Prof. Dr. Erol Kilic Inonu University

Prof. Dr. Eva López Sanjuán University of Extremadura

Prof. Dr. F. Nejat Ekmekci Ankara Univeristy

Prof. Dr. Gabil Adilov Akdeniz University

Prof. Dr. Hans-Peter Kunzi University of Cape Town

Prof. Dr. H. Hilmi Hacisalihoglu

Bilecik Seyh Edebali Univeristy

Prof. Dr. Hamid Vaezi University of Tabriz,

Prof. Dr. Hector Luna Garcia Universidad Autonoma Metropolitana

Prof. Dr. Hikmet Ozarslan Erciyes University

Prof. Dr. Ilham Aliyev Akdeniz University

Prof. Dr. Ismail Yaslan Pamukkale University

Prof. Dr. István Juhász Alfred Renyi Institute of Mathematics

Prof. Dr. Jose Juan Pena Gil Universidad Autónoma Metropolitana Azc.

Prof. Dr. Jun Kawabe Shinshu University


358

Prof. Dr. Kamile Sanli Kula Ahi Evran University

Prof. Dr. Karoly Bezdek University of Calgary

Prof. Dr. Kazim Ilarslan Kirikkale University

Prof. Dr. Kenjiro Yanagi Josai University

Prof. Dr. Keziban Orbay Amasya University

Prof. Dr. Léandre Rémi Université de Bourgogne-Franche-Comté

Prof. Dr. Levent Kula Ahi Evran University

Prof. Dr. Ljubisa Kocinac University of Nis

Prof. Dr. Lyudmila Romakina Saratov State University

Prof. Dr. Mahir Rasulov Beykent University

Prof. Dr. Mahmut Ergut Namik Kemal University

Prof. Dr. Manaf Manafli Adiyaman University

Prof. Dr. Mehmet Ali Gungor Sakarya University

Prof. Dr. Mehmet Gurdal Suleyman Demirel University

Prof. Dr. Melike Bildirici Yildiz Technical University

Prof. Dr. Metin Orbay Amasya University

Prof. Dr. Mikail Et Firat University

Prof. Dr. Mohammad Mursaleen

Aligarh Muslim University

Prof. Dr. Murat Altun Uludag University

Prof. Dr. Murat Tosun Sakarya University

Prof. Dr. Mustafa Caliskan Gazi University

Prof. Dr. Mustafa Kemal Yildiz Afyon Kocatepe University


359

Prof. Dr. Nazmiye Yahnioglu Yildiz Technical University

Prof. Dr. Necdet Batir Nevsehir HBV University

Prof. Dr. Nezihe Caliskan Gazi University

Prof. Dr. Nihal Yilmaz Ozgur Balikesir University

Prof. Dr. Perihan Dinc Artut Cukurova University

Prof. Dr. Praveen Agarwal Anand ICE

Prof. Dr. Sadik Keles Inonu University

Prof. Dr. Salih Aytar Suleyman Demirel University

Prof. Dr. Salih Rizgar University of Raparin

Prof. Dr. Sang Eon Han Chonbuk National University

Prof. Dr. Shigeru Furuichi Nihon University

Prof. Dr. Shilpi Jain Poornima College of Engineering

Prof. Dr. Sibel Yalcin Tokgoz Uludag University

Prof. Dr. Szymon Dolecki Burgundy University

Prof. Dr. Veli Kurt Akdeniz University

Prof. Dr. Victor Martinez-Luaces

Fing Udelar

Prof. Dr. Xiangfan Piao Hannam University

Prof. Dr. Yuri Melnikov Middle Tennessee State University

Assoc. Prof. Dr. Alp Arslan Kirac Pamukkale University

Assoc. Prof. Dr. Anna Malinova University of Plovdiv Paisii Hilendarski

Assoc. Prof. Dr. Antonio Miguel Márquez Durán

Pablo de Olavide University


360

Assoc. Prof. Dr. Ayse Sandikci Ondokuz Mayis University

Assoc. Prof. Dr. Aysegul Cetinkaya Ahi Evran University

Assoc. Prof. Dr. Aysegul Saglam Arslan

Karadeniz Technical University

Assoc. Prof. Dr. Ayten Pinar Bal Cukurova University

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Assoc. Prof. Dr. Bulent Yilmaz Marmara University

