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ABSTRACT BOOK OF ICRAPAM 2014
INTERNATIONAL CONFERENCE on RECENT ADVANCES inINTERNATIONAL CONFERENCE on RECENT ADVANCES inPURE and APPLIED MATHEMATICS
6-9 November 2014, ANTALYA, TURKEYwww.icrapam.org
i
INTERNATIONAL CONFERENCE on RECENT
ADVANCES in PURE and APPLIED MATHEMATICS
(ICRAPAM 2014) 6-9 November 2014, ANTALYA, TURKEY
BOOK OF ABSTRACT
BOOK OF ABSTRACT ISBN: 978-975-00211-1-4 ISBN: 978-975-00211-1-4
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INTERNATINAOL CONFERENCE on RECENT ADVANCES in PURE and APPLIED MATHEMATICS (ICRAPAM 2014) 6-9 November 2014, ANTALYA, TURKEY
ORGANIZING COMMITTEE
Prof.Dr. Ekrem Savaş
Istanbul Ticaret University, Turkey
Prof.Dr. Feyzi Basar
Fatih University, Turkey
Prof.Dr. I. Naci Cangul
Uludag University, Turkey
Prof.Dr. M. Mursaleen
Aligarh Muslim University, India
Prof.Dr. Billy E. Rhoades
Indiana University, USA
Prof.Dr. R. Patterson
North Florida University, USA
Prof. Dr. Ants Aasma
Tallinn University of Technology, Estonia
Prof.Dr. Yılmaz Altun
Artvin Çoruh University, Turkey
Prof.Dr. Mehmet Dik
Rockford University, USA
Prof.Dr. Fatih Nuray
Afyon Kocatepe University, Turkey
Assoc. Prof. Dr. Bunyamin Aydin
Necmettin Erbakan University, Turkey
Assoc. Prof. Dr. Necip Simsek
Istanbul Ticaret University, Turkey
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INTERNATINAOL CONFERENCE on RECENT ADVANCES in PURE and APPLIED MATHEMATICS (ICRAPAM 2014) 6-9 November 2014, ANTALYA, TURKEY
SCIENTIFIC COMMITTEE
Prof.Emine Mısırlı, Turkey Prof.Huseyin Cakalli, Turkey Prof.Pratulananda Das, India Prof.Mikail Et, Turkey Prof.Metin Basarir, Turkey Prof.Agron Tato, Albania Prof.S. A. Mohiuddine, S. Arabia Prof.T. A. Chishti, India Prof.Billy E. Rhoades, USA Prof.Cihan Orhan, Turkey Prof.Ayhan Serbetci, Turkey Prof.Bilal Altay, Turkey Prof.Ljubisa Kocinac, Serbia Assoc.Prof.Ismail Ekincioglu, Turkey Prof.Ivana Djolovic, Serbia Prof.A. Sinan Cevik, Turkey Prof.Leiki Loone, Estonia Prof.H. M. Srivastava, Canada Prof.Seyit Temir, Turkey Prof.Halit Orhan, Turkey Prof.Huseyin Aydin, Turkey Prof.Vatan Karakaya, Turkey Assoc.Prof.Hamdullah Sevli, Turkey Prof.Hasan Akın, Turkey Assoc.Prof.Mehmet Gurdal, Turkey Prof.Amir Khosravi, Iran Prof.Seifedine Kadry, Kuwait Prof.Ali M. Akhmedov, Azerbaijan Prof.Gangaram S. Ladde, USA
Prof.Claudio Cuevas, Brazil Prof.Reza Saadati, Iran Prof.Allaberen Ashyralyev, Turkey Prof.Ram Mohapatra, USA Prof.Charles Swartz, USA Prof.Mujahid Abbas, Pakistan Prof.Aref Jeribi, Tunisia Prof.Yusuf Yayli, Turkey Prof.Husamettin Coskun, Turkey Prof.Abdullah Aziz Ergin, Turkey Prof.Cemil Tunc, Turkey Prof.Maria Zeltser, Estonia Prof.Salih Celebioglu, Turkey Prof.Kamalmani Baral, Nepal Prof.Ants Aasma, Estonia Prof.G. Das, India Prof.Ismail N. Cangul, Turkey Prof.Naim Braha, Republic of Kosova Dr.Lejla Miller Van-Wieren, Bosnia Prof.Murat Tosun, Turkey Prof.Yilmaz Simsek, Turkey Prof.Harry Miller, Bosnia Assoc. Prof.Bunyamin Aydin, Turkey Assoc.Prof.M. Kemal Karacan, Turkey Prof.Ali Fares, France Assoc.Prof.Ibrahim Canak, Turkey Assoc.Prof.M. Tamer Kosan, Turkey Prof.Fariq M. Bhatti, Pakistan
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Dear Colleaques May I offer you a warm welcome to the International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2014) . Conferences generally provide the opportunities for professional and practitioners of common and related interest to exchange the ideas, knowledge and experience objective of expanding their horizons and raising their potentials in a highly competitive world. I am sure that many of you will meet old friends and also will make new ones. In the deliberations to follow, you will hear some familiar mathematics as well as some that are quite new. Mathematics develops faster and faster, many outstanding problems are solved and new areas demand new approaches. New applications make new demands, and surprising connections are discovered between apparently unrelated fields. In this exciting process the world of mathematics unites different parts of the globe making knowledge transfer easier and more pertinent One advantage of a congress like this one is bringing together mathematicians and scientists with different interests under one roof to share their ideas and exchange progress on problems of varied interest. Many times, we get to see how interconnected are our subjects, and how the most interesting results often occur between quite different areas. I hope this will allow us to see more clearly the unity of mathematics and will encourage us to meet new challenges with determination and skill. I would like to take this opportunity to thank those who are involved in making this conference possible, especially to the hardworking members of the international conference committee. The most important people of this conference are you, the delegates. It is your contribution in paper presentation that brings synergy and vitality. Thank you all for coming from far and near to participate in this conference, and may the conference be an enjoyable and beneficial one for each of us. Prof. Dr. Ekrem SAVAS Chair of the conference
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Contents
Entanglement of Nanoresonator Interacting with Two Qubit Mahmoud Abdel-Aty
1
Boundary Value Problems for Delay Differential Equations Ravi P Agarwal
2
Determining Functionals for Damped Nonlinear Wave Equations Varga K. Kalantarov
3
Calderon-Zygmund theory for Divergence Form Operators in Generalized Morrey Spaces Lubomira Softova
4
On the Fine Spectra of n-th Band Triangular Matrices R. Birbonshi and P.D. Srivastava
5
The Navier-Stokes Equations-A Never Ending Challenge? Werner Varnhorn
6
The effect of the Terrorism on Tourism in Algeria Abdel Kader Boudi and Zouaoui Chikr el Mezouar
7
An Estimate on Volumes of Trajectory -Balls for Kaehler Magnetic Fields Toshiaki Adachi
8
Maximum Principle of Stochastic Switching Systems with Constraints Charkaz Aghayeva
9
A Note on Delay Parabolic Equations Deniz Agirseven
10
Analogies between Electromagnetism and Gravitation in General Relativity Zafar Ahsan
11
Existence of Solutions for a Three -Point Second-Order Boundary Value Problems with Integral Boundary Conditions Ummahan Akcan and Nuket Aykut Hamal
12
Necessary and Sufficient Conditions for the Solvability of Inverse Problem for a Class of Dirac Operators Ozge Akcay and Khanlar R. Mamedov
13
Integral Inequalities for Log -Convex Functions via Riemann - Liouville Fractional Integrals Ahmet Ocak Akdemir, Erhan Set and M. Emin Ozdemir
14
Generalization of Hausdorff Matrices F. Aydin Akgun and B.E. Rhoades
15
New Indices on Special Graphs Nihat Akgunes, Ahmet Sinan Cevik and Ismail Naci Cangul
16
The New Method of the Fine Spectrum of a Class of Operator Matrix in Some Sequences Spaces Ali M. Akhmedov
17
On Compactness of the Hardy Operator in the Weighted (.)(0, )pL ¥ Space Lutfi Akin and Yusuf Zeren
18
On the Behavior of a Class of p-Adic Dynamical Systems Hasan Akin, Farrukh Mukhamedov and Mutlay Dogan
19
Sharp Markov -Type Inequalities for Rational Functions on Several Intervals Mehmet Ali Akturk and Alexey Lukashov
20
Computing Fresnel Integrals via Modified Trapezium Rules M. Alazah, S.N. Chandler-Wilde and S. La Porte
21
Animation and Graphics to Understand Mathematics Abir Alharbi and Fairouz Tchier
22
On Certain Estimates for the Littlewood-Paley Operator Along Surfaces of Revolution Mohammed Ali
23
Stabilized Mixed Finite Element Method Q1-Q0 for the Generalized Stokes Problem Chibani Alima and Nasserdine Kechkar
24
Oscillation Theorem for One Spectral Problem Ziyatkhan S. Aliyev and Humay Sh. Rzayeva
25
Taylor Series Solution of Ordinary Differential Equations Fathi M Allan
26
Numerical Solution of an Integral Equation for Perpetual Bermudan Options Ghada Alobaidi
27
Bounds for Oscillatory Singular Integrals on Rⁿ Hussain Al Qassem, Leslie Cheng, Ayako Fukui and Yibiao Pan
28
vi
On Lacunary Difference Sequence of Fuzzy Numbers Hifsi Altinok, Mikail Et and Rifat Colak
29
Some Sequence Spaces Defined by the Weighted Mean Method and Modulus Function Selma Altundag and Bayram Sozbir
30
Weighted Statistical Convergence of Double Sequences Selma Altundag and Bayram Sozbir
31
On the Fine Spectrum of the Operator ( , , )B r s t over the Class of Convergent Series Selma Altundağ and Merve Abay
32
Exponential Extinction for Discrete Nicholson’s Blow Flies Model with Nonlinear Mortality and Patch Structure Terms Jehad Alzabut
33
Numerical Approach for Solving Fractional Pantograph Equation Ayse Anapali, Yalcin Ozturk and Mustafa Gulsu
34
Discovering Knowledge in Mathematics and Dynamic Modeling Halil Ardahan
35
Fejer type inequalities for Logarithmically Convex Functions Merve Avcı Ardıc, M. Emin Özdemir and Alper Ekinci
36
Stability in nonlinear neutral differential equations with infinite delay Abdelouaheb Ardjouni and Ahcene Djoudi
37
On a Canonical Form for Maxwell Equations by Geometric Calculus and Convergence of Finite Element Schemes for a Vlasov-Maxwell System Mohammed Asadzadeh
38
A Survey of Results in the Investigation of the Structure of Fractional Spaces Generated by Positive Operators in Banach Spaces Allaberen Ashyralyev
39
Fractional Spaces Generated by the Positive Difference Operator in the Half-Line ℝ+ and Their Applications Allaberen Ashyralyev and Sema Akturk
40
A Second Order of Accuracy Difference Scheme for a Fractional Schrödinger Differential Equation Allaberen Ashyralyev and Betul Hicdurmaz
41
Application of Summability Process on Baskakov-Type Korovkin Theory Ismail Aslan and Oktay Duman
42
On a Kirk–Type New Iteration Process Yunus Atalan, Vatan Karakaya, Kadri Dogan and Nour El Houda Bouzara
43
2-Quadratic Modules of Algebras Hasan Atik
44
Generalized Plane Deformation Solutions of the Shallow Located Cavity at the Anisotropic Rock Massif under the Actions of Elastic SH-Waves Lyazzat Atymtayeva and Bagdat Yagaliyeva
45
Peiffer Ideals, Precrossed Modules and Crossed Modules of R-Algebroids Osman Avcioglu and Ilker Akca
46
Two-Interval Sturm-Liouville Differential Operators in Direct Sum Spaces Kadriye Aydemir and Oktay Mukhtarov
47
The Variational Principle and Complexity of 𝑍𝑍𝑛𝑛Actions Bunyamin Aydin
48
On The Rough Statistically Cauchy Sequences Salih Aytar
49
Coincidence of Crisp and Fuzzy Functions Maliha Rashid, Akbar Azam and Nayyar Mehmood
50
An Iterative Method for Solving a Combined Inverse Problem of Reservoir Parameters Identification Aliya Azhibekova
51
Numerical Solution of System of Nonlinear Fredholm and Volterra Integral Equations Using Haar Wavelet Imran Aziz
52
A Study on Involute-Evolute Curves in Euclidean 3–Space Vildan Bacak and Nihat Ayyildiz
53
Bipas Flow with Dominant Poloidal Component A. A. Bachtiar and R. Kosasih
54
Large Sample Variance of Simulation Using Refined Descriptive Sampling 55
vii
Leila Baiche and Megdouda Ourbih-tari Best Proximity Points for a New Proximal Amitabh Banerjee
56
On the Domain of Riesz Mean in the Space 𝓛𝓛𝒔𝒔 Feyzi Basar
57
Theorems on Generalized Bose-Einstein and Fermi-Dirac Functions Abdelmejid Bayad
58
Non-Linear Characterization of Some Banach Spaces Maatougui Belaala
59
Numerical Solutions of Boundary Integral Equation by Using the Galerkin Method Menad Bendehiba
60
Weak Solution and Exact Controllability of the Caputo’s Fractional Cauchy Problem of Order α with 1 < α< 2 R. Boukhamla and S. Mazouzi
61
A Delay Second Order Set-Valued Differential Equation with Hukuhara Derivative Wafiya Boukrouk
62
Maximum Norm Analysis of a Nonmatching Grid Method for Semilinear Elliptic Variational Inequalities Messaoud Boulbrachene and Abida Harbi
63
Existence of Fixed Points for Continuous Operators in Banach Spaces Using Measure of Noncompactness and Under an Integral Condition N.H. Bouzara, V. Karakaya, Y. Atalan and K. Dogan
64
Some Geometric Properties Related to the Second Order Cesàro Operators Naim L. Braha and Valdete Loku
65
Lacunary Statistically Upward Continuity Huseyin Cakalli
66
Topological Indices of Subdivision Graphs Ismail Naci Cangul, Aysun Yurttas, Muge Togan, Ahmet Sinan Cevik
67
Existence of Solutions for Nonlinear Fractional Differential Equations with m-Point Integral Boundary Conditions Tugba Senlik Cerdik, Nuket Aykut Hamal andFulya Yoruk Deren
68
On Reciprocity Law of the Y(h,k) Sums Associated with the Two and Three-Term Polynomial Relations Elif Cetin, Yilmaz Simsek and Ismail Naci Cangul
69
On the Solvability and Maximal Regularity of Complete Abstract Differential Equations of Elliptic Type with General Robin Boundary Conditions in Holder Spaces Mustapha Cheggag, Angelo Favini, Rabah Labbas, Stéphane Maingot and Ahmed Medeghri
70
A Generalization of Gershgorin Circles Mao-Ting Chien
71
Implementation of Public-Key Cryptosystems on Embedded Devices Noureddine Chikouche and Walid Tayoub
72
Convergence Analaysis of Strang Splitting Method for Burgers-Huxley Equation Yesim Cicek and Gamze Tanoglu
73
Computing Homology Groups of Complexes of Matchings İsmet Cinar and Ismet Karaca
74
Deferred Statistical Convergence of Order α Muhammed Cinar, Mikail Et, Fatih Temizsu and Murat Karakas
75
Some Fixed Point Theorems for a Hybrid Type of Bogin-Popescu Mappings in Complete Metric Spaces L. Ciric, P. Promsilpchai and N. Petrot
76
On the Integral of Products of Higher-Order Bernoulli and Euler Polynomials M. Cihat Dagli and Mumin Can
77
Algebraic Invariant and Stability of Differential System Dahira Dali
78
Approximation by Chlodowsky Type q-Jakimovski-Leviatan Operators Ozge Dalmanoglu and Sevilay Kirci Serenbay
79
Some Large Sets in Countable Integral Domains Dibyendu De
80
Matrix Transformation of Statistically Convergent Sequences of Interval Numbers Shyamal Debnath and Subrata Saha
81
viii
A Study on Solvability of Functional Equations Arising in Dynamic Programming Deepmala and Arup Kumar Das
82
On the Fixed Points for Pointwise Lipschitzian Semigroups in Modular Function Spaces Buthinah A. Bin Dehaish
83
Monotone Iterative Technique and Existence Results for Fractional Functional Differential Equations Fulya Yoruk Deren, Nuket Aykut Hamal and Tugba Senlik Cerdik
84
Best Proximity Point Theorem for F-Contraction in a Complete Metric Space Lakshmi Kanta Dey
85
Relative and Tate Homology with Respect to Semidualizing Modules Zhenxing Di
86
A Differential Equation Model for the Dynamics of Youth Gambling Tae Sug Do and Young S Lee
87
On The Existence of Positive Solutions for the One-Dimensional 𝒑𝒑-Laplacian Boundary Value Problems on Time Scales Abdulkadir Dogan
88
On a Kirk-MP Iteration Process Kadri Dogan and Vatan Karakaya
89
On n - normed Cesàro Sequence Space ,nCes f Cenap Duyar and Oguzhan Kanber
90
Some Families of Generating Functions for Laguerre and Charlier Types d-Orthogonal Polynomials Duriye Korkmaz Duzgun and Esra Erkus Duman
91
Traveling Wave Solutions for Some Nonlinear Evolution Equations Serife Muge Ege and Emine Misirli
92
New Integral Inequalities of Ostrowski Type for Quasi-Convex Functions with Applications A.Ekinci, M. E. Ozdemir and E. Set
93
On Boundedness of Multilinear Singular Integral Operators in Lorentz Spaces İsmail Ekincioglu, Cansu Keskin and Ozgun Gurmen Alansal
94
Abstract Harmonic Analysis on Spacetime Poincare Group Kahar El-Hussein
95
On the Fine Spectra of a New Matrix Operator Over the Sequence Spaces 0c and c Sumeyra Elmaci and Vatan Karakaya
96
Eigenvalue Asymptotics and a Trace Formula for the Linear Damped Wave Equation Ahu Ercan and Etibar Panakhov
97
On Some Topological Properties of Generalized Sequence Spaces of Non-Absolute Type Sinan Ercan and Cigdem A. Bektas
98
A One-Sided Theorem for the Product of Abel and Cesàro Summability Methods Yilmaz Erdem and Ibrahim Canak
99
On the Spectrum of the Product Operator 𝑾𝑾 on the Sequence Space 𝒃𝒃𝒃𝒃 Ezgi Erdogan and Vatan Karakaya
100
Infinite Matrices and Invariant Means Rahmet Savas Eren
101
Groups and Irreducible Character Degrees Temha Erkoc
102
Energy Conservation for the 3-Coupled Nonlinear Schrödinger Equation by Using the Average Vector Field Method Sevim Ertug and Ayhan Aydin
103
On Lacunary Statistical Convergence of Order α of Difference Sequences Mikail Et
104
Colorings of Cycle Graph with Topological Approach Seher Fisekci and İsmet Karaca
105
Estimation of the Presicion Matrix of an Elliptically Symmetric Distribution Dominique Fourdrinier, Fatiha Mezoued and Martin T. Wells
106
Description of Bloch Spaces and Their Invariant Subspaces, and Related Questions Mubariz T. Garayev, Mehmet Gurdal and Ulas Yamancı
107
Common Fixed Points of Almost Generalized Contraction on Modular Spaces Ekber Girgin and Mahpeyker Ozturk
108
Functions Represented into a Newton Interpolating Series and Applications Ghiocel Groza
109
Numerical Approach for Magneto-Hydrodynamic Flow Passed Through a Wedge 110
ix
Murat Gubes and Galip Oturanc An Algorithm for Some Small Dimensional Representations Kubra Gul, Nurullah Ankaralioglu and Abdullah Cagman
111
Distribution of First Passage Times for Lumped States in Markov Chains Murat Gul and Salih Celebioglu
112
On the Generalized Bernoulli, Euler and Genocchi Polynomials Esra Guldogan and Esra Erkus Duman
113
On Boundedness of Singular Integral Operators Generated by Bessel Generalized Shift Operator in Weighted Beppo-Levi Spaces Serap Guner and Ismail Ekincioglu
114
Continuity of Superposition Operators on the Double Sequence Spaces of Maddox ( )pL Nihan Gungor and Birsen Sagir
115
Weyl Type Theorems for Unbounded Hyponormal Operators Anuradha Gupta and Karuna Mamtani
116
Composition Operators on Lorentz-Karamata-Bochner Spaces Anuradha Gupta and Neha Bhatia
117
A Note on Integral Inequalities for n-Time Differentiable Mappings Mustafa Gurbuz, Abdullah Yaradılmıs and M. Emin Ozdemir
118
Statistical Convergence and Some Questions of Operator theory Mehmet Gurdal and Ulas Yamanci
119
Statistical Convergence and C* -Operator Algebras Mehmet Gurdal and Mualla Birgul Huban
120
New Runge-Kutta Methods for Numerical Solutions of Multiplicative Initial Value Problems Yusuf Gurefe and Emine Misirli
121
On the Stability of Picard-Kirk-S Iterative Method Faik Gursoy and Vatan Karakaya
122
Time series modeling a temperature data in Bechar South West of Algeria Lahmar Habib and Zouaoui Chikr el Mezouar
123
On the Solutions of Some System of Difference Equation Nabila Haddad and Nouressadat Touafek
124
On the Exact Values of Wavelet Functions Mohamed Ali Hajji
125
Nonlinear Boundary Value Problems of p- Laplacian Fractional Differential Systems Nuket Aykut Hamal, Fulya Yoruk Deren and Tugba Senlik Cerdik
126
The Study of Existence and Uniqueness of Certain Fractional Differential Equations Boulares Hamid
127
Some Applications of Clifford Algebras and Octonions to Differential Geometry Hideya Hashimoto
128
A Bayesian Estimation of Traffic Intensity in M/M/∞ Queue Under Different Loss Functions Braham Hayette
129
Ideally Slowly Oscillating Continuity Bipan Hazarika
130
Equivalent Cauchy Sequences on Generalized Metric Spaces Elida Hoxha and Sidite Duraj
131
Generalization of Fixed Point Theorems in Quasi Cone Metric Space Relating to the Diameter of Orbits by Using a Comparison Function Elida Hoxha and Eriola Sila
132
Riesz Idempotent and Generalized Weyl’s Theorem for k-Quasi Class *nA Operators
Ilmi Hoxha and Naim L. Braha
133
A Morgan-Voyce Collacation Method for Numerical Solution of Generalized Pantograph Equations Ozgul Ilhan
134
Numerical Solution of Burger Equation by Using General Frechet Derivatives Combined with Differential Quadrature N. Imamoglu, G.Guraslan and G. Tanoglu
135
Lacunary Statistical Convergence of Order α in Probabilistic Normed Spaces Mahmut Isik
136
Evolution Modeling of NPZ and SIR Models with and without Diffusion Siraj-ul-Islam and Saeedullah Jan
137
x
Inverse Problem of Finding Diffusion Coefficient in Heat Equation with Nonlocal Boundary Conditions Mansur I. Ismailov and Bulent Ogur
138
Improved R-order of convergence for solving nonlinear equations Jai Prakash Jaiswal
139
On the Basis Number of the Wreath Product of Paths with Wheels and Some Related Problems M.M. M. Jaradat, M.S. Bataineh and M.K. Al-Qeyyam
140
Soft Completely Simple Semigroups Mehmet Kalkan and Haci Aktas
141
Determination of a Time Dependent Diffusion Coefficient of a Quasilinear Parabolic Equation in the Case of Nonlocal Conditions Fatma Kanca
142
On the Metric Dimension of Uniform Fuzzy Graphs Vasantha Kandasamy W.B and Regin Thangaraj
143
Some Common Coupled Fixed Point Theorems for Rational Expressions in Complex Valued 𝑮𝑮𝒃𝒃-Metric Spaces Neslihan Kaplan and Mahpeyker Ozturk
144
Stability and Accuracy of Time-Stepping Schemes for a Second-Order Wave Equation Samir Karaa
145
On H(θ)-Open Sets and Modifications on Hereditary Generalized Topological Spaces Umit Karabiyik
146
On Positive Solutions for Fourth-Order Four-Point Boundary Value Problems with Alternating Coefficient on Time Scales Ilkay Yaslan Karaca
147
A New Difference Sequence Set of Order a and Its Geometrical Properties Vatan Karakaya, Yunus Atalan and Mikail Et
148
Numerical Solution of Non-Linear Equations via Multiplicative Calculus Tolgay Karanfiller
149
Notes on Banach Contraction Principle Erdal Karapinar
150
Gröbner-Shirshov Bases of Some Exceptional Braid Groups Eylem G. Karpuz, Nurten Urlu and A. Sinan Cevik
151
Scalarization Methods in Multiobjective Optimization Refail Kasimbeyli
152
Some Convergence Theorems of Nonlinear Integral Functionals Jun Kawabe
153
Numerical and Exact Solutions for Time Fractional a Nonlinear Equation Dogan Kaya and Asıf Yokus
154
Common Fixed Point Results for (Ϝ,ψ)-Contractions in Ordered Partial Metric Spaces Meltem Kaya, Mahpeyker Ozturk and Hasan Furkan
155
Inverse Nodal Problem for p-Laplacian Sturm Liouville Equation H. Kemaloglu(Koyunbakan) and E. Yilmaz
156
On the Uniform Convergence of the Fourier Series for One Spectral Problem with a Spectral Parameter in a Boundary Condition Nazim B. Kerimov and Emir Ali Maris
157
Benjamin-Bona-Mahony Equation with Variable Coefficients: Conservation Laws Chaudry Masood Khalique
158
Excess of Retro Banach Frames G.Khattar and L.K. Vashisht
159
On Binomial Transformations of a Product of Rising and Falling Factorials Emrah Kilic and Halit Ozturk
160
An examinationtion on the positions of Frenet ruled surfaces along the evolute-involute curves, according to their normal vector fields in Euclidean 3-space Seyda Kilicoglu
161
Iterative Schemes and Common Fixed Point Problems on a Complete Geodesic Space Yasunori Kimura
162
Weighted Lacunary Statistical Convergence in a Locally Convex Topological Vector Space Sukran Konca and Metin Basarir
163
Numerical Solution for a Telegraph Equation via Bernestein Polynomials A. Kouadri and D. Belakroum
164
xi
Probability Theory on a Topological Monoid of Strings and Its Application to Statistical Machine Learning Hitoshi Koyano and Morihiro Hayashida
165
On generalized absolute Cesàro summability of orthogonal series Xhevat Z. Krasniqi
166
q-Barnes Multiple Bernoulli Polynomials Veli Kurt
167
Multi-Spectral Image Classification by Nonparametric Regression Splines Semih Kuter, Zuhal Akyurek and Gerhard-Wilhelm Weber
168
Controlling of Period-2 Unstable Orbits in a Bouncer System Using Patterns of H-Ranks Mantas Landauskas and Minvydas Ragulskis
169
Approximation by q-Balazs-Szabados Operators Nazim Mahmudov
170
Existence of Solutions of Fractional Boundary Value Problems with p-Laplacian Operator Nazım I.Mahmudov, Sinem Unul
171
Some Paranormed Difference Sequence Spaces of Order m-Derived by Generalized Means and Compact Operators Amit Maji and P. D. Srivastava
172
Fuzzy and Set-Valued Stochastic Differential Equations Marek T. Malinowski
173
A New Curvaturelike Tensor Field in an Almost Contact Riemannian Manifold Koji Matsumoto
174
Probability-Theoretic Foundations of International Trade Models Andrei Matveenko and Taras Hrendash
175
Asymptotic Behaviour of Solutions to Initial Boundary Value Problems for Marine Riser Equations Muge Meyvaci
176
Co-integration Analysis between Oil price and Algeria Inflation Zouaoui Chikr el Mezouar
177
New Shift-Compactness Results Harry I. Miller
178
Some Statistical Cluster Point Theorems Leila Miller, Van Wieren and Harry I. Miller
179
Homogenization of the Membrane Model of Perforated Shell Ait Yahia Mohamed and Mourad Lhannafi
180
A New Approach to Differential Evolution Algorithm for Solving Stochastic Programming Problems Ali Wagdy Mohamed
181
On the Travelling Wave Solutions of a Generalized Zakharov-Kuznetsov Equation Dimpho Millicent Mothibi and Chaudry Masood Khalique
182
Mixed d Semi Prefuzzy Topological Spaces and Some Results of Separation Axioms in Mixed d Semi Prefuzzy Topological Spaces Anjan Mukherjee
183
Second-Order Operator-Differential Equations with Transmission Conditions Oktay Sh. Mukhtarov, Kadriye Aydemir and Hayati Olgar
184
General Stability in Memory-Type Thermoelasticity with Second Sound Muhammad I. Mustafa
185
Time Series Forecasting Using Box-Jenkins Methodology Application on Census Data in Iraq Qais Mustafa
186
Fixed Point Theorems for Weakly T-Chatterjea and Weakly T-Kannan Contractions in b-Metric Spaces Zead Mustafa, Jamal Rezaei Roshan, Vahid Parvaneh and Zoran Kadelburg
187
Nonlinear three point boundary value problem Farid Nouioua
188
A Note on the Solutions of the Nonlinear Fractional Differential Equations Meryem Odabasi and Emine Misirli
189
Charge or Dipole Simulation Method for Approximations of Complex Analytic Functions Hidenori Ogata
190
On New Cesàro-Orlicz Double Difference Sequence Space Oguz Ogur and Cenap Duyar
191
Positive Solutions for a Singular Semipositone Dynamic System on Time Scales Arzu Denk Oguz and S. Gulsan Topal
192
xii
G2-Congruence Classes of Curves in Purely Imaginary Octonions Misa Ohashi
193
On the Riesz Basis Property of Eigenfunctions of One Boundary-Value Problem with Transmission Condition Hayati Olgar and Oktay Sh. Mukhtarov
194
Cauchy Type Problem for Nonlinear Fractional Integro-Differential Equations with Multivariate Mittag-Leffler Function in the Kernel Mehmet Ali Ozarslan
195
On a Double Integral Equation Including a Set of Two Variables Polynomials Suggested by Laguerre Polynomials Mehmet Ali Ozarslan and Cemaliye Kurt
196
On Some Fixed Point Results for Hybrid Rational a-Type Contractive Mappings Mahpeyker Ozturk
197
New Integral Operator for Solution of Differential Equations Ali Ozyapici
198
The Differential Geometry of Regular Curves on Regular Time-Like Surface Emin Ozyilmaz and Yusuf Yayli
199
Lacunary Invariant Statistical Convergence of Sequences of Sets with Respect to a Modulus Function Nimet Pancaroglu and Fatih Nuray
200
Stochastic Modelling and Inference of Model Parameters in Complex Biochemical Systems Vilda Purutcuoglu
201
Nash Equilibrium for Binary Convexities Taras Radul
202
Approximate Solutions to a Problem of Two Moving Boundaries Governed with Fractional Time Derivative in Drug Release Devices Rajeev and M.S. Kushwaha
203
Carleman Estimate for a One-Dimensional System of m-Coupled Parabolic PDEs with BV Diffusion Coefficients Hichem Ramoul
204
Waterloo Numbers and Their Relation to Pascal Triangle and Polygons Wajdi Mohamed Ratemi and Otman Basir
205
A Note on Derivations in Rings and Banach Algebras Mohd Arif Raza
206
On Global Solutions of Fractional Evolution System Samira Rihani and Amor Kessab
207
Risk Modeling Insurance IARD by Decision Trees and V-TEST Khadidja Sadi, Nora Lounnici and Hanya Kherchi
208
Boundedness of Superposition Operators on the Double Sequence Spaces of Maddox ( )pL Birsen Sagir and Nihan Gungor
209
Some Convergence Results for Nonexpansive Mappings in Uniformly Convex Hyperbolic Spaces Aynur Sahin and Metin Basarir
210
A Multi-Criteria Neutrosophic Group Decision Making Metod Based TOPSIS for Supplier Selection Ridvan Sahin and Muhammed Yigider
211
Existence and stability of a damped wave equation with two delayed terms in boundary Zitouni Salah and Amiar Rachida
212
Lp Boundedness for Marcinkiewicz Integrals and Extrapolation Salti Samareh
213
Approximation Properties of Max-Product Operators Engin Sari
214
Some Inequalities Associated with the Hermite-Hadamard-Fejér Type for Convex Function Mehmet Zeki Sarikaya, Hatice Yaldiz and Samet Erden
215
Equivalent Norms in Nikol’skij-Triebel-Morrey Spaces and Lizorkin-Triebel-Morrey Spaces Merey Sautbekova
216
Ideal Statistical Quasi Cauchy Sequences Ekrem Savas and Huseyin Cakalli
217
SBT - Hausdorff Space Guzide Senel and Naim Cagman
218
Rate of Convergence of Generalized Favard-Szàsz Type Operators 219
xiii
Sevilay Kirci Serenbay Sufficient Tauberian Conditions for the Weighted Mean Summability of Sequences of Fuzzy Numbers Sefa Anil Sezer, Ibrahim Canak and Zerrin Onder
220
On Fine Spectra and Subspectrum (Approximate Point, Defect and Compression) of Operator with Periodic Coefficients Necip Simsek and Vatan Karakaya
221
Pretopologies for Structuring the Digital Plane Josef Slapal
222
Linear Parabolic Boundary-Value Problems in Generalized Morrey Spaces Lubomira Softova
223
Main Chaos-Based Image Encryption Algorithms Ibtissem Talbi
224
The Way-Below Soft Set Relation Bekir Tanay and Gozde Yaylali
225
A New Operator Splitting Method for Non-Linear Systems and Its Abstract Analysis Gamze Tanoglu and Sila Ovgu Korkut
226
A New Three Step Iteration and Convergence of Pseudocontractive Mappings Balwant Singh Thakur
227
Contrastings on Textual Entailmentness and Algorithms of Syllogistic Logics Selcuk Topal and Tahsin Oner
228
Some Tauberian Conditions for the Weighted Mean Method of Summability Umit Totur and Ibrahim Canak
229
A Tauberian Theorem for the Power Series Method of Summability Umit Totur and Ibrahim Canak
230
Some Tauberian Theorems for the Logarithmic Integrability Method Umit Totur and Muhammet Ali Okur
231
Eigenvalues for Kaehler Graphs of Connected Product Type Yaermaimaiti Tuerxunmaimaiti
232
Generalized Weighted Norlund Mean and Ideal Convergence Orhan Tug and Feyzi Basar
233
On Convergence Methods of Functions Defined on Time Scales Ceylan Turan and Oktay Duman
234
Classification of Integral Curves of a Linear Vector Field on (2n+1) Dimensional Semi-Euclidean Space Tunahan Turhan and Nihat Ayyildiz
235
More Results on the Upper Solution Bounds of the Continuous Algebraic Riccati Matrix Equation Zubeyde Ulukok
236
Global Attractors for Quasilinear Parabolic-Hyperbolic Equations Governing Longitudinal Motions of Nonlinearly Viscoelastic Rods Suleyman Ulusoy
237
()-boundedness of localization operators involving watson transform associated to regular representations and wavelet multipliers S.K.Upadhyay
238
A Maximum Modulus Estimate for the Steady Stokes Equations Werner Varnhorn
239
Approximation Properties of Chlodowsky-Durrmeyer Type q-Bernstein-Schurer-Stancu Operators Tuba Vedi and Mehmet Ali Ozarslan
240
On Relative Homology Groups of Khalimsky Spaces Tane Vergili and İsmet Karaca
241
Stabilization in a n-Species Chemotaxis System with a Logistic Source Wang Wenjia
242
Hermite-Hadamard-Fejer Type Inequalities Hatice Yaldiz
243
A Note on a Functional Identity on Lie Ideals Nihan Baydar Yarbil
244
On the Higher Order Gaussian Curvatures in Lorentz Space Ayse Yasar Yavuz and F. Nejat Ekmekci
245
On Soft Dual Space of Soft Normed Spaces Murat Ibrahim Yazar, Yilmaz Altun and Tunay Bilgin
246
On Some Properties of the B-Convex Functions 247
xiv
İlknur Yesilce and Gabil Adilov Statistical Convergence of Multiple Sequences on a Product Time Scale Emrah Yilmaz, Yavuz Altin and Hikmet Koyunbakan
248
Optimality Conditions for Non-Lipschitz Optimization Nurullah Yılmaz and Ahmet Sahiner
249
Groups and Graphs Utku Yilmazturk
250
Center Coloring and the Other Colorings Zeynep Ors Yorgancioglu and Pinar Dundar
251
Dual Transformations and One-Paremeter Motions Gulsum Yuca and Yusuf Yayli
252
On Boundedness of Fractional Maximal Operator in the Weighted (.)(0,1)pL Space Yusuf Zeren and Lutfi Akin
253
Global Existence and Asymptotic Properties of the Solution to a Two-Species Chemotaxis System Qingshan Zhang
254
One Dimensional Model of Biodegradable Elastic Curved Rods Bojan Zugec
255
Convergence Analysis and Numerical Solution of Benjamin-Bona-Mahony Equation by Lie-Trotter Splitting Fatma Zurnaci, Nurcan Gucuyenen andMuaz Seydaoglu and Gamze Tanoglu
256
1
Entanglement of Nanoresonator Interacting with Two Qubit Mahmoud Abdel-Aty
Vice President African Academy of Science University of Science and Technology at Zewail City Egypt [email protected]
Abstract: In this communication we discuss different aspects of multi-qubit system interacting with nanoemchanical resonators. Information dynamics of charge qubits coupled to a nanomechanical resonator under influence of both a phonon bath in contact with the resonator and irreversible decay of the qubits is considered. The focus of our analysis is devoted to multi-particle entanglement and the effects arising from the coupling to the reservoir. Even in the presence of the reservoirs, the inherent entanglement is found to be rather robust. Due to this fact, together with control of system parameters, the system may therefore be especially suited for quantum information processing. Our findings also shed light on the evolution of open quantum many-body systems. For instance, due to intrinsic qubit-qubit couplings our model is related to a driven XY spin model.
2
Boundary Value Problems for Delay Differential Equations Ravi P Agarwal
Texas A&M–Kingsville [email protected]
Abstract: We develop an upper and lower solution method for second order boundary value problems for nonlinear delay differen-tial equations on an infinite interval. Sufficient conditions are imposed on the nonlinear term which guarantees the existence of a solution between a pair of lower and upper solutions, and triple solutions between two pairs of upper and lower solu-tions. An extra feature of our existence theory is that the obtained solutions may be unbounded. Two examples which show how easily our existence theory can be applied in prac-tice are also illustrated.
3
Determining Functionals for Damped Nonlinear Wave Equations
Varga K. Kalantarov Koc University, Istanbul [email protected]
Abstract: The problem of determining by finitely many functionals of asymptotic behavior of solutions as t → ∞ to initial boundary value problems for strongly damped semi-linear wave equations and wave equations with nonlinear damping terms will be discussed.
4
Calderon-Zygmund theory for divergence form operators in
Generalized Morrey spaces
Lubomira Softova
Department of Civil Engineering, Design, Construction and Environment, Second University of Naples, Italy
Abstract: We study the regularity properties of the solutions of the Cauchy-Dirichlet problem
for divergence form parabolic equations with measurable data in non-smooth domains.
Problems like these arise in the modeling of composite materials and in the mechanics of
membranes and films of simple non-homogeneous materials which form a linear laminated
medium. Assuming partial BMO smallness of the coefficients and Reifenberg flatness of the
boundary of the underlying domain, we develop a Calderon-Zygmund type theory for such
parabolic operators in the settings of the generalized Morrey under various conditions on the
weight. As consequence of the main result, we get regularity in parabolic Morrey scales for the
spatial gradient of the weak solutions to the considered problem.
Key words: Calderon – Zygmund estimates, generalized Morrey spaces, weak solutions,
Cauchy-Dirichlet problem
References:
[1] S. Byun, L. Softova, Gradient estimates in generalized Morrey spaces for parabolic operators, submitted
[2] V. Guliyev, L. Softova, Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients,
submitted
.
5
On the Fine Spectra of n-th Band Triangular Matrices R. Birbonshi and P.D. Srivastava
Department of Mathematics, Indian Institute of Technology Khargapur, India [email protected], [email protected]
Abstract: Bilgic and Furkan [4] have studied the fine spectra of the operator B(r,s) over the sequence spaces 𝑙𝑙𝑝𝑝 and 𝑏𝑏𝑏𝑏𝑝𝑝; (1 < 𝑝𝑝 < ∞) which are generalized by Furkan, Bilgic and Basar [7] to B(r,s,t). The fine spectra of the operator B(r,s) are studied by Altay and Basar [1] and Furkan, Bilgic and Kayaduman [5] over the sequence spaces 𝑐𝑐0 ,𝑐𝑐, 𝑙𝑙1 & 𝑏𝑏𝑏𝑏 respectively. Furkan, Bilgic and Altay [6], Bilgic and Furkan [3] generalized these result to B(r,s,t) over the sequence space 𝑐𝑐0 , 𝑐𝑐, 𝑙𝑙1 & 𝑏𝑏𝑏𝑏 respectively. The fine spectra of the upper triangular double band matrices over the sequence spaces 𝑐𝑐0, 𝑐𝑐 have been studied by Karakaya and Altun [8]. Later on, Altun [2] has examined the fine spectra of triangular Toeplitz operator over the sequence spaces 𝑐𝑐0 & 𝑐𝑐. Here we have derived some general results for finding the spectrum and fine spectrum of n th band lower and upper triangular matrix. Our results include the corresponding results of [1, 3, 4, 5, 6, 7, 8] as well as the results of [2]. References: [1] B. Altay, F. Basar, “On the fine spectrum of the generalized difference operator B(r,s) over the sequence spaces 𝑐𝑐0 and 𝑐𝑐” , Int. J. Math. Sci. 18 (2005) 3005-3013. [2] M. Altun, “On the fine spectra of tranglar Toeplitz operator”, App. Math. Comput. 217 (2011) 8044-8051. [3] H. Bilgic, H. Furkan, “On the fine spectrum of the operator B(r,s,t) over the sequence spaces 𝑙𝑙1 and 𝑏𝑏𝑏𝑏”, Math. Comput. Model. 45 (7-8) (2007) 883-891. [4] H. Bilgic, H. Furkan, “On the fine spectrum of the generalized difference operator B(r,s) over the sequence spaces 𝑙𝑙𝑝𝑝 and 𝑏𝑏𝑏𝑏𝑝𝑝, 1 < 𝑝𝑝 < ∞”, Nonlinear Anal. 68 (3) (2008) 499-506. [5] H. Furkan, H. Bilgic, K. Kayaduman, “On the fine spectrum of the generalized difference operator B(r,s) over the sequence spaces 𝑙𝑙1 and 𝑏𝑏𝑏𝑏”, Hokkaido Math. J. 35 (2006) 897-908. [6] H. Furkan, H. Bilgic, B. Altay, “On the fine spectrum of the operator B(r,s,t) over 𝑐𝑐0 and 𝑐𝑐”, Comput. Math. Appl. 53 (2007) 989-998. [7] H. Furkan, H. Bilgic, F. Basar, “On the fine spectrum of the operator B(r,s,t) over the sequence spaces 𝑙𝑙𝑝𝑝 and 𝑏𝑏𝑏𝑏𝑝𝑝; 1 < 𝑝𝑝 < ∞”, Comput. Math. Appl. 60 (2010) 2141-2152. [8] V. Karakaya, M. Altun, “Fine spectra of upper triangular double-band matrices”, J. Comput. Appl. Math. 234 (2010), 1387-1394.
6
The Navier-Stokes Equations – A Never Ending Challenge? Werner Varnhorn
Institute of Mathematics, Kassel University, Germany [email protected]
Abstract: We consider the nonstationary nonlinear three-dimensional Navier-Stokes equations 𝑏𝑏𝑡𝑡 − 𝜈𝜈 ∆𝑏𝑏 + 𝑏𝑏 • 𝛻𝛻𝑏𝑏 + 𝛻𝛻𝑝𝑝 = 𝑓𝑓,
∇ • v = 0, (N) v|∂Ω = 0, 𝑏𝑏|𝑡𝑡 = 0 = 𝑏𝑏0 . These equations describe the motion of a viscous incompressible fluid flow: The vector function v = v(t, x) = (v1 (t, x), v2 (t, x), v3(t, x)) denotes the velocity and the scalar function p = p(t, x) the pressure of the fluid at time t > 0 in x = (x1, x2 , x3) ∈ Ω. Here the constant ν > 0 represents the kinematic viscosity, the vector function f = (f1 (t, x), f2(t, x), f3 (t, x)) is the given external force density, and the steady vector function v0 = (v01 (x), v02 (x), v03 (x)) denotes the prescribed initial velocity at time t = 0. In the following we consider
the fluid flow always in a bounded domain Ω ⊂ R3 with smooth boundary ∂Ω of class C 2,µ (0 < µ ≤ 1). The system (N) occupies a central position in the study of nonlinear partial differential equations, dynamical systems, scientific computation, and classical fluid dynamics. Because of the complexity and variety of fluid dynamical phenomena on the one hand, and the simplicity and exactitude of the equations’ shape on the other hand, a strong depth and beauty is expected in the mathematical theory. It is a source of pleasure and fascination that many of the most important questions in the theory remain yet to be answered. So the famous American Clay Mathematics Institute created the Navier-Stokes Millennium Price Problem and offered one Million Dollar for its solution, stating: Although the Navier-Stokes equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory, which will unlock the secrets hidden in the Navier-Stokes equations. The modern mathematical theory of the Navier-Stokes equations (N) started with the pioneer- ing work of Jean Leray in 1933 - 34. Leray was the first to use methods of functional analysis for the treatment of partial differential equations. He developed the concept of weak solutions for the Navier-Stokes Cauchy problem and proved their existence globally in time long before the theory of distributions was established by Schwartz and even before Sobolev systematically introduced the spaces which bear his name. Leray has laid the basis of the mathematical theory for (N) as we know it today, and he has introduced many tools and ideas used constantly since then. The lecture introduces the Navier-Stokes equations from a historical and physical point of view, touches some fundamental mathematical problems of viscous incompressible fluid flow and ends up with recent regularity results on strong solutions.
7
The effect of the Terrorism on Tourism in Algeria Abdel Kader Boudi and Zouaoui Chikr el Mezouar
Laboratory of Study Economics and Local Development in South West of Algeria
University of Bechar
E-mail : [email protected] and [email protected]
Abstract: The objective of this research is to show the effect of terrorism on tourism in Algeria and to show also
how terrorism event has damaged the tourism sector in Algeria and how it has deeply affected the number of foreign
arrival tourists to Algeria using mathematical model.
References:
[1] Boudi Abdel Kader (2006) “The importance of marketing tourism in the development of tourism in Algeria”
PhD theses, University of Algeria.
[2] Chatfield, C. (2004) “The Analysis of Time Series: An Introduction”. 6th Edn., New Jersey:Chapman & Hall.
[3] Neter, J. , Wasserman, W. and Kutner, M.H. (1989) “ Applied Regression Models ” 2nd Edn,IRWIN.
[4] Wei, W.W.S. (1990) “Time series analysis: Univariate and multivariate methods”. Addison-Wesley Publishing
Company, Inc.
8
An Estimate on Volumes of Trajectory-Balls for Kaehler Magnetic Fields Toshiaki Adachi
Department of Mathematics, Nagoya Institute of Technology, Nagoya, Japan [email protected]
Abstract: Let M be a Kaehler manifold with complex structure J. We call constant multiples of the Kaehler form BJ on M Kaehler magnetic fields. Under the action of a Kaehler magnetic field Bk = kBJ, the motion of an electric charged particle is expressed as a smooth curve γ which is parameterized by its arclength and satisfies the equation ∇𝛾𝛾′𝛾𝛾′ = 𝑘𝑘𝑘𝑘𝛾𝛾′. We say such curves to be trajectories for Bk. As trajectories for the trivial magnetic field B0 are geodesics, we can regard them as natural extended objects of geodesics. Since geodesics play quite an important role in the study of Riemannian manifolds (cf. [3]), we consider that trajectories give us some clues to study Kaehler manifolds from the Riemannian geometric point of view. In this paper, we study trajectory-harps which consist of trajectories and geodesics and compare those on a general Kaehler manifolds and those on a complex space form. By use of a result corresponding to the Toponogov’s comparison theorem on triangles, we give an estimate on volumes of trajectory-balls from below. Keywords: Kaehler magnetic fields; Trajectories; Kaehler manifolds; Comparison theorems; Trajectory-harps; Trajectory-balls. References: [1] T. Adachi, “A theorem of Hadamard-Cartan type for Kaehler magnetic fields”, J. Math. Soc. Japan 64(2012), 969-984. [2] P. Bai, “Volume densities of trajectory-balls and trajectory-spheres for Kaehler magnetic fields”, Prospects of Differential Geometry and Related Fields, World Scientific (2013), 115-128. [3] J. Cheeger, D. G. Evin, “Comparison Theorems in Riemannian Geometry”, 1975, North-Holland Pub. Co.
9
Maximum Principle of Stochastic Switching Systems with Constraints Charkaz Aghayeva
Department of Industrial Engineering, Anadolu University, Eskisehir, Turkey [email protected]
Abstract: A lot of theoretical and numerical advances have recently been realized in the field of stochastic control. Necessary conditions satisfied by an optimal solution, play an important role for investigation of optimization and optimal control problems. This paper is devoted to stochastic optimal problem of switching control systems. Dynamics of these processes are governed by the collection of stochastic differential equations with control terms in the drift and diffusion coefficients. Necessary conditions of optimality for described systems with the restrictions in each interval are obtained. The constraints on the transitions are described by the set of functional inclusions. Ekeland's variational principle are applied to prove maximum principle in general form. The necessary conditions developed in this study can be viewed as a stochastic analogues of the problems that are formulated in [1,2,3]. And main result, the maximum principle for considered problem, is a natural evolution of the results given in [4,5,6]. Keywords: Stochastic Control Systems; Necessary Condition of Optimality; Optimal Switching Systems; Condition of Transversality. References: [1] S. Bengea, A. Raymond, “Optimal Control of Switching systems”,Autom., 41(2005),11-27. [2] D. Capuzzo, L. Evans, “Optimal Switching for ordinary differential equations”, SIAM,Journal on Con. and Optimization, 22(1984), 143-161. [3] T. Seidmann, “Optimal control for switching systems”, Proceedings of the 21st Ann. Con. in Formations Science and Systems ,(1987),485-489. [4] Ch. Aghayeva, Q.Abushov,“The maximum principle for the nonlinear stoch- astic optimal control problem of switching systems”, JOGO,56(2013), 341-352. [5] Q. Abushov, Ch. Aghayeva, “Stochastic maximum principle for the nonlinear optimal control problem of switching systems”, CAM, 259(2014),371-376. [6] Ch. Aghayeva, Q. Abushov, “Stochastic maximum principle for switching systems”,Proc. of the 4th Int.Conf: Prob. of Cyb.and Infor.,3(2012), 198-201.
10
A Note on Delay Parabolic Equations Deniz Agirseven
Department of Mathematics, Trakya University, Edirne, Turkey [email protected]
Abstract: In this study, two main theorems on well-posedness of the initial value problem for delay differential equations with unbounded operators acting on delay terms.are established. The coercive stability estimates in Hölder norms for solutions of delay parabolic equations are obtained. Theorems on well-posedness of first and second order of accuracy of difference schemes for approximate solutions of the initial value problem for delay differential equations with unbounded operators acting on delay terms are established. The coercive stability estimates for the solution of difference schemes of delay parabolic equations are obtained. Finally the illustrative numerical experiment for numerical solution of delay parabolic equations is given. Keywords: Delay parabolic equations, fractional spaces, stability, well-posedness.
References: [1] A. Ashyralyev, D. Agirseven, “On Convergence of Difference Schemes for Delay Parabolic Equations”, Computers and Mathematics with Applications, 66(7), 1232-1244, 2013. [2] A. Ashyralyev, D. Agirseven, “Well Posedness of Delay Parabolic Difference Equations”, Advances in Difference Equations, 2014:18, 2014. [3] A. Ashyralyev, D. Agirseven, “Well Posedness of Delay Parabolic Equations with Unbounded Operators Acting on Delay Terms”, Boundary Value Problems, 2014:126, 2014.
11
Analogies between Electromagnetism and Gravitation in General Relativity Zafar Ahsan
Department of Mathematics, Aligarh Muslim University, Aligarh, India [email protected]
Abstract: The correspondence between electromagnetism and gravitation is very rich and detailed. Some of these correspondence are still uncovered while some of them are further developed. This correspondence is reflected in the Maxwell-like form of the gravitational field tensor (the Weyl tensor), the super-energy-momentum tensor (the Bel-Robinson tensor) and the dynamical equations (the Bianchi identities). In this talk, we shall discuss some of the analogies between electromagnetism and gravitation. It is known that electromagnetic field has two invariants and the vanishing of these invariants is the criterion for the existence of electromagnetic radiation. Since the gravitational field is truly characterized by the Riemann curvature tensor, based on the analogy with electromagnetism, we shall make a study of the invariants of Riemann curvature tensor. The Riemann curvature tensor has fourteen invariants. There are four invariants of the Weyl curvature tensor, three invariants of the Einstein curvature tensor, six invariants of the mixed Weyl and Einstein curvature tensors and there is a Ricci scalar. The study of these invariants is important in general theory of relativity since they allow a manifestation of coordinate invariant characterization of certain geometrical properties of the space-times. In empty space-time, Riemann tensor reduces to Weyl tensor and thus there are four invariants of the Riemann tensor. Here, we have obtained a criterion for the existence of gravitational radiation in terms of these invariants and it is found that the vanishing of these invariants provides the existence of gravitational radiation. This assertion is then verified for some well known metrics of general relativity. Also, it is known that a physical field is always produced by a source, which is termed as its charge. Manifestation of fields when charges are at rest is called electric and magnetic when the charges are in motion. This general feature is exemplified by the Maxwell's theory of electromagnetism from where the terms of electric and magnetic are derived. This decomposition can be adapted in general relativity and the Weyl tensor can be decomposed into electric and magnetic parts. Based on this decomposition, we have made a classification of the space-times and have established the conditions under which a given space-time is purely electric or purely magnetic. It is found that the Petrov types III and N space times are neither purely electric nor purely magnetic. This classification scheme is then applied to number of known space-time solutions of general relativity. Moreover, the decomposition of Riemann tensor involves certain irreducible tensors. In empty space-time, pure gravitational radiation field is described by the Weyl tensor. However, the Weyl tensor is still pertinent when the gravitational waves propagate through matter. It was thought by Lanczos, in 1962, that the Weyl tensor can also be derivable from a simpler tensor field. This tensor field is now known as Lanczos potential. Moreover, it is known that an electromagnetic field can be generated by a potential, the question then arises that whether or not it is possible to generate the gravitational field through a potential. The answer is in affirmative - this indeed can be done through Lanczos potential. Using the methods of general observers, we have made a study of Lanczos potential. The kinematical quantities such as expansion, shear, rotation, etc., have been translated into the language of the tetrad formalism given by Newman and Penrose (known as spin coefficient formalism); and the Lanczos potential for perfect fluid space-times have been obtained in terms of the spin coefficients. The gravitational potentials for the Godel cosmological solution and Kasner metric has also been obtained. Using another tetrad formalism, known as compacted spin coefficient formalism, a potential for a Petrov type D space-time has been obtained. These results are then applied to a Kerr black hole.
12
Existence of Solutions for a Three-Point Second-Order Boundary Value Problems with Integral Boundary Conditions Ummahan Akcan and Nuket Aykut Hamal
Department of Mathematics, Faculty of Science, Anadolu University, Eskisehir, Turkey Department of Mathematics, Ege University, Bornova, Izmir, Turkey
[email protected], [email protected]
Abstract: In this study, we make use of the monotone iterative technique to verify the existence of concave symmetric positive solutions of a second-order three-point boundary value problem with integral boundary conditions. The interesting point here is that the nonlinear term f depends on the first-order derivative explicitly. An example which supports our result is also indicated. Keywords: Boundary value problems; Symmetric positive solution; Integral bıundary condition; Monotone iterative technique. References: [1] V. A. Il'in, E.I. Moiseev, “Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator”, Differ. Eqn. 23 (1987), 979-987. [2] F. Hao, “Existence of symmetric positive solutions for m-point boundary value problems for second-order dynamic equations on time scales”, Math. Theory Appl. (Changsha) 28 (2008), 65-68. [3] W. Feng, “On an m-point boundary value problem”, Nonlinear Anal. 30 (1997) , 5369-5374. [4] C.P. Gupta, “A generalized multi-point boundary value problem for second order ordinary differential equations”, Appl. Math. Comput. 89 (1998), 133-146. [5] Y. Sun, X. Zhang, “Existence of symmetric positive solutions for an m- point boundary value problem”, Bound. Value Probl. Art. ID 79090 (2007) 14 pp. [6] H. Pang, Y. Tong, “Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions”, Boundary Value Problem. (2013):150 doi:10.1186/1687-2770-2013-150.
13
Necessary and Sufficient Conditions for the Solvability of Inverse Problem for a Class of Dirac Operators
Ozge Akcay and Khanlar R. Mamedov Mathematics Department, Mersin University,
Mersin, Turkey [email protected] and [email protected]
Abstract: In this work, the boundary value problem generated by one dimensional Dirac differential equations system ' ( ) ( ) , 0By x y x y xlr p+ W = < < with boundary condition 1 1(0) ( ) 0y y p= = is considered, where
0 11 0
B æ ö= ç ÷-è ø
, ( ) ( ) ( )( ) ( )
p x q xx
q x p xæ ö
W = ç ÷-è ø, ( ) ( )
( )1
2
y xy x
y xæ ö
= ç ÷è ø
,
( ), ( )p x q x are real valued functions, 2( ) (0, )p x L pÎ , 2( ) (0, )q x L pÎ , l is a spectral parameter,
( )1,
,xr
aì
= íî
0 ,
,x a
a x p£ £< £
and 1 0a¹ > . A complete solution of the inverse spectral problem for a class of Dirac operators is given by spectral data. The main theorem on the necessary and sufficient conditions for the solvability of this inverse problem is proved and a solution algorithm of the inverse problem is given. Keywords: Dirac operator; inverse problem; necessary and sufficient condition References: [1] M. G. Gasymov and B. M. Levitan, “The inverse problem for the Dirac system”, Dokl. Akad. Nauk SSSR 167(1966), 967-970. [2] H. M. Huseynov and A. R. Latifova “On eigenvalues and eigenfunctions of one class of Dirac operators with discontinuous coefficients”, Trans. Natl. Acad. Sci. Azerb. 24(2004), 103-112.
14
Integral Inequalities for Log-Convex Functions via Riemann-Liouville Fractional Integrals Ahmet Ocak Akdemir, Erhan Set and M. Emin Ozdemir
Department of Mathematics, Agrı İbrahim Cecen University, Agrı, Turkey Department of Mathematics, Ordu University, Ordu, Turkey
Department of Mathematics Education, Atatürk University, Erzurum, Turkey [email protected], [email protected], [email protected]
Abstract: In this paper, we established some new Hadamard-type integral inequalities for functions whose derivatives of absolute values are log-convex functions via Riemann-Liouville fractional integrals. Acknowledgement: This study was supported by Ağrı İbrahim Çeçen University BAP with the project number FEF.14.011. Keywords: Log-convex functions; Riemann-Liouville fractional integral. References: [1] S. S. Dragomir, “Refinements of the Hermite-Hadamard integral inequality for log-convex functions”, Aust. Math. Soc. Gaz., 28 (3), 129-134 (2001). [2] J. Pečarić, F. Proschan, Y. L. Tong, “Convex Functions, Partial Orderings and Statistical Applications”, Academic Press, Inc., 1992.
15
Generalization of Hausdorff Matrices F. Aydin Akgun and B.E. Rhoades
Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey Department of Mathematics, Indiana University, Bloomington, USA
[email protected], [email protected]
Abstract: In 1926 W.A. Hurwitz showed that a row finite matrix is totally regular if and only if it has at most a finite number of diagonals with negative entries. He also proved that a regular Hausdorff matrix is totally regular if and only if it has all nonnegative entries. In 1921 Hausdorff proved that the Hölder and Cesaro Matrices are equivalent for each 𝛼𝛼 > −1. Basu, in 1949, compared the matrices totally. In this paper we investigate these theorems of Hurtwitz, Hausdorff, and Basu for the E-J and H-J generalized matrices. Keywords: Generalized Hausdorff matrices; Totally regular; Totally equivalence. References: [1] S. K. Basu, “On the total relative strength of the Hölder and Cesaro methods”, J. London Math. Soc. 50, 51- 59, (1948- 49). [2] F. Hausdorff, “Summationsmethoden und Momentfolgen”, I, Math. Z. 9, 74-109, (1921). [3] W. A. Hurwitz, “Some properties of methods of evaluation of divergent sequences”, Proc. London Math. Soc. 26, 231- 248, (1926).
16
New Indices on Special Graphs Nihat Akgunes, Ahmet Sinan Cevik and Ismail Naci Cangul
Department of Mathematics-Computer Sciences, Faculty of Science, Necmettin Erbakan University,, Konya-Turkey Department of Mathematics, Faculty of Science, Selcuk University, Campus, Konya-Turkey
Department of Mathematics, Faculty of Arts and Science, Uludag University, GorukleCampus, Bursa, Turkey [email protected] , [email protected], [email protected]
Abstract: In this talk, for a simple graph G with n vertices and m edges, we will introduce two new indices, namely Harmonic Mean Index and Average Harmonic Mean Index. By considering some special type of graphs, we will compare these indices with Harmonic Index (cf. [2, 3, 4]). Keywords: Topological indices; Harmonic index; Graph parameters. References: [1] N. Akgunes, K.Ch. Das, A.S. Cevik, “Topological indices on a graph of monogenic semigroups”, In: Ivan Gutman (Ed.), Topics in Chemical Graph Theory, Mathematical Chemistry Monographs, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac (2014). ISBN 978-86-6009-027-2. [2] R. Chang, Z. Yan, “On the harmonic index and the minimum degree of a graph”, Romanian Journal of Information Science and Technology, 15(4) (2012), 335-343. [3] R. Wu, Z. Tang, H. Deng, H, “A lower bound for the harmonic index of a graph with minimum degree at least two”, Filomat, 27(1) (2013) , 51-55. [4] L. Zhong, “The harmonic index for graphs”, Applied Mathematics Letters, 25(3) (2012), 561-566.
17
The New Method of the Fine Spectrum of a Class of Operator-Matrix in Some Sequences Spaces
Ali M. Akhmedov Faculty of Mechanical-Mathematics, Baku StateUniversity, Baku, Azerbaijan
Abstract: In this paper, we give the new method for the investigation of fine spectrum of a class generalized difference-operator matrixs acting in some sequences spaces. According to it we probe the convergence of the iterative process for an infinite family of bounded linear operators on a Banach space. We apply the main result presented in this paper for certain problem in summability theory. The obtained results generalize some similar consequences of the spectral properties of known works ([1]-[3]). Keywords: Iterative process; Convergence; Bounded linear operator; Banach space. References: [1] P. D. Srivastava, S. Kumar, “On the fine spectrum of the generalized difference operator over the sequence
space ”, Commun. Math. Anal., 2009, 6, no.1, p. 8-21. [2] A.M. Akhmedov, S.R. El-Shabrawy, “On the spectrum of the generalized difference operator over the
sequence space ”, Baku Univ. News J., Phys. Math. Sci. Ser., 2010, no. 4, p. 12-21.
[3] A.M. Akhmedov, S.R. El-Shabrawy, “On the fine spectrum of the operator over the sequence space c”, Comput. Math. Appl., 2011, 61, p. 2994-3002.
vD
0c
,a bD
0c,a bD
18
On Compactness of the Hardy Operator in the Weighted (.)(0, )pL ¥ Space
Lutfi Akin and Yusuf Zeren Yildiz Technical University, Departmant of Mathematics, Turkey
[email protected], [email protected]
Abstract: The last century, several changes of mechanical structures discovered in a private environment . In this project, which is close to their concrete problems linking species been investigated the effects of natural environments and fluids processes shape the conditions of growth of non-standard types of non-linear parabolic and elliptic partial differential equations leads to the examination. There are solutions to the equations of this type are collected in the natural functional space 𝑝𝑝(𝑥𝑥) index variable type space 𝐿𝐿𝑝𝑝(𝑥𝑥). A lot of the characteristics of these spaces, for example, the structure of joint space, reflection, smooth functions wherever compactness, continuity and compact embedding question adequately studied.
Let (0, ) :nB x R x¥ = Î < ¥ be in Euclidean space nR .
Let : (0, ) (1, ) and : (0, ) ( , )r B u B¥ ® ¥ ¥ ® -¥ +¥ be measurable functions. Denote by (.), ( (0, ))r uL B ¥ the
space of all measurable functions : (0, ) nf B R¥ ® such that ( )( ) ( )r y
y x
f y u y dy<
< ¥ò . This is a Banach space
with norm (.),
( )
( (0, ))(0, )
( )inf 0: ( )r u
r y
L BB
f yf u y dyl
l¥¥
ì üæ öï ï= > < ¥í ýç ÷è øï ïî þ
ò .
Hardy operator ( ) ( )y x
Hf x v y dy<
= ò to be compact from (.), ( (0, ))p wL B ¥ to space (.), ( (0, ))q vL B ¥ it is necessary
and sufficient that (0)(0)
0(0, ) (0, )sup ( ) sup ( ) ( ) and lim ( ) 0
qp
tx B x BA A t V x W x A t¢
®Î ¥ Î ¥= = < ¥ = .
Keywords: Compactness; Measurable functions; Hardy operator; Lebesgue space; Variable exponent. References: [1] D. Cruz Uribe, A. Fiorenza. “Variable Lebesgue Spaces: Foundations and Harmonic Analysis (Applied and Numerical Harmonic Analysis)" , Birkhauser,2013 [2] D. Cruz Uribe, A. Fiorenza. “Variable Lebesgue spaces: Foundations and Harmonic Analysis(Applied and numerical analysis)”, Birkhauser, 2013.
19
On the Behavior of a Class of p-Adic Dynamical Systems Hasan Akin, Farrukh Mukhamedov and Mutlay Dogan,
Department of Mathematics, Faculty of Education, Zirve University, Kizilhisar Campus, Gaziantep, 27260, Turkey Department of Computational Theoretical Sciences, Faculty of Science, International Islamic University Malaysia,P.O. Box, 141, 25710, Kuantan,Pahang, Malaysia Department of Mathematics, Faculty of Education, Zirve University, Kizilhisar Campus, Gaziantep, 27260, Turkey
[email protected], [email protected], [email protected]
Abstract: In this paper, we consider a rational p-adic dynamical system corresponding to p-adic non-linear function
of the form2 2
2 2
( 1)( )( )
a ab xf xax b
+=
+ in p¤ . We investigate the dynamics of non-linear function f . We find its fixed
points of the rational function .f Not that the function f has been investigated for real case in Ganikhodjaev et al. [1]
in real setting. Recently, Mukhamedov et al. [2] have obtained the p-adic dynamical system 2 2
, 2 2
1( ) .a bb xg x ax b
æ ö+= ç ÷+è ø
We will show that whether the non-linear function f is conjugate to the non-linear function .g We study the behavior of trajectories of one class of rational p-adic dynamical systems. Attractors of such dynamical systems are investigated. We also find the basin of the attractor of the system. On extreme Gibbs measures of the Vannimenus model”, J. Stat. Mech-Theory E., (2011) P03025
[2] Mukhamedov F., Dogan M. and Akın H., “Phase
Keywords: Dynamical systems, p-adic numbers, attractors, Siegel disk.
References:
[1] Ganikhodjaev N., Akın H., Uguz S., Temir S., “
transition for the p-adic Ising-Vannimenus model on the Cayley tree”, J. Stat. Mech-Theory E., scheduled for
September 2014.
20
Sharp Markov-Type Inequalities for Rational Functions on Several Intervals Mehmet Ali Akturk and Alexey Lukashov
Department of Engineering Sciences, Istanbul University, Turkey
Department of Mechanics and Mathematics, Saratov State University, Russia [email protected], [email protected]
Abstract: In this study, we give sharp Markov-type inequalities for rational functions on several intervals. We combine methods of works [1],[3],[4]. As a particular case we obtain Markov type inequalities for polynomials on inverse images of intervals which improve results of [2],[5]. Keywords: Inequalities in approximation; Approximation by rational functions References: [1] D.Benko, P. Dragnev, V. Totik, “Convexity of harmonic densities”, Rev. Mat. Iberoam., 28(4), 1–14, 2012. [2] P. Borwein, “Markov’s and Bernstein’s inequalities on disjoint intervals”, Canad. Math.J., 38(1), 201-209, 1981. [3] A.L. Lukashov, “Inequalities for Derivatives of Rational Functions on Several Intervals”, Izv. Math., 68(3), 543-565, 2004. [4] V.N. Rusak, “Rational Functions as an Apparatus of Approximation”, Belorus. State Univ. Press, Minsk, 1979. [5] V. Totik, “Polynomial Inverse Images and Polynomial Inequalities”, Acta Math., 187(1), 139--160, 2001.
21
Computing Fresnel Integrals via Modified Trapezium Rules M. Alazah, S.N. Chandler-Wilde and S. La Porte
University of Reading, UK Brunel University, UK
Abstract: We propose methods for computing Fresnel integrals based on modified trapezium rule approximations to integrals on the real line. Our approximations are exponentially convergent as a function of N, the number of quadrature points, with an explicit error bound which shows that accuracies of 10−15 uniformly on the real line are achievable with N = 12, this confirmed by numerical computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement. Keywords: Fresnel integrals; Modified trapezium rule. References [1] M. Alazah, S. N. Chandler-Wilde, S. La Porte, “Computing Fresnel Integrals via Modified Trapezium Rules”, Numerische Mathematik (published online April 2014), 1–27 doi 10.1007/s00211-014-0627-z [2] F. Matta, A. Reichel, “Uniform computation of the error function and other related functions”, J. Math. Phys. 34, 298–307 (1956) [3] D. B.Hunter, T. Regan, “A note on evaluation of the complementary error function”, Math. Comp. 26, 539–54 (1972).
22
Animation and Graphics to Understand Mathematics Abir Alharbi and Fairouz Tchier
King Saud University Mathematics department
[email protected], [email protected]
Abstract: Mathematics is the foundation of many core concepts in science and engineering. A successful career in science and engineering is impossible without a solid mathematical base [1]. However, mathematics is always considered a hard subject because many mathematical concepts in calculus, multivariate calculus, algebra, trigonometry, and discrete mathematics are difficult to explain to students on whiteboards or on paper. Without a thorough understanding students are bound to lose their interest in this very important subject. Even if the students get some understanding, it is extremely difficult to visualize many graphs and curves of various equations. The goal of our education system is to keep our students motivated, enhance their learning capabilities, and develop the ability to reason precisely and think logically. We realize that our younger generation is technology smart and feel that these goals are achievable by introducing technological methods and software in teaching mathematics [2]. In the past we relied almost entirely on chalk and blackboard, and examples that could be done by hand, to convey the key concepts of the above-mentioned subjects. Longhead calculations do not appeal to impatient young minds and may deter some students from fully appreciating the beautiful structure of the subject. Now the advent of powerful convenient software makes it imperative to reconsider how to teach these subjects. The greatest impact of software is in the areas of graphics and computation [3]. Colored graphics makes it possible to display curves, surfaces, and solids in two and three dimensions in a way that is both more effective and more engaging for the student. This is especially important for today’s students, who have not had much experience with solid geometry [4]. Color can also be used as a fourth dimension to help locate extreme points and to display quantities such as temperatures, density, and fluid velocity. Color animated graphics also allows us to deal seriously with computations. Fortunately, today’s software allows us to bridge the gap between understanding mathematics and visualizing its basic principles. Students will see and implement some basic animated graphics in their studies and instructors can rely on such graphics to explain these subjects. In this paper, we will show how to understand mathematics using maple animations. Keywords: Maple Animations, Graphics Animations, Visualize Mathematics References: [1] Baldwin, Doug and Peter Henderson. “The Importance of Mathematics to the Software Practicioner.” IEEE Software. March/April 2002: 22-24. [2] Beaubouef, Theresa. “Why Computer Science Students Need Math.” Inroads: SIGCSE Bulletin. December 2002: 57-59. [3] Bruce, Kim, et al. “Why Math?” Communications of the ACM. September 2003: 41-44. [4] “Computational Mathematics.” University of Waterloo. December 2005 <http://www.math.uwaterloo.ca/navigation/CompMath/>
23
On Certain Estimates for the Littlewood-Paley Operator Along Surfaces of Revolution Mohammed Ali
Jordan University of Science and Technology, Jordan [email protected]
Abstract: This work is primarily concerned with the proof of Lp boundedness for parabolic Littlewood-Paley operator with rough kernels. Our result and extrapolation arguments essentially improve and extend some known results in parabolic Littlewood-Paley operators.
24
Stabilized Mixed Finite Element Method Q1-Q0 for the Generalized Stokes Problem Chibani Alima and Nasserdine Kechkar
Department of Mathematics, Constantine1 university, Constantine, Algeria [email protected]
Abstract: In this paper, two stabilization techniques for the unstable mixed finite element method Q1-Q0 (bilinear velocity and constant pressure approximations) are discussed. The first one is based on stabilizing discontinuous pressure approximations via the local jump formulation, whereas the second one consists in a reduction of the local jump formulation with similar stability properties. To illustrate the effectiveness of these two approaches, some obtained results for a standard test Stokes problem are presented. Among others, It is shown that efficient iterative solution solvers of conjugate gradient type can be applied to the discrete Stokes system. Keywords: Stabilisation, mixed finite element method, Iterative methods.
25
Oscillation Theorem for One Spectral Problem Ziyatkhan S. Aliyev and Humay Sh. Rzayeva
Baku State University, IMM NAS Azerbaijan, Baku, Azerbaijan, Ganja State University, Ganja, Azerbaijan [email protected], [email protected]
Abstract: We consider the following boundary value problem for Dirac system .0)()(,)()( 222121112121112 =++¢-=++¢ yxpyxpyyyxpyxpy l (1)
,0sin)0(cos)0( 12 =+ aa yy (2)
,0sin)(cos)( 12 =+ bpbp yy (3)
CÎl is a spectral parameter, 2,1,),( =kixpik ( ))()( 1221 xpxp º are real valued, continuous functions on
the interval ],,0[ p a and b are real constants, and .,0 pba <£ It is known (see [1, Ch. 1]) that eigenvalues of the boundary value problem (1)-(3) are real, simple and the values range from −∞ to +∞ and can be numerated in increasing order. In our opinion, so far hardly been studied oscillation properties of eigenvector-functions of the problem (1)-(3) which are subject of this present work. We denote by )(gs the number of zeros of the function
( )RCxg ];,0[)( pÎ in the interval .),0( p In addition, we introduce the function: ,0)( =xs if ,0£x
,1)( =xs if .0>x The main result of this work is the following
Theorem The eigenvalues ,, ZÎnnl of the problem (1)-(3) can be numbered in ascending order on the real axis
so that the eigenvector-functions ( ) ( )Tnn
Tnn xyxyxyxy ),(),,()(),( 21,2,1 ll= have, with a suitable
interpretation, the following oscillation properties: if 0>n and ba ³= ,0n (except the cases 0== ba
and 2pba == ), then ÷÷ø
öççè
æ+-
-+-+-=÷÷
ø
öççè
æa
bpspassgn1
)2()2(1)()(
,2
,1
nn
ysys
n
n;
if 0<n and ,,0 ba <=n then
.sgn1||
)2()2(1||)()(
,2
,1
÷÷ø
öççè
æ+-
-+-+-=÷÷
ø
öççè
æb
pbsapsn
nysys
n
n
References: [1] B.M. Levitan, I.S. Sargsjan, Introduction to spectral theory: Selfadjoint ordinary differential operators, in Translation of mathematical Monographs, 39, AMS Providence, Rhode Island, 1975
26
Taylor Series Solution of Ordinary Differential Equations Fathi M Allan
United Arab Emirates University, Department of Mathematical Sciences, Al Ain, UAE [email protected]
Abstract: In this presentation, a new approach to solve initial and boundary value problems using the series expansion method will be discussed. The method is based on expanding the solution at different choices of the expansion point. Detailed analysis of the suggested method will be discussed and different linear and nonlinear initial value problems will be presented to demonstrate the efficiency of the method. Keywords: Series solution of differential equations; Taylor sereis expansion; Homotopy Analysis method.