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Assoc. Prof. Dr. Emrah Evren Kara Duzce University

Assoc. Prof. Dr. Fazil Kayikci Yildiz Technical University

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Assoc. Prof. Dr. Filiz Tascan Guney Eskisehir Osmangazi University

Assoc. Prof. Dr. Filiz Yildiz Hacettepe University

Assoc. Prof. Dr. Gul Karadeniz Gozeri Istanbul University

Assoc. Prof. Dr. Gullazata Dairbayeva Al-Farabi Kazakh National University

Assoc. Prof. Dr. Gultekin Tinaztepe Akdeniz University

Assoc. Prof. Dr. Gunay Ozturk Kocaeli University

Assoc. Prof. Dr. H. Terence Liu Tatung University

Assoc. Prof. Dr. Handan Cerdik Yaslan

Pamukkale University

Assoc. Prof. Dr. Hatice Gul Ince Ilarslan

Gazi University


361

Assoc. Prof. Dr. Hifsi Altinok Firat University

Assoc. Prof. Dr. Hulya Bagdatli Yilmaz

Marmara University

Assoc. Prof. Dr. I. Onur Kiymaz Ahi Evran University

Assoc. Prof. Dr. Luca Guerrini Polytechnic University of Marche

Assoc. Prof. Dr. Mahmut Akyigit Sakarya University

Assoc. Prof. Dr. Memet Kule Kilis 7 Aralik University

Assoc. Prof. Dr. Messaoud Boulbrachene

Sultan Qaboos University

Assoc. Prof. Dr. Murat Kirisci Istanbul University

Assoc. Prof. Dr. Mustafa Ozdemir Akdeniz University

Assoc. Prof. Dr. Naceri Mostepha Oran School of Economics

Assoc. Prof. Dr. Nural Yuksel Erciyes University

Assoc. Prof. Dr. Oznur Ozkan Kilic Baskent University

Assoc. Prof. Dr. Rahmet Savas Istanbul Medeniyet University

Assoc. Prof. Dr. Resat Kosker Yildiz Technical Univesity

Assoc. Prof. Dr. Resat Yilmazer Firat University

Assoc. Prof. Dr. Selahattin Arslan Karadeniz Technical University

Assoc. Prof. Dr. Sergey Borodachev Ural Federal University

Assoc. Prof. Dr. Serpil Halici Pamukkale University

Assoc. Prof. Dr. Soley Ersoy Sakarya University

Assoc. Prof. Dr. Tulay Kesemen Karadeniz Technical University

Assoc. Prof. Dr. Violeta Vasilevska Utah Valley University

Assist. Prof. Dr. Abdulkadir Karakas Siirt University


362

Assist. Prof. Dr. Afsin Kursat Gazanfer

Bulent Ecevit University

Assist. Prof. Dr. Ali Bashan Bulent Ecevit University

Assist. Prof. Dr. Ali Hikmet Deger Karadeniz Technical University

Assist. Prof. Dr. Arzu Akgul Kocaeli University

Assist. Prof. Dr. Arzu Guleroglu Trakya University

Assist. Prof. Dr. Arzu Unal Ankara University

Assist. Prof. Dr. Banu Gunturk Baskent University

Assist. Prof. Dr. Bilge Inan Kilis 7 Aralik University

Assist. Prof. Dr. Bozidar Popovic University of Montenegro

Assist. Prof. Dr. Bulent Altunkaya Ahi Evran University

Assist. Prof. Dr. Cahit Aytekin Ahi Evran University

Assist. Prof. Dr. Cigdem Topcu Guloksuz

Bartin University

Assist. Prof. Dr. Erdal Bayram Namik Kemal University

Assist. Prof. Dr. Evrim Guven Kocaeli University

Assist. Prof. Dr. Ganimet Mulazimoglu Kizilirmak

Ahi Evran University

Assist. Prof. Dr. Gozde Ozkan Tukel Suleyman Demirel University

Assist. Prof. Dr. Gulistan Kaya Gok Hakkari University

Assist. Prof. Dr. Gulseli Burak Pamukkale University

Assist. Prof. Dr. Hakan Simsek Antalya International University

Assist. Prof. Dr. Hidayet Huda Kosal Sakarya University

Assist. Prof. Dr. Idris Oren Karadeniz Technical University


363

Assist. Prof. Dr. Ilknur Koca Mehmet Akif Ersoy University

Assist. Prof. Dr. Lucjan Sapa AGH University of Science and Technology

Assist. Prof. Dr. Luiz Guerreiro Lopes University of Madeira

Assist. Prof. Dr. Mehmet Baki Yagbasan


Assist. Prof. Dr. Mehmet Eyup Kiris Afyon Kocatepe University

Assist. Prof. Dr. Mehmet Gulbahar Siirt University

Assist. Prof. Dr. Merve Avci Ardic Adiyaman University

Assist. Prof. Dr. Mevlut Ersoy Suleyman Demirel University

Assist. Prof. Dr. Muge Meyvaci Mimar Sinan Fine Arts University

Assist. Prof. Dr. Murat Demirer Uskudar University

Assist. Prof. Dr. Mustafa Yildiz Bulent Ecevit University

Assist. Prof. Dr. Neriman Akdam Selcuk University

Assist. Prof. Dr. Nuray Caliskan Dedeoglu

Sakarya University

Assist. Prof. Dr. Nuri Celik Bartin University

Assist. Prof. Dr. Omer Unsal Eskisehir Osmangazi University

Assist. Prof. Dr. Onder Gokmen Yildiz Bilecik Seyh Edebali University

Assist. Prof. Dr. Osman Palanci Suleyman Demirel University

Assist. Prof. Dr. Osman Zeki Okuyucu Bilecik Seyh Edebali University

Assist. Prof. Dr. Rabia Cakan Akpinar Kafkas University

Assist. Prof. Dr. Selda Calkavur Kocaeli University


364

Assist. Prof. Dr. Serife Yilmaz Irtem Mehmet Akif Ersoy University

Assist. Prof. Dr. Serpil Sahin Amasya University

Assist. Prof. Dr. Seyda Kilicoglu Baskent University

Assist. Prof. Dr. Sukran Konca Bitlis Eren University

Assist. Prof. Dr. Tevfik Sahin Amasya University

Assist. Prof. Dr. Tulay Erisir Erzincan University

Assist. Prof. Dr. Yasemen Ucan Yildiz Technical University

Assist. Prof. Dr. Yasemin Kiymaz Ahi Evran University

Lecturer Dr. Meryem Odabasi Ege University

Lecturer Dr. Seda Karateke Istanbul Arel University

Lecturer Dr. Serdar Soylu Giresun University

Lecturer Furkan Aydin Kahramanmaras Sutcu Imam University

Dr. Abdessamad Barbara IMB, Université de Bourgogne

Dr. Anna Poskrobko Bialystok University of Technology

Dr. Anna Poskrobko Bialystok University of Technology

Dr. Boban Karapetrovic University of Belgrade

Dr. Burak Kurt Akdeniz University

Dr. Dojin Kim Kyungpook National University

Dr. Evgeny Gershikov Braude Academic College

Dr. Fatma Coban Yildiz Technical Univesity

Dr. Hosook Kim Korea Science Academy of KAIST


365

Dr. Huseyin Oguz Dumlupinar University

Dr. Ivana Kuzmanovic University of Osijek

Dr. Jasbir Singh Manhas Sultan Qaboos University

Dr. Jelena Stanojevic University of Belgrade

Dr. Jelena Stanojevic University of Belgrade

Dr. Jeongho Park UNIST

Dr. Kismet Kasapoglu Trakya University

Dr. Marcin Lawnik Silesian University of Technology