27
Numerical Solution of an Integral Equation for Perpetual Bermudan Options
Ghada Alobaidi American University of Sharjah, Department of Mathematics and Statistics,
Abstract: We consider perpetual Bermudan options, which have no expiration and can be exercised every T time units. We use the Green’s function approach to write down an integral equation for the value of a perpetual Bermudan call option on an expiration date; this integral equation leads to a Wiener-Hopf problem. We discretize the integral in the integral equation to convert the problem to a linear algebra problem, which is straightforward to solve, and this enables us to find the location of the free boundary and the value of the perpetual Bermudan call. Finally we compare our results to earlier studies which used other numerical methods. Keywords: Bermudan options; integral equation; Green’s function, Wiener-Hopf problems. References: [1] Y.Z. Bergman, B.D. Grundy, and Z, Wiener, “General properties of option prices”, J. Finance 51 (1996), pp.1573-1610. [2] P. Bjerksund and G. Stensland, “American exchange options and a put-call transformation”: A note, J. Bus., Finance Acc. 20 (1993), pp. 761-764. [3] F. Black and M. Scholes, “The pricing of options and corporate liabilities”, J. Political Econ. 81 (1973), pp. 637-659. [4] S. I. Boyarchenko and S.Z. Levendorskii, “Pricing of perpetual Bermudan options”, J. Quant. Finance 2 (2002), pp. 432-442. [5] H. Brunner, “Collocation Methods for Volterra integral and related functional equations”, Cambridge University Press, Cambridge, 2004.
28
Bounds for Oscillatory Singular Integrals on Rⁿ Hussain Al Qassem, Leslie Cheng, Ayako Fukui and Yibiao Pan
Department of Mathematics and Physics,Qatar University, Doha-Qatar Department of Mathematics, Bryn Mawr College Bryn Mawr, U.S.A. Department of Mathematics, Bryn Mawr College, Bryn Mawr, U.S.A.
Department of Mathematics, University of Pittsburgh, Pittsburgh, U.S.A. [email protected], [email protected], [email protected], [email protected]
Abstract: We establish uniform bounds for two types of oscillatory singular integrals. In each case we allow the singular kernel to be given by a function in the Hardy space 𝐻𝐻1(𝑆𝑆𝑛𝑛−1), while such results were known previously only for kernels in 𝐿𝐿𝑙𝑙𝐿𝐿𝐿𝐿𝐿𝐿(𝑆𝑆𝑛𝑛−1) a proper subspace of 𝐻𝐻1(𝑆𝑆𝑛𝑛−1). Keywords: Oscillatory singular integral; Rough kernel; Singular integral; Orlicz spaces; L^p boundedness. References: [1] R. Coifman and G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. [2] L. Colzani, Hardy spaces on spheres, Ph.D. Thesis, Washington University, St. Louis, 1982. [3] Y. Jiang and S. Lu, Oscillatory singular integrals with rough kernel, Harmonic Analysis in China, 135--145, Kluwer Academic Publishers, 1995. [4] S. Lu and Y. Zhang, Criterion on L^p-boundedness for a class of oscillatory singulars with rough kernels, Rev. Mat. Iberoamericana, 8 (1992), 201--220. [5] M. Papadimitrakis and I. Parissis, Singular oscillatory integrals on Rⁿ, Math. Z., 266 (2010)169- [6] F. Ricci and E. Stein, Harmonic analysis on nilpotent groups and singular integrals. I: Oscillatory integrals, J. Func. Anal. 73 (1987), 179--194. [7] E.M. Stein, Oscillatory integrals in Fourier Analysis, Beijing Lectures in Harmonic Analysis, 307--355, Ann. Math. Studies, Princeton University Press, 1986. [8] E.M. Stein, Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.
29
On Lacunary Difference Sequence of Fuzzy Numbers Hifsi Altinok, Mikail Et and Rifat Colak
Department of Mathematics, Firat University, Elazıg, Turkey [email protected], [email protected], [email protected]
Abstract: The notion of difference sequence was introduced by Kızmaz [5] and the theory of sequences of fuzzy numbers was first introduced by Matloka [6]. In this paper, we define the class of sequence bvθ(Δ,F) for sequences of fuzzy numbers using the difference operator Δ and a lacunary sequence and examine some of its properties like solidity, symmetricity, convergence free, etc. Keywords: Fuzzy number; Difference operator; Lacunary sequence References: [1] H. Altinok, Y. Altin, M. Et, “Lacunary almost statistical convergence of fuzzy numbers”, Thai Jour. Math. 2 (2), (2004), 265-274. [2] F. Basar, B. Altay, M. Mursaleen, “Some generalizations of the spaces bvp of p-bounded variation sequences”, Nonlinear Analysis, TMA, 68, (2008), 273-287. [3] M. Et, R. Colak, “On some generalized difference sequence spaces”, Soochow J. Math. 21 (4), (1995), 377-386. [4] J. A. Fridy, C. Orhan, “Lacunary statistical convergence”, Pacific J. Math. 160 (1), (1993), 43-51. [5] H. Kizmaz, “On certain sequence spaces”, Canad. Math. Bull. 24 (2), (1981), 169-176. [6] M. Matloka, “Sequences of fuzzy numbers”, BUSEFAL 28, (1986), 28-37. [7] F. Nuray, “Lacunary statistical convergence of sequences of fuzzy numbers”, Math. Slovaca 45 (3), (1995), 269-273. [8] E. Savas, V. Karakaya, “Some new sequence spaces defined by lacunary sequences”, Math. Slovaca 57 (4), (2007), 393-399. [9] B. C. Tripathy, A. J. Dutta, “Bounded variation double sequence space of fuzzy real numbers”, Comput. Math. Appl. 59 (2), (2010), 1031-1037.
30
Some Sequence Spaces Defined by the Weighted Mean Method and Modulus Function Selma Altundag and Bayram Sozbir
Department of Mathematics, Sakarya University, Sakarya, Turkey [email protected], [email protected]
Abstract: In this paper, using by modulus function, we introduce the concept of [ , , ]nN p fl -summability as an extension of the concept of [ , ]nN pl -summability which is defined in [6] and we give the relation between [ , ]nN pl -summability and [ , , ]nN p fl -summability. We also study some connections between weighted statistically l -convergence (or NS
l-convergence) and [ , , ]nN p fl -summability.
Keywords: Weighted statistical convergence; Modulus function; Summability. References: [1] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2 (1951) 241-244. [2 ] J. A. Fridy, “On statistical convergence”, Analysis 5 (1985) 301-313. [3] J. S. Connor, “On strong matrix summability with respect to a modulus and statistical convergence”, Canad. Math. Bull., 32 (1989) 194-198. [4] H. Nakano, “Concave modular”, J. Math. Soc. Japan, 5 (1953) 29-49. [5] M. Mursaleen, V. Karakaya, M. Ertürk, F. Gürsoy, “Weighted statistical convergence and its application to Korovkin type approximation theorem”, Appl. Math. Comput. 218 (2012) 9132-9137. [6] C. Belen, S. A. Mohiuddine, “Generalized weighted statistical convergence and application”, Appl. Math. Comput. 219 (2013) 9821-9826. [7] E. Savaş, “On some generalized sequence spaces defined by a modulus”, Indian J. Pure Appl. Math. 30 (1999) 459-464. [8] I. J. Maddox, “Sequence spaces defined by a modulus”, Math. Proc. Camb. Philos. Soc. 100 (1986) 161-166.
31
Weighted Statistical Convergence of Double Sequences Selma Altundag and Bayram Sozbir
Department of Mathematics, Sakarya University, Sakarya, Turkey [email protected], [email protected]
Abstract: In this paper, we introduce the concepts of weighted statistical convergence and , ,[ ]N q pm n -
summability for double sequences. We also establish the relations between our new methods and we give some inclusions theorems. Keywords: Weighted statistical convergence; Riesz mean; double sequence; summability. References: [1] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2 (1951) 241-244. [2] A. Pringsheim, “Zur Ttheorie der zweifach unendlichen Zahlenfolgen”, Math. Ann. 53 (1900) 289–321. [3] M. Mursaleen, O. H. H. Edely, “Statistical convergence of double sequences”, J. Math. Anal. Appl. 288 (2003) 223–231. [4] F. Moricz, C. Orhan, “Tauberian conditions under which statistical convergence follows from statistical summability by weighted means”, Studia Sci. Math. Hungar. 41 (2004) 391–403. [5] V. Karakaya, T.A. Chishti, “Weighted statistical convergence”, Iran. J. Sci. Technol. Trans. A Sci. 33 (2009) 219–223. [6] M. Mursaleen, V. Karakaya, M. Ertürk, F. Gürsoy, “Weighted statistical convergence and its application to Korovkin type approximation theorem”, Appl. Math. Comput. 218 (2012) 9132–9137. [7] A. Alotaibi, C. Çakan, “The Riesz convergence and Riesz core of double sequences”, J. Inequal. Appl., 56 (2012) doi:10.1186/1029-242X-2012-56.
32
On the Fine Spectrum of the Operator ( , , )B r s t over the Class of Convergent Series Selma Altundag and Merve Abay
Department of Mathematics, Sakarya University, Sakarya, Turkey [email protected], [email protected]
Abstract: The main aim of this paper is to determine the fine spectrum of the operator ( , , )B r s t on g of all convergent series. Also, we study the approximate point spectrum, defect spectrum and compression spectrum of the matrix operator ( , , )B r s t on g . Keywords: Spectrum of an operator; Fine spectrum; Goldberg's classification; Approximate point spectrum; Defect spectrum; Compression spectrum. References: [1] B. Altay, F. Basar, “On the fine spectrum of the difference operator Δ on c0, c” Inform. Sci., 168(2004), 217-224. [2] B. Altay, F. Basar, “On the fine spectrum of the generalized difference operator B(r,s) over the sequence spaces c0 and c”, Int. J. Math. Math. Sci., 18(2005), 3005-3013. [3] S. Goldberg, “Unbounded Linear Operator”, Dover publications, Inc. New York, (1985). [4] F. Basar, N. Durna, M.Yildirim, “Subdivisions of the spectra for the triple band matrix over certain sequence Spaces”, Gen. Math. Notes, 4(1) (2011), 35-48. [5] H. Furkan, H. Bilgic, B. Altay, “On the fine spectrum of the operator B(r,s,t) over c0, c”, Comput. Math. Appl., 53(2007), 989-998. [6] A. J. Dutta, B. C. Tripathy, “Fine spectrum of the generalized difference operator B(r,s) over the class of convergent series”, International Journal of Analysis, (2013), 1-4.
33
Exponential Extinction for Discrete Nicholson’s Blow Flies Model with Nonlinear Mortality and Patch Structure Terms
Jehad Alzabut Department of Mathematics and Physical Sciences, Prince Sultan University, Riyadh, Saudi Arabia
Abstract: Under appropriate conditions, we establish sufficint criteria to guarantee the exponential extinction of solutions of discrete Nicholson’s blowflies model with nonlinear mortality and patch structure terms. Illustartive examples are demonstrated to show the validity of the proposed results. Keywords: Exponential extinction; Nicholson’s blowflies model. References: [1] W. S. C. Gurney, S. P. Blythe, R. M. Nisbet “Nicholson’s blowflies revisited”, Nature287, 17—21, 1980.
34
Numerical Approach for Solving Fractional Pantograph Equation Ayse Anapali, Yalcin Ozturk and Mustafa Gulsu
Department of Mathematics, Faculty of Science, Mugla Sitki Kocman University, Mugla, Turkey [email protected]
Abstract: In this study we give a numerical approach for solving fractional Pantograp equation
( ) ( ) ( ) ( )* 0
miD y x f x p x y q xr rra = + å
=, bxa ££ , 1n m na- £ < (1)
under the conditions ( )*
iD y c il= , 0,1, ..., 1i n= - , a c b£ £ , 0,1, ..., 1i n= - (2)
where 0 1qr< < and il is a appropriate constant. The pantograph equation is a kind of delay differential equation which is used different fields of pure and applied mathematics such as number theory, dynamical systems, probability, quantum mechanics and electrodynamics[1]. We investigate the approximate solution of Eq.(1) under the conditions Eq.(2) with the fractional Taylor series as
( ) ( , ]kD y x C a baa Î [2],
( )( ) ( )
0( ) ( )
1Ni
iN x a iy x D y aai
aa
a=
-= å
G + (3)
In this article, we have investigated a Taylor collocation method, which is based on collocation method for solving fractional pantograph equation. This method is based on first taking the truncated fractional Taylor expansions of the solution function in the mathematical model and then substituting their matrix forms into the equation. Keywords: Fractional pantograph equation; Fractional differential equation; Collocation method. References: [1] Z. Fan, M. Liu, W. Cao, “Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations”, J. Math. Anal. Appl. 325(2007) 1142-1159. [2] Z. Odibat, N.T. Shawagfeh, “Generalized Taylor’s Formula”, Appl. Math. Comput. 186 (2007)286-293. [3] M. Sezer, M. Gulsu, “Polynomial solution of the most general linear Fredholm-Volterra integro Differential-difference equation by means of Taylor collocation method”, Appl. Math. Comput., 185(2007) 646-657.
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Discovering Knowledge in Mathematics and Dynamic Modeling Halil Ardahan
Department of Mathematics Education, Faculty of Education, Necmettin Erbakan University, Konya [email protected]
Abstract: Establishing and improving the quality and standards in mathematics education needs to be designed as generally depending on the quality of the Mathematics content knowledge, pedagogic content knowledge and technologic content knowledge. Especially, learning process has to be designed based on the Information Technologies (IT), Inquiry Based Learning and Problem Solving Model (IBLPS) [1]. Inquiry Based Learning Model (IBPSM) consists of five critical sequenced steps as presented in Picture1 [2] .
Picture 1. Inquiry-Based Discovery Learning Model
Stratford explains the necessary cognitive strategies for students in order to form a dynamic model in case of any fact or problem as analyzing, relational reasoning, synthesizing, Testing and correcting, Explanation [3]. Also, mathematical modeling is a process that while trying to find a solution for daily life problems, representing the process with mathematical terms [4]. Mathematical modeling provides the opportunity of presenting the information in different ways [5]. However, there aren’t seen enough modeling application on the programs [6]. Thinking mathematically and expressing with the mathematics language have the first degree positive effect on qualified learning via dynamic modeling [7]. If we describe the learning concept as “learning is a mental transformation of a suitable model of the problem to the dual coded mental image” then we can see the importance and the role of the dynamic mathematical modeling. The samples of the study consist of 123 prospective teachers randomly selected from the N.E.U. Faculty of Education Department of Mathematics Education. Data was collected using the Dynamic Modeling Activity Scale (DMAS) with 5-point Likert type ten questions and 7 open ended items. Reliability coefficients of scale (Cronbach Alpha) is found as 0,896 and 0,879 respectfully by the pretest and post test. The following factors are affecting the meaningful and permanent learning: IBLPS model and the dynamic modeling predict ML in 0,665 percent, finding a pattern and describing symbolically predict ML in 0,655 percent, dynamic modeling predicts ML in 0,613 percent, and mathematical thinking and reasoning predict ML in 0,583 percent. From the statistical finding above, it can be suggested that prospective teachers ought to have competencies on dynamic modeling. Teacher Training Programs have to be reviewed and revised. Key words: Inquiry learning model; Dynamic modeling; Meaningful learning; Classroom activities. References: [1] H. Ardahan and Y. Ersoy(2001). Issues on Integrating CAS in Teaching Mathematics: A Functional and Programming Approach to Some Questions. ICTM-5: Special Group1. 6-10 Aug 2001. University of Klagenfurt. Austria. [2] H. Ardahan, “Mathematical Modelling Activities in Primary Mathematics Education”, Further Education in The Balkan Countries. Vol. II. s.1367. Eğitim Kitapevi. Konya. 2008 [3] S.J. Stratford, J. Krajcik, E. Soloway, “Secondary students’ dynamic modeling processes: analyzing reasoning about synthesizingand testing models of stream ecosystems”, Journal of Science Education and Technology. 7(3). 215–234, 1998. [4] A.K. Cheng, “Teaching Mathematical Modelling in Singapore Schools”, The Maths. Educator. Vol. 6(1), p.63-75, 2001. [5] NCTM, “Principles and standards for school mathematics”, Reston. VA: NCTM, (2000). [6] MEB-TTKB, “Secondary Mathematics (9,10,11ve 12 classes) Curriculum”, p.9, Ankara, Turkey, 2005. [7] H. Ardahan, “An Innovative Approach to Learning Process: Effect of Dynamic Modeling on Teaching of Mathematics”, 16th Asian Technology Conference in Mathematics , AIBU, September 19-23, 2011, Bolu, Turkey, 2011.
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Fejer type inequalities for Logarithmically Convex Functions Merve Avcı Ardıc, M. Emin Ozdemir and Alper Ekinci
Adıyaman University, Faculty of Science and Arts, Department of Mathematics, Adıyaman, Turkey
Atatürk University, K.K. Education Faculty, Department of Mathematics, 25240, Erzurum, Turkey
Agrı Ibrahim Cecen University, Faculty of Science and Arts, Department of Mathematics, 04100, Ağrı, Turkey
[email protected],[email protected], [email protected]
Abstract: In this paper, we proved some new integral inequalities of Fejer type by using an integral identity and
Riemann-Liouville fractional integrals. We also gave some reduced results by selecting special values of the
parameters and functions.
Keywords: log- Convex Functions, Fejer Inequality.
References:
[1] S.S. Dragomir, “Refinements of the Hermite-Hadamard integral inequality for log-convex functions”, Aust.
Math. Soc. Gaz., 28 (3), 129-134 (2001).
[2] J. Pečarić, F. Proschan and Y.L. Tong, “Convex Functions, Partial Orderings and Statistical Applications”,
Academic Press, Inc., 1992.
[3] C.E.M. Pearce, J. Pečarić and V. Šimić, “Stolarsky means and Hadamard's inequality”, J. Math. Anal. Appl., 220,
99-109 (1998).
[4] G.S. Yang and D.Y. Hwang, “Refinements of Hadamard's inequality for r-convex functions”, Indian Journal Pure
Appl. Math., 32 (10), 1571-1579, (2001).
[5] X. Zhanga and W. Jiang, “Some properties of log-convex function and applications for the exponential function”,
Computers and Mathematics with Applications, 63 (2012), 1111-1116.
[6] C.P. Niculescu, “The Hermite-Hadamard inequality for log-convex functions”, Nonlinear Analysis, 75 (2012), 662-669.
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Stability in nonlinear neutral differential equations with infinite delay Abdelouaheb Ardjouni ¹ and Ahcene Djoudi ²
¹ Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria, ² Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, University of Annaba, P.O. Box 12, Annaba 23000,
Algeria [email protected], [email protected]
Abstract: In this paper we use the contraction mapping theorem to obtain asymptotic stability results of a nonlinear
neutral differential equation with infinite delay. An asymptotic stability theorem with a necessary and sufficient
condition is proved, which improves and generalizes some results in the literature. Keywords: Fixed points, Stability, Neutral differential equations, Infinite delay.
References:
[1] S. Althubiti, H. A. Makhzoum, Y. N. Raffoul, Periodic solution and stability in nonlinear neutral system with
infinite delay, Applied Mathematical Sciences, Vol. 7, 2013, no. 136, 6749-6764.
[2] A. Ardjouni and A. Djoudi, Fixed points and stability in linear neutral differential equations with variable delays,
Nonlinear Analysis 74 (2011), 2062-2070.
[3] A. Ardjouni and A. Djoudi, Stability in nonlinear neutral integro-differential equations with variable delay using
fixed point theory, J. Appl. Math. Comput. (2014) 44:317-336.
[4] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New
York, 2006.
38
On a Canonical Form for Maxwell Equations by Geometric Calculus and Convergence of Finite Element Schemes for a Vlasov-Maxwell System
Mohammed Asadzadeh Department of Mathematics, Chalmers University of Technology and the University of Gothenburg, Goteborg, Sweden
Abstract. This work is a swift introduction to a few concepts in geometric calculus leading to a simple representation of Maxwell equations. We then approximate a “one and one-half” dimensional relativistic Vlasov-Maxwell system using streamline diffusion (SD) and discontinuous Galerkin (DG) finite element methods. In this part we derive optimal convergence rates in Lp-norm, p = 1, 2, ∞ due to the maximal available regularity of the exact solution. Geometric calculus unifies, simplify and generalize many fields of mathematics that involve geometric concepts. The scope and power of geometric calculus are overwhelming. This is illustrated for linear algebra, real and complex analysis, as well as several geometries: Euclidean, non-Euclidean, projective and conformal. Here geometric objects: points, lines, planes, circles … are represented by members of an algebra, a geometric algebra, rather than by equations relating coordinates. Geometric relations on objects: rotate, translate, intersect, project, construct a circle through three points,.. are represented by the algebraic operations on the objects. In a single sentence: geometric algebra is coordinate free. The presentation below is in the simplest possible way, without worrying too much about the rigor. It concerns, e.g., a novel approach to a simple and general formulation for the Maxwell equation and unifying its approximation strategies where we construct and analyze SD and DG methods for both Maxwell equations as well as Vlasov-Maxwell system and prove stability and convergence theorems.
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A Survey of Results in the Investigation of the Structure of Fractional Spaces Generated by Positive Operators in Banach Spaces
Allaberen Ashyralyev Department of Mathematics, Fatih University, Istanbul, Turkey
Department of Applied Mathematics, ITTU, Ashgabat, Turkmenistan [email protected]
Abstract: The role played by positivity property of differential and difference operators in Hilbert and Banach spaces in the study of various properties of boundary value problems for partial differential equations, of stability of difference schemes for partial differential equations, and of summation Fourier series is well-known (see, [1]-[3]). This is a review paper on results for fractional spaces generated by positive operators. Its scope ranges from theory of differential and difference operators in a space to operators with local and nonlocal boundary conditions. We also discuss their applications to partial differential equations and theory of difference schemes for partial differential equations. Keywords: Fractional spaces; Positive operators; Differential and difference operators; Banach spaces; Interpolation spaces; Stability. References: [1] A. Ashyralyev, P. E. Sobolevskii, “Well posedness of Parabolic Difference Equations”, Operator Theory Advances and Applications, Birkhauser Verlag, Basel, Boston, Berlin, 1994. [2] P. E. Sobolevskii, “A New Method of Summation of Fourier Series Converging in C-norm”, Semi-group Forum, 71(2005), 289-300. [3] A. Ashyralyev, P. E. Sobolevskii, “New Difference Schemes of Parital Differential Equations”, Operator Theory Advances and Applications, Birkhauser Verlag, Basel, Boston, Berlin, 2004.
40
Fractional Spaces Generated by the Positive Difference Operator in the Half-Line ℝ+ and Their Applications
Allaberen Ashyralyev and Sema Akturk
Department of Mathematics, Fatih University, Istanbul, Turkey [email protected], [email protected]
Abstract: In this study, we investigate the structure of fractional spaces generated by the difference operator 𝐴𝐴ℎ𝑥𝑥
𝐴𝐴ℎ𝑥𝑥𝜑𝜑ℎ(𝑥𝑥) = (−1)𝑚𝑚ℎ−2𝑚𝑚 𝑎𝑎𝑟𝑟,𝑠𝑠∆+
𝑟𝑟 ∆−𝑠𝑠
2𝑚𝑚≤𝑟𝑟+𝑠𝑠≤𝑆𝑆𝑠𝑠≤𝑚𝑚
𝜑𝜑ℎ(𝑥𝑥) + 𝛿𝛿𝜑𝜑ℎ(𝑥𝑥), 𝑥𝑥𝑥𝑥ℝℎ1+
which acts on functions defined on the entire space ℝℎ1+ . Here 𝑠𝑠 ∈ ℝ+ is a nonnegative integer
∆𝑘𝑘∓𝑓𝑓ℎ(𝑥𝑥) = ∓𝑓𝑓ℎ(𝑥𝑥 ∓ ℎ) − 𝑓𝑓ℎ(𝑥𝑥).
The definition of 𝜑𝜑ℎ(𝑥𝑥) is extended to 𝜑𝜑 ℎ(𝑥𝑥) defined on ℝℎ1+ and additionally on the points
𝑥𝑥 = 𝑘𝑘ℎ, 𝑘𝑘 = −1, −2, … , −(𝑚𝑚 − 1),
and
𝜑𝜑ℎ(0) = 0, ℎ−𝑘𝑘 𝛼𝛼𝑠𝑠,𝑘𝑘𝜑𝜑ℎ(𝑠𝑠ℎ) = 0,−𝑘𝑘≤𝑠𝑠≤𝑠𝑠𝑘𝑘
𝑘𝑘 = 1, 𝑚𝑚 − 1, 𝛼𝛼𝑠𝑠,𝑘𝑘 ≠ 0.
Keywords: Fractional spaces; Positive operators; Green’s function. References: [1] A. Ashyralyev, P.E. Sobolevskii, “Well-Posedness of Parabolic Difference Equations”, Operator Theory Advances and Applications, Birkhauser Verlag, Basel, Boston, Berlin, 1994. [2] P.E. Sobolevskii, “A new method of summation of Fourier series converging in C-norm”, Semigroup Forum vol. 71, pp. 289-300, 2005.
41
A Second Order of Accuracy Difference Scheme for a Fractional Schrödinger Differential Equation
Allaberen Ashyralyev and Betul Hicdurmaz Department of Mathematics, Fatih University, Buyukcekmece, Istanbul, Turkey
Department of Mathematics, Gebze Institute of Technology, Cayirova, Kocaeli, Turkey Department of Mathematics, Istanbul Medeniyet University, Kadikoy, Istanbul, Turkey
[email protected], [email protected]
Abstract: A new second order of accuracy difference scheme is presented for a fractional Schrödinger differential equation with a self-adjoint positive definite operator. The stability estimates for the solution of the difference scheme are established. The theoretical statements are supported by some numerical experiments. Results of the numerical experiments are interpreted. Keywords: Finite Difference Method; Fractional Schrödinger differential equation; Stability. References: [1] A. Ashyralyev, “Well-Posedness of the Boundary Value Problem for Parabolic Equations in Difference Analogues of Spaces of Smooth Functions”, Mathematical Problems in Engineering, vol. 2007, Article ID 90815, 16 pages, 2007. doi:10.1155/2007/90815. [2] Z. Cakir, "Stability of dİfference schemes for fractional parabolic PDE with the Dirichlet-Neumann conditions", Abstract and Applied Analysis, vol. 2012 Article ID 463746, 17 pages, 2012. doi:10.1155/2012/463746.
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Application of Summability Process on Baskakov-Type Korovkin Theory Ismail Aslan and Oktay Duman
Department of Mathematics, TOBB Economics and Technology University, Ankara, Turkey [email protected]
Abstract: In this study, motivating the results in [1] and [2], we study the summability process introduced by Bell [3] to approximate a function and its derivatives by means of a wider class of linear operators than a family of positive linear operators. Our results improve not only Baskakov's idea in [2] but also the Korovkin theory based on positive linear operators [4]. In order to verify it we display a specific sequence of approximating operators by plotting their graphs. Keywords: Korovkin theory; Baskakov-type approximation; Summability Process.
References: [1] G. A. Anastassiou, O. Duman, “A Baskakov type generalization of statistical Korovkin theory”, J. Math. Anal. Appl. 340 (2008) 476-486. [2] V. A. Baskakov, “Generalization of certain theorems of P. P. Korovkin on positive operators (in Russian)”, Mat. Zametki 13 (1973) 785-794. [3] H. T. Bell, “Order summability and almost convergence”, Proc. Amer. Math.Soc. 38 (1973), 548-552. [4] P. P. Korovkin, “Linear Operators and Theory of Approximation”, Hindustan Publ. Corp., Delhi. 1960.
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On a Kirk-Type New Iteration Process Yunus Atalan, Vatan Karakaya, Kadri Dogan and Nour El Houda Bouzara Department of Mathematics, Yildiz Technical University, Istanbul, Turkey
Abstract: In this presentation we introduce Kirk-type new iterative scheme which we called Kirk- DPM scheme and we study the convergence of this iterative scheme by using weak contraction mappings. Furthermore, we obtain this iteration process is T-stable. Keywords: Kirk-multistep iteration; Strong convergence; Stability; Weak contraction mappings. Acknowledgment: The authors would like to thank Yıldız Technical University Scientific Research Projects Coordination Department under Project no. BAPK 2014-07-03-DOP02 for financial support during the preparation of this paper. References: [1] H. Akewe, G. A. Okeke, A. F. Olayiwola. "Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators." Fixed Point Theory and Applications 2014.1 (2014): 45. [2] N. Hussain, R. Chugh, V. Kumar, A. Rafiq, “On the Rate of Convergence of Kirk-Type Iterative Schemes,” Journal of Applied Mathematics, vol. 2012, Article ID 526503, 22 pages, 2012. doi:10.1155/2012/526503. [3]W. A. Kirk, "On successive approximations for nonexpansive mappings in Banach spaces." Glasgow Mathematical Journal 12.01 (1971): 6-9. [4] F. Gursoy, V. Karakaya, B. E. Rhoades, “Some Convergence and Stability Results for the Kirk Multistep and Kirk-SP Fixed Point Iterative Algorithms,” Abstract and Applied Analysis, vol. 2014, Article ID 806537, 12 pages, 2014. doi:10.1155/2014/806537.
44
2-Quadratic Modules of Algebras Hasan Atik
Department of Mathematics, Istanbul Medeniyet University, Uskudar, Istanbul, Turkey [email protected]
Abstract: In this work, we give the notion of 2-quadratic module for commutative algebras and we explore the connections among this structure and 2-crossed modules and simplicial algebras in terms of hypercrossed complex pairings. Keywords: 2-Quadratic Modules; Simplicial Algebras; 3-Crossed Modules References: [1] Z. Arvasi, T. S. Kuzpınarı, E. O. Uslu, “Three-crossed Modules”, Homology Homotopy and Application, 11, 161-187, (2009). [2] M. Artin, B. Mazur, “On the Van Kampen Theorem”, Topology, 5, 179-189, (1966). [3] Z. Arvasi, T. Porter, “Higher Dimensional Peiffer Elements in Simplicial Commutative Algebras”, Theory and Applications of Categories, Vol. 3, No. 1, pp 1-23, (1997). [4] Z. Arvasi, T. Porter, “Freeness Conditions for 2-Crossed Module of Commutative Algebras”, Applied Categorical Structures, 6, pp 455-477, (1998). [5] H. J. Baues, “Algebraic Homotopy”, Cambridge Studies in Advanced Mathematics, 15, 450 pages, (1998). [6] H. J. Baues, “Combinatorial Homotopy and 4-Dimenional Complexes”, Walter de Gruyter, 15, 380 pages,(1991). [7] R. Brown, N. D. Gilbert, “Algebraic Models of 3-Types and Automorfizm Structures for Crossed Modules”, Proc. London Math. Soc. (3) 59, pp 51-73, (1989). [8] P.Carrasco, “Complejos hipercruzados, cohomologia y extensions”, Ph.D. Thesis, Univ. de Granada, (1987). [9] P. Carrasco, A. M. Cegarra, “Group-Theoretic Algebraic Models for Homotopy Types”, Journal of Pure and Applied Algebra, 75, pp 195-235, (1991). [10] D. Conduche, “Modules Crois´es G´en´eralis´es de Longueur 2”, Journal of Pure and Applied Algebra, 34, pp 155-178, (1984). [11] G. J.Ellis, “Higher dimensional crossed modules of algebras”, Journal of Pure and Applied Algebra 52 (1988) 277-282. [12] G. J. Ellis, R. Steiner, “Higher dimensional crossed modules and the homotopy groups of (n+1)-ads.”, Journal of Pure and Applied Algebra, 46, pp 117-136, (1987). [13] A.R. Grandjean, M.J. Vale, “2-Modulos Cruzados en la Cohomologia de Andre Quillen”, Memorias de la Real Academia de Ciencias, 22, 1-28, (1986). [14] D. Guin-Walery, J-L. Loday, “Obsruction´a L’excision en K-theories Alg´ebrique”, In: Friedlander, E.M.,Stein, M.R.(eds.) Evanston conf. on algebraic K-Theory 1980, (Lect. Notes Math., vol.854, pp 179-216), Berlin Heidelberg New York: Springer (1981).
45
Generalized Plane Deformation Solutions of the Shallow Located Cavity at the Anisotropic Rock Massif under the Actions of Elastic SH-Waves
Lyazzat Atymtayeva and Bagdat Yagaliyeva Department of Management of Information Systems, Kazakh-British Technical University, Almaty, Kazakhstan
[email protected], [email protected] Abstract: In this paper we study the stress-strain state of a horizontally elongated non-circular cavity shallow located in transversal-isotropic massif with an inclined plane of isotropy under the generalized plane deformation conditions at stationary diffraction of elastic SH-waves. Based on the theory of Bessel functions[1-2] we create the basic equations for the solution of the first and second fundamental problems of the theory of elasticity[3-4]. With the help of affine transformations[5] the motion equations (in displacements) are reduced to the Helmholtz equation having a solution in the form of series of cylindrical Bessel and Hankel functions. As a result of complex transformations on the boundary contour expressions of the cavity and boundary expressions on the free border line we get a system of linear algebraic equations that to be solved by the reduction method[6-8]. Keywords: Anisotropic (transversal-istropic) massif, generalized plane deformation, shallow located cavities, diffraction, SH-wave References: [1] A.N.Guz, V.D.Kubenko, M.A.Cherevko, “Diffraction of elastic waves”, Kiev, Publishing House “Naukova dumka”, 1978, Russian G.I. [2] A.S.Kosmodamianski, V.I.Storozhev, “Dynamical tasks of elasticity theory for anisotropic environments”, Kiev, Publishing House Naukova dumka”, 1985, Russian [3] Zh.S.Erzhanov, Sh.M.Aitaliyev, Zh.K.Masanov, “Seismically stressed state of underground constructions in anisotropic layered massif”, Alma-Ata, Publishing House “Nauka”, 1980, Russian [4] S.G. Lechnitsky “Theory of elasticity for the anisotropic media”. Moscow, 1951 (in Russian) [5] N.I. Muskhelishvili, “Some basic problems of the mathematical theory of elasticity”, Moscow, 1977, Russian [6] L. Atymtayeva, Zh. Masanov, B. Yagaliyeva, G. Myrzakhmetova “Dynamics of transtropic elastic continuums with non-homogeneities” //Proceedings of World Congress on Engineering 2010, the 2010 International Conference of Mechanical Engineering, London, UK, 30 June-2 July, 2010. - P. 1024-1027. [7] L. Atymtayeva, Zh. Masanov, B. Yagaliyeva, G. Myrzakhmetova “Stationary diffraction of elastic waves on the mine tunnels in the anisotropic layered massif” // Proceedings of the International conference “Modern problems of applied mathematics and mechanics: theory, experiment and practice”, Novosibirsk, Russia, May 30 – June 4 , 2011, http://conf.nsc.ru/niknik-90/reportview/36661 [8] L. B. Atymtayeva , Zh. K. Masanov, and .B. E. Yagaliyeva “Diffraction of Elastic PP-, SV-and SH-Waves on the Arbitrary Deep-founded Cavities in the Anisotropic Rock Massif” // Proceedings of the World Congress on Engineering 2012, Vol III , WCE 2012, July 4 - 6, 2012, London, U.K., pp.1907-1912
46
Peiffer Ideals, Precrossed Modules and Crossed Modules of R-Algebroids Osman Avcioglu and Ilker Akca
Department of Mathematics, Faculty of Science and Literature, Usak University, Turkey Department of Mathematics and Computer Sciences, Faculty of Science and Letters, Osmangazi University, Turkey
[email protected], [email protected]
Abstract: In this paper we define the Peiffer commutators and Peiffer Ideal of an R-algebroid. Then we construct a crossed module from a given precrossed module using Peiffer Ideal. Further, we introduce the functor (−)cr from the category of precrossed modules of R-algebroids to that of crossed modules of R-algebroids and analyse its properties. Keywords: Algebroids; Crossed Modules; Peiffer Ideals. References: [1] S. M. Amgott, “Saperable Categories”, J. Pure Appl. Algebra 40 (1986) 1–14. [2] R. Brown, J. Huebschmann, “Identities Among Relations, Lowdimensional topology”, (Bangor, 1979), London Math. Soc. Lecture Note Ser., Volume 48. Cambridge Univ. Press, Cambridge (1982), 153–202. [3] R. Brown, P.J. Higgins, R. Sivera, “Nonabelian Algebraic Topology”, Version 09 Jan, (2010). [4] H. Baues, J. D. Conduche, “The central series for Peiffer commutators in groups with operators”, J. Algebra, 133 (1), (1990), 1–34. [5] M. Gerstenhaber, “On the deformation of rings and algebras”, Annual of Mathematics 84 (1996) 1-19. [6] A. S. T. Lue, “Nonabelian cohomology of associative algebra”, Mathematische Zeitschrift 121 (1971) 220 239. [7] S. Lichtenbaum, M. Schlessinger, “The Cotangent Complex of a Morphism”, Transection American Mathematics Society, 128, (1967), 41–70. [8] B. Mitchell, “Some applications of module theory to functor categories”Bull. Amer. Math. Soc.84(1978)867-885.
47
Two–Interval Sturm–Liouville Differential Operators in Direct Sum Spaces Kadriye Aydemir and Oktay Mukhtarov
Department of Mathematics, Gaziosmanpasa University, Tokat, Turkey [email protected], [email protected]
Abstract: In [1] Everitt and Zettl characterized all self-adjoint realizations of Sturm–Liouville problems on two intervals in the direct sum of Hilbert spaces associated with each of the intervals. Mukhtarov and Yakubov [3] observed that the set of two-interval self-adjoint realizations can be further enlarged by using different multiples of the usual inner products associated with each of the intervals. The purpose of this paper is to provide an operator theoretic framework for the study of one differential operator on two disjoint intervals. This study is partly motivated by the occurrence of Sturm-Liouville problems with coefficients that have a singularity in the interior of the basic interval. Many physical problems leading to differential equations of Sturm-Liouville type can be found in the applied mathematics and mathematical physics literature(See, [3,4]). Keywords: Sturm-Liouville differential operator; Interior singularity; Transmission conditions. References: [1] W. N. Everitt and A. Zettl, “Sturm-Liouville differential operators in direct sum space”, Rocky Mountain J. of Mathematics, (1986), 497-516. [2] O. Sh. Mukhtarov and S. Yakubov, “Problems for ordinary differential equations with transmission conditions”,
Appl. Anal., 81(2002), 1033-1064. [3] A.N. Tikhonov and A.A. Samarskii,“Equations of Mathematical Physics”,Oxford and New York, Pergamon, 1963. [4] J. D. Pryce, “Numerical Solution of Sturm-Liouville Problems”, Oxford University Press, 1993.
48
The Variational Principle and Complexity of Zn Actions Bunyamin Aydin
Education of Faculty, Necmettin Erbakan University, Konya-Turkey [email protected]
Abstract: The complexity of afinite object was introduced by A.Kolmogoorow and V. Tihomirov in [1] and it was conjevtured that for actions the coplexity coindes with topolojical entropy, [1,2,3]. In the present paper we introduce complexity for Zn actions and prove the Kolmogorow assertion for continious actions of Z. After We study variational principle and complexity for Zn actions. In dynamical systems and ergodic theory, the topolojical entropy describes the com,plexity of a system. Topolojical entropy is an invariant for equivalent homeomorphisms [1-9,10]. Recently Lewis Bowen İntroduced a collection of entropy indariants for measure-preserving actions of countable sofic group a standard probality space admitting a generating ratition with finite entropy. [3-5]. Given Bowen’s work, it is natural to ask whether there exist analogous invariants for continuous actions of a countable sofic group on a compact metrzable space, and if so whether they are connected to Bowen’s measure entropy via a variational principle. It is well known that topolojical entropy is an invariant of topolojical conjugay. If the topoojical entropy is positive, the system is complexity and chaotic.If the is zero, the system is rather simle. Howerer from the theory and application, there still exists relatively comlex and chaotic behavior. Therefore, for more general research on complexity of a system. This idea was firstly introduced in the research of ergodic theory [2-1], and then in symbolic dynamical systems [2-2] by Frerenczi. The entropy theory of daynamical systems originated in the papers of A.N.Kolmogorov in the fifties. Topolojik entropy which is the analog of metric entropy in topolojical Dynamics was introduced by Adler, Konheim, and McAndrew [4-1]. The conjecture was made there that the topolojical entropy coincides with the least upper bound of the metricentrıpies over the set of all invariant Borel probability measure. This assertion which has been xalled the variational principle (VP) was proved by Dinaburg for homeomomorhisms of finite dimensional compacta [4-2]. Theorem Let be a symbolic dynamical systsem on . Then
References: [1]A. Kolmogorow, V. Tihomirov, V.s-Entropy and s-Capacity of sets in function spaces. Uspehi. Math., nauk 14, No.2(86), English trasl., Amer. Co., Trans., 1996,3-86. [2] L. Levin., Universal Sortg problems. Problemy Predaci ınformacii, (in Russian) English trans. In prolems of ınformation transmission, 9, 1973, 265-266. [3] r Cldebank (Ed). Different Aspects of Coding Theory. Amer.math. soc.Short Cours., january 2-3, San Francisco, California, 1995. [4] M.stepin, A.T.Tagi-zade, combinatorinal Interpretations of the entropy Symbolic Systems. Math. Note. Vol.46.No.3, 1989, 653-658. [5] Bunyamin, Entropy and the compleity for Zn Actionns, Informatinés technologijos R Valdymas, 2004, Nr.2(31), 65-67.
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On The Rough Statistically Cauchy Sequences Salih Aytar
Department of Mathematics, Süleyman Demirel University, Isparta Turkey [email protected]
Abstract: We have two ideas to define the concept of rough statistically Cauchy sequence. First definition is that
A sequence )( ix is said to be r -statistically Cauchy provided that, for every 0>e there exists a number
NÎ= )(eNN such that the set e+³-Î rxxi Ni:N has natural density zero. Comparing any ix with a fixed Nx is far away from the proper spirit of classical Cauchy sequence and of rough Cauchy sequence [1]. If all consequential results are correct, one could assume that the defined term is completely different from the classical Cauchy sequence and the appearance of Cauchy's name in this definition is a bad choice of language. For this reason we have to give second definition as follows: A sequence )( ix is said to be r -statistically Cauchy provided that, for every 0>e there exists an index set
NÍK with ( ) 1=Kd such that er +<- ji xx holds for every Kji Î, .
The main idea of this talk is to compare above definitions, and using the second definition to determine the set of positive real numbers for which a sequence is rough statistically convergent, provided that a rough statistically Cauchy degree is given for this sequence. Keyword: Cauchy degree, rough statistically Cauchy Reference: [1] H.X. Phu (2001). Rough convergence in normed linear spaces, Numer. Funct. Anal. and Optimiz. 22:201-224.
50
Coincidence of Crisp and Fuzzy Functions Akbar Azam, Maliha Rashid and Nayyar Mehmood
Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Pakistan Department of Mathematics, International Islamic University, Islamabad, Pakistan
Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Pakistan [email protected], [email protected], [email protected]
Abstract: We prove a common coincidence point theorem for a non-fuzzy mapping and a pair of fuzzy mappings under a φ-contraction condition on a metric space in connection with the Hausdorff metric on the family of fuzzy sets. Further, we establish a common coincidence point theorem on a metric space with the - metric on fuzzy sets, which extends a number of recent results. As applications, we obtain a common coincidence point theorem for a single-valued mapping and a pair of set-valued mappings and apply it to achieve some existence and uniqueness theorems of solution for a class of nonlinear integral equations. Keywords: Coincidence point; Set-valued mapping; Fuzzy mapping; Integral equation.
51
An Iterative Method for Solving a Combined Inverse Problem of Reservoir Parameters Identification Aliya Azhibekova
Kazakh-British Technical University Almaty, Kazakhstan [email protected]
Abstract: There is a wide literature concerned with the problem of determining an unknown coefficients entering into a parabolic problem [1, 2]. Methods based on the minimization of a residual functional have been used quite widely. The minimization is generally realized using gradient-based methods [2]. The unique existence of a solution, as well as the convergence of iterative process presents major difficulties in the application of these methods. In some particular cases, these difficulties have been overcome [3, 4]. In this paper, we shell solve the inverse combined problem. We obtain some theoretical estimates for a direct and adjoint problem. Using these estimates we prove monotonicity of the objective functional and the convergence of iteration sequences. Keywords: Inverse Problem; Iterative method; Adjoint Problem. References: [1] M.M. Lavrentiev, V.G. Romanov, S.P. Shishatskii, “Ill-Posed Problems of Mathematical Physics and Analysis”, Moscow: Nauka, 1980. [2] O.M. Alifanov, E.A. Artyukhin, S.V. Rumyantsev, “Extreme Methods for Solving Ill-Posed Problems”, Moscow: Nauka, 1988. [3] V. Isakov, S. Kindermann, “Identifications of the Diffusion Coefficient in a One-Dimensional Parabolic Equation”, Inv. Prob., 16(2000), 665–680. [4] S. I. Kabanikhin, “Inverse and Ill-Posed Problems”, Novosibirsk, 2009.
52
Numerical Solution of System of Nonlinear Fredholm and Volterra Integral Equations Using Haar Wavelet
Imran Aziz [email protected]
Abstract: The main focus of this work is to propose a new and computationally efficient algorithm for a system of non-linear Fredholm and Voterra integral equations (FVIEs) of second kind. The new algorithm is based on Haar wavelet and is an extension of the methods [1–3]. The proposed algorithm is generic and focused on providing a single platform for numerical solution of system of nonlinear FVIEs of second kind. The computational cost of the new algorithm is considerably reduced by using special properties of Haar wavelet which remained unexplored in the present context so far. The use of Broyden’s method instead of Newton’s method also contributes to computational efficiency of the present algorithm as the new method avoids calculation of Jacobians at each iteration. Most of the existing numerical methods designed for numerical solution of FVIEs in the literature rely on numerical quadrature to evaluate the integrals of kernel function of FVIEs. The present method adopts a different route by performing exact integration instead of numerical evaluation of integrals. The payoff of the exact integration returns in terms of better accuracy and less computational cost. The method is tested on a number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that the accuracy of the obtained solutions is quite high even when the number of collocation points is small. References: [1] I. Aziz, Siraj-ul-Islam, “New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets”, J. Comp. Appl. Math. 239 (2013) 333–345. [2] Siraj-ul-Islam, I. Aziz, M. Fayyaz, “A new approach for numerical solution of integro- differential equations via haar wavelets”, Int. J. Comp. Math. 90 (2013) 1971–1989. [3] Siraj-ul-Islam, I. Aziz, A. Al-Fhaid, “An improved method based on Haar wavelets for nu- merical solution of nonlinear and integro-differential equations of first and higher orders”, J. Comp. Appl. Math. 260 (2014) 449–469.
53
A Study on Involute-Evolute Curves in Euclidean 3-Space Vildan Bacak and Nihat Ayyildiz
Department of Mathematics, Selcuk University, Konya, Turkey Department of Mathematics, Suleyman Demirel University, Isparta, Turkey
[email protected], [email protected]
Abstract: In this work, Euclidean 3-space curves with positive curvature k , torsion t and spherical tangential parameter s such that the radius of curvature 1k- which satisfies an eigenvalue equation are examined. We delineate the equations in terms of evolutes and involutes of a space curve c and supply relations between evolutes and involutes. Keywords: Curvature function; Eigenvalue equation; Euclidean space curve; Evolute; involute; Spherical tangential image; Support function. References: [1] R. P. Encheva and G. H. Georgiev, “Similar frenet curves”, Results Math. 55 (2009), 359-372. [2] R. Encheva and G. Georgiev, “Shapes of space curves”, J. Geom. Graph. 7 (2003), no. 2, 145-155. [3] A. Karger and J. Novak, “Space kinematics and Lie groups”, Gordon and Breach Science Publishers, 1985. [4] C. Y. Kim, J. H. Park, and Y. Sinsuke, “Curves on the unit 3-sphere 3 (1)S in Euclidean 4-sphere 4
¡ ”, Bull. Korean Math. Soc., 50 (2013), no. 5, 1599-1622. [5] S. Müller, A. Schwenk-Schellschmidt and U. Simon, “Eigenvalue equations in curve theory. Part II: Evolutes and involutes”, Results Math. 50 (2007), 109-124. [6] B. O'Neill, “Elementary differential geometry”, Academic Press; 2 edition, (2006). [7] M. C. Romero-Fuster and E. Sanabria-Codesal, “Generalized evolutes, vertices and conformal invariants of curves in 1n+
¡ ”, Indag. Mathem. N.S. 10 (1999), no. 2, 297-305. [8] T. Sato, “Pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz-Minkowski space”, J. Geom. 103 (2012), 319-331. [9] A. Schwenk-Schellschmidt, U. Simon and M. Wiehe, “Eigenvalue equations in curve theory. Part I: Characterizations of conic sections”, Results Math. 40 (2001), 273-285. [10] R. C. Yates, “A handbook on curves and their properties”, Edward Brothers Inc., (1947).
54
Bipas Flow with Dominant Poloidal Component A. A. Bachtiar and R. Kosasih
Department of Mathematics, University of Indonesia Department of Mathematics University of Indonesia and University of Gunadarma
Abstract: The Planar Velocity Anti-dynamo Theorem (PVT) precludes the existence of a self excited dynamo when the velocity is perpendicular to a certain plane [4]. However, Bachtiar et al [1] showed that PVT fails if the velocity occupies a sphere. In order to provide more evidences, Bachtiar [2] proposed several new planar flows, namely BiPAS and QuasiPAS flows. Recently, Bachtiar et al [3] reported that QuasiPAS flow with dominant poloidal part improves the magnetic growth's rate. In this work, we will show that BiPAS has similar behavior as QuasiPAS. Keywords: Self excited dynamo; PVT theorem; Planar Flows; BiPAS flow. References: [1] A. A. Bachtiar, D.J. Ivers and R.W. James,”Planar velocity dynamos in a sphere”. Proc. R. Soc. Lond. A 462 (2006), 2439-2456 [2] A. A. Bachtiar,”A study of planar velocity dynamos and related issues”. Ph.D. thesis(2009), University of Sydney. [3] A. A. Bachtiar, D. Riminarsih and Y. Satria,”A Review on QuasiPAS Flow. Far East Journal of Mathematical Sciences”, Vol 85(2014), No. 1, 87-97. [4] Ya. B. Zeldovich,”The magnetic field in the two-dimensioned motion of a conductiong turbulent liquid”. JETP, 4(1957).460.
55
Large Sample Variance of Simulation Using Refined Descriptive Sampling Leila Baiche and Megdouda Ourbih-tari
Laboratoire de Mathématiques Appliquées, Faculté des Sciences Exactes,Université de Bejaia, Bejaia, Algérie [email protected], [email protected]
Abstract: Derived from Descriptive Sampling [1] as a better approach to Monte Carlo simulation [2], Refined Descriptive sampling [3] is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables. In this paper, the asymptotic variance of such an estimate in case of independent variables is obtained and it was shown that asymptotically, the variance is less than that obtained using simple random sampling. Keywords: Simulation; Monte Carlo Methods; Sampling Theory; Estimation. References: [1] E. Saliby, “Descriptive sampling: a better approach to monte carlo simulation”, J. Operat. Res. Soc. 41(1990), 1133-1142. [2] I. T. Dimov, “Monte Carlo methods for applied scientists”, World Scientific Publishing Co. Pte, Ltd Singapore. (2008). [3] M. Tari, A. Dahmani, “Refined descriptive sampling: a better approach to Monte Carlo simulation”, Simulation Modelling Practice and Theory. 14(2006), 143-160.
56
Best Proximity Points for a New Proximal Amitabh Banerjee
Govt. Sukhram Nage College, Nagri-Sihawa, District-Dhamtari, Chhattisgarh, India [email protected]
Abstract: In this paper, we define a new proximal contraction and prove best proximity point theorems. We also provide an example to illustrate the theorem. Keywords: Best proximity point; Proximal contraction; Fixed points.
57
On the Domain of Riesz Mean in the Space 𝓛𝓛𝒔𝒔 Feyzi Basar
Department of Mathematics, Fatih University, Istanbul, Turkey [email protected] , [email protected]
Abstract: The space ℒ𝑠𝑠 of absolutely s-summable double sequences was defined by Başar and Sever [1], where 1 ≤ 𝑠𝑠 < ∞. The reader can refer to Başar [2] and Mursaleen and Mohiuddine [3] for relevant terminology and required details on the double sequences and releted topics. Let 0 < s < 1. In this study, as a natural continuation of [4, 5] we introduce the double sequence space 𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑠𝑠) as the domain of four dimensional Riesz mean 𝑅𝑅𝑞𝑞𝑡𝑡 in the space ℒ𝑠𝑠 and establish that the inclusions ℒ𝑠𝑠 ⊂ 𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑠𝑠) and 𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑠𝑠) ⊂ 𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑟𝑟) strictly hold with 1 ≤ 𝑠𝑠 < ∞ and 1 < s < r < 1, respectively. Furthermore, we show that 𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑠𝑠) is a Banach space, a barrelled space and solid when 1 ≤ 𝑠𝑠 < ∞ and 𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑠𝑠) is not a barrelled space when 0 < s < 1. We determine the 𝛼𝛼-, 𝛾𝛾- and 𝛽𝛽(𝜗𝜗)-duals of the spaces ℒ𝑠𝑠, (0 < s < 1) and 𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑠𝑠), (0 < s < 1). Finally, we characterize the classes (ℒ𝑠𝑠 ∶ 𝑀𝑀𝑢𝑢), (ℒ𝑠𝑠 ∶ 𝐶𝐶𝑏𝑏𝑝𝑝), (ℒ𝑠𝑠 ∶ 𝑀𝑀𝑢𝑢), (𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑠𝑠) : 𝑀𝑀𝑢𝑢) and (𝑅𝑅𝑞𝑞𝑡𝑡(ℒ𝑠𝑠): 𝐶𝐶𝑏𝑏𝑝𝑝) of four dimensional matrices, in the cases both 0 < s < 1 and 1 ≤ 𝑠𝑠 < ∞ together with some direct consequences. In the special case q = t = e, since the Riesz mean 𝑅𝑅𝑞𝑞𝑡𝑡 is reduced to the four dimensional Cesàro mean C of order one, our results are much more general and comprehensive than the corresponding results given by Mursaleen and Başar [5] for the space ℒ𝑠𝑠. Keywords: Double sequence space; Alpha-, beta- and gamma-duals; Matrix mappings. References: [1] F. Basar, Y. Sever, “The space ℒ𝑠𝑠 of double sequences”, Math. J. Okayama Univ. 51 (2009), 149-157. [2] F. Basar, “Summability Theory and Its Applications”, Bentham Science Publishers, e-books, Monographs, Istanbul-2012. [3] M. Mursaleen, S.A. Mohiuddine, “Convergence Methods For Double Sequences and Applications”, Springer, New Delhi - Heidelberg - New York - Dordrecht - London, 2014. [4] B. Altay, F. Basar, “Some new spaces of double sequences”, J. Math. Anal. Appl. 309 (1) (2005), 70-90. [5] M. Mursaleen, F. Basar, “Domain of Cesàro mean of order one in some spaces of double sequences”, Stud. Sci. Math. Hungar. 51 (3) (2014), 335-356.
58
Theorems on Generalized Bose-Einstein and Fermi-Dirac Functions Abdelmejid Bayad
D´epartement de math´ematiques, Universit´e d’Evry Val d’Essonne, France [email protected]
Abstract: In this talk, by means of integral representation, we investigate the generalized Hurwitz–Lerch zeta functions of arbitrary complex order. For these functions, we establish reduction formulas and its associated dual formulas. We investigate analytic continuations to the whole complex plane and special values. As applications of this study we generalized Bose-Einstein and Fermi-Dirac zetas functions. We deduce for them reduction and dual formulas. We also obtain nice and useful formulae for the Bernoulli–N¨orlund and Apostol–Euler–N¨orlund polynomials. Our formulas are the arithmetical nature and can be applied to physics. This is a joint work with J. Chikhi ([email protected])
59
Non-Linear Characterization of Some Banach Spaces Maatougui Belaala
Department of Mathematics, M’sila University, M’sila, Algeria
Abstract: In this contribution, we study the non-linear extending theorems of characterization of some Banach spaces. Our work consists of three sections
1) Reflexive space and characterization, James theorem; using the multilinear form
and the polynomials
)(...)()( ,)()()(
1,...,1xxxxxPxxxP m
m
xxm
x m
*** == ***
2) Multilinear and polynomial characterization of ¥L -space.
3) Polynomial and multilinear version of Kwpień theorem. Keywords: James theorem, Kwpień theorem, p-summing multilinear operators.
References:
[1] D. Achour and L. Mezrag, “ On the Cohen strongly p-summing multilinear operators”, J. Math. Anal. Appl.
327.1(2007), 550-563.
[2] G. Botelho, “Ideals of polynomials generated by weakly compact operators”, Note Mat, 25 (2005), 69-102.
[3] V. Dimant, “Strongly p-summing multilinear operators”, J. Math. Anal. Appl. 278(2003), 182-193.
[4] S. Kwpień, “A linear topological characterization of inner product space”, Studia Mathematica, T(1970), 277-
278.
60
Numerical Solutions of Boundary Integral Equation by Using the Galerkin Method Menad Bendehiba
Laboratory of mathematic and its applications (LAMAP), University of Oran, Algeria [email protected]
Abstract: In this paper the exact solution of boundary integral equation (BIE) of the second kind with degenerate kernel is presented. The Galerkin method is applied to get the approximate solution of BIE. The numerical solutions are given and compared with the exact solution. Solve numerical example throw MATLAB.Grafic representation of boundary integral equation (BIE) by using the Galerkin method. Keywords: Approximation; Galerkin method; Integral equation; Orthogonal polynomials. References: [1] K.E. Atkinson “The numerical of integral equation of the second kind”, Cambridge press, 1997. [2] F.D. Gakhov “Boundary Value Problems”, Pergamon Press.1966. [3] N.I Muskhelishvili “Singular integral equations”, Noordhoff, Holland, 1953. [4] A. Palamara “Orsi Product integration for Volterra integral equations of the second kind with weakly singular kernels”, Math. Comp. 65, pp. 1201–1212, (1996).
61
Weak Solution and Exact Controllability of the Caputo’s Fractional Cauchy Problem of Order α with 1 < α < 2 R. Boukhamla and S. Mazouzi
University of Annaba, Department of Mathematics, Annaba 23000, Algeria. Laboratoire de Mathématiques & informatique University of Souk Ahras, Algeria.
Abstract: The theory of fractional differential equations or simply fractional systems has become an important area of research in recent years. Linear fractional problems, meanwhile, are fundamental in most branches of applied mathemat-ics, and phenomena studied, for instance in control theory, physics, biology, engineering, population dynamics, science and technology, etc [1,2]. In this work we study exact controllability of abstract Cauchy problem for the fractional evolution equation of order α with Caputo fractional derivative in the case 1<α≤2. We give a fundamental theory and basic properties of α-time fractional evolution operators (α-time fractional resolvent operators), a generalization and extension of the resolvent operators of standard differential equations to arbitrary α-time fractional differential equations with α is the positif real numbers, which is important in the calitative theory of α-time fractional differential equations [3,5]In order to estabilsh representation formula for classical solution of the abstract fractional Cauchy problem for the fractional evolution equation. By analogy, the existence and uniqueness of a Cα mild solution of abstract Cauchy problem for the fractional evolution equation with infinite impulses based on fixed point theorem "fixed point problem for an appropriate operator in a Cα function space" [4,7,9].Sufficient conditions are established for the existence of weak solution (on α fractional version) in a some suitable Hilbert space [6].We introduce the notion (on α fractional version) of the null controllability of the initial state. We generalize some results of controllability obtained for classical evolution equations to the abstract α-time fractional Cauchy problem for the fractional impulsive evolution equation with .nite impulses in a Hilbert space with 1<α<2. These results extend certain aspects the standard theory of null controllability for the first order evolution equation. It takes an equation from an initial point to a final point in a state space, in a given finite time interval [8]. Keywords: Fractional differential equations; α-time fractional differential equations; Controllability of the Caputo’s fractional; Cauchy problem. References: [1] K. Balachandran and J.Y. Park, Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Analysis: Hybrid Systems, (2009), doi:10.1016/j.nahs.2009.01.014. [2] R. Boukhamla, Study of the controllability of differentail equations under impulsive conditions, phd, University of Annaba, 2008. [3] Y.Q. Chen, H.S. Ahu and D. Xue, Robust controllability of interval frac- tional order linear time invariant systems, Signal Processing, 86 (2006), 2794.2802. [4] Jin Cheng, Junichi Nakagawa, Masahiro Yamamoto and Tomohiro Ya- mazaki, Uniqueness in an inverse problem for one-dimensional fractional diffusion equation, UTMS 2009.4 May 7, 2009. [5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. [6] X. Li and C. Xu , Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Di¤usion Equation and a Spectral Method Approximation, Commun. Comput. Phys., 8 (2010), pp. 1016-1051. [7] Changpin Li, Deliang Qian, and YangQuan Chen, On Riemann-Liouville and Caputo Derivatives, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 562494, 15 pages. [8] Brahima Mbodje, Wave energy decay under fractional derivative controls IMA Journal of Mathematical Control and Information 2006 23(2):237-257;doi:10.1093/imamci/dni056. [9] Nakagawa, J., Sakamoto, K., Yamamoto, M.: Overview to mathemati- cal analysis for fractional di¤usion equations .new mathematical aspects motivated by industrial collaboration. J. Math-for-Industry 2 (2010A-10), 99.108.
62
A delay second order set-valued differential equation with Hukuhara derivatives
Dalila Azzam-Laouir and Wafiya Boukrouk Jijel University, Algeria [email protected]
Abstract: The notion of Hukuhara derivative was introduced by M. Hukuhara in [3] . This notion is strictly connected with the theory of set differential equations. This theory has been started by De Blasi and Iervolino in 1969 ([1]). In this paper, we prove the existence of solutions for a second order set differential equation with three-point boundary conditions with the notion of Hukuhara derivative. The existence and uniqueness of a solution is proved under a Lipschitz condition on the right hand side in the second and third variables. Keywords: Boundary conditions, Fixed point, Hukuhara derivatives.
References:
[1] F. S. De Blasi, F. Iervolino, Equazioni differenziali con soluzioni a valore compatto convesso, boll. Unione Mat.
Ital. 2, No. 4-5, (1969), 491-501.
[2] P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, London Sydney (1967).
[3] M. Hukuhara, Intégration des applications mesurables dont la valeur est un compact convexe, Funkcial. Ekvac.
10 (1967), 205-223.
[4] M. Marek, T. Malinkowski, Second type Hukuhara differentiable solutions to the delay setvalued differential equations, Appl. Math. Comput. 218 (2012) 9427-9437. [5] M. Piszczek, On a multivalued second order differential problem with Hukuhara Derivative, em J. Appl. Math.
Comput. (2008) 151-161.
63
Maximum Norm Analysis of a Nonmatching Grid Method for Semilinear Elliptic Variational Inequalities
Messaoud Boulbrachene and Abida Harbi Department of Mathematics, Statistics, Sultan Qaboos University, Muscat, Oman
Department of Mathematics, University Badji Mokhtar, Annaba, Algeria [email protected], [email protected]
Abstract: Extensive mathematical analysis of Schwarz alternating method for nonlinear elliptic boundary value problems has been achieved in the last there decades (c.f., e.g., [1],[2],[3]). In this paper, we provide a maximum norm analysis of an overlapping Schwarz finite element method on nonmatching grids for elliptic variational inequalities (VI) with nonlinear source terms. More precisely, we consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The grid points on the subdomain boundaries need not match the grid points from the other subdomain. Under a Lipschitz asssumption on the nonlinearity, we establish, on each subdomain, an optimal error estimate in the maximum norm between the discrete Schwarz sequence and the exact solution of the VI. This work can be regarded as an extension of [4] to semilinear variational inequalities. Keywords: Semilinear variational inequalities; Domain decomposition; Schwarz method; Finite elements; Nonmatching grids; Error estimate. References: [1] P.L. Lions, “On the Schwarz Alternating Method II, Proc. 2nd Int. Symp. on domain decomposition methods for partial Differential Equations”, S.I.A.M. Philadelphia, pp. 47-70 (1989). [2] S-H Lui, “On Monotone and Schwarz Alternating Methods For Nonlinear Elliptic PDEs.”, Mathematical Modelling and Numerical Analysis, M2AN, Vol. 35, 1, 1-15 (2001) [3] S-H Lui, “On Linear Monotone Iteration and Schwarz Methods For Nonlinear Elliptic PDEs.”, Numerish Mathematik, pp.109-129(2002) [4] A. Harbi, M. Boulbrachene, “Maximum Norm Analysis of a Nonmatching Grids Method for Elliptic PDEs.”, Journal of Applied Mathematics, vol (2011), Article ID 605140 (18 pages).
64
Existence of Fixed Points for Continuous Operators in Banach Spaces Using Measure of Noncompactness and Under an Integral Condition
N.H. Bouzara, V. Karakaya, Y. Atalan, and K. Dogan Department of Mathematics, Yildiz Technical University, Istanbul, Turkey
Abstract: Our aim in this talk is to present results of existence of fixed points for continuous operators in Banach spaces using measure of noncompactness and under an integral condition. This results are generalisation of results gived by A. Aghajania and M. Aliaskaria in [1] which are generalisation of Darbo’s fixed point theorem. As application we use these results to solve integral equations in Banach spaces. References: [1] A. Aghajania, M. Aliaskaria, "Generalization of Darboís Öxed point theorem and application", Int. J. Nonlinear Anal. Appl, No.2, 2011, pp 86-95.
65
Some Geometric Properties Related to the Second Order Cesàro Operators
Naim L. Braha and Valdete Loku Department of Mathematics and Computer Sciences, University of Prishtina, Kosova
[email protected]; [email protected]
Abstract: In this paper it is proved that for any (1, )p Î ¥ the Cesaro second order sequence space 2 ( )Ces p has
( )b - property, respectively, is (k-NUC), for 2.k ³ Has Kadec-Klee property, rotund property and for any natural number $k,$ it has the Uniform Opial property. Moreover, it is proved that it possesses (L) property, uniform convexity, is reflexive, has fixed point and is calculated packing constant (C). Keywords: Cesaro second order sequence spaces; Normed sequence spaces; Rotund property; Kadec-Klee property; Uniform Opial property; ( )b - property; kNUC-property. References: [1]N.L.Braha, “Some geometric properties of $N(\lambda,p)$ spaces”, J. Inequal. Appl. 2014, 2014:273. [2]N.L.Braha, E. Malkowsky, V. Krasniqi, “Some properties of the second order Cesaro sequence spaces” (submitted). [3] S. Chen, “Geometry of Orlicz spaces”, Dissertationes Math. 356 (1996) 1-224. [4] Y.A. Cui, H. Hudzik, R. Pluciennik, “Banach-Saks property in some Banach spaces”, Annales Math. Polonici 65, 193-202 (1997). [5] Y.Cui, H. Hudzik, “On the uniform Opial property in some modular sequence spaces”, Funct. Approx. Comment. Math. 26 (1998), 93-102 [6] Cui, Yunan; Meng, Chenghui; Pluciennik, Ryszard. “Banach-Saks property and property $(\beta)$ in Cesàro sequence spaces”, Southeast Asian Bull. Math. 24 (2000), no. 2, 201-210. [7] J. Diestel, “Geometry of Banach Spaces - Selected Topics”, Springer - Verlag, 1984. [8] R. Goebel, W.A. Kirk, “Topics in Metric Fixed Point Theory”, Cambridge University Press, 1990. [9] V. I. Gurarii, “Differential properties of the convexity moduli of Banach spaces”, Matematicheskie Issledovaniya, vol. 2, pp. 141-148, 1967. [10] C.A. Kottman, “Packing and reflexivity in Banach spaces”, Trans. Amer. Math. Soc. 150 (1970), 565-576. [11] I.J. Maddox, “Spaces of strongly summable sequences”, Quart. J. Math. Oxford 18 (2)(1967) 345-355. [12] H. Nakano, “Modulared sequence spaces”, Proc. Japan Acad. 27 (2) (1951) 508-512. [13] H. Nergiz, F.Basar, “\textitSome geometric properties of the domain of the double sequential band matrix $B(\tilder, \tildes)$ in the sequence space $l(p)^*$”, Abstr. Appl. Anal. 2013, Art. ID 949282, 10 pp. [14] S. Prus, “Banach spaces with uniform Opial property”, Nonlinear Anal. 8 (1992), 697-704. [15] Savas, Ekrem; Karakaya, Vatan; Simsek, Necip. “Some $l(p)$-type new sequence spaces and their geometric properties”, Abstr. Appl. Anal. 2009, Art. ID 696971, 12 pp. [16] S. Simons, “The sequence spaces $l(p_v)$ and $m(p_v)$”, Proc. London Math. Soc. 15 (3) (1965) 422-436. [17] J.S. Shiue, “On the Cesaro sequence space”, Tamkang J. Math. 2(1970) 19-25.
66
Lacunary Statistically Upward Continuity Huseyin Cakalli
Mathematics Division, Maltepe University, Maltepe, Istanbul, Turkey [email protected]
Abstract: A real valued function defined on a subset E of IR is lacunary statistically upward continuous if it preserves lacunary statistically upward half quasi-Cauchy sequences. We investigate lacunary statistically upward continuity, and prove interesting theorems. Keywords: Lacunary Statistical Convergence; Continuity. References: [1] D. Burton, J. Coleman, “Quasi-Cauchy sequence”, Amer. Math. Monthly, 117.4(2010) 328-333. [2] H. Cakalli, Forward continuity”, J.Comput. Anal. Appl., 13.2(2011) 225-230. [3] H. Cakalli, E. Savas, “Statistical convergence of double sequences in topological groups”, J. Comput. Anal. Appl. 12.2(2010) 421-426. [4] J. Fridy, C. Orhan, “Lacunary statistical convergence”, Pac. J. Math. 160.1(1993) 43-51. [5] H. Cakalli, “Lacunary statistical convergence in topological groups”, Indian J. Pure Appl. Math. 26.2(1995) 113-119. [6] H. Cakalli, C. Aras, A. Sonmez, “On lacunary statistically quasi-Cauchy sequences”, The Algerian-Turkish International days on Mathematics, Abstract Book, Istanbul, Turkey, (2013) p. 42. [7] F.J. Palladino, “On half Cauchy sequences”, ArXiv e-prints 1102.4641. [8] H. Cakalli, “Half quasi-Cauchy sequences”, ArXiv e-prints 1205.3674. [9] L. D. Kocinac, “Selection properties in fuzzy metric spaces”, Filomat, 26.2 (2012) 305-312. [10] D.Djurcic, Ljubisa D.R. Kocinac and Malisa R. Zizovic, “Double sequences and selections”, Abstr. Appl. Anal. 2012(2012) 6. doi:10.1155/2012/497594.
67
Topological Indices of Subdivision Graphs Ismail Naci Cangul, Aysun Yurttas, Muge Togan, Ahmet Sinan Cevik,
Department of Mathematics, Uludag University, Gorukle, Bursa, Turkey Department of Mathematics, Selcuk University, Konya, Turkey
[email protected], [email protected],[email protected], [email protected]
Abstract: Topological indices of graphs are deÖned and intensively studied in last 40 years. Amongst these, Örst
and second Zagreb indices can be count easily and they have nice applications in Chemistry and Mathematics.
Several other versions of these indices were deÖned in terms of vertex degrees, minimal and maximal vertex degrees
and similar data. Here we determine several types of Zagreb indices of the subdivision graphs of certain graph types
and also give some relations and inequalities between them.
Keywords: Topological Graph Theory, Imbeding, Enumeration of Graphs and Maps
68
Existence of Solutions for Nonlinear Fractional Differential Equations with m-Point Integral Boundary Conditions
Tugba Senlik Cerdik, Nuket Aykut Hamal, Fulya Yoruk Deren, Department of Mathematics, Ege University, Bornova, Izmir, Turkey
Abstract: In this paper, we consider m-point integral boundary value problems for fractional differential equations
involving the Riemann Liouville fractional derivative. The existence results of solutions are established via the
application of fixed point theorems.
Keywords: Fixed Point Theorem, m-Point Integral Boundary Conditions, Riemann Liouville Derivative.
References:
[1] R. P. Agarwal, S. K Ntouyas, B. Ahmad, M. S. Alhothuali, Existence of solutions for integro-differential
equations of fractional order with nonlocal three-point fractional boundary conditions, Advances in Difference
Equations 2013, 2013:128.
[2] B. Ahmad, S. K. Ntouyas, and A. Alsaedi, New Existence Results for Nonlinear Fractional Differential Equations
with Three-Point Integral Boundary Conditions, Advances in Difference Equations, Volume 2011, 107384, 11 p.
[3] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).
[4] K.Deimling, Nonlinear Functional Analysis. Springer, Berlin, 1985.
[5] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in:
North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V, Amsterdam, 2006.
69
On Reciprocity Law of the Y(h,k) Sums Associated with the Two and Three-Term Polynomial Relations
Elif Cetin, Yilmaz Simsek and Ismail Naci Cangul Department of Mathematics, Uludag University, Bursa, Turkey;
Department of Mathematics, Akdeniz University, Antalya, Turkey [email protected],[email protected],[email protected]
Abstract: In [9], by using the two-term polynomial relations, we found a new finite sum which is related to the Hardy-Berndt sums and the Simsek's sum Y(h,k). Now, by using this two-term polynomial relation again we give another poof of reciprocity law of Simsek's sum Y(h,k). Our method is different from that of Simsek's (On Analytic properties and character analogs of Hardy Sums, Taiwanese J. Math. 13 (1) (2009), 253-268).Then we give some useful relations and remarks on these sums. Keywords: Hardy-Berndt Sums; Dedekind Sums; Three-term polynomial relations; Greatest Integer Function; Y(h,k) sum. References: [1] T. M. Apostol, “Modular functions and Dirichlet Series in Number Theory”, Springer-Verlag(1976). [2] T. M. Apostol, T. H. Vu, “Elementary Proofs of Berndt's Reciprocity Laws”, Pasific J. Math. 98 (1982), 17-23. [3] M. Beck, “Geometric proofs of polynomial reciprocity laws of Carlitz, Berndt, and Dieter, M. Beck, in Diophantine analysis and related fields 2006”, Sem. Math. Sci. 35, Keio Univ., Yokohama, 2006, pp. 11--18. [4] B. C. Berndt, “Analytic Eisenstein Series, Theta-functions, and Series relations in the spirit of Ramanujan”, J. Reine Angew. Math. 303/304(1978), 332-150. [5] B. Berndt, U. Dieter, “Sums involving the greatest integer function and Riemann Stieltwes integration”, J. Reine Angew. Math. 337, 208-220 (1982). [6] Berndt, B. C., and Goldberg, L. A., Analytic Properties of Arithmetic Sums arising in the theory of the classical Theta-functions, SIAM., J. Math. Anal. 15(1984),. 143-150. [7] M. Can, V. Kurt, “Character analogues of certain Hardy-Berndt sums”, International Journal of Number Theory 10 (3) (2014), 737-762. [8] L. Carlitz, “Some polynomials associated with Dedekind Sums”, Acta Math. Sci. Hungar, 26 (1975), 311-319. [9] E. Cetin, Y. Simsek, I. N. Cangul, “ome special finite sums related to the three-term polynomials relations and their applications”, preprint. [10] L. A. Goldberg, “Transformation of Theta-functions and analogues of Dedekind sums”, Thesis, University of Illinois Urbana(1981). [11] G. H. Hardy, “On certain series of discontinues functions, connected with the modular functions”, Quart. J. Math. 36(1905), pp. 93-123 (= Collected papers, vol.IV, pp. 362-392. Clarendon Press Oxford (1969)). [12] M. R. Pettet, R. Sitaramachandraro, “Three-Term relations for Hardy sums”, J. Number Theory 25(1989), 328-339. [13] Y. Simsek, “On Generalized Hardy Sums s₅(h,k)”, Ukrainian Math. J., 56(10) (2004), 1434-1440. [14] Y. Simsek, “Theorems on Three-Term Relations for Hardy sums”, Turkish J. Math. 22(1998), 153-162. [15] Y. Simsek, “A note on Dedekind sums”, Bull. Cal. Math. Soci. 85(1993) 567-572. [16] Y. Simsek, “On Analytic properties and character analogs of Hardy Sums”, Taiwanese J. Math. 13 (1) (2009), 253-268. [17] R. Sitaramachandrarao, “Dedekind and Hardy sums”, Acta Arith. XLVIII (1978).
70
On the Solvability and Maximal Regularity of Complete Abstract Differential Equations of Elliptic Type with General Robin Boundary Conditions in Holder Spaces
Mustapha Cheggag, Angelo Favini, Rabah Labbas, Stéphane Maingot and Ahmed Medeghri Department of Mathematics and ComputerSciences, Polytechnic National school of Oran, Algérie;
Dipartimento di Matematica, Università di Bologna, Italia; Laboratoire de Mathématiques Appliquées, Université du Havre, France;
Laboratoire de Mathématiques Pures et Appliquées,Université de Mostaganem, Algérie [email protected], [email protected], [email protected],
[email protected], [email protected]
Abstract: In this work we give some new results on complete operational second order differential equations of elliptic type with general Robin boundary coefficient-operator conditions in the case of Hölder spaces. Necessary and sufficient conditions of compatibility are proved to obtain a strict solution and also to ensure that the strict solution has the maximal regularity property. This work completes the ones studied by Favini et al (see [4] and [5]) under Dirichlet boundary conditions and Cheggag et al, see [1] and [2]. Keywords: ADE; Robin; Semigroups; Interpolation spaces; Hölder spaces; Strict solution; PDE. References: [1] M. Cheggag, A. Favini, R. Labbas, S. Maingot, A. Medeghri, “Sturm-Liouville Problems for an Abstract Differential Equation of Elliptic Type in UMD Spaces”, Differential and Integral Equations, 21, no. 9-10 (2008), 981-1000. [2] M. Cheggag, A. Favini, R. Labbas, S. Maingot, A. Medeghri, “Abstract Differential Equations of Elliptic Type with General Robin Boundary Conditions in Hölder Spaces”, Applicable Analysis, 91, no. 8, August 2012, 1453-1475. [3] G. Da Prato, P. Grisvard, “Sommes d'Opérateurs Linéaires et Equations Différentielles Opérationnelles”, J. Math. Pures Appl. IX Ser., 54 (1975), 305-387. [4] A. Favini, R. Labbas, S. Maingot, H. Tanabe, A. Yagi, “On the Solvability and Maximal Regularity of Complete Abstract Differential Equations of Elliptic Type”, Funkc. Ekv., 47 (2004), 423-452. [5] A. Favini, R. Labbas, S. Maingot, H. Tanabe, A. Yagi, “Necessary and Sufficient Conditions in the Study of Maximal Regularity of Elliptic Differential Equations in Hölder Spaces”, Discrete and Continuous Dynamical Systems, 22 (2008), 973-987. [6] E. Sinestrari, “On the Abstract Cauchy Problem of Parabolic Type in Spaces of Continuous Functions”, J. Math. Anal. App., 66 (1985), 16-66.
71
A Generalization of Gershgorin Circles Mao-Ting Chien
Department of Mathematics, Soochow University, Taipei, Taiwan [email protected]
Abstract: The Gershgorin circle thorem shows that the spectrum of a matrix is contained in the Gershgorin region of the matrix. The Gershgorin circle theorem has many applications, for instance, in locating the zeros of a olynomial (cf. [1, 3]), in stability theory (cf. [2]) and controller design (cf. [4]). There are many generalized Gershgorin types of inclusion regions. In this talk, we give a new inclusion type of Gershgorin circles, and show how this generalization compares with the regular Gershgorin circle theorem. Keywords: Gershgorin circles; Eigenvalues; Polynomial roots. References: [1] Y. A. Alpin, M. T. Chien, L. Yeh, “The numerical radius and bounds for zeros of a polynomial”, Proc. Amer. Math. Soc. 131(2003), 725-730. [2] P. F. Curran, “On a variation of the Gershgorin circle theorem with applications to stability theory”, IET Irish Signals and Systems Conference, Dublin, 2009. [3] A. Melman, “Generalizations of Gershgorin disks and polynomial zeros”, Proc. Amer. Math. Soc. 138(2010), 2349-1327. [4] D. Rosinova, N. Q. Thuan, V. Vesely, L. Marko, “Robust decentralized controller design: Subsystem approach”, J. Elect. Eng. 63(2012), 28-34.
72
Implementation of Public-Key Cryptosystems on Embedded Devices Noureddine Chikouche and Walid Tayoub
Department of Computer Science, University of M’sila, M’sila, Algeria [email protected] and [email protected]
Abstract: The security of Public Key Cryptography (PKC) is based on difficulty of mathematical problems. The public key cryptosystem is RSA cryptosystem [1] which is based on integer
factorization problem. The Elliptic Curve Cryptography [2, 3] is relied on discrete a logarithm that is harder to
face at the same key size. The NTRU cryptosystem [4] was presented in 1996 and published in 1998, and its
security is based on two difficult problems, SVP (Shortest Vector Problem) and CVP (Closest Vector Problem),
and his domain calculation is the ring of polynomials Z [X] = (XN -1).
In addition, the key size of Elliptic Curve Cryptography and NTRU are smaller than RSA. These two last cryptosystems need less bandwidth, less storage space and less computing time, compared
with the other schemes. This allows applying encryption in restricted platforms, such as smart cards, wireless
devices, and thin clients. The objectives of this paper are: (1) to implement the RSA, ECC and NTRU
cryptosystems (key generation, encryption, decryption and signature algorithms) on different number of platforms of
Android mobile phones, and (2)discus the results obtained.
Keywords: RSA Crytptosystem, ECC Cryptosystem, NTRU Cryptosystem, performance. References: [1] R. L. Rivest, A. Shamir, and L. Adleman, “A Method for Obtaining Digital Signatures and Public-Key
Cryptosystems”, Communications of the ACM, vol.
21(1978), 120-126.
73
Convergence Analaysis of Strang Splitting Method for Burgers-Huxley Equation Yesim Cicek and Gamze Tanoglu
Department of Mathematics, Izmir Institute of Technology, Urla, Izmir, Turkey [email protected]
Abstract: We provide an error analysis of the operator splitting method of the operator splitting of the Godunov and Strang type applied to the Burgers-Huxley equation, 𝑢𝑢𝑡𝑡 + 𝛼𝛼𝑢𝑢𝑢𝑢𝑥𝑥 − 𝑥𝑥𝑢𝑢𝑥𝑥𝑥𝑥 = 𝛽𝛽(1 − 𝑢𝑢)(𝑢𝑢 − 𝛾𝛾)𝑢𝑢. The major task is to prove the convergence rates for the two splitting methods in Sobolev spaces. We split the equations into linear and nonlinear parts and show that the operator splitting methods have the correct convergence rates in 𝐻𝐻𝑠𝑠(ℝ), where 𝐻𝐻𝑠𝑠(ℝ) is the Sobolev space and 𝑠𝑠 is an arbitrary non-negative integer. We numerically apply the operator splitting methods to the Burgers-Huxley equation for the split step size ∆t. Keywords: Operator splitting; Burgers-Huxley equation; Nonlinear PDEs References: [1] C. A. Ambrosetti, G. Prodi, “A Primer of Nonlinear Analysis”, Cambridge UP, Cambridge, 1995. [2] H. Holden, C. Lubich, N. H. Risebro, “Operator splitting for partial differential equations with Burger nonlinerity”, Mathematics of Computation. 82 (2013) 173–185. [3] H. Holden, C. Lubich, N. H. Risebro, T. Tao, “Operator splitting for the KDV equation”, Mathematics of Computation. 80 (2011) 821–846. [4] A. Kolmogorov, I. Petrovskii, N. Piskunov, Moscow Univ. Bull. Math., Moscow Univ. Bull. Math, 1937 [5] E. B. Nilsen, “On Operator Splitting for the Viscous Burgers’ and the Korteweg-de Vries Equations”, Master of Science in Physics and Mathematics. (2011) [6] G. I. Marchuk, “Methods of splitting”, Nauka, Moscow, 1988. [7] G. Strang, “On the construction and comparison of different splitting schemes”, SIAM J. Numer. Anal. 5 (3) (1968) 506–517. [8] M. Mimura, T. Nakaki, K. Tomeada , “A numerical approach to interface curves for some nonlinear diffusion equations”, Japan. J. Appl. MAth. 1 (1968) 93–139. [9] R. S. Marinova, C. I. Christov, T. T. Marinov, “A fully coupled solver for incompressible Navier-Stokes equations using operator splitting”, Int. J. Comput. Fluid Dyn. 17 (5) (2003) 71–385. [10] K. Hvistendahl Karlsen, A. Lie, H. F. Nordhaug and H. K. Dahle , “Operator splitting methods for systems of convection-diffusion equations: Nonlinear error mechanisms and correction strategies”, J. Comput. Phys. 173 (2001) 636–663. [11] C. I. Christov, R. S. Marinova , “Implicit vectorial operator splitting for incompressible Navier-Stokes equations in primitive variables”, J. Comput. Technol. 6 (4) (2001) 92–119. [12] J. G. Verwer, B. Sportisse, “A note on operator splitting in a stiff linear case”, CWI, Amsterdam, Netherlands, MAS-R9830 1998.
74
Computing Homology Groups of Complexes of Matchings Ismet Cinar and Ismet Karaca
Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected] , [email protected]
Abstract: This poster is related to the topology of graph complexes [4], the emphasis being placed on homology,
homotopy type [1], connectivity degree [2], and Euler characteristic. We introduce some of monotone graph
properties [5]. One can recall that monotone graph properties are complexes on such a graph satisfying the additional
condition. They are invariant under permutations of V (i.e., the set of V is vertex set). Some well known monotone
graph properties that we research in this poster are complexes of matchings [3,6] and we determine the homology
groups of complexes of matchings.
Keywords: simplicial complex, monotone graph property, homotopy type, connectivity degree,complexes of
matchings
References:
[1] J. R. Munkres, “Elements of Algebraic Topology”, Persues Books Publishing, (1984).
[2] D. B. Karaguezian, “Homology of complexes of degree one graph”, PhD thesis, Stanford University, (1994).
[3] V. Reiner and J. Roberts, “Minimal resolutions and homology of chessboard
and matching complexes”, J. Algebraic Combin., 11(2000), 135-154.
[4] M. L. Wachs, “Topology of matchings, chessboard and general bounded degree graph complexes”, Alg.
Universalis, 49(4)(2003), 345-385.
[5] J. Jonsson, “Matching complexes on grid”, Manuscript, (2005).
[6] J. Shareshian and M. L. Wachs, “Torsion in the matching complex and chessboard complex”, Adv. Math.,
212(2)(2007), 525-570.
75
Deferred Statistical Convergence of Order α
Muhammed Cinar, Mikail Et, Fatih Temizsu and Murat Karakas Department of Mathematics, Mus Alparslan University, Mus, Turkey Department of Mathematics, Firat University, 23119 Elazig, Turkey
Department of Mathematics, Bingol University, Bingol, Turkey [email protected], [email protected], [email protected], [email protected]
Abstract: In this article, we introduce the concepts of deferred statistical convergence of order and strongly r-deferred Cesaro summability of order of complex (or real) sequences. Also, some relations between deferred statistical convergence of order and strongly r-deferred Cesaro summability of order are given. Keywords: Statistical convergence; Summability of sequences; Cesaro summability References: [1] M. Yilmazturk, M. Kucukaslan, “On strongly deferred Cesaro summability and deferred statistical convergence of the sequences”, Bitlis Eren Univ. J. Sci. and Technol. 3, 22-25 (2013).
76
Some Fixed Point Theorems for a Hybrid Type of Bogin-Popescu Mappings in Complete Metric Spaces
L. Ciric, P. Promsilpchai and N. Petrot Faculty of Mechanical Engineering, Kraljice Marije, Belgrade, Serbia
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, Thailand Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, Thailand
[email protected], [email protected], [email protected]
Abstract: By using a concept of generalized commuting mappings, we study a new class of hybrid type mappings. Some fixed point theorems and an interesting example are considered and discussed on such an introduced class. The presented results in this work are generalizations and improvements of many important results, in the sense that we are providing more choices of tool implements to check whether a fixed point of considered mapping exists. Keywords: Fixed point; Common fixed point; Coincidence point; Commuting mappings; Weakly compatible mappings.
77
On the Integral of Products of Higher-Order Bernoulli and Euler Polynomials M. Cihat Dagli and Mumin Can
Department of Mathematics, Akdeniz University, Konyaaltı, Antalya, Turkey [email protected], [email protected]
Abstract: In this study, we extend known results on the integrals of products of the ordinary Bernoulli polynomials ([1, 2, 3]) to the higher-order Bernoulli polynomials. By the same way, similar relations are obtained not only for r higher-order Euler polynomials but also for l higher-order Bernoulli polynomials and r higher-order Euler polynomials. Moreover, we relate these results to the reciprocity formulas for 𝑠𝑠3,𝑝𝑝(ℎ, 𝑘𝑘) and 𝑠𝑠4,𝑝𝑝(ℎ, 𝑘𝑘) Hardy-Berndt sums ([4]) and 𝑇𝑇𝑚𝑚(ℎ, 𝑘𝑘) generalized Dedekind sums ([5]). Keywords: Bernoulli polynomials; Bernoulli numbers; Integrals; Recurrence relations; Dedekind sums. References: [1] L. Carlitz, “Note on the integral of the product of several Bernoulli polynomials”, J. London Math. Soc., 34(1959), 361-363. [2] T. Agoh and K. Dilcher,“Integrals of products of Bernoulli polynomials”, J. Math. Anal.Appl., 381(2011), 10-16. [3] S. Hu, D. Kim, M.-S. Kim, “On the integral of the product of four and more Bernoulli polynomials”, Ramanujan J., 33(2014), 281-293. [4] M. Can, M. Cenkci, V. Kurt, “Generalized Hardy-Berndt sums”, Proc. Jangjeon Math. Soc., 9(2006), 19-38. [5] Min-Soo Kim, Jin-Woo Son, “On generalized Dedekind sums involving quasi-periodic Euler functions”, J. Number Theory, 144(2014), 267-280.
78
Algebraic Invariant and Stability of Differential System Dahira Dali
Department of Mathematics, University of Sciences and Technology Houari Boumedien Bab-Ezzouar, Algiers
Abstract: One consider differential polynomial differential system of finite dimension and of degree at most k with real coefficients and the action ofthe general linear group. One give an algorithmic method to characterize the local stability of a given cubic system using invariant theory with thehelp of algebraic and semi-algebraic relations [1]. Key words: differential system, invariants, linear transformations, singular points, stability. References: [1] D. Dali: Grobner basis of algebraic invariant in polynomial differential system, Matematische. Vol XLIII (2008), FSasc. II, 16–21. [2] D. Hilbert: theory, Cambridge university press. (1193). [3] C. S. Siberskii: Introduction to the algebraic theory of invariants of differential equations, Nonlinear science , theory, and application, Manchester university press (1988).
79
Approximation by Chlodowsky Type q-Jakimovski-Leviatan Operators Ozge Dalmanoglu and Sevilay Kirci Serenbay
Department of Mathematics, Baskent University, Faculty of Education, Ankara, Turkey [email protected], [email protected]
Abstract: Chlodowsky type generalization of Jakimovski-Leviatan operators are defined by
÷÷ø
öççè
æ÷÷ø
öççè
æ -= å
¥
=
-
nnkk
xb
n
n bkfx
bnp
gexfP
n
0
*
)1();(
in [4]. In this paper, we introduce a q-generalization of Chlodowsky type Jakimovski-Leviatan operators. We
establish approximation properties and rate of convergence results for these new operators. We also give a statement
about the rate of convergence of r-th q-derivative of the operator via divided differences and q- differences. Keywords: Jakimovski-Leviatan operators, Chlodowsky operators, rate of convergence, q-calculus, q-differences,
divided differences.
References:
[1] C. Atakut and İ. Büyükyazıcı, “ The rate of convergence of the q-analogue of Favard-Szasz type operators”,
Journal of Comp. Ana. and App., 13, 4 (2011), 673-682.
[2] A. Aral, “A generalization of Szasz-Mirakyan operators based on q-integers”, Mathematical and Computer
Modelling, 47, (2008), 1052-1062.
[3] A. Aral and V. Gupta, The q-derivative and applications to q-Szasz Mirakyan operators, Calcolo, 43, (2006),
151-170.
80
Some Large Sets in Countable Integral Domains Dibyendu De
Department of Mathematics, University of Kalyani, West Bengal-741235, India [email protected]
Abstract: Famous Szameredi's Theorem states that any subset of N with positive additive upper Banach density contains arithmatic progression of arbitrary length. Furstenberg ergodic theoritic proof of Szameredi's Theorem was so powerful that it opened a new branch in research, called ``Ergodic Ramsey Theory''. Vitaly Bergelson was the first person who investigated combinatorial structures of subsets of N with positive multiplicative density. Using various ergodic multiple recurrence theorems, Bergelson proved that multiplicatively large sets that is sets with positive multiplicative density have a rich combinatorial structure. He proved that for any multiplicatively large subset 𝐴𝐴 of N and any 𝑘𝑘 ∈ 𝑵𝑵 there exists 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑, 𝑒𝑒, 𝑞𝑞 ∈ 𝑁𝑁 such that 𝑞𝑞𝑗𝑗(𝑎𝑎 + 𝑖𝑖𝑏𝑏): 𝑖𝑖 ≤ 𝑖𝑖, 𝑗𝑗 ≤ 𝑘𝑘 ⊆ 𝐴𝐴 and 𝑏𝑏(𝑐𝑐 + 𝑖𝑖𝑒𝑒)𝑗𝑗: 0 ≤ 𝑖𝑖, 𝑗𝑗 ≤ 𝑘𝑘 ⊆ 𝐴𝐴. This shows that multiplicatively large subsets of 𝑁𝑁 contains combined arikthmatic and geometric progressions. In this presentation we to extend these results for countable integral domains in particular the the integral domain of Gaussian integers Z[i]. Keywords: Ergodic Ramsey theory; Folner density. References: [1] V. Bergelson, “Multiplicatively Large Sets and Ergodic Ramsey Theory”, Israel Journal of Mathematics 148 (2005), 23-40. [2] H. Furstenberg, “Recurrence in Ergodic theory and Combinatorial Number Theory”, Princeton University Press, Princeton NJ 1981. [3] E. Szemeredi, “On sets of integers containing no k elements in arithmetic progression”, Acta Arith. 27 (1975), 299-345.
81
Matrix Transformation of Statistically Convergent Sequences of Interval Numbers Shyamal Debnath and Subrata Saha
Department of Mathematics, Tripura University, India [email protected]
Abstract: The concept of statistical convergence was introduced by Steinhaus [10] and Fast [8] and then reintroduced by Schoenberg [12] independently. Later on, Statistical convergence turned out to be one of the most active areas of research in summability theory after the works of Fridy [13] and Šalát [20]. For some very interesting investigations concerning statistical convergence, one may consult the papers of Cakalli [7], Miller [9], Maddox [11] and many others, where more references on this important summability method can be found. Recently the sequence of interval numbers and usual convergence of sequences of interval numbers are studied by Chiao [14]. Later, Şengönül and Eryilmaz [15] introduced and studied bounded and convergent sequence spaces of interval numbers and showed that these spaces are complete metric space. In the recent days, Esi [1, 4] introduced and studied strongly almost l- convergence and statistically almost l- convergence of interval numbers and lacunary sequence spaces of interval numbers, respectively. For more information about interval numbers one may refer to Debnath et al. [19], Dwyer [16 ], Moore [17], Moore and Yang [18 ], Esi [2, 3]. The main objective of this paper is to determine necessary and sufficient conditions for the matrix of interval number 𝐴 = (𝑎𝑎𝑛𝑛𝑘𝑘) such that 𝐴𝐴 -transform of 𝑥 = 𝑥𝑘𝑘 belongs to c0
S(i) ∩ l∞𝑖𝑖 and cS(i) ∩ l∞
𝑖𝑖 , where 𝑥 ∈ c0S(i) ∩ l∞
𝑖𝑖 and 𝑥 ∈ cS(i) ∩ l∞
𝑖𝑖 respectively. Also establish some results related to this new type of sequence spaces. Keywords: Matrix transformations; Interval number; Statistical convergence. References: [1] A. Esi, “Strongly almost l- convergence and statistically almost l-convergence of interval numbers”, Scientia Magna, 7(2), 117-122 (2011). [2] A. Esi, “Lacunary Sequence Spaces of Interval Numbers”, Thai J. Mathematics, 10(2), 445-451, (2012). [3] A. Esi, “A new class of interval numbers”, J. of Qafqaz University, 31, 98-102 (2011). [4] A. Esi, “l- Sequence spaces of interval numbers”, Appl. Math. & Inform. Sci., 8(3), 1099-1102 (2014). [5] B. C. Tripathy, “Matrix transformation between some classes of sequences, J. Math. Anal. Appl. 206, 448-450, (1997). [6] E. Savas and M. Mursaleen, “Matrix transformations in some sequence spaces, Istanbul Univ. Fen Fak. Mat. Der. 52, 1-5, (1993). [7] H. Cakalli, “A study on statistical convergence”, Funct. Anal. Approx. Comput. 1(2), 19-24 (2009). [8] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2, 241-244 (1951). [9] H. I. Miller, “a measure theoretical subsequence characterization of statistical convergence”, Trans. Am. Math. Soc. 347(5), 1811-1819 (1995). [10] H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique”, Colloq. Math. 2, 73-74 (1951). [11] I. J. Maddox, “On strong almost convergence”, Math. Proc. Camb. Philos. Soc. 85(2), 345-350 (1979). [12] I. J. Schoenberg, “The integrability of certain functions and related summability methods”, Am. Math. Mon. 66, 361-375.
82
A Study on Solvability of Functional Equations Arising in Dynamic Programming Deepmala and Arup Kumar Das
SQC & OR Unit, Indian Statistical Institute, India. [email protected]
Abstract: This paper deals with the existence, uniqueness and iterative approximations of solutions for the functional equations arising in dynamic programming of multistage decision making processes in Banach spaces. Our main results extend and generalize the results due to several authors. Thus, we can say that the method described in our main section is an important procedure which is helpful for researchers to find the existence of the solutions of functional equations arising in dynamic programming. An example is also given to demonstrate the advantage of our results than existing ones in the literature. Keywords: Optimization; Dynamic programming; Functional equations.
83
On the Fixed Points for Pointwise Lipschitzian Semigroups in Modular Function Spaces Buthinah A. Bin Dehaish
Department of Mathematics Faculty of Science For Girls, King Abdulaziz University Jeddah, Saudi Arabia. [email protected], [email protected]
Abstract: We study the existence of common fixed points for pointwise Lipschitzian semigroups of nonlinear mappings :Tt C C® i.e. a family such that
0( ) , ( ) ( )T f f Ts t f Ts Tt f= + = o and ( ( ) ( )) ( ) ( )T f T g t f f gr a r- £ - where ( )
tlimsup 1t fa
®¥£ for every f CÎ , where C is a ρ-bounded, ρ-closed, convex subset of a modular function
space Lρ. Keywords: Fixed point; Modular function space; Nonexpansive mapping; Orlicz space; Pointwise Lipschitzian mapping; Pointwise nonexpansive mapping; Semigroup; Uniform convexity. References: [1] W. M. Kozlowski, “Modular Function Spaces”, Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 122. Dekker, New York (1988) [2] M. A. Khamsi, W. M. Kozlowski, S. Reich, “Fixed point theory in modular function spaces”, Nonlinear Anal. 14, 935-953(1990) [3] M. A Khamsi, W. M. Kozlowski, C. Shutao, “Some geometrical properties and fixed point theorems in Orlicz spaces”, J. Math. Anal. Appl. 155(2), 393-412 (1991) [4] M. A. Khamsi, “Fixed point theory in modular function spaces”, In: Proceedings of the Workshop on Recent Advances on Metric Fixed Point Theory, Sevilla, September 1995, pp. 31-35 (1996). MR1440218 (97m:46044) [5] W. M. Kozlowski, “Advancements in fixed point theory in modular function”, Arab. J. Math. (2012). doi:10.1007/s40065-012-0051-0 [6] M. A. Khamsi, W. M. Kozlowski, “On asymptotic pointwise contractions in modular function spaces”, Nonlinear Anal. 73,2957-2967 (2010) [7] M. A. Khamsi, W. M. Kozlowski, “On asymptotic pointwise nonexpansive mappings in modular function spaces”, J. Math.Anal. Appl. 380(2), 697-708.
84
Monotone Iterative Technique and Existence Results for Fractional Functional Differential Equations
Fulya Yoruk Deren, Nuket Aykut Hamal and Tuğba Senlik Cerdik, Department of Mathematics, Ege University, Bornova, Izmir, Turkey
Abstract: This paper is concerned with the existence of positive solutions for boundary value problems of fractional
functional differential equations involving the Caputo fractional derivative. The proof is based on the monotone
iterative technique.
Keywords: Monotone Iterative Technique, Positive Solution, Fractional Functional Differential Equation.
References:
[1] X. Li, S. Liu, W. Jiang, Positive Solutions for Boundary Value Problem of Nonlinear Fractional Functional
Differential Equations, Applied Mathematics
and Computation 217 (2011) 9278-9285.
[2] Y. Li, S. Sun, Z. Han, and H. Lu, The Existence of Positive Solutions for Boundary Value Problem of the
Fractional Sturm-Liouville Functional Differential Equation, Abstract and Applied Analysis, 2013, 301560, 20 p.
[3] Y. Zhao, H. Chen, L. Huang, Existence of Positive Solutions for Nonlinear Fractional Functional Differential
Equation, Computers and Mathematics with Applications 64 (2012) 3456-3467.
[4] C.Bai, J. Ma, Li Huang, Eigenvalue Criteria of Multiple Positive Solutions to Boundary Value Problems of
Second Order Delay Differential Equations, Journal of Mathematical Analysis and Applications 301 (2005) 457-476.
[5] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
[6] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in:
North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V, Amsterdam, 2006.
85
Best Proximity Point Theorem for F-Contraction in a Complete Metric Space Lakshmi Kanta Dey
Department of Mathematics, National Institute of Technology Durgapur, Durgapur, India [email protected]
Abstract: In the article, we prove the existence and uniqueness of best proximity point using P-property of a new type of contraction namely F-contraction, perceived to be one of the most general nonlinear contraction, which is recently introduced in [Wardowski, D.: Fixed point of a new type of contractive mappings in complete metric space; Fixed Point Theory and Applications, 2012, 2012:94]. Suitable examples are presented to substantiate our main result. Keywords: Best proximity point; F-contraction, P-property.
86
Relative and Tate Homology with Respect to Semidualizing Modules Zhenxing Di
Department of Mathematics, Southeast University, Nanjing, Jiangsu, China
Abstract: We introduce and investigate in this paper a kind of Tate homology of modules over a commutative coherent ring based on Tate -resolutions,CF where C is a semidualizing module. We show firstly that the class of
modules admitting a Tate -resolutionsCF equal to the class of modules of finite ( )-projectiveC dimension. Then an Avramov-Martsinkovsky type exact sequence is constructed to connect such Tate homology functors and relative homology functors. Finally, motivated by the idea of Sather-Wagstaff et al. [2], we establish a balance result for such Tate homology over a Cohen-Macaulay ring with a dualizing module by using a good conclusion provided in [1]. Keywords: Semidualizing module, finite Gorenstein dimension, Tate resolution, Tate homology, balance.
References: [1] E. E. Enochs, S. E. Estrada and A. C. Iacob, “Balance with unbounded complexes”, Bull. London Math. Soc. 44(2012), 439–442. [2] S. Sather-Wagstaff, T. Sharif and D. White, “Comparison of relative cohomology theorem with respect to semidualizing modules”, Math. Z. 264(2010), 571–600.
87
A Differential Equation Model for the Dynamics of Youth Gambling Tae Sug Do and Young S Lee,
Kwandong University, South Korea Manchester University, USA [email protected], [email protected]
Abstract: We create a simple mathematical model to study dynamics of youth gambling. The population of this study is youth ages 16-24. One of the main goals is to see how prevalence rates of at-risk gambling and problem gambling change as adolescents enter into young adulthood. A threshold condition that spreads gambling is identified through stability analysis. We estimate all model parameters using a longitudinal prevalence study by Winters, Stinchfield, and Botzet. The model indicates: problem gambling is present in an endemic state among the youth with a steady prevalence rate of about 4-5%; the prevalence rate of problem gambling in young adults ages 18 to 24 is lower than that in adolescents ages 16 to 18; at-risk gambling among young adults has increased. The parameters to which the system is most sensitive correspond to primary prevention. Prevention and control strategies are discussed. Numerical solutions are also exploited. In our knowledge, this is the first mathematical modeling approach to youth gambling behaviors.
88
On The Existence of Positive Solutions for the One-Dimensional 𝒑𝒑-Laplacian Boundary Value Problems on Time Scales
Abdulkadir Dogan Department of Applied Mathematics,Abdullah Gul University, Kayseri,Turkey
Abstract: In this paper, we study the following 𝑝𝑝-Laplacian boundary value problems on time scales
𝜙𝜙𝑝𝑝 𝑢𝑢Δ(𝑡𝑡)∇
+ 𝑎𝑎(𝑡𝑡)𝑓𝑓𝑡𝑡, 𝑢𝑢(𝑡𝑡), 𝑢𝑢∆(𝑡𝑡) = 0, 𝑡𝑡 ∈ [0, 𝑇𝑇]Τ, 𝑢𝑢(0) − 𝐵𝐵0𝑢𝑢∆(0) = 0, 𝑢𝑢∆(𝑇𝑇) = 0, where 𝜙𝜙𝑝𝑝 (𝑢𝑢) = |𝑢𝑢|𝑝𝑝−2𝑢𝑢, for 𝑝𝑝 > 1. We prove the existence of triple positive solutions for the one-dimensional 𝑝𝑝-Laplacian boundary value problem by using the Leggett-Williams fixed point theorem. The interesting point in this paper is that the non-linear term 𝑓𝑓 is involved with first-order derivative explicitly. An example is also given to illustrate the main result. Keywords: Time scales; Boundary value problem; 𝑝𝑝-Laplacian; Positive solutions; Fixed point theorem.
89
On a Kirk-MP Iteration Process Kadri Dogan and Vatan Karakaya
Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey [email protected]
Abstract: In this presentation, we introduce Kirk-type new iterative scheme which we called Kirk- MP scheme and we study the convergence of this iterative scheme by using weak contraction mappings. Furthermore, we obtain this iteration process is T-stable. Keywords: Kirk-MP iteration process; Strong convergence; Stability; Weak contraction mappings. Acknowledgment: The authors would like to thank Yıldız Technical University Scientific Research Projects Coordination Department under Project no.BAPK 2014-07-03-DOP02 for financial support during the preparation of this paper. References: [1] H. Akewe, G. A. Okeke, A. F. Olayiwola. "Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators." Fixed Point Theory and Applications 2014.1 (2014): 45. [2] K. Dogan, V. Karakaya, “On some results of MP iteration procedure for weak contraction operator in Banach spaces”, Submited.
90
On n - normed Cesàro Sequence Space ,nCes f Cenap Duyar and Oguzhan Kanber
Department of Mathematics, Ondokuz Mayıs University, Atakum, Samsun, Turkey [email protected]
Abstract: In this paper, it is introduced and studied n-normed cesaro sequence with its some algebraical and topological properties, like having a vector space, Banach space, conditions not to be empty, etc. Keywords: Cesaro Sequence space; Cesaro-Orlicz Sequence Space; n-norm; n-normed space. References: [1] S. Gähler, Linear 2-normierte, Räume. Math. Nachr, 28, 1-45 (1965). [2] J.S. Shiue, On the Cesaro sequence spaces, Tamkang J. Math., 1, 19-25 (1970). [3] D. Kubiak, A note on Cesaro-Orlicz sequence spaces, J. Math. Anal. Appl., 349, 291-296 (2009). [4] H.Gunawan, On n-inner product, n-norms and the Cauchy-Schwartz inequality, Scientiae Mathematicae Japonicae, Online 5, 47-54 (2001) [5] H. Gunawan H, M. Mashadi, On n-normes spaces, Int. J. Math. Sci., 27, 631-639 (2001).
91
Some Families of Generating Functions for Laguerre and Charlier Types d-Orthogonal Polynomials
Duriye Korkmaz Duzgun and Esra Erkus Duman Department of Mathematics, Gazi University, Teknikokullar, Ankara, Turkey
[email protected], [email protected] Abstract: This paper is mainly connected with the theory of generating functions for some d-orthogonal
polynomials. In this study, we present some results giving multilateral and multilinear generating functions for the
Laguerre and Charlier types generalized hypergoemetric d-orthogonal polynomials. We also derive their
miscellaneous properties. We mainly use the following references: [1-5]. Keywords: Laguerre type d-orthogonal polynomials, Charlier type d-orthogonal polynomials, generating function,
recurrence relation.
References:
[1] Y. Ben Cheikh and K. Douak, “On the classical d-orthogonal polynomials defined by certain generating
functions II”, Bull. Belg. Math. Soc. 8(2001), 591-605.
[2] R. N. Jain, “A generalized hypergeometric polynomials”, Ann. Polon. Math. 19(1967), 177-184.
[3] Y. Ben Cheikh and K. Douak, “A generalized hypergeometric d-orthogonal polynomial set”, C. R. Acad. Sci.
331(2000), 349-354.
[4] A. Altın and E. Erkuş, On a multivariable extension of the Lagrange-Hermite polynomials, Integral Transform.
Spec. Funct. 17 (2006), 239-244.
[5] I. Lamiri and A. Ouni, “d- Orthogonality of Hermite type polynomials”, Appl. Math. and Comput. 202(2008),
24-43.
92
Traveling Wave Solutions for Some Nonlinear Evolution Equations Serife Muge Ege and Emine Misirli
Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected], [email protected]
Abstract: Many physical and mathematical models are defined by nonlinear partial differential equations. In this study, we construct the traveling wave solutions of some nonlinear evolution equations by using the modified Kudryashov method. This method is powerful, efficient and can be used as an alternative to establish new solutions of different type of evolution equations applied in mathematical physics. Through applicable structure of the method, it can be used for finding traveling solutions of many nonlinear evolution equations with the help of Mathematica. Keywords: Nonlinear differential equations; Modified Kudryashov method; Traveling wave solutions. References: [1] N. A. Kudryashov, “One method for finding exact solutions of nonlinear differential equations”, Commun. Nonl. Sci. Numer. Simul., 17(2012), 2248-2253. [2] N. A. Kudryashov, M. B. Kochanov, “Quasi-exact solutions of nonlinear differential equations”, Appl. Math. Comp., 219(2012), 1793–1804. [3] M. M. Kabir, “Modified Kudryashov method for generalized forms of the nonlinear heat conduction equation”, Int. J. Phys. Sci., 6(2011), 6061-6064. [4] A. Stakhov, B. Rozin, “On a new class of hyperbolic functions”, Chaos, Solitons Fract., 23(2005), 379-389. [5] S. M. Ege, E. Misirli, “The modified Kudryashov method for solving some evolution equations”, AIP Conf. Proc., 1470(2012), 244-246.
93
New Integral Inequalities of Ostrowski Type for Quasi-Convex Functions with Applications A.Ekinci, M. E. Ozdemir, and E. Set Department of ·Mathematics Faculty of Science and Letters, Agrı Ibrahim Cecen University, Agrı, Turkey
Department of Mathematics, K.K. Education Faculty, Ataturk University, Erzurum, Turkey Department of Mathematics, Faculty of Science and Letters, Ordu University, Ordu, Turkey
[email protected], [email protected], [email protected]
Abstract: In this paper some new Ostrowski-type inequalities for functions whose derivatives in absolute values are quasi-convex are established. Some applications to special means of real numbers and applications for P.D.F.ís are
given. We also give some applications of our obtained results to get new error bounds for the sum of the midpoint formula. References: [1] T-Y. Zhang, F. Qi, “Integral inequalities of Hermite- Hadamard type for m-AH convex functions”, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 3, 60-64.
94
On Boundedness of Multilinear Singular Integral Operators in Lorentz Spaces Ismail Ekincioglu and Cansu Keskin and Ozgun Gurmen Alansal
Department of Mathematics, Dumlupınar University, Kutahya, Turkey
[email protected], [email protected], [email protected]
Abstract:In this paper, we prove an O’Neil inequality for the k-linear convolution operator in the Lorentz spaces. As an application, we obtain the necessary and sufficient conditions on the parameters for the boundedness of the k-sublinear fractional maximal operator 𝑀𝑀α(f ) and the k-linear fractional integral operator 𝐼𝐼α(f) and k-linear Riesz transform 𝑅𝑅α(f) from the spaces 𝐿𝐿𝑝𝑝1𝑟𝑟1x𝐿𝐿𝑝𝑝2𝑟𝑟2x…x𝐿𝐿𝑝𝑝𝑘𝑘𝑟𝑟𝑘𝑘to 𝐿𝐿qs, where n/(n +α) ≤ p< q < ∞, 0 < r≤ s < ∞, p is the harmonic mean of, 𝑝𝑝1 , 𝑝𝑝2,…, 𝑝𝑝𝑘𝑘>1 and r is the harmonic mean of, 𝑟𝑟1, 𝑟𝑟2,…, 𝑟𝑟𝑘𝑘>0. Keywords: Bessel equation, generalized shift operator, mean value theorem.
References:
[1] S.Barza, L.E. Persson and J. Soria, “Sharp weighted multidimensional integral inequalities for monotone
functions”, Math. Nachr., 210(2000), 43-58.
[2] C. Bennett and R. Sharpley, “Interpolation of operators”, Academic Press, Boston (1988).
[3] J.S. Bradley, “The Hardy’s inequalities with mixed norms”, Canad. Math. Bull., 21 4(1978), 405-408.
[4] R. Coifman and L. Grafakos, “Hardy spaces estimates for multilinear operators I”, Rev. Math. Iber., 8(1992), 45-
68.
[5] A. Gogatishvili and V.D. Stepanov, “Reduction thorems for operators on the cones of monotone functions”, Ins.
of Math. AS CR, Prague, 4(2012), 1-25.
[6] L. Grafakos, “On multilinear fractional integrals”, Studia Math., 102(1992), 49-56.
[7] L. Grafakos, “Hardy spaces estimates for multilinear operators”, Rev. Math. Iber., 8(1992), 69-92.
[8] L. Grafakos and N. Kalton, “Some remarks on multilinear maps and interpolation”, Math. Ann., 319(2001), 49-
56.
[9] V.S. Guliyev and Sh.A. Nazirova, “A rearrangement estimate for the rough multilinear fractional integrals”, Sib.
Math. Jour., 48 3(2007), 1-12.
95
Abstract Harmonic Analysis on Spacetime Poincare Group
Kahar El-Hussein Department of Mathematics, Faculty of Science, Al Furat University, Dear El Zore, Syria Department of Mathematics, Faculty of Arts Science, Al Quryyat, Al-Jouf University, KSA
Abstract: Let G=SL(2,R) be the 2x2 connected complex Lie group and let P=R4x SL(2,R) be the Poincare group (space time). In mathemat ics, the PoincarÈ group (spacetime), named after Henri Poincare is the group of isometries of Minkowski spacetime, introduced by Hermann Minkowski. It is a non-abelian Lie group with 10 generators. Space time, in physical science, single concept that recognizes the union of space and time, posited by Albert Einstein in the theories of relativity (1905, 1916). One of the interesting problems for Mathematicians and Physicists is. Can we do the Fourier analysis on P. The purpose of this paper is to deÖne the Fourier transform in order to obtain the Plancherel formula for G, and then we establish the Plancherel theorem for Spacetime(Poincare group).
Keywords: Semidirect Product of Two Lie Groups; Spacetime (Poincare Group); Fourier Transform; Plancherel Theorem. References: [1] G. S. Chirikjian, A. Kyatkin, A, “Engineering Applications in Non-commutative Harmonic Analysis”, Johns Hopkins University, Baltimore, Maryland, CRC Press., (2000). [2] A. Cerezo, F. Rouviere, "Solution elemetaire d'un operator differentielle lineare invariant agauch sur un group de Lie reel compact" Annales Scientiques de E.N.S. 4 serie, tome 2, n^o4,p 561-581, 1969. [3] A. Einstein "Space--Time", Encyclopedia Britannica, 13th ed(1926),. [4] K. El- Hussein, “Operateurs Differentiels Invariants sur les Groupes de Deplacements”, Bull. Sc. Math. 2e series 113,. p. 89-117(1989). [5] K. El- Hussein.,“Eigendistributions for the Invariant Differential operators on the Affine Group”, Int. Journal of Math. Analysis, Vol. 3, no. 9, 419-429(2009). [6] K. El- Hussein., “Fourier transform and invariant differential operators on the solvable Lie group G4”, in Int. J. Contemp. Maths Sci. 5. No. 5-8, 403-417(2010). [7] K. El- Hussein., “On the left ideals of group algebra on the affine group”, Int. Math Forum, Int, Math. Forum 6, No. 1-4, 193-202(2011). [8] K. El- Hussein., “Non Commutative Fourier Transform on Some Lie Groups and Its Application to Harmonic Analysis”, International Journal of Engineering Research & Technology (IJERT) Vol. 2 Issue 10, 2429- 2442(2013). [9] K. El- Hussein.), “Abstract Harmonic Analysis and Ideals of Banach Algebra on 3-Step Nilpotent Lie Groups”, International Journal of Engineering(2013).
96
On the Fine Spectra of a New Matrix Operator Over the Sequence Spaces 0c and c
Sumeyra Elmaci and Vatan Karakaya Department of Mathematics, Yildiz Technical University, Istanbul, Turkey
Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey [email protected]
Abstract: Several authors have studied the spectrum and fine spectrum of linear operators defined by some particular limitation matrices over some sequence spaces. The spectrum of weighted mean operator studied by Rhoades[1]. Altay and Başar [2] studied the spectrum of difference operator D over the sequence spaces 0,c c .
Srivastava and Kumar [3,4] determined the fine spectrum of the operator vD over the sequence spaces 1l and 0c . Recently, generalizing these results, Erdogan and Karakaya [5] investigated the fine spectra of the matrix operators on a difference sequence space of weighted means. In this paper, we shall study the point spectrum, the continuous spectrum, and the residual spectrum of a new matrix operator, which is the multiplication of weighted mean matrix and generalized difference matrix of order m on sequence spaces 0c and c . Keywords: Fine spectra; Spectrum of an operator; Weighted mean matrix; Generalized Difference matrix of order m, Sequence space. References: [1] B. E. Rhoades, “The fine spectra for weighted mean operators”, Pacific J. Math. 104(1)(1983), 219-230.
[2] B. Altay, F. Başar, “On the fine spectrum of the difference operator D on 0c and c ”, Inform. Sci. 168(2004),
217-224. [3] P. D. Sivastava, S. Kumar, “Fine spectrum of the generalized difference operator vD on sequence space 1l ”, Thai. J. Math. 8(2)(2010), 221-223. [4] P. D. Sivastava, S. Kumar, “ On the fine spectrum of the generalized difference operator vD over the sequence
space 0c ”, Commun. Math. Anal. 6(1)(2009), 8-21.
[5] E. Erdogan, V. Karakaya, “On spectral properties of a new operator over sequence spaces 0c and c ”, Acta Math. Sci. (2014)34B(5), 1481-1494.
97
Eigenvalue Asymptotics and a Trace Formula for the Linear Damped Wave Equation Ahu Ercan and Etibar Panakhov
Department of Mathematics, Firat University, Elazig, Turkey [email protected], [email protected]
Abstract: In this study, we determined the general form of the asymptotics for Dirichlet eigenvalues of the one dimensional linear damped wave operator. The eigenvalue problem associated with the one dimensional linear damped wave equation on the interval (0,1)is given (1.1)–(1.2).We derive a trace formula for the problem of (1.1) –(1.2) 𝑢𝑢𝑥𝑥𝑥𝑥 − (𝛾𝛾2 + 2𝛾𝛾𝑝𝑝 − 𝑞𝑞)𝑢𝑢, 𝑥𝑥 ∈ (0,1) (1,1) 𝑢𝑢(0) = 𝑢𝑢(1) = 0 (1,2) Keywords: Asymptotics; Eigenvalues; Trace formula. References: [1] D. Borisov, P. Freitas, “Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation”, J. Differential Equations, 2009 ,247 3028-3039.
98
On Some Topological Properties of Generalized Sequence Spaces of Non-Absolute Type Sinan Ercan and Cigdem A. Bektas
Department of Mathematics, Fırat University, Elazig, Turkey [email protected], [email protected]
Abstract: M. Mursaleen and A. K. Noman [3] introduced the sequence spaces l¥l , cl and 0cl as the sets of all
l -bounded, l -convergent and l -null sequences. They also generalized sequence spaces cl and 0cl , identifying
( )cl D and 0 ( )cl D [4].
In this paper, we introduce the sequence spaces ( )ml¥ Dl , 0 ( )mcl D and ( )mcl D with notion of difference
operator mD for mÎ¥ . We proved that these spaces are BK-spaces of non-absolute type and we obtained that these spaces are linearly isomorphic to the spaces ¥l , 0c and c , respectively. Moreover, we examined some topological properties of these spaces. We give some inclusion relations involving these sequence spaces. Keywords: Sequence spaces of non-absolute type; BK- spaces; Difference sequence spaces. References: [1] C. Bektas, G. Atici, “On some topological properties of new type of difference sequence spaces”, Acta Universitatis Apulensis, 32 (2012), 61-67. [2] I. J. Maddox, “Elements of Functional Analysis”, 2nd ed., The University Press, Cambridge, 1988. [3] M. Mursaleen, A.K. Noman, “On the spaces of -convergent and bounded sequences”, Thai J. Math. (2) (2010) 311-329. [4] M. Mursaleen, A.K. Noman, “On some new difference sequence spaces of non-absolute type”, Math.Comput. Mod. (52) (2010) 603-617.
99
A One-Sided Theorem for the Product of Abel and Cesàro Summability Methods Yilmaz Erdem and Ibrahim Canak
Department of Economy and Finance, Adnan Menderes University, Cakmar-Kocarlı, Aydin, Turkey, Department of Mathematics, Ege University, Bornova, Izmir, Turkey
[email protected], [email protected]
Abstract: In this paper, a one-sided condition is given to recover (C, α) summability of a sequence from its (A)(C, α+1) summability. Our result extends and generalizes well known classical Tauberian theorems given for Abel and Cesàro summability methods. Keywords: Abel summability; Cesàro summability; (A)(C,α)$ summability; (C, α) summability; Tauberian conditions and theorems; Product methods. References: [1] N. H. Abel, “Recherches sur la series ”, J. für Math. 1 311-339, 1826. [2] A. Amir, “On a converse of Abel's theorem”, Proc. Am. Math. Soc., 3 (1952), 244-256.
[3] J. Boos, “Classical and modern methods in summability”, New York, NY: Oxford University Press, 2000. [4] D. Borwein, “Theorems on some methods of summability”, Q. J. Math., Oxf. II. Ser.,9 (1958), 310-316. [5] I. Canak, Y. Erdem, “On Tauberian theorems for (A)(C;α) summability method”, Appl. Math. Comput., 218 (2011), No. 6, 2829-2836. [6] I. Canak, Y. Erdem, U. Totur, “Some Tauberian theorems for (A)(C; α), summability method”, Math. Comput. Modelling, 52 (2010), No. 5-6, 738-743. [7] Y. Erdem, I. Canak, “A Tauberian theorem for (A)(C;α) summability”, Comput. Math. Appl., 60 (2010), No. 11, 2920-2925. [8] G. H. Hardy, “Theorems relating to the summability and convergence of slowly oscillating series”, Proc. Lond. Math. Soc. (2), 8 (1910), 301-320. [9] G. H. Hardy, J. E. Littlewood, “Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive”, Proc. Lond. Math. Soc. (2), 13 (1914), 174-191. [10] E. Kogbetliantz, “Sur les séries absolument sommables par la méthode des moyennes arithmétiques”, Bull. Sci. Math., II. Sér., 49 (1925), 234-251.
L+++ - 22.11
)1(11 xx mmm
100
On the Spectrum of the Product Operator 𝑾𝑾 on the Sequence Space 𝒃𝒃𝒃𝒃 Ezgi Erdogan and Vatan Karakaya
Department of Mathematics, Marmara University, Istanbul, Turkey Department of Mathematical Engineering, Yıldız Technical University, Istanbul, Turkey
[email protected] , [email protected]
Abstract: Spectral theory for linear operators is one of the most important topics of functional analysis and operator theory. In fact, much information about a linear operator depends on its spectrum, and thus knowing the spectrum means knowing a large part of the properties of the operator. Several authors have studied the spectrum and fine spectrum of linear operators defined by some particular limitation matrices over some sequence spaces. Weighted mean matrices as operators on the sequence space ℓ𝑝𝑝 have been investigated by Cartlidge [1] and Rhoades studied the fine spectrum of the weighted mean operators [2]. The fine spectrum of the difference operator D over the sequence spaces ℓ𝑝𝑝 and 𝑏𝑏𝑏𝑏𝑝𝑝 have been studied by Akhmedov and Basar [3], [4]. In this paper, our aim is to determine the spectrum and the fine spectrum of the product operator 𝑊𝑊 which is multiplication of weighted mean and difference matrices over the sequence space 𝑏𝑏𝑏𝑏. The product operator 𝑊𝑊 over 𝑏𝑏𝑏𝑏 is defined by (𝑊𝑊𝑥𝑥)𝑘𝑘 = ∑ 𝑢𝑢𝑘𝑘
∞𝑖𝑖=0 𝑏𝑏𝑖𝑖(𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖−1) with 𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏, and (𝑢𝑢𝑛𝑛) and (𝑏𝑏𝑛𝑛) are two bounded, either constant or
strictly decreasing sequences of positive real numbers satisfying certain conditions. Keywords: Spectrum of an operator; Sequence space; Matrix transformation. References: [1] P. J. Cartlidge, “Weighted mean matrices as operators on ℓ𝑝𝑝”, Ph. D. Dissertation, Indiana University. [2] B. E. Rhoades, “The fine spectra for weighted mean operators”, Pasific J. Math. 104 (1) (1983) 219-230. [3] A.M. Akhmedov, F. Basar, “On the spectra of the difference operator ∆ over the sequence space ℓ𝑝𝑝”, Demonstratio Math. 39(3)(2006), 585-595. [4] A. M. Akhmedov, F. Basar, “On the fine spectra of the difference operator Δ over the sequence space 𝑏𝑏𝑏𝑏𝑝𝑝 (1 ≤𝑝𝑝 < ∞), Acta Math. Sin. Eng. Ser. 23(10)(2007), 1757-1768.
101
Infinite Matrices and Invariant Means Rahmet Savas Eren
Department of Mathematics, Istanbul Medeniyet University, Uskudar, Istanbul, Turkey [email protected]
Abstract: Let s be a one-to-one mapping from the set of natural numbers into itself. A continous linear functional f on l¥ is said to be an invariant mean or a s - mean provided that
(i) ( ) 0xf ³ when the sequence x = ( kx ) is such that 0kx ³ for all k,
(ii) ( ) 1ef = where e = (1,1,1,…) , and
(iii) ( )(x) ( )kxsf f= for all x l¥Î .
We denote by Vs the space of s - convergent sequences. It is known that x VsÎ if and only if
( )
1
1k
m
nk
x am s
=
®å limit
as m ® ¥ , uniformly in n. Here ( )k ns denotes the k th iterate of the mapping s at n. A s -mean extends the limit functional on c in the sense that ( ) limx xf = for all x cÎ if and only if s has
no finite orbits, that is to say, if and only if , for all n > 0 , k ³ 1, ( )k n ns ¹ .
In this paper, we define the spaces 0( )V pl
s and ( )V pls by using de laValee poussin and invariant mean.
Furthermore we characterize some matrix transformations which will up a gap in the existing literature. Keywords: Infinite matrices, de la Vallee Poussin, invariant mean, matrix transformations. References: [1] S. Banach, Theorie des Operations linearies, Warszawa,1932. [2] G. Das and S. K. Mishra, Banach limits and lacunary strong almost convergence, J. Orissa Math. Soc. 2(2)(1983), 61-70. [3] V. Karakaya, sq -sumable sequences and some matrix transformations, Tamkang J. Math. 35(4)(2004), 313-320. [4] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta. Math.80(1948),167-190. [5] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. 18(1967), 345-355. [6] I. J. Maddox, and J. W. Roles Absolute convexity in certain topological linear spaces,Proc. Cambridge Philos. Soc.66(1969), 541-545. [7] I. J. Maddox, Elements of functional analysis, Camb. Univ. Press (1970). [8] M. Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford, 34(1983), 77-86. [9] M. Mursaleen, A. M. Jarrah and S. A. Mohiuddine, Bounded linear operators for some new matrix transformations, Iranian J. Sci. Tech. Trans. A, Vol.33(A2), 169-177. [10] Mursaleen, On in_nite matrices and invarient means , Indian J.Pure.Appl.Math.10(4) (1979),457-460. [11] M. Mursaleen, Invariant means and some matrix transformations, Tamkang Jour.Math.,10 (1979), 181-184. [12] Mursaleen, A. K. Gaur and T. A. Chishti, On some new sequence spaces of invariant means, Acta Math. Hung., 75(1997),185-190.
102
Groups and Irreducible Character Degrees Temha Erkoc
Department of Mathematics, Istanbul University, Fatih, Istanbul, Turkey [email protected]
Abstract: Let G be a finite group and let Irr(G) be the set of irreducible complex characters of G. The set of irreducible complex character degrees of G is denoted by cd(G) i.e. cd(G)=χ(1) | χ∈Irr(G). It is well known that the set cd(G) can be used to obtain information about the structure of the group G. For example, if the cardinality of cd(G) is 2, then the commutator subgroup of G is abelian [1]. Ito-Michler’s Theorem says that if a prime p divides no character degrees of a finite group G, then G has a normal abelian Sylow p-subgroup by [2], [3], [1] and [4]. In this talk, we will obtain some results on the relation between the structure of a group G and its irreducible complex character degrees. Keywords: Character degrees; Finite groups. References: [1] I.M. Isaacs, “Character Theory of Finite Groups”, AMS Chelsea Publishing, Providence, RI, 2006 [2] N. Ito, “Some Studies of Group Characters”, Nagoya Math. J., 2 (1951), 17-28. [3] N. Ito, “On the Degrees of Irreducible Representations of a Finite Group”, Nagoya Math. [4] G. Michler, “A Finite Simple Group of Lie-type has p-Blocks with Different Defects, p≠2”, J. Algebra, 104 (1986), 220-230
103
Energy Conservation for the 3-Coupled Nonlinear Schrödinger Equation by Using the Average Vector Field Method
Sevim Ertug and Ayhan Aydin Department of Mathematics, Atilim University, Incek, Ankara, Turkey
[email protected], [email protected]
Abstract: In this paper, we construct an energy conserving numerical scheme for the 3-CNLS equation [1] by using the AVF method [2,3]. The numerical results show that the proposed energy conserving scheme has an excellent performance in simulating the plane waves, periodic solutions, single and colliding soliton solutions of the 3-CNLS equation during long-time integration. The ability of capturing low- and high-frequency waves is examined for the proposed scheme. Moreover, the scheme preserves the other invariants of the 3-CNLS equation as well as the discrete energy of the system. Acknowledgments: This work is supported by The Scientific and Technological Research Council of Turkey (TUBITAK) with the project number 114F020. Keywords: Coupled nonlinear Schrödinger equation; Energy preserving scheme; Average vector field method. References: [1] A. Aydin, “Multisymplectic integration of N-coupled nonlinear Schrödinger equation with destabilized periyodic wave solutions”, Chaos, Solitons & Fractals, 41(2009), 735-751. [2] R.I. McLachlan, G.R.W. Quispel, N. Robidoux, “Geometric integration using discrete gradients”, Phil. Trans. Roy. Soc. A, 357(1999), 1021-1046. [3] E. Celledoni, V. Grimm, R.I. McLachlan, D.I. McLaren, D. O’Neale, B. Owren, G.R.W. Quispel, “Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method”, J. of Computational Physics, 231(2012), 6770-6789.
104
On Lacunary Statistical Convergence of Order α of Difference Sequences Mikail Et
Department of Mathematics, Firat University, Elazig, Turkey [email protected]
Abstract: The idea of difference sequence spaces was ntroduced by Kızmaz [ Canad. Math. Bull. 24(2) (1981), 169–176 ] and was generalized by Et and Çolak [ Soochow J. Math. 21(4) (1995), 377–386 ]. In this paper using generalized difference operator Dm, we introduce the concept of lacunary statistical convergence of order α of difference sequences and give some relations about this concept. References: [1] J. S. Connor, “The statistical and strong p-Cesaro convergence of sequences”, Analysis 8 (1988), 47-63. [2] R. Colak, “Statistical convergence of order α Modern Methods in Analysis and Its Applications”, New Delhi, India: Anamaya Pub, 2010: 121–129. [3] G. Das, S. K. Mishra, “Banach limits and lacunary strong almost convergence”, J. Orissa Math. Soc. 2 (1983), 61-70. [4] M. Et, R. Colak, “On generalized di¤erence sequence spaces”, Soochow J. Math. 21(4) (1995), 377-386. [5] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2 (1951), 241-244. [6] A. R. Freedman, “Sember J. J. and Raphael, M. Some Cesaro-type summability spaces”, Proc. Lond. Math. Soc. 37(3) (1978), 508-520. [7] J. Fridy, “On statistical convergence”, Analysis 5 (1985), 301-313. [8] J.A. Fridy, C. Orhan, “Lacunary statistical convergence”, Paci…c J. Math. 160 (1993), 43-51. [9] H. Kizmaz, “On certain sequence spaces”, Canad. Math. Bull. 24(2) (1981), 169-176.965), 375-387.
105
Colorings of Cycle Graph with Topological Approach Seher Fisekci and Ismet Karaca
Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected], [email protected]
Abstract: In this Poster, graph colorings, chromatic number and cycle graphs[6] are examined in the graph theory and search obstructions to graph colorings[1] in the following direct way: take a graph, associate to it a topological space[2], and then look for obstructions to colorings of the graph[1] by studying the algeabric invariants[5] of this space. The construction of such a space has been suggested by L. Lovasz[4]. The invariants of this space its cohomology groups, and, when it can be viewed as a 2Z space, its Stiefel – Whitney characteristic classes[2]. This connection[3] made with the help of Lovasz conjecture has been proved by Bobson and Kozlov[2]. Keywords: cycle graphs, colorings graph, cohomology groups. References: [1] E. Babson and D. N. Kozlov, “Topological obstructions to graph colorings”, Electron. Res. Announc. Amer. Math. Soc. 9(2003), 61-68. [2] E. Babson and D. N. Kozlov, “Proof of the Lovasz conjecture”, Annals of Mathematics, Second Series, Vol. 165, No. 3(May, 2007), pp 965-1007. [3] S. Lj. Cukic and D. N. Kozlov, “Higher connectivity of graph coloring complexes”, Internat. Math. Res. Notics 25(2005), 1543-1562. [4] A. Engstrom, “A short proof of a conjecture on the connectivity of graph coloring com plexes”, Proc. Amer. Math. Soc. 134(2006), 3703-3705. [5] A. Hatcher, “Algebraic Topology”, Cambridge Univ. Press, Cambridge, (2002). [6] James A. Anderson, “Discrete Mathematicts with Combinatorics”, 2(2004), 227-250.
106
Estimation of the Presicion Matrix of an Elliptically Symmetric Distribution Dominique Fourdrinier, Fatiha Mezoued and Martin T. Wells
Université de Rouen, France, Ecole Nationale de Statistique et d'Economie Appliquée, Alger, Algeria Cornell University, USA
[email protected], [email protected], [email protected]
Abstract: Let ), . . . ,,(),( 1 nUUXUX =
be 1+n random vectors having an elliptically contoured distribution with joint density of the form [ ]))) ((( 112/)1( SXXt rf tn --+- S+--SS qq
Where X and iU ’s are 1´p vectors and tUUS = and both q and S are unknown. We provide estimators of
the precision matrix 1-S . According to the fact that the matrix S is invertible or not, natural estimators are of the form 1-aS or +aS where a is a positive constant and where 1-S and +S are respectively the inverse and the
Moore-Penrose inverse of S . We propose improved estimators under the loss 211 )ˆ( -- S-St r . So here, we
generalize the works in [2], [3],[4] ,[5] which dealt with the Gaussian case. To this end, a new Stein-Haff lemma is derived. Keywords: Elliptically symmetric distributions; Precision matrix; Berger class. References: [1] D. Chételat, M. T. Wells, “Improved multivariate normal mean estimation with unkown covariance when p is greater than n”, Annals of statistics, pages 3137-3160, 2012. [2] D. K. Dey, “Improved estimation of multinormal precision matrix”, Statistics and probability letters, 6:125 128,1987. [3] T. Kubokawa, M. S. Srivastava, “Estimation of the precision matrix of a singular wishart distribution and its application in high-dimentional data”, Journal of multivariate Analysis, 99:1906-1928,2008. [4] H. Tsukuma, Y. Konno, “On improved estimation of normal precision and discriminant coefficients”, Journal of multivariate Analysis, 97(1):1477-1500,2006. [5] H. Tsukuma, “Estimating the inverse matrix of scale parameters in an elliptically contoured distribution”, J. Japan. Statist. Soc, 35(1):21-39,2005.
107
Description of Bloch Spaces and their Invariant Subspaces, and Related Questions Mubariz T. Garayev, Mehmet Gurdal and Ulas Yamancı
Department of Mathematics, King Saud University, Riyadh, Saudi Arabia Department of Mathematics, Suleyman Demirel University, Isparta, Turkey
[email protected], [email protected] and [email protected]
Abstract: Let D be the unit disc of complex plane C , and ( )DolHH = the class of functions analytic in D . Recall
that an ( )Dolf HÎ is said to belong to the Bloch space ( )DBB = if ( ) ( ) .1sup: 2 +¥<-=¢
Îzfzf
z DB
With the norm
( ) ,0B
fff += B is Banach space. Let ( )D00 BB = be the space which consists of all BÎf satisfying
( ) ( ) .01lim 2
1=-
¢
®zfz
zThis space is the so-called little Bloch space. For more informations Bloch spaces, see Zhu
[1,2]. Here we will give a new description of Bloch spaces in terms of Berezin symbols of diagonal operators associated with the Taylor coefficients of its functions. We also give in terms of Berezin symbols the multiple shift
invariant subspaces of these Bloch spaces. Some Other questions also will be discussed in this talk.
Keywords: Berezin Symbol, Invariant Subspace, Bloch Space. References: [1] K. Zhu, “Operator Theory in Function Spaces”, Marcel Dekker Ins., 1990. [2] K. Zhu, “Bloch type spaces of analytic functions”, Rocky Mountain J. Math., 23(1993), 1143-1177.
108
Common Fixed Points of Almost Generalized Contraction on Modular Spaces Ekber Girgin and Mahpeyker Ozturk
Department of Mathematics, Sakarya University, Sakarya, Turkey [email protected], [email protected]
Abstract: The purpose of this paper is to establish new common fixed point theorems for generalized almost contractions on modular spaces. As an application, we derive some common fixed point results for such contractions on modular spaces with a directed graph. The existence of fixed points of mappings satisfying contractive condition of integral type is also obtained in such spaces. Keywords: Common Fixed Point; Almost Contraction; Directed Graph; Modular Space. References: [1] V. Berinde, “Approximation fixed points of weak contractions using the Picard iteration”, Nonlinear. Anal. Forum 9:1, 2004, 43-53. [2] L. Ciric, M. Abbas, R. Saadati, N. Hussain, “Common fixed points of almost generalized contractive mappings in ordered metric spaces”, Appl. Math. Comput. 217 (2011), 5784-5789. [3] C. Mongkolkeha, P. Kumam, “Some fixed point results for generalized weak contraction mappings in modular spaces”, International Journal of Analysis, (2013), (2013), 6, Article ID 247378. [4] M. Ozturk, M. Abbas, E. Girgin, “Fixed Points of Mappings Satisfying Contractive Condition of Integral Type in Modular Spaces endowed with a Graph”, Submitted. [5] E. Savas, M. Ozturk, E. Girgin, “Common fixed point theorems on modular space involving a graph”, Submitted.
109
Functions Represented into a Newton Interpolating Series and Applications Ghiocel Groza
Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, Romania [email protected]
Abstract: Let 1³kka be a sequence of real numbers from an interval I and the polynomials ,1)(0 =xu
( ) .,...2,1,)(1
=-= Õ=
ixxui
kki a A series of the form
,,)(0
R댴
=i
iii axua (1)
is called a Newton interpolating series with real coefficients ia at 1³kka . Consider a function R®If : . We say that f is represented into a Newton interpolating series at 1³kka if there exists a series of the form (1) which converges uniformly to f on I. Newton interpolating series, which is a natural generalization of Taylor series, has useful applications in different
areas of mathematics. Thus, for example, a proof of well-known result of Lindemann on the transcendency of ge , where g is an algebraic number (see [7], Theorem 6, Ch. 2, Sec. 3) is based on the expansion of the exponential function into a series of the form (1), where 1³kka is a purely periodic integer sequence. These particular Newton interpolating series are studied in [2]. Since the partial sums of a Newton interpolating series are Newton interpolation polynomials they are a useful tool in the interpolation theory and can be used to approximate solutions of boundary value problems for differential equations (see, for example, [2], [3], [5]). There are extensions of Newton interpolation series for different valued fields. Some results in the case of complex field can be found in [4], [6], and for non-archimedean fields, where they generalize Mahler series, for example, in [1]. In this paper we study real functions which are represented into a Newton interpolating series and we give some applications. Keywords: Newton interpolating series; Analytic function; Boundary value problems. References: [1] G. Groza, A. Haider, “On the ring of Newton interpolating series over a local field”, Math. Rep., Bucur. 9(59) (2007), No. 4, 343-356. [2] G. Groza, N. Pop, “Approximate solution of multipoint boundary value problems for linear differential equations by polynomial functions”, J. Difference Equ. Appl., 14 (2008), No. 12, 1289-1309. [3] G. Groza, N. Pop, “A numerical method for solving of the boundary value problems for ordinary differential equations”, Result. Math., 53(2009), No. 3-4, 295-302. [4]. G. Groza, A. Haider, S. M. Ali Khan, “Interpolation of entire functions”, Bol. Soc. Mat. Mex., III. 17 (2011), No. 2, 127-133. [5] G. Groza, S. M. Ali Khan, N. Pop, “Approximate solutions of boundary value problems for ODEs using Newton interpolating series”, Carpathian J. Math., 30(2014), No. 3 (to appear). [6] Y. Martin, “Sur les séries d’interpolation”, Ann. Sci. ´ Ecole Norm. Sup., 66 (1949), sér. 3, 311-366. [7] A. Shidlowski, “Transcendental Numbers”, de Gruyter, 1989.
110
Numerical Approach for Magneto-Hydrodynamic Flow Passed Through a Wedge Murat Gubes and Galip Oturanc
Department of Mathematics, Karamanoglu Mehmetbey University, Yunus Emre Campus, Karaman, Turkey Department of Mathematics, Selcuk University, Selcuklu, Konya, Turkey
[email protected], [email protected]
Abstract: In this study, we consider the magnetohydrodynamics (MHD) laminar boundary layer flow passed through a wedge. This system of equations is expressed a group of partial differential equations which occured well-known continuity equation, energy equation and momentum equation. For simplicity and compatibility, a suitable similarity transformation is applied this differential system to convert the nonlinear ordinary differential equations called dimensionless form. We solve the dimensionless form of the equations numerically via modified differential transform method (DTM). Skin friction coefficient, temperature and concentration gradients are calculated by this method and compared Maple 13 numerical procedure and previous study in literature. Also, The radiation, thermal transfer, heat source and magnetic parameters, pressure gradient, Reynold, Lewis and Prandtl numbers effects on the variations of velocity, temperature and concentration profiles are displayed graphically. Keywords: Modified Differential Transform Method, Pade Approximant, Thermal Radiation, Heat generation, Nonlinear Ordinary Differential Equations. References: [1] HT Lin, LK Lin, Similarity solutions for laminar forced convection heat transfer from wedges to fluids of any Prandtl Number. Int. J. Heat Mass Tran. 30, 1111-1118 (1987). [2] K Vajravelu, A Hadjinicalaou, Heat transfer in a viscous dissipation and internal heat generation. Int. Comm. Heat Mass Trans. 20, 417-430 (1993). [3] WA Khan, I Pop, Boundary layer flow past a wedge moving in a nanofluid. Math Probl. Eng., 1-7 (2013). [4] MS Khan, I Karim, MS Islam, M Wahiduzzaman, MHD boundary layer radiative, heat generating and chemical reacting flow past a wedge moving in a nanofluid. Nano Convergence 1 (20) (2014). [5] FM White, Viscous Fluid Flow, 2nd edn. (McGraw-Hill, New York, NY, USA, 1991). [6] KA Yih, Uniform suction/blowing effect on forced convection about a wedge: uniform heat lux. Acta Mech. 128,173-181 (1998). [7] NA Yacob, A Ishak, I Pop, Falker-Skan problem for a static or moving wedge in nanofluids. Int. J. Therm. Sci. 50, 2011. [8] MA Seddeek, Effects of radiation and variable viscosity on a MHD free convection flow past a semi-infinite flat plate with an aligned magnetic field in the case of unsteady flow. Int. J. Heat Mass Transfer 45, 931-935 (2002).
111
An Algorithm for Some Small Dimensional Representations Kubra Gul, Nurullah Ankaralioglu and Abdullah Cagman
Mathematics Department, Faculty of Science, Ataturk University, 25240, Erzurum, Turkey Mathematics Department, Faculty of Science and Letters, Agri Ibrahim Cecen University, 04100, Ağrı, Turkey
[email protected] , [email protected] , [email protected]
Abstract: We present an algorithm to construct a -dimensional representation from the irreducible representation which is a subspace of tensor products of symmetric powers for the special linear group with dimension . Keywords: Special linear group, irreducible representation. References: [1] Fulton W, Harris J. Representation Theory A First Course. Graduate Texts in Mathematics, Springer. 1999:551. [2] Magaard K, O’Brien E, Seress A. Recognition of small dimensional representations of general lineer groups. J. Avustralian Math. Soc., 2008;85, 229-250. [3] Corr BP. Estimation and computation with matrices over finite fields. Phd Thesis, The University of Western Australia Department of Mathematics. January, 2014;180.
112
Distribution of First Passage Times for Lumped States in Markov Chains Murat Gul and Salih Celebioglu
Department of Statistics, Giresun University, Gazipaşa, Giresun, Turkey Department of Statistics, Gazi University, Teknikokullar, Ankara, Turkey
[email protected], [email protected]
Abstract: First passage time in Markov chains is defined as the first time that a chain passes a specified state or lumped states. This state or lumped states may indicate first passage time of an interesting, rare and amazing event. In this study, obtaining distribution of the first passage time relating to lumped states which are constructed by gathering the states through lumping method for a irreducible Markov chain whose state space is finite was deliberated. Thanks to lumping method the chain’s Markov property has been preserved. Another benefit of lumping method in the way of practice is reduction of the state space thanks to gathering states together. As the obtained first passage distributions are continuous, it may be used in many fields such as reliability and risk analysis Keywords: Markov Chain; Distribution of First Passage Time; Lumped States. References: [1] A.DasGupta, “Fundamentals of Probability: A first course”, Springer Texts in Statistics, (2010), 145-158. [2] E.Cinlar, “Introduction to Stochastic Processes”, Englewood Cliffs, New Jersey, (1997), 106-277. [3] J.G. Kemeny and J.L. Snell, “Finite Markov Chains”, Springer, Berlin, (1976), 127-148. [4] L. Zhang “ Lumpability Approximation Methods for Markov Models” , PhD Thesis, Purdue University, (2006), 6-16, 27-39. [5] M.U. Thomas and D.R. Barr, “An approximate test of Markov chain lumpability”, Journal of American Statistics Association, 72 (1977), 175-179. [6] R. Syski, "Passage Times for Markov chains", IOS, Amsterdam (1992), 3-15. [7] S.Ross and Z. Schechner, "Using Simulation to estimate first passage distribution", Management Science, 31.2(1985), 224-234. [8] Web: K. Sigman, “Simulation of Markov Chain, Columbia University” http://www.columbia.edu/~ks20/4703-Sigman/Monte-Carlo-Sigman.html (2009).
113
On the Generalized Bernoulli, Euler and Genocchi Polynomials Esra Guldogan and Esra Erkus Duman
Department of Mathematics, Gazi University, Teknikokullar, Ankara, Turkey [email protected] , [email protected]
Abstract: In the present paper, we deal with generalized Euler and Bernoulli, and unification of the Bernoulli, Euler and Genocchi polynomials. We obtain theorems giving multilinear and multilateral generating functions for these polynomials. We display some special cases and derive their recurrence relations. We mainly use the following references: [1-5] Keywords: Multilinear and multilateral generating functions; Addition formula; Recurrence relation. References: [1] H. M. Srivastava, M. Garg, S. Choudhary “Some New Families of Generalized Euler and Genocchi Polynomials”, Taiwanese J. Math 15(2011), 283-305. [2] H. Ozden, Y. Simsek, H.M. Srivastava “A Unified Presentation of the Generating Functions of the Generalized Bernoulli, Euler and Genocchi Polynomials”, Comput. Math. Appl., 60(2010), 2779-2787. [3] H. M. Liu, W. P. Wang “Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums”, Discrete Math., 309(2009), 3346-3363. [4] E. Erkus, A. Altın “A note on the Lagrange polynomials in several variables”, J. Math. Anal. Appl., 310(2005), 338-341. [5] H. M. Srivastava, S. Handa “Some general theorems on generating functions and their applications”, Appl. Math. Lett., 4(1988), 391-394.
114
On Boundedness of Singular Integral Operators Generated by Bessel Generalized Shift Operator in Weighted Beppo-Levi Spaces
Serap Guner and Ismail Ekincioglu Mathematics Department, Science Faculty, Dumlupinar University, Kutahya, Turkey
[email protected], [email protected]
Abstract: Classical Riesz Transformations are important and main singular integral operators in singular integral theory.This operator’s boundedness problems are studied a lot of researchers in several spaces. T. Kurokawa [20], studied classical Riesz transformations and Riesz potantials in Beppo-Levi Spaces and got a relation between Riesz transforms and Riesz potantials, to help this relation, studied boundedness problems of classical Riesz transformations. In this paper, instead of classical Riesz transformations, Riesz-Bessel translation operator generated by Bessel generalized shift operator are taken. After the boundedness problem of Riesz Bessel translations is studied in weighted Beppo-Levi spaces with the help derivates of Bessel potantial operator generated by Bessel generalized shift operator. Keywords: BesselGeneralized Shift Operator; B-Potantial; B-Riesz Translation; Weighted Beppo-Levi Spaces. References: [1] I. Ekincioglu, “Riesz Transformations Generated by a Generalized Shift Operator”, PhD. Thesis, (1994). [2] T. Kurukawa, “Riesz potentials, higher Riesz transforms and Beppo-Levi spaces”, Hiroshima Math. J. 18(1988), 541-597. [3] Y. Mizuta, “Integral represantations of Beppo-Levi functions of higher order”, Hiroshima Math. J. 4(1974), 375-396. [4] Y. Mizuta, “Potential Theory in Euclidean Spaces”, Mathematical Sciences and applications,Vol.6,341pp,199 [5] S.G. Samko, “On spaces of Riesz potentials”, Math. USSR Izv. 10 (1976). 1089-1117. B.M. Levitan, “Generalized shift operatörs”. Moscow Nauka, 1973.
115
Continuity of Superposition Operators on the Double Sequence Spaces of Maddox ( )L p
Nihan Gungor and Birsen Sagir Department of Mathematical Engineering, Gumushane University, Gumushane, Turkey
Department of Mathematics, Ondokuz Mayıs University, Kurupelit, Samsun, Turkey [email protected], [email protected]
Abstract: Chew & Lee [1] and Petranuarat & Kemprasit [2] have characterized continuity of the superposition operator acting from the sequence space pl into ql where 1 ,p q£ < ¥ . Sağır & Güngör [3] have defined the
superposition operator gP by ( )( ) ( ), ,g ks ksP x g k s x= for all real double sequences ( )ksx where 2:g ´ ®¥ ¡ ¡ . Nevertheless, we characterized the continuity of the superposition operator acting from the double sequence spaces
pL into qL for 1 ,p q£ < ¥ . In this study, we have generalized gP acting from Maddox double sequence spaces,
( ) ( ), 1
kspks ks
k sp x x x
¥
=
ì üï ï= = < ¥í ýï ïî þ
åL into qL where ( )ksp p= is a bounded sequence of positive numbers and
1 q£ < ¥ . The main aim of this study is constructing the necessary and sufficient conditions for the continuity of
( ):g qP p ®L L for 1 q£ < ¥ . Keywords: Superposition Operators; Continuity; Double Sequence Spaces; Pringsheim’s convergent. References: [1] T. S. Chew, P. Y. Lee, “Orthogonally Additive Functionals On Sequence Spaces”, Southeast Asian Bull. Math. 9 (1985), 81-85. [2] S. Petranuarat, Y. Kemprasit, “Superposition Operators On pl And 0c Into ql (1 , )p q£ < ¥ ”, Southeast Asian Bull. Math., 21 (1997), 139-147. [3] B. Sagir, N. Gungor, “Continuity Of Superposition Operators On The Double Sequence Spaces pL ”, (2014) to accepted Filomat.
116
Weyl Type Theorems for Unbounded Hyponormal Operators Anuradha Gupta and Karuna Mamtani
Department of Mathematics, Delhi College of Arts and Commerce, University of Delhi, Netaji Nagar, New Delhi, India Department of Mathematics, University of Delhi, India
[email protected], [email protected]
Abstract: In [1], Weyl asserts that if T is a bounded normal operator then T satisfies Weyl’s theorem and in [2], Berkani gave the generalization of the same. Further, in [3] Berkani proved this generalized version of classical Weyl's theorem for bounded hyponormal operators. In this paper, we shall consider closed linear unbounded hyponormal operator T on an infinite dimensional hilbert space H with resolvent set ρ(T) ≠ ɸ and show that T satisfies Weyl's theorem, generalized Weyl's theorem, Browders' theorem and generalized Browder's theorem. For such T, it is shown that a-Weyl's theorem is equivalent to generalized a-Weyl's theorem and that a-Browder's theorem holds for T as well as T*. Moreover, equivalence of two of the variants of Weyl's theorem, property (w) [4] and property (b) [5] is established. Keywords: Unbounded hyponormal operators; Weyl's theorem, a-Weyl's theorem; Browder's theorem, a-Browder's theorem; Generalized Weyl's theorem; Generalized a-Weyl's theorem; Generalized Browder's theorem; Property (w), Property (b). Refrences: [1] H. Weyl, “Über beschränkte quadatische Formen, deren Differenz Vollstetig ist” , Rend. Circ. Mat. Palermo, 27(1909), 373-392 . [2] M. Berkani, “Index of B-Fredholm operators and generalization of a Weyl's theorem”, Proc. Amer. Math. Soc., 130(2002), 1717-1723. [3] M. Berkani, “Generalized Weyl's theorem and hyponormal operators”, J. Aust. Math. Soc., 76.2(2004), 291-302. [4] V. Rakocevic, “On a class of operators”, Mat. Vesnik., 37(1985), 423-426. [5] M. Berkani and H. Zariouh, “Extended Weyl type theorems”, Mathematica Bohemica, 134.4(2009), 369-378.
117
Composition Operators on Lorentz-Karamata-Bochner Spaces Anuradha Gupta and Neha Bhatia
Delhi College of Arts and Commerce Department of Mathematics, University of Delhi, Delhi, India Department of Mathematics, University of Delhi, Delhi, India
[email protected], [email protected]
Abstract: The Lorentz-Karamata-Bochner space is a rearrangement invariant-Bochner space defined by means of slowly varying functions [2]. Composition operator CT [3] map f to f ο T where T is a non-singular measurable transformation. I. Erylimaz [1] discussed about the multiplication operators on these spaces. In this paper, we shall study about the composition operators on these spaces. We discussed about the boundedness and closed range of the composition operators in terms of the measure. We also discussed about the condition under which CT becomes an isometry. Also, we have discussed about the invertibility and essential range of the composition operators on Lorentz-Karamata-Bochner spaces. Keywords: Slowly varying function; Lorentz-Karamata-Bochner spaces; Composition operators; Closed range, Essential range. References: [1] I. Eryilmaz, “Multiplication operators on Lorentz-Karamata-Bochner space”, Math. Slovaca 62(2012), no. 2, 293-300. [2] D. E. Edmunds, W. D. Evans, “Hardy operators, function spaces and embeddings”, Springer Monographs in Mathematics Springer-Verlag , Belin, 2004. [3] R. K. Singh, J. S. Manhas, “Composition operators on function spaces”, North Holland Math. Studies 179, North Holland, Amsterdam, 1993.
118
A Note on Integral Inequalities for n-Time Differentiable Mappings Mustafa Gurbuz, Abdullah YARADILMIS M and . Emin OZDEMIR
Department of Mathematics, Agri Ibrahim Cecen University, Agri, Turkey K.K.E.F. Department of Mathematics, Ataturk University, Erzurum, Turkey
[email protected], [email protected], [email protected]
Abstract: In this paper, we obtained some new generalizations for different kinds of convex functions by using an integral identity which is established for n-time differentiable mappings by Ozdemir and Yildiz in [1]. We also give some special cases of our results that strengths the theorems in [1]. Keywords: convex, convex, convex, Power Mean, Hölder Inequality. References: [1] M. E. Ozdemir and C. Yildiz, “A new generalization of the midpoint formula for n-time differentiable mappings which are convex”, arXiv:1404.5128v1. [2] S.-P. Bai, S.-H. Wang and F. Qi, “Some Hermite-Hadamard type inequalities for n-time differentiable convex functions”, Jour. of Ineq. and Appl., 2012, 2012:267. [3] P. Cerone, S. S. Dragomir and J. Roumeliotis and J. Sunde, “A new generalization of the trapezoid formula for n-time differentiable mappings and applications”, Demonstratio Math., 33 (4), (2000), 719-736. [4] D.-Y. Hwang, “Some inequalities for n-time differentiable mappings and applications”, Kyung. Math. Jour., 43 (2003), 335-343. [5] W.-D. Jiang, D.-W. Niu, Y. Hua and F. Qi, “Generalizations of Hermite-Hadamard inequality to n-time differentiable function which are convex in the second sense”, Analysis (Munich), 32 (2012), 209-220. [6] M. E. Ozdemir and C. Yildiz, “New inequalities for n-time differentiable functions”, arXiv:1402.4959v1. [7] S. H. Wang, B.-Y. Xi and F. Qi, “Some new inequalities of Hermite-Hadamard type for n-time differentiable functions which are convex”, Analysis (Munich), 32 (2012), 247-262.
119
Statistical Convergence and Some Questions of Operator Theory Mehmet Gurdal and Ulas Yamanci
Department of Mathematics, Suleyman Demirel University, Isparta, Turkey [email protected], [email protected]
Abstract: We use the concepts of statistical convergence and Berezin symbols for solving of some problems of operator theory. Namely, we prove that under some conditions the weak statistical limit of compact operators is compact. We also use statistical convergence for the solving of similar problem for the sequence of operators from Schatten-Neuman class. Some related questions are also discussed. Keywords: Statistical Convergence, Compact operator, Schatten-Neumann class, Berezin Symbol. References: [1] N. Aronzajn, “Theory of reproducing kernels”, Trans. Amer. Math. Soc., 68(1950), 337-404. [2] J. Connor, M. Ganichev, V. Kadets, “A Characterization of Banach spaces with separable duals via weak statistical convergence”, J. Math. Anal. Appl. 244 (2000) 251--261. [3] H. Fast, “Sur la convergence statistique”, Colloq. Math., 2(1951), 241-244. [4] J. A. Fridy, “On statistical convergence”, Analysis, 5(1985), 301-313. [5] M.T. Karaev, “Berezin Symbols and Schatten-von Neumann Classes”, Math. Notes, 72(2)(2002), 185-192; translated from Mat. Zametki, 72(2)(2002), 207-215. [6] S. Pehlivan, M.T. Karaev, “Some results related with statistical convergence and Berezin symbols”, J. Math. Anal. Appl., 299(2)(2004), 333-340.
120
Statistical Convergence and *C -Operator Algebras Mehmet Gurdal and Mualla Birgul Huban
Department of Mathematics, Suleyman Demirel University, Isparta, Turkey [email protected], [email protected]
Abstract: In this paper we define in terms of Berezin symbols, reproducing kernels and statistical radial convergence the notion of generalized Englis algebras, which is a *C -operator algebra on the Hardy space ( ),2 DH and study its some properties Keywords: Berezin Symbol; Englis algebra; Statistical Convergence; Toeplitz operators; Hardy space. References: [1] C.A. Berger and L.A. Coburn, “Toeplitz operators and quantum mechanics”, J. Funct. Anal., 68(1986), 273-299. [2] R. Douglas, “Banach algebra techniques in operator theory”, Academic press, 1972. [3] M. Englis, “Toeplitz operators and the Berezin transform on H^2,” Lin. Alg. Appl., 223/224(1995), 171-204. [4] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2(1951), 241-244. [5] J. A. Fridy, “On statistical convergence”, Analysis, 5(1985), 301-313. [6] S. Pehlivan and M.T. Karaev, “Some results related with statistical convergence and Berezin symbols”, J. Math. Anal. Appl., 299(2004), 333-340. [7] M.T. Karaev, “Berezin smybols and Schatten-von Neumann Classes”, Math. Notes, 72(2002), 188-192. [8] K. Zhu, “Operator Theory in Function Spaces”, Marcel Dekker Ins., 1990.
121
New Runge-Kutta Methods for Numerical Solutions of Multiplicative Initial Value
Problems Yusuf Gurefe and Emine Misirli
Department of Econometrics, Usak University, Usak, Turkey Department of Mathematics, Ege University, Bornova, Izmir, Turkey
[email protected],[email protected]
Abstract: Multiplicative calculus, which is defined in a manner analogous to the concepts in the classical calculus, has become important in recent years. Firstly, multiplicative Runge-Kutta (R-K) methods are introduced by Aniszewska [1]. Then, Bashirov et al. give some basic properties with respect to the multiplicative calculus [2]. Also, multiplicative finite difference methods and multiplicative Adams Bashforth-Moulton methods are respectively developed in [3,4]. The new R-K methods are proposed in order to solve the multiplicative initial value problems (MIVPs) and also the stability conditions of these techniques are analyzed [5]. Keywords: Multiplicative Runge-Kutta methods, Multiplicative differential equations, Multiplicative stability. References: [1] D. Aniszewska, “Multiplicative Runge-Kutta Methods”, Nonlinear Dyn. 50.1-2(2007), 262-272. [2] A. E. Bashirov, E. Mısırlı Kurpınar and A. Özyapıcı, “Multiplicative Calculus and its Applications”, J. Math. Anal. Appl. 337.1(2008), 36-48. [3] M. Rıza, A. Özyapıcı and E. Mısırlı, “Multiplicative Finite Difference Methods”, Q. Appl. Math. 67.4(2009), 745-754. [4] E. Misirli, Y. Gurefe, “Multiplicative Adams Bashforth-Moulton Methods”, Numer. Algorithms, 57.4(2011), 425-439. [5] Y. Gürefe, “Multiplikatif Diferansiyel Denklemler ve Uygulamaları Üzerine”, Doktora Tezi, Ege Üniversitesi, 2013.
122
On the Stability of Picard-Kirk-S Iterative Method Faik Gursoy and Vatan Karakaya
Department of Mathematics, Adiyaman University, Adiyaman, Turkey Department of Mathematical Engineering, Yildiz Technical University, Esenler, Istanbul, Turkey
[email protected], [email protected]
Abstract: A quick review for the literature of fixed point iterative methods reveals that they have been successfully used for a wide range of applications ranging from economics to medicine. For this reason, it is now indisputable that fixed point iteration methods are among most important mathematical tools. In recent years, much attention has been given to introduce and study the qualitative features of fixed point iterative methods, see e.g., ([1]-[6]) to cite few of them. In this presentetion, continuing this trend, we introduce a new iterative method called Picard-Kirk-S by
⎩⎪⎨
⎪⎧𝑥𝑥0 ∈ 𝑋𝑋,
𝑥𝑥𝑛𝑛+1 = 𝑇𝑇𝑦𝑦𝑛𝑛 , 𝑦𝑦𝑛𝑛 = 𝛼𝛼𝑛𝑛,0𝑇𝑇𝑥𝑥𝑛𝑛 + ∑ 𝛼𝛼𝑛𝑛,𝑖𝑖1𝑇𝑇𝑖𝑖1𝑧𝑧𝑛𝑛
𝑠𝑠1𝑖𝑖1=1 ,
𝑧𝑧𝑛𝑛 = ∑ 𝛽𝛽𝑛𝑛,𝑖𝑖2𝑇𝑇𝑖𝑖2𝑥𝑥𝑛𝑛𝑠𝑠2𝑖𝑖2=0 , 𝑛𝑛 ∈ ℕ,
(1)
where 𝑋𝑋 is a normed linear spaces, 𝑇𝑇 is a self-map of 𝑋𝑋, 𝛼𝛼𝑛𝑛,𝑖𝑖1 , 𝛽𝛽𝑛𝑛,𝑖𝑖2 are sequences in [0,1] satisfying 𝛼𝛼𝑛𝑛,𝑖𝑖1 ≥ 0, 𝛽𝛽𝑛𝑛,𝑖𝑖2 ≥ 0, 𝛼𝛼𝑛𝑛,𝑖𝑖0 ≠ 0, 𝛽𝛽𝑛𝑛,𝑖𝑖0 ≠ 0, ∑ 𝛼𝛼𝑛𝑛,𝑖𝑖1
𝑠𝑠1𝑖𝑖1=0 = 1, ∑ 𝛽𝛽𝑛𝑛,𝑖𝑖2
𝑠𝑠2𝑖𝑖2=0 = 1, and 𝑠𝑠1, 𝑠𝑠2 are fixed integers with 𝑠𝑠1 ≥ 𝑠𝑠2.
Here, we analyze some convergence and stability results for Picard-Kirk-S iterative method (1). Keywords: Convergence; Stability; Iterative Methods; Normed Linear Space. References: [1] E. Picard, “Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives”, Journal de Mathématiques pures et appliquées, 6 (1890), 145–210. [2] F. Gursoy, V. Karakaya, and B. E. Rhoades, “Some convergence and stability results for the Kirk multistep and Kirk-Sp fixed point iterative algorithms”, Abstract and Applied Analysis, 2013.2013(2013),1-12. [3] F. Gursoy, V. Karakaya, "Some Convergence and Stability Results for Two New Kirk Type Hybrid Fixed Point Iterative Algorithms", Journal of Function Spaces, 2014.2014 (2014), 1-8. [4] F. Gursoy, V. Karakaya, “A Picard-S hybrid type iteration method for solving a differential equation with retarded argument”, http://arxiv.org/abs/1403.2546. [5] M. Abbas, T. Nazir, “A new faster iteration process applied to constrained minimization and feasibility problems”, Matematicki Vesnik, 66.2(2014), 223. [6] M.A. Noor, “New approximation schemes for general variational inequalities”, Journal of Mathematical Analysis and Applications, 251.1(2000), 217–229. [7] M.O. Olatinwo, “Some stability results for two hybrid fixed point iterative algorithms in normed linear space”, Matematicki Vesnik, 61.4(2009), 247–256. [8] R.P. Agarwal, D. O Regan, D. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings”, Journal of Nonlinear and Convex Analysis, 8(2007), 61–79. [9] S. Ishikawa, “Fixed points by a new iteration method”, Proceedings of the American Mathematical Society, 44 (1974),147–150. [10] W.R. Mann, “Mean value methods in iteration”, Proceedings of the American Mathematical Society 4 (1953), 506–510.
123
Time series modeling a temperature data in Bechar
South West of Algeria Lahmar Habib, Zouaoui Chikr el Mezouar,
Laboratory of Study Economics and Local Development in South West of Algeria
University of Bechar
E-mail : [email protected]
Abstract: The main purpose of time series analysis is to find a best fit to a data set that can be
defined by a model that can be used for forecasting. In this paper, a temperature of the Bechar
South West of Algeria was modeled by SARIMA or seasonal Autoregressive integrated moving
average. References:
[1] Amiri A. 2004. Studying the probability climate change and fitting a suitable ARIMA model on temperature and
rain data of Gilan province, IRIMO, Gilan Met. Vol. 5–6: pp.78-125. [2] Chatfield, C. (2004) “The Analysis of Time
Series: An Introduction”. 6th Edn., New Jersey:Chapman & Hall.
[3] Neter, J. , Wasserman, W. and Kutner, M.H. (1989) “ Applied Regression Models ” 2nd Edn,IRWIN.
[4] Wei, W.W.S. (1990) “Time series analysis: Univariate and multivariate methods”. Addison-Wesley Publishing
Company, Inc.
124
On the Solutions of Some System of Difference Equations Nabila Haddad and Nouressadat Touafek
Department of Mathematics, Jijel University, Algeria [email protected]
Abstract: Difference equations are of paramount importance in many fields, for example they enter as approximations of continous problems and as models describing life phenomena in genetics, biology, control theory and so foth. So studying properties of difference equations allow to undenstand such phenomena. For some works in this subject, see, [1]-[5] and references cited therein. Recently there has been a great interest in solving nonlinear difference equations. In this talk contributes, we get the form of the solutions of some systemof difference equations. Keywords: Difference equations, System of difference equations. References: [1] E. M. Elsayed, M. Mansour and M. M. El-Dessoky, “Solutions of fractional systems of difference equations”, Ars Combinatoria, 110: 469--479, 2013. [2] X. Liu, “A note on the existence of periodic solution in discrete predator-prey models”, Appl. Math. Model., 34(2010), 2477-2483. [3] S. Stevic, “On a solvable system of difference equation of fourth order”, Appl.Math.Compu., 219(2013), 5706-5716. [4] I. Yalcinkaya, “On the global asymptotic behavior of a system of two nonlinear difference equations”, ARS Combinatoria, 95(2010), 151-159.
125
On the Exact Values of Wavelet Functions Mohamed Ali Hajji
Department of Mathematical Sciences, College of Science, United Arab Emirates University
Abstract: Many wavelet based methods for the numerical solutions of differential equations rely on the values of the wavelet bases elements. The famous Daubechies’ wavelet basis functions do not have a closed formula; they are defined recusively via refinement equations. Having at hand accurate approximate values of such wavelet basis functions is crucial to obtaining accurate numerical solution to differential equations. In this paper, we propose an efficient and accurate algorithm for the calculation of the exact values of compactly-supported refin able wavelet functions. The proposed algorithm is iterative, performing a single convolution operation at each step. It requires solving, at the first step only, an eigenvalue problem of a relatively small size. The novelty of the algorithm is that once the values at dyadic points at a certain level j are calculated they do not need to be updated at the next step; they are exact. At each step, only values at odd dyadic points are calculated. We find that this algorithm is superior to and more accurate than the cascade algorithm proposed by Ingrid Daubechies. We apply the algorithm to find the exact values of Daubechies’ scaling and wavelet functions as well as the values of their autocorrelation functions, which are of paramount importance in the numerical solution of differential equations. Comparison with the values obtained using Matlab is made. We found that the proposed algorithm is more accurate.
126
Nonlinear Boundary Value Problems for p- Laplacian Fractional Differential Systems
Nuket Aykut Hamal, Fulya Yoruk Deren, Tugba Senlik Cerdik, Department of Mathematics, Ege University, Bornova, Izmir, Turkey
Abstract: In this paper, we study the existence results of positive solutions for p-Laplacian fractional differential systems by means of fixed point theorems on cones. As an application, an example is given to demonstrate our main results. Keywords: Fixed Point Theorem, Positive Solution, Fractional Differential System. References: [1] K. R. Prasad, B. M. B. Krushna, Multiple Positive Solutions for a Coupled System of p- Laplacian Fractional Order Two Point Boundary Value Problems, International Journal of Differential Equations, 2014, Article ID 485647. [2] S. Li, X. Zhang, Y. Wu, L. Caccetta, Extremal Solutions for p-Laplacian Differential Systems via Iterative Computation, Applied Mathematics Letters, 26 (2013) 1151-1158. [3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego(1999). [4] K.Deimling, Nonlinear Functional Analysis. Springer, Berlin, 1985. [5] Z. Wei, C. Pang, Y. Ding, Positive Solutions of Singular Caputo Fractional Differential Equations with Integral Boundary Conditions, Communications in Nonlinear Science and Numerical Simulation, 17 (2012) 3148-3160.
127
The Study of Existence and Uniqueness of Certain Fractional Differential Equations Boulares Hamid
Department of Mathematics, 8 mai 1945 University, Guelma, Algeria [email protected]
Abstract: In this study, we prove the existence of a general solution for the problem Dαx (t) = f (x(t)) g (Dβx(t)), 1<α<2, 0<β<1, t Î [0, T], T>0, x(0)= x0, Dβx (0)=x1. (1.1) f and g are not necessarily continuous or monotone [1]. Moreover neither f nor g need be bounded around, respectively x0 and x1 [2], give to the singularity of the problem (1.1) [3]. There are others topics which study in our framework as uniqueness and continuation. Keyword: Fractional differential equations; Existence; Uniqueness; Continuation. References: [1] LinLi Lv, JingRong Wang, Wei Wei, “Existence and uniqueness results for fractional differential equations with boundary value conditions”, Opuscula Mathematica, 31(2011), 629-643. [2] E.R. Kaufmann, E. Mboumi, “Positive solutions of a boundary value problem for a nonlinear fractional differential equation”, Electronic Journal of Qualitative Theory of Differential Equations, 3(2008), 1-11. [3] G. Samko, A.A. Kilbas, O.I. Marichev, “Fractional Integrals and Derivatives: Theory and Applications”, Gordon and Breach Science Publishers, 1993.
128
Some Applications of Clifford Algebras and Octonions to Differential Geometry Hideya Hashimoto
Department of Mathematics, Meijo University, Japan [email protected]
Abstract: This talk is a joint work with Prof. Misa Ohashi. In [1], Atiyah, Bott and Shapiro construct Clifford algebra 𝐶𝐶𝑙𝑙𝑛𝑛 with respect to the canonical inner product of n-dimensional Euclidean space ℝ𝑛𝑛. It is well-known that 𝐶𝐶𝑙𝑙2 is isomorphic to the quaternions ℍ and 𝐶𝐶𝑙𝑙3 is isomorphic to a direct sum of two quaternions ℍ⨁ℍ. Bott and Milnor porved that the following famous Theorem; The vector space ℝ𝑛𝑛 posseces a bilinear product opearation without zero divisors only for 𝑛𝑛 = 1, 2, 4 and 8. If we identify ℝ8 with octonions 𝕆𝕆, then 𝕆𝕆 does not have the zero divisors. Also we note that Hurwitz proved that the normed algebra isomorphic to ℝ, ℂ, ℍ or 𝕆𝕆. The 𝐶𝐶𝑙𝑙3 is isomrhic to ℝ8 as a vector space and admits bilinear product operation with zero divisors. From thses properties, we want to explain some construction of 𝐺𝐺2 and Spin(7) fibre bundle structure on some 3-dimensional manifold. Keywords: Clifford algebra; Octonions; Zero divisor, 3-dimensional manifold. References: [1] M.F. Atiyah, R. Bott, A. Shapiro, “Clifford moddules”, Topology 3(1964),3-38.
129
A Bayesian Estimation of Traffic Intensity in M/M/∞ Queue Under Different Loss Functions
Braham Hayette Department of Mathematics, Bejaia University, Bejaia, Algeria
Abstract: Bayesian analysis of queueing systems is a relatively recent research area; it combines the prior knowledge of the random variations in the parameter distribution with observed data. Some useful references are [1] and [2]. The problem of estimation of the queueing process, such as the arrival rate, service rate and traffic intensity is important from the point of view of design of queues and comparison of the performances of queueing systems. In this paper we derive bayesian estimators of traffic intensity in the density of steady state of the M/M/∞ queue, under three different loss functions [3]. Simulation study is given to compare the three estimators given by the three cases. Keywords: M/M/∞ queue; Bayesian estimator; Prior and posterior density; Loss function. References: [1] A. Armero, M. J. Bayarri, “Bayesian prediction in M/M/1 queues”, Queueing Systems,. 15(1994), 401-417. [2] K. K. Sharma, V. Kumar, “Inference on M/M/1 queue systems”, Operation Research 36 (1) (1999), 26-34. [3] H. Ren, G. Wang, “Bayes Estimation of Traffic Intensity in M/M/1 Queue under a Precautionary Loss Function” Procedia Engineering 29(2012), 3646-3650.
130
Ideally Slowly Oscillating Continuity
Bipan Hazarika Department of Mathematics, Rajiv Gandhi University, Rono Hills, India
Abstract: In this paper, we introduce the notion of ideally slowly oscillating sequences, which is lying between ideal convergent and ideal quasi Cauchy sequences, and study on ideally slowly oscillating continuous functions in topological vector space valued cone metric space. Keywords: Ideal convergence; Cone metric; Continuity; Quasi Cauchy sequence; Slowly oscillating sequences.
131
Equivalent Cauchy Sequences on Generalized Metric Spaces Elida Hoxha and Sidite Duraj
Department of Mathematics, Tirana University, Tirane, Albania [email protected]
Abstract. This paper consider the problem of defining Cauchy sequence and equivalent Cauchy sequences in generalized metric space such as dislocated metric space, quasi dislocated and quasi pseudo metric space. This paper contains a study of equivalent Cauchy sequences in quasi pseudo metrics (the concept of a quasi pseudo metric was introduced by Wilson 1931, Albert 1941 and Kelly 1963). Our basic contribute is a characterization of equivalent Cauchy sequences under mildly restrictive conditions.
132
Generalization of Fixed Point Theorems in Quasi Cone Metric Space Relating to the Diameter of Orbits by Using a Comparison Function
Elida Hoxha and Eriola Sila Department of Mathematics, Tirana University, Tirane, Albania
[email protected], [email protected]
Abstract: Huang and Zhang[7] have introduced the concept of cone metric spaces replacing the set of real numbers by an ordered Banach space. They have proved some theorems about fixed points in cone metric space s for contractive mappings and their results generalize some fixed point theorems in metric space. Abdeljaward and Karapinar[4] have given the definition of quasi cone metric space and there are many authers who have worked in this direction. Babu[1] proved some fixed theorems in metric space relating to the diameter of orbits using a control function. In this paper we prove some theorems in quasi cone metric space relating to the diameter of orbits by using a comparison function, which generalize the results of Babu[1] and Jachymski[2]. Also, we have given an example to illustrate the main result. Keywords: Fixed point; Quasi cone metric space; Comparison function. References: [1] G.V.R. Babu, “Generalization of fixed point theorems relating to the diameter of orbits by using a control function”, Tamkang Journal of Mathematics, Vol 35, Number 2, Summer 2004. [2] Jacek Jachymski, “ A generalization of the theorem by rhoades and Watson for contractive type mappings”, Math, Japonica 38, No. 6 (1993), 1095-1102. [3] P.Raja and S. M. Vaezpour,”Some extentions of Banach’s contraction Principle in complete cone metric space, Hindawi publishing Coorporation, Fixed point theory and Applications, Vol 2008, Article ID 768294, 11 pages. [4] T. Abdeljaward and E. Karapinar, “Quasi cone metric spaces and generalization of Caristi Kirk’s Theorem”, Fixed Point Theory and Application, Vol.2009, no 1, article ID 574387 [5] F. Shaddad and Noorani, “Fixed point results in quasi cone metric spaces” Abstract and Applied Analysis, vol 2013, Article ID 303626, 7 pages. [6] B.E. Rhoades, “A Comparison of various defintions of contractive mappings”, Transactions of American Mathematical Society, Vol 226,pp 257-290,1977. [7] L. G. Huang and X.Zhang, “Cone metric spaces and fixed point theorems of contractive mappings.” J. Math. Anal. Appl.332 (2007), 1468-1476.
133
Riesz Idempotent and Generalized Weyl’s Theorem for k−Quasi Class *nA Operators
Ilmi Hoxha and Naim L. Braha Department of Mathematics and Computer Sciences, University of Prishtina, Kosova
[email protected], [email protected]
Abstract: An operator ( )T L HÎ , is said to belong to the k−quasi class *nA operator if
2 2* 1 *1k n knT T T T O+ +
æ ö÷ç ÷- ³ç ÷ç ÷çè ø
for positive integers n and k. First, there will be seen some basic properties for a k−quasi class *nA operator,
examples and inclusions. Second, it will be shown that the Riesz idempotent Pl of every k−quasi class *nA operator
T with respect to each isolated point 0l ¹ of its spectrum ( )Ts is self-adjoint and satisfies *( ) ker( ) ker( )P H T Tl l l= - = - ,
and if 0l = , then 1( ) ker( )kP H Tl+= . Finally, it will be proved the generalized Weyl’s theorem for T and
for *T , if T is a k−quasi class *nA operator. Also we proved if *T is a k−quasi class *
nA then T satisfies a−Weyl’s theorem. Keywords: k−quasi class *
nA operators; Riesz idempotent; Generalized Weyl’s theorem; Generalized a−Weyl’s theorem. References: [1] S. C. Arora, J. K. Thukral, “On a class of operators", Glasnik Math. 21(1986) 381-386. [2] M. Berkani, “Index Of B−Fredholm operators and generalization of a Weyl theorem”, Proceedings of the American Mathematical Society, Volume 130, no.6(2001), 1717-1723. [3] M. Berkani, “On a class of quasi−Fredholm operators”, Integr. Equ. Oper. Theory 34(1999), 244-249. [4] M. Berkani, A. Arroud, “Generalized Weyl’s theorem and hyponormal operators”, J. Austra. Math. Soc, 76(2004), 291-302. [5] M. Berkani, J. Koliha, “Weyl type theorems for bounded operators”, Acta Sci.Math. 69(2003), no. 1-2, 359-376. [6] L. A. Coburn, “Weyl’s theorem for nonnormal operators”, Michigan Math. J. 13(1966), 285-288. [7] J. B. Conway, “A course in Functional Analysis”, Springer-Verlag, New York, 1985. [8] B. P. Dugall, I. H. Jeon, I. H. Kim, “On *−paranormal contractions and properties for *−class A operators”, Linear Algebra Appl. 436 (2012), no. 5, 954-962. [9] T. Furuta,” On The Class of Paranormal Operators”, Proc. Jap. Acad. 43(1967), 594-598. [10] T. Furuta, M. Ito, T. Yamazaki, “A subclass of paranormal operators including class of log−hyponormal and several classes”, Sci. Math. 1(1998), no. 3, 389-403.
134
A Morgan-Voyce Collacation Method for Numerical Solution of Generalized Pantograph Equations
Ozgul Ilhan Department of Mathematics, Faculty of Science, Mugla Sitki Kocman University Mugla, Turkey
Abstract: In this paper, we develop a method which is called the Morgan-Voyce matrix method for solving the generalized pantograph equations
1
( ) ( )
0 0( ) ( ) ( ) ( )
J mm k
jk jk jkj k
y t P t y t g tl m-
= =
= + +åå 0 t b£ £ (1)
with the initial conditions
1
( )
0(0) , 0,1,..., 1.
mk
ik ik
c y i ml-
=
= = -å
Here ( )jkP t and ( )g t are continuous functions defined in the interval 0 ; , ,ik i jkt b c l l£ £ , and jkm are real or complex constants. We want to find an approximate solution of (1) expressed in the truncated Morgan-Voyce series form
0
( ) ( ) 0 t bN
n nn
y t a B t=
= £ £å
in here , 0,1, 2,...,na n N= are unknown Morgan-Voyce coefficients. N is chosen any positive integer such that
1N ³ and ( ), 0,1, 2,...,nB t n N= are the Morgan-Voyce polynomials [1] defined by
0
1( ) .
-1
nk
nk
n kB x x
n=
+ +æ ö= ç ÷
è øå
Keywords: Morgan-Voyce Polynomials; Generalized pantograph equations; Collocation method. References:
[1] M.N.S. Swamy, ”Further properties of Morgan-Voyce polynomials”, The Fibonacci Quarterly 6.2 (1968) 167-175. [2] M. Z. Liu, D. Li, “Properties of analytic solution and numerical solution and multi-pantograph equation”, Appl Math Comput 155 (2004), 853-871. [3] Y. Muroya, E. Ishiwata, H. Brunner, “On the attainable order of collocation methods for pantograph integro-differential equations”, J Comput.
135
Numerical Solution of Burger Equation by Using General Frechet Derivatives Combined with Differential Quadrature N. Imamoglu, G.Guraslan and G. Tanoglu
Department of Mathematics, Izmir Institute of Technology, Turkey Department of Civil Engineering, Pamukkale University, Turkey
Abstract: A differential quadrature rule combined with general Frechet derivative is proposed for the first time to solve Burger-type equation. By using the Frechet derivative, the nonlinear equation is converted into a set of linear algebraic equations which are solved iteratively. The effectiveness of this method has been demonstrated through a number of examples. Moreover this new method can be applied to new range of nonlinear partial differential equation. Keywords: Frechet derivative; Differential quadrature formula; Linearization; Burgers’ equation; Nonlinear partial differential equation References: [1] E. Birger Nilsen, “On operator splitting for the viscous Burgers’ and the Korteweg-ge Vries eauations”, Master of Science in Physics and Mathematics,2011. [2] G. R. Liu, T. Y. Wu, “A differential quadrature as a numerical method to solve differential equations”, Computational Machanics,24(1999), 197-205. [3] G. R. Liu, T. Y. Wu, “Differential quadrature solutions of eight-order boundary-value differential equations”, Journal of Computational and Applied Mathematics, 145(2002), 223-235. [4] R. Jiwar, R. C. Mittal, K. K. Sharma, “A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation”, Applied Mathematics and Computation,219(2013), 6680-6691. [5] Z. Zong, Y. Zhang, “Advenced differential quadrature methods”,(2009).
136
Lacunary Statistical Convergence of Order α in Probabilistic Normed Spaces Mahmut Isik
Department of Statistics, Firat University, Elazig, Turkey [email protected]
Abstract: In this paper, we intend to make a new approach to introduce the notion of lacunary statistical convergence of order α, in probabilistics normed spaces where α is a real number such that 0<α≤1 and prove some results. Keywords: Statistical convergence; T-norm; Probabilistic normed space. References: [1] A. Alotaibi, “Generalized statistical convergence in probabilistic normed spaces”, Open Math. J. 1 (2008), 82-88. [2] C. Alsina, B. Schweizer, A. Sklar, “On the definition of a probabilistic normed space”, Aequationes Math. 46 (1993), no. 1-2, 91-98. [3] R. Colak, “Statistical Convergence of Order α”, Modern Methods in Analysis and Its Applications, NewDelhi, India: Anamaya Pub, (2010), 121-129. [4] J.A. Fridy, C. Orhan, “Lacunary statistical convergence”, Pacific J. Math. 160 (1993), no.1, 43-51 [5] B. Lafuerza-Guillén, M. Rafi, “Statistical convergence in strong topology of probabilistic normed spaces”, Note Mat. 29 (2009), no. 1, 79-88. [6] H. Sengul, M. Et, “On lacunary statistical convergence of order α”, Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), no.2, 473-482.
137
Evolution Modeling of NPZ and SIR Models with and without Diffusion Siraj-ul-Islam and Saeedullah Jan
Department of Basic Science University of Engineering and Technology Peshawar, Pakistan [email protected]
Abstract: In this paper, the two biological models i.e. Nitrogen, Phytoplankton and Zooplankton (NPZ) and whooping cough SIR models [1] are being modified and solved numerically by finite difference and meshless methods. Diffusion process has been added to the existing models [1] so that a unidimensional movement of three species can be incorporated in the models. The effects of diffusion have been studied in both the models. An operator splitting method coupled with the meshless and finite difference procedures is being considered for numerical solution of the two biological models with and without diffusion. A one step explicit meshless procedure is also applied for the numerical solution of the nonlinear models. The NPZ model contains the concentration of Nitrogen, Phytoplankton and Zooplankton and the whooping cough model contains susceptible, infected, and recovered classes of the population. Equilibrium points of both models have been investigated. Stability of equilibrium points regarding SIR model has been studied. The basic reproduction number of SIR model is also determined. Due to nonavailability of the exact solution, the numerical results obtained are mutually compared and their correctness is being verifed by the theoretical results as well. References: [1] G. Parra, A.J. Arenas, B. M. C. Charpentier, “Combination of nonstandard schemes and Richardson's extrapolation to improve the numerical solution of population models”, Math. Comp. Modelling, 52, 1030-1036, 2013.
138
Inverse Problem of Finding Diffusion Coefficient in Heat Equation with Nonlocal Boundary Conditions
Mansur I. Ismailov and Bulent Ogur Department of Mathematics, Gebze Institute of Technology, Gebze, Kocaeli, Turkey
[email protected], [email protected]
Abstract: Denote 𝜋𝜋𝑇𝑇 = (𝑥𝑥, 𝑡𝑡): 0 < 𝑥𝑥 < 1, 0 < 𝑡𝑡 ≤ 𝑇𝑇. Consider the inverse problem of determining 𝑢𝑢(𝑥𝑥, 𝑡𝑡) and 𝑎𝑎(𝑡𝑡) in heat equation 𝑢𝑢𝑡𝑡 = 𝑎𝑎(𝑡𝑡)𝑢𝑢𝑥𝑥𝑥𝑥 + 𝑓𝑓(𝑥𝑥, 𝑡𝑡)
(𝑥𝑥, 𝑡𝑡) ∈ 𝜋𝜋𝑇𝑇 (1.1)
with the following conditions: 𝑢𝑢𝑥𝑥(0, 𝑡𝑡) + 𝛼𝛼𝑢𝑢(0, 𝑡𝑡) = 0,
𝑢𝑢(0, 𝑡𝑡) = 𝑢𝑢(1, 𝑡𝑡),
0 ≤ 𝑡𝑡 ≤ 𝑇𝑇 (1.2)
𝑢𝑢(𝑥𝑥, 0) = 𝜑𝜑(𝑥𝑥),
0 ≤ 𝑥𝑥 ≤ 1 (1.3)
𝑢𝑢(𝑥𝑥, 𝑡𝑡)𝑑𝑑𝑥𝑥
1
0= 𝐸𝐸(𝑡𝑡),
0 ≤ 𝑡𝑡 ≤ 𝑇𝑇 (1.4)
where 𝛼𝛼 is a given real number and 𝑓𝑓(𝑥𝑥, 𝑡𝑡), 𝜑𝜑(𝑥𝑥), 𝐸𝐸(𝑡𝑡) are given functions. The pair 𝑎𝑎(𝑡𝑡), 𝑢𝑢(𝑥𝑥, 𝑡𝑡) from the class 𝐶𝐶[0, 𝑇𝑇] × (𝐶𝐶2,1[𝜋𝜋𝑇𝑇] ∩ 𝐶𝐶1,0[𝜋𝜋𝑇𝑇]) which are satisfying the condition (1.1) −(1.4) with 𝑎𝑎(𝑡𝑡) > 0 on [0, 𝑇𝑇] is called the classical solution of the inverse problem (1.1) − (1.4). In the case of 𝛼𝛼 = 0, the inverse problem is studied in [1]. Our goal is to study the inverse problem in the nontrivial case, i.e, 𝛼𝛼 ≠ 0. A similar inverse coefficient problem is considered in [2]. For the classical solution, the following consistency conditions are necessarily satisfied 𝜑𝜑′(0) + 𝛼𝛼𝜑𝜑(0) = 0, 𝜑𝜑(0) = 𝜑𝜑(1) and ∫ 𝜑𝜑(𝑥𝑥)𝑑𝑑𝑥𝑥1
0 = 𝐸𝐸(0). In the present paper, we present the existence, uniqueness of solution of (1.1) − (1.4) and Lipschitz continuous dependence of the solution upon the data 𝑓𝑓, 𝜑𝜑, 𝐸𝐸 under some regularity and nonnegativity conditions on the the data. Keywords: Heat equation; Integral overdetermination condition; Time dependent diffusion coefficient; Non-local boundary condition. References: [1] F. Kanca, M. I. Ismailov, “The inverse problem of finding the time-dependent diffusion coefficient of the heat equation from integral overdetermination data”, Inverse Problems in Science and Engineering, 20(4) (2012), 463-476. [2] N. B. Kerimov, M. I. Ismailov, “An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions”, J. Math. Anal. Appl. 396 (2012), 546-554.
139
Improved R-order of convergence for solving nonlinear equations
Jai Prakash Jaiswal
Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, M.P., India-462051
Abstract :
The present paper is devoted to improve the R-order of convergence of iterative method for solving nonlinear equations. The theoretical proof of the R-order convergence is also given here. Finally some numerical results are given to confirm theoretical results.
140
On the Basis Number of the Wreath Product of Paths with Wheels and Some Related Problems
M.M.M. Jaradat, M.S. Bataineh and M.K. Al-Qeyyam Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar.
Abstract: For a given graph G, the set E of all subsets of E(G) forms an |E(G)|-dimensional vector space over Z₂ with vector addition X⊕Y=(X\Y)∪(Y\X) and scalar multiplication 1.X=X and 0.X=∅ for all X,Y∈E. The cycle space, C(G), of a graph G is the vector subspace of (E,⊕,.) spanned by the cycles of G. A basis B of G is called a d-fold if each edge of G occurs in at most d cycles of the basis B. The basis number, b(G), of G is the least non-negative integer d such that C(G) has a d-fold basis; a required basis of C(G) is a basis for which each edge of G belongs to at most b(G) elements of B. A basis B is called a minimum cycle basis (MCB) if its total length ∑ |𝐷𝐷|𝑛𝑛
𝐷𝐷∈𝐵𝐵 is minimum among all bases of C(G). The wreath product 𝐺𝐺𝐺𝐺𝐻𝐻 has the vertex set V(GρH)=V(G)×V(H) and the edge set E(GρH)=(u₁,v₁)(u₂,v₂)|u₁=u₂ and v₁v₂∈E(H), or u₁u₂∈G and there is 𝛼𝛼∈Aut(H) such that 𝛼𝛼(v₁)=v₂. In this paper, we investigate the basis number for the wreath product of wheels with paths. Also, as a related problem, we construct a minimum cycle basis of the same. Keywords: Cycle space; Minimum cycle basis; Basis number; Wreath product. References: [1] A.A. Ali and G.T. Marougi, “The basis number of Cartesian product of some graphs”, J. of the Indian Math. Soc. 58, 123-134 (1992). [2] M.K. Al-Qeyyam and M.M.M. Jaradat, “On the basis number and the minimum cycle bases of the wreath product of some graphs II”. J. Com- bin. Math. Combin. Comput. 72, 65-72 (2010). [3] A.S. Alsardary and J. Wojciechowski, “The basis number of the powers of the complete graph”, Discrete Math. 188, no. 1-3, 13-25 (1998). [4] F. Berger, “Minimum cycle bases in graphs”, PHD thesis, Munich 2004. [5] Z. Bradshaw and R. Hammack, “Minimum cycle bases of the direct products of graphs with cycles”, Ars Mathematica Contemporanea 2, 101-119 (2009). [6] Z. Bradshaw and M.M.M. Jaradat, “Minimum cycle bases for direct products of K2 with complete graphs”. Australasian Journal of Combi- natorics 43, 127–131 (2009). [7] R. Hammack, “Minimum cycle bases of direct products of bipartite Graphs”, Australasian Journal of Combinatorics 36, 213–221, (2006). [8] R. Hammack, “Minimum cycle bases of direct products of complete Graphs”, Information Processing Letters 102, 214-218, (2007). [9] W. Imrich and P. Stadler, “Minimum cycle bases of product graphs”, Australasian Journal of Combinatorics 26, 233-244 (2002) .
141
Soft Completely Simple Semigroups Mehmet Kalkan and Haci Aktas
Department of Mathematics, Nevsehir Hacı Bektas Veli University, Nevsehir, Turkey Department of Mathematics, Erciyes University, Kayseri, Turkey
[email protected], [email protected]
Abstract: In 1940, the notion of completely simple semigroups (CSS) was introduced by Ree’s[1]. In Rees theorem “CSSs are isomorphic to some M(G;I,Λ; P)” which are known as Ree’s matrix semigroups. Many developments have been studied on this notion[2-4]. Araujo and Konieczny proved that the Molaei’s generalised groups are equivalent to the CSSs[5-8]. In all studies, it has been agreed that the proof of the Ree’s theorem is correct. But in 2002, J.M. Howie in his talk said “His statement and proof look clumsy as we look back, but this was completely new ground, and his achievement was remarkable.” In 1999, Molodtsov defined the notion of soft sets for solving problems dealing with uncertainties. In this talk, first the proof of Ree’s theorem will be given, then the definition of the soft completely simple semigroup will be defined. Keywords: Rees Theorem, Soft Sets, Soft Completely Simple Semigroups. References: [1] D. Rees. On semi-groups. Proc. Cambridge Philos. Soc., 36, (1940) 387-400. [2] P.A.Grillet, “Semigroups: An introduction to the structure theory”, Tulane University, New Orleans, Louisiana, (1995), 44-58. [3] A.H.Clifford and G.B.Preston “The algebraic theory of semigroups”,Vol II, Math. Survey of the American Math.Society, no.7 Pro., R.I.,(1967), 214-226. [4] J.M. Howie, “Fundamentals of Semigroup Theory”, Oxford University Press, New York, 1995. [5] J. Araujo and J. Konieczny “Molaei's generalized groups are completely simple semigroups”, Bul. Inst. Politeh. Iasi. Sect. I. Mat. Mec. Teor. Fiz., 48(52)(1-2): (2003), 1-5. [6] M. R. Molaei “Generalized groups”, Bul. Inst. Politeh. Iasi. Sect. I. Mat. Mec. Teor.Fiz., 45(49)(3-4): (2001), 21-24. [7] J.M. Howie, “Semigroup, past, present and future”, Proceedings of the International Conference on Algebra and Its Applications (2002), 6-20. [8] M. Kalkan, “Completely simple semigroups”, Master thesis, AİBÜ, Bolu, (2007). [9] D. Molodtsov, “Soft set theory - First results”, Computers and mathematics with applications. 37, (1999), 19-31.
142
Determination of a Time Dependent Diffusion Coefficient of a Quasilinear Parabolic Equation in the Case of Nonlocal Conditions
Fatma Kanca Department of Management Information Systems, Kadir Has University,34083, Istanbul, Turkey
Abstract: This paper investigates the inverse problem of finding the time-dependent diffusion coefficient in a quasilinear parabolic equation with the nonlocal boundary and integral overdetermination conditions. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown. Some considerations on numerical procedure of this problem are presented. Keywords: Heat equation, Inverse problem, Nonlocal boundary condition, Integral overdetermination condition, Time-dependent diffusion coefficient. References: [1] F. Kanca, “The inverse coefficient problem of the heat equation with periodic boundary and integral overdetermination conditions”, Journal of Inequalities and Applications, 108 (2013), 1-9. [2] F. Kanca, “Inverse Coefficient Problem of the Parabolic Equation with Periodic Boundary and Integral Overdetermination Conditions”, Abstract and Applied Analysis, 2013 (2013) 1-7. [3] M. Ismailov, F. Kanca, “An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions”, Mathematical Methods in the Applied Science, 34 (2011), 692-702. [4] F. Kanca, I. Baglan, “Continuous dependence on data for a solution of the quasilinear parabolic equation with a periodic boundary condition”, Boundary Value Problems, 28 (2013).
143
On the Metric Dimension of Uniform Fuzzy Graphs Vasantha Kandasamy W.B and Regin Thangaraj
Department of Mathematics, Indian Institute of Technology Madras, Chennai, India. [email protected]
Abstract: For an ordered set B = v1,v2,…,vk of nodes in a connected graph G and a node v of G, the metric representation of v with respect to B is the k-vector r(v|B) = (d(v,v1),d(v,v2),…,d(v, vk)). The set B is a resolving set for G if r(u|B) = r(v|B) implies that u = v for all pairs u,v of nodes of G. The metric dimension dim (G) of G is the minimum cardinality of a resolving set. This article concerns with the study of metric dimension for uniform fuzzy graphs. The notion of fuzzy graphs was first introduced by Rosenfeld in 1975. In an attempt to extend the idea of metric dimension of graphs to its equivalent in fuzzy graphs, we have defined metric dimension of connected fuzzy graphs using fuzzy metric, which is induced by the ρ-distance in fuzzy graphs and results are obtained in case of some standard uniform fuzzy graphs. Keywords: Metric dimension; Metric representation; Fuzzy graphs; Uniform fuzzy graphs. References: [1] G. Chartrand, L. Eroh, M. Johnson, O.R. Oellermann, “Resolvability in graphs and the metric dimension of a graph”, Discrete Appl. Math. 105 (2000), 99-113. [2] F. Harary, R.A. Melter. “On the metric dimension of a graph”, Ars Combin. 2 (1976), 191-195. [3] J. N. Mordeson, P. S. Nair, “Fuzzy Graphs and Fuzzy Hypergraphs”, Physica-Verlag, New York, 2000. [4] O. Kaleva, S. Seikkala, “On fuzzy metric spaces”, Fuzzy sets and systems 12 (1984), 215-229. [5] P. Erdos, F.Harary, W. T. Tutte, “On the dimension of a graph”,Mathematika 12 (1965), 118-122.
144
Some Common Coupled Fixed Point Theorems for Rational Expressions in Complex Valued bG -Metric Spaces
Neslihan Kaplan and Mahpeyker Ozturk Department of Mathematics, Sakarya University, Sakarya, Turkey
[email protected], [email protected]
Abstract: The purpose of this study is to introduce the notion of complex valued bG -metric spaces, by using the
setting of bG -metric spaces and complex valued metric spaces. We also demonstrate the existence and uniqueness of the common coupled fixed points in these spaces along with contractive conditions involving rational expressions. Our results extend and improve several fixed point theorems in the literature. Keywords: Common coupled fixed point; Complex valued Gb-metric spaces; Rational expressions. References: [1] A. Azam, B. Fisher, M. Khan “Common fixed point theorems in complex valued metric spaces”, Numerical Functional Analysis and Optimization, 32.3(2011), 243-253. [2] S. Bhatt, S. Choukiyal, R.C. Dimri, “Common fixed point of mappings satisfying rational inequality in complex valued metric spaces”, International Journal of Pure and Applied Mathematics, 73.2(2011), 159-164. [3] M. A. Kutbi, A. Azam, J. Ahmad, C. D. Bari, “Some common coupled fixed point results for generalized contraction in complex-valued metric spaces”, Journal of Applied Mathematics , 2013(2013), 10 pages. [4] S. Sedghi, N. Shobkolaei, J. R. Roshan, W. Shatanawi, “Coupled fixed point theorems in Gb-metric spaces”, MATEMATИЧKИ BECHИK, 66.2(2014), 190-201.
145
Stability and Accuracy of Time-Stepping Schemes for a Second-Order Wave Equation Samir Karaa
Department of Mathematics, Sultan Qaboos University, Muscat, Sultanate of Oman [email protected]
Abstract: We investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications [1, 2, 3]. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. These new results are more accurate than those in the literatue, e. g., [3, 4]. A convergence analysis is carried out and optimal a priori error estimates are obtained. Keywords: Discontinuous Galerkin method; Wave equation; Newmark scheme; Energy method; Stability condition; Optimal error estimates. References: [1] G. Cohen, P. Joly, “Construction and analysis of fourth-order nite difference schemes for the acoustic wave equation in nonhomogeneous media”, SIAM J. Numer. Anal., 33 (1996), 1266-1302. [2] H. Lim, S. Kim, J. Douglas, “Numerical methods for viscous and nonvis-cous wave equations”, Appl. Numer. Math., 57 (2007), 194-212. [3] G. Shubin, J. Bell, “A modified equation approach to constructing fourth-order methods for acoustic wave equations”, SIAM J. Sci. Stat. Comput. 8 (1987), 135–151. [4] P. A. Raviart, J. M. Thomas, “Introduction à l’Analyse Numérique des Equations aux Dérivés Partielles”, Masson, Paris (1983).
146
On H(θ)-Open Sets and Modifications on Hereditary Generalized Topological Spaces Umit Karabiyik
Department of Mathematics-Computer Science, Necmettin Erbakan Unıversity, Meram, Konya, Turkey [email protected]
Abstract. We intruduce the H(θ)-open sets modification given by a hereditary class and a generalized topology μ and investigate some properties and characterizations. With the help of H(θ)-open sets, we define new classes of sets H(θ)-pre-open, H(θ)-semiopen, H(θ)-α-open, H(θ)-β-open in hereditary generalized topological spaces and study the relation between such sets and θ-open sets on generalized topologies. Keywords:H(θ)-open sets; H(θ)-pre-open; H(θ)-semiopen; H(θ)-α-open; H(θ)-β-open. References: [1] A. Csaszar, “Generalized topology, generalized continuity”, Acta Math.Hungar. 2002; 96(4):351-357 . [2] A. Csaszar, “Generalized open set in generalized topologies”, Acta Math.Hungar. 2005; 106(1-2): 53-66. [3] A. Csaszar, “Modi_cation of generalized topologies via hereditary classes”, Acta Math.Hungar. 2007; 115:29-36. [4] A. Csaszar, “δ and θ modifications of generalized topologies”, Acta Math.Hungar. 2008;120(3), 275-279. [5] Y.K. Kim, W.K. Min, “On operations induced by hereditary classes on generalized topological spaces”, Acta Math.Hungar., 137, No-s:1-2(2012), 130-138. [6] Y.K. Kim, W.K. Min, “H(θ)- open sets induced by hereditary classes on generalized topological spaces”, Acta Math.Hungar.,2014; 93(3), 307-315. [7] R.D. Sarma, “On extremally disconnected generalized topologies”, Acta Math.Hungar. 2012; 134(4):583-588. [8] D. Jankovic, T.R. Hamlett, “New topologies from old via ideals”, Amer Math.Monhly, 1990(4); 295-310.
147
On Positive Solutions for Fourth-Order Four-Point Boundary Value Problems with Alternating Coefficient on Time Scales
Ilkay Yaslan Karaca Department of Mathematics, Ege University, Bornova, Izmir, Turkey
Abstract: In this paper, by using four functionals fixed point theorem and five functionals fixed point theorem, we study the existence of at least one positive solution and three positive solutions respectively of a fourth-order four point boundary value problem with alternating coefficient on a time scale. Examples are also included to illustrate our results. References: [1] D.R. Anderson, C. Zhai, “Positive solutions to semi-positone second-order three-point problems on time scales”, Appl. Math. Comput. 215 (2010), no. 10, 37133720. [2] D.R. Anderson, I.Y. Karaca, “Higher-order three-point boundary value problem on time scales”, Comput. Math. Appl. 56 (2008) 2429-2443. [3] R.I. Avery,“A generalization of the Leggett-Williams fixed point theorem”,Math. Sci.Res.Hot-Line, 2(1998)9 14. [4] R. Avery, J. Henderson, D. O’Regan, “Four functionals fixed point theorem”, Math. Comput. Modelling 48 (2008)1081-1089. [5] M. Bohner, A. Peterson,“Dynamic Equations on time scales, An Introduction with Appli cations”, Birkhauser, 2001. [6] M. Bohner, A. Peterson, “Advances in Dynamic Equations on Time Scales”, Birkhauser, Boston, 2003. [7] Z. Fang, C. Li, C. Bai, “Multiple positive solutions of fourth-order four-point boundary value problems with changing sign coefficient”, Electron. J. Differential Equations 2008 (2008) no.159 1-10. [8] L.G. Hu, X.F. Zhou, “Positive solutions of the singular eigenvalue problem for a higher-order differential equation on time scales”, Math. Comput. Modelling 53 (2011), no. 5-6, 667677. [9] Y. Liu, H. Shi, X. Liu, “Multiple positive solutions of time scale boundary value problems on infinite intervals (II)”, Math. Sci. Res. J. 14 (2010), no. 12, 271292. [10] H. Pang, W. Ge, “Existence results for some fourth-order multi-point boundary value problem”, Math. Comput. Modelling 49 (2009), no. 7-8, 13191325. [11] T.T. Sun, L.L. Wang, Y.H. Fan, “Existence and positive solutions to a nonfocal boundary value problem with p.. laplacian on time scales”, Adv. Difference Equ. 2010 Art. ID 809497, 15 pp. [12] J. Zhao, L. Wang, W. Ge, “Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems”, Nonlinear Anal. 72 (2010), no. 2, 822835.
148
A New Difference Sequence Set of Order 𝛼𝛼 and Its Geometrical Properties Vatan Karakaya, Yunus Atalan and Mikail Et
Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey
Department of Mathematics, Yildiz Technical University, Istanbul, Turkey Department of Mathematics, Firat University, Elazıg, Turkey
Abstract: In this presentation we introduce a new class of sequences named as 𝑚𝑚𝛼𝛼 (Δ𝑟𝑟, 𝜙𝜙, 𝑝𝑝) and, for this space, we study some inclusion relations, topological properties, and geometrical properties such as order continuous, the Fatou property, and the Banach-Saks property of type p. Keywords: Sequence spaces; Order continuous; Banach-Saks property References: [1] V. Karakaya and F. Altun, “On some geometric properties of a new paranormed sequence space,” Journal of Function Spaces, vol. 2014, Article ID 685382, 8 pages, 2014. [2] H. Hudzik, V. Karakaya, and M. Mursaleen, “Banach-Saks type and GGurarii modulus of convexity of some Banach sequence spaces,” Abstract and Applied Analysis, vol. 2014, Article ID 427382, 9 pages, 2014. [3]M. Et, M. Karakas V. Karakaya, “Some geometric properties of a new difference sequence space defined by de la Vall´ee Poussinmean,” Applied Mathematics and Computation, vol. 234, pp. 237–244, 2014.
149
Numerical Solution of Non-Linear Equations via Multiplicative Calculus Tolgay Karanfiller
Department of Education, Cyprus International University, Lefkosa [email protected]
Abstract: Recent studies related to the multiplicative numerical methods demonstrate their effectiveness in numerical approximations. Since numerical solution of non-linear equations arise widely in science and engineering, multiplicative based numerical solution of nonlinear functions can be used with more accurate results in many applications. Thus, in this study multiplicative numerical methods for solution of nonlinear equations are derived and some numerical results are compared with some well-known ordinary methods. Keywords: Non-linear equations; Multiplicative calculus; Root-finding methods. References: [1] D. Aniszewska, “Multiplicative Runge-Kutta method”, Nonlinear Dynamics 50 (1-2)(2007) 265-272. [2] A. E. Bashirov, Emine Misirli, Yucel Tandogdu, Ali Ozyapici. "On modeling with multiplicative differential equations", Applied Mathematics- A Journal of Chinese Universities, Volume 26, Number 4, (2011)pages 425-428, [3] A. Bashirov, E. Kurp¬nar, A. Özyap¬c¬, “Multiplicative calculus and its applications”, Journal of Mathematical Analysis and Its Applications 337 (1)(2008) 36-48. [5] F. Córdova-Lepe, “The multiplicative derivative as a measure of elasticity in economics”, TEMAT-Theaeteto Atheniensi Mathematica 2(3) (2006), online. [6] J. Englehardt, J. Swartout, C. Loewenstine, “A new theoretical discrete growth distribution with veri.cation for microbial counts in water”, Risk Analysis 29 (6) (2009) 841.856. [9] M. Grossman, R. Katz, R., “Non-Newtonian Calculus”, Pigeon Cove, Lee (1972)Press, Massachusats,. [10] M. Grossman, “Bigeometric Calculus: A System with a Scale-Free Derivative, Archimedes Foundation”, Rockport, (1983) Massachusats. [11] W. Kasprzak, B. Lysik, M. Rybaczuk, M., “Dimensions, Invariants Models and Fractals”, Ukrainian Society on Fracture Mechanics, SPOLOM, (2004) Wroclaw-Lviv, Poland,. [12] E. Misirli, Yusuf Gurefe. "Multiplicative Adams Bashforth Moulton methods", Numerical algorithms, DOI: 10.1007/s11075-010-9437-2, Volume 57, Number 4, (2011) pages 425 – 439. [14] M. Pakdemirli, H.Boyaci, H.A. Yurtsever, "Perturbative Derivation and Comparisons of Root-Finding Algorithms with Fourth Order Derivatives",Mathematical and Computational Applicaitons,Volume 12, Number 2, (2007)pages 117-124,. [15] A. Ozyapici, M.Riza, B.Bilgehan,A.Bashirov, “On multiplicative minimization methods”, Numerical Algorithms,online .rst, DOI 10.1007/s11075-013-9813-9. [16] A. Ozyapici, E. Misirli, “Exponential Approximations on Multiplicative Calculus”, PJMS (Proceedings of the Jangjeon Mathematical Society), vol:12, No. 2, (2009)227-236. [19] M. Riza, A. Ozyapici, E. Kurp¬nar, “Multiplicative .nite di¤erence methods”, Quarterly of Applied Mathematics, Vol: 67, No: 4, (2009)pages: 745-754. [20] D. Stanley, “A multiplicative calculus”, Primus Vol: IX, No:4, (1999) pages 310.326.
150
Notes on Banach Contraction Principle Erdal Karapinar
Department of Mathematics, Atilim University, Incek, Ankara, Turkey [email protected]
Abstract: In this talk, we aim to discuss the wondrously improvement in fixed point theory, in particular Banach contraction mapping by giving some examples from the literature. Keywords: Fixed Point References: [1] R. P. Agarwal, H. Alsulami, E.Karapinar and F.Khojasteh, Remarks on some recent fixed point results in quaternion-valued metric spaces, Abstract and Applied Analysis, (2014) Article Id: 171624. [2] E.Karapinar, Discussion on Contractions on Generalized Metric Spaces, Abstract and Applied Analysis Volume 2014 (2014), Article ID 962784, 7 pages [3] E. Karapinar and B.Samet, A note on psi-Geraghty type contractions, Fixed Point Theory and Applications, (2014), 2014:26. [4] E. Karapinar and R.P. Agarwal, A note on "Coupled fixed point theorems for alpha-psi contractive type mappings in partially ordered metric spaces", Fixed Point Theory and Applications, (2013) 2013:216. [5] B. Samet, E. Karapinar, H. Aydi and V. C. Rajic, Discussion on some coupled fixed point theorems, Fixed Point Theory and Applications, 2013, 2013:50.
151
Gröbner-Shirshov Bases of Some Exceptional Braid Groups Eylem G. Karpuz, Nurten Urlu and A. Sinan Cevik
Department of Mathematics, Karamanoglu Mehmetbey University, Karaman, Turkey Department of Mathematics, Selcuk University, Konya, Turkey
[email protected], [email protected] , [email protected] Abstract: The Gröbner-Shirshov basis theory was developed by A. I. Shirshov for Lie algebras [1] and B. Buchberger for commutative algebras [2]. It was also generalized by G. M. Bergman and L. A. Bokut to the case of associative algebras [3]. This theory is very useful in the study of presentations of associative algebras, Lie algebras, semigroups and groups by generators and defining relations. It is also a powerful tool to solve many problems; normal form, word problem, embedding theorems, etc. In this paper, we study exceptional braid groups and obtain Gröbner-Shirshov bases of these groups. Keywords: Gröbner-Shirshov basis, exceptional braid group. References: [1] A. I. Shirshov, “Some algorithmic problems for Lie algebras”, Siberian Math. J. 3(1962), 292-296. [2] B. Buchberger, “An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal”, Ph.D. Thesis, University of Innsbruck, 1965. [3] L. A. Bokut, “Imbedding into simple associative algebras”, Algebra Logic 15(1976), 117-142.
152
Scalarization Methods in Multiobjective Optimization Refail Kasimbeyli
Department of Industrial Engineering, Anadolu University, Iki Eylul Campus, Eskişehir, Turkey [email protected]
Abstract: In general, scalarization means the replacement of a multi-objective optimization problem by a suitable scalar optimization problem which is an optimization problem with a real valued objectivefunctional. Since the scalar optimization theory is widely developed, scalarization turns out to be of great importance for the multi-objective optimization theory. One way of constructing a single-objective optimization problem is through scalarizing functions involving possibly some parameters or additional constraints. Multi-objective optimization methods utilize different scalarizing functions in different ways. This paper presents the conic scalarization method for nonlinear multi-objective optimization problems. We introduce a special class of monotonically increasing sublinear scalarizing functions and show that the zero sublevel set of every function from this class is a convex closed and pointed cone which contains the negative ordering cone. It is shown that the scalar optimization problem constructed by using these functions enables to characterize the complete set of efficient and properly efficient solutions of multi-objective optimization problems without convexity and boundedness conditions. Keywords: Separation theorem, Multiobjective optimization, Conic scalarization method.
153
Some Convergence Theorems of Nonlinear Integral Functionals Jun Kawabe
Faculty of Engineering, Shinshu University, Wakasato, Nagano, Japan [email protected]
Abstract: In nonadditive measure theory, several types of nonlinear integrals have been proposed. Among them, the Choquet integral [1], the Sugeno integral [2], and the Shilkret integral [3] are typical and widely used in theory as well as its practical applications. The Lebesgue integral may be viewed as a linear functional, based on the Lebesgue measure, on the space of measurable functions. By contrast, the Choquet, the Sugeno, and the Shilkret integrals are nonlinear functionals since they are based on nonadditive measures. Therefore, in order to discuss some convergence theorems regardless of the type of such integrals and develop a unifying approach to the proof, it is reasonable to formulate them for a general nonlinear integral functional. In this formulation, a key concept is the perturbation of functional, which manages the small change of the functional value by adding a small term to a measurable function as well as to a measure in the domain of the functional. In this paper, by using the perturbation of functional, we formulate some fundamental convergence theorems of a nonlinear functional, which yield those of the Choquet, the Sugeno, and the Shilkret integrals in nonadditive measure theory besides the Lebesgue integral. Keywords: Nonadditive measure; Nonlinear integral functional; Choquet integral; Sugeno integral; Shilkret integral; Perturbation of functional. References: [1] G. Choquet, “Theory of capacities”, Ann. Inst. Fourier (Grenoble) 5 (1953-1954), 131-295. [2] M. Sugeno, “Theory of fuzzy integrals and its applications”, Ph.D. Dissertation, Tokyo Institute of Technology, Tokyo, 1974. [3] N. Shilkret, “Maxtive measure and integration”, Indag. Math. 33 (1971), 109-116.
154
Numerical and Exact Solutions for Time Fractional a Nonlinear Equation
Doğan Kaya and Asıf Yokus
Istanbul Commerce University, Department of Mathematics, ISTANBUL, TURKEY Firat University, Department of Actuary, Elazig, TURKEY
[email protected], [email protected]
Abstract: In this study, an expansion method and fractional differential operators are defined and analyzed. We have
implemented this method for finding traveling wave solutions of nonlinear time fractional Korteweg–de Vries
equation. The equation with initial condition is considered for the extended finite difference method which is based
on Caputo formula is used. This equation is discussed with an example and error estimates obtained for the Finite
difference method. The numerical methods have been applied to solve a numerical example and results are compared
with the exact solution. We also pointed out that this expansion method could implement the other nonlinear
equations. These results are presented in tables using the Mathematica software package when it is needed.
Key Words: Time Fractional Korteweg–de Vries Equation; An Expansion Method; Finite Difference Method;
Caputo Formula; Fractional Partial Differential Equation; Numerical Solutions; Exact Solutions.
References
[1] M. Wang, X. Li and J. Zhang, The ( )GG¢ -expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372 (2008) 417-423.
[2] S. Guo and Y. Zhou, The extended ( )GG¢ -expansion method and its applications to the Whitham–Broer–Kaup–Like equations and coupled Hirota–Satsuma KdV equations. Appl. Math. Comput. 215 (2010) 3214–3221.
[3] A. Yokus, Solutions of Some Nonlinear Partial Differential Equations and Comparison of Their Solutions, Ph.D. Thesis, Firat University, 2011.
[4] P.A. Clarkson, New Similarity Solutions for the Modified Boussinesq Equation. J. Phys. A: Gen. 22 (1989) 2355-2367.
[5] A. Wazwaz, The tanh method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd–Bullough equations, Chaos, Solitons & Fractals 25 (2005) 55-63.
155
Common Fixed Point Results for (Ϝ,ψ)-Contractions in Ordered Partial Metric Spaces
Meltem Kaya, Mahpeyker Ozturk and Hasan Furkan Department of Mathematics, Kahramanmaras Sutcu Imam University, Kahramanmaras, Turkey
Department of Mathematics, Sakarya University, Sakarya, Turkey [email protected], [email protected], [email protected],
Abstract: The purpose of this article is to present some fixed and common fixed point theorems for nondecreasing and weakly increasing mappings satisfying certain (Ϝ,ψ)-contraction conditions in partial ordered complete partial metric space. Furthermore, we give necessary and sufficient conditions for uniqueness of common fixed point. Keywords: Partial ordered complete partial metric space; Fixed point; Common fixed point. References: [1] S. G. Matthews, “Partial metric topology”, Proc. 8th Summer Conference on General Topology and Applications, Ann New York Acad Sci 728, 183-197, (1994). [2] S. Oltra, O. Valero, “Banach' s fixed point theorem for partial metric spaces”, Rendiconti dell' Istituto di Matematica dell'Universita di Trieste, vol. 36, no. 1-2, pp. 17-26, (2004). [3] S. Romeguera, “Matkowski' s type theorems for generalized contractions on (ordered) partial metric spaces”, Applied General Topology, Universidad Politecnica de Valencia, Volume 12, no. 2, 2011, pp. 213-220. [4] E. Karapınar, W. Shatanawi, “On Weakly C, ψ, ϕ-Contractive Mappings in Ordered Partial Metric Spaces”, Hindawi Publishing Corporation Abstract and Applied Analysis, Volume 2012, Article ID 495892, 17 pages, doi:10.1155/2012/495892 [5] H. Aydi, “Some Fixed Point Results In Ordered Partial Metric Spaces”, J. Nonlinear Sci. Appl. 4 (2011), no. 3, 210-217.
156
Inverse Nodal Problem for p-Laplacian Sturm Liouville Equation H. Kemaloglu(Koyunbakan), E. Yilmaz
Department of Mathematics, Firat University, Elazıg, Turkey [email protected], [email protected]
Abstract: In this study, inverse nodal problem is solved for p-Laplacian Sturm-Liouville problem with the Drichlet conditions. Asymptotic form of Eigenvalues, nodal points and especially, an explicit formula are given for the problem. First results on inverse nodal problem for Classical Sturm-Liouville problem were given in [1], [2]. Keywords: Inverse problem, Sturm-Liouville Equation, Nodal point References: [1] Law, C.K., Lian, W.C. Wang, W.C. “Inverse nodal problem and Ambarzumyan theorem for the p-Laplacian”, Proc. R. Soc. Edinb. A 139, 1261-1273 (2009). [2] Wang, W.C., Cheng, Y.H. and Lian, W.C., Inverse nodal problem for the p-Laplacian with eigenparameter dependent boundary conditions, Math. Comp. Model., 54, 2718-2724 (2011)
157
On the Uniform Convergence of the Fourier Series for One Spectral Problem with a
Spectral Parameter in a Boundary Condition Nazim B. Kerimov and Emir Ali Maris
Department of Mathematics, Mersin University, Merkez, Mersin, Turkey [email protected], [email protected]
Abstract: Consider the spectral problem −𝑦𝑦′′ + 𝑞𝑞(𝑥𝑥)𝑦𝑦 = 𝜆𝜆𝑦𝑦, 0 < 𝑥𝑥 < 1, (1)
𝑏𝑏0𝑦𝑦(0) = 𝑑𝑑0𝑦𝑦′(0), (𝑎𝑎1𝜆𝜆 + 𝑏𝑏1)𝑦𝑦(1) = (𝑐𝑐1𝜆𝜆 + 𝑑𝑑1)𝑦𝑦′(1) (2) where 𝜆𝜆 is a spectral parameter, 𝑞𝑞(𝑥𝑥) is a real-valued continuous function on the interval [0,1] and 𝑎𝑎1, 𝑏𝑏0, 𝑏𝑏1, 𝑐𝑐1, 𝑑𝑑0, 𝑑𝑑1 are real constants which satisfy the following conditions:
|𝑏𝑏0| + |𝑑𝑑0| ≠ 0, 𝜎𝜎 = 𝑎𝑎1𝑑𝑑1 − 𝑏𝑏1𝑐𝑐1 > 0. In this paper, we study the uniform convergence of the expansions for the functions which belong to 𝐶𝐶[0,1] in terms of eigenfunctions of the boundary