Dr. Marija Miloloza Pandur

University of Osijek

Dr. Onder Sener Ondokuz Mayis University

Dr. Pranas Katauskis Vilnius University

Dr. Selma Kirac Pamukkale University

Dr. Snezana S. Djordjevic University of Nis

Rsc. Assist. Dr. Fulya Ozaksoy Dogus University

Rsc. Assist. Dr. Mehmet Alper Ardic Adiyaman University

Rsc. Assist. Dr. Mustafa Ozkan Trakya University

Rsc. Assist. Dr. Omer Faruk Dogan Namık Kemal University

Rsc. Assist. Dr. Ozlem Ersoy Hepson Eskisehir Osmangazi University

Rsc. Assist. Dr. Tuba Gulsen Firat University

Rsc. Assist. Dr. Zehra Guzel Ergul Ahi Evran University

Rsc. Assist. Adnan Karatas Pamukkale University

Rsc. Assist. Bagdagul Kartal Erciyes University


366

Rsc. Assist. Deniz Pinar Sunaoglu Ankara University

Rsc. Assist. Dilek Varol Bayram Pamukkale University

Rsc. Assist. Ebru Aydogdu Kocaeli University

Rsc. Assist. Emrah Gurlek Ahi Evran University

Rsc. Assist. Erkan Agyuz Gaziantep University

Rsc. Assist. Esra Sengun Ermeydan

Ankara Yildirim Beyazit University

Rsc. Assist. Fadime Gokce Pamukkale University

Rsc. Assist. Fatih Aylikci Yildiz Technical University

Rsc. Assist. Fatma Ates Ankara University

Rsc. Assist. Fatma Bulut Bitlis Eren University

Rsc. Assist. Furkan Semih Dundar

Bogazici University

Rsc. Assist. Gulsah Aydin Sekerci Suleyman Demirel University

Rsc. Assist. Hilal Doganay Kati Bursa Technical University

Rsc. Assist. Kemal Taskopru Bilecik Seyh Edebali University

Rsc. Assist. Kubra Ozlu Deger Karadeniz Technical University

Rsc. Assist. Melek Sofyalioglu Gazi University

Rsc. Assist. Merve Ilkhan Duzce University

Rsc. Assist. Muhammet Sahal Yildiz Technical Univesity

Rsc. Assist. Sahsene Altinkaya Uludag University

Rsc. Assist. Esra Selcen Yakici Topbas

Gazi University

Rsc. Assist. Sibel Koparal Kocaeli University


367

Rsc. Assist. Sibel Sevinc Cumhuriyet University

Rsc. Assist. Tugba Baran Kaya Kirikkale University

Rsc. Assist. Zeliha Bedir Cumhuriyet University

Student Aditya Sivakumar Beaverton High School

Student Aysun Tok Onarcan Eskisehir Osmangazi University

Student Caglar Zeki Odabasi Erciyes University

Student Cemile Duygu Colak Kocaeli University

Student Gordeeva Irina Viktorovna

Perm National Research Polytechnic University

Student Ivan Tarasyuk Belarusian State Univrsity

Student Kemal Eren Sakarya Univerity

Student Koksal Kizilirmak Ahi Evran University

Student Miranda Gabelaia Iv. Javakhishvili Tbilisi State University

Student Mualla Birgul Huban Suleyman Demirel University

Student Roksana Brodnicka University of Silesia in Katowice, Poland

Student Veysel Kivanc Karakas


Student Yonghyeon Jeon Kyungpook National University

Student Zubeyde Er Ministry of Education of Turkey

Other Hasan Bayram Uludag University


368

6th - iecmsa · mahir rasulov..... 3 on some applications ... some results for matrix...

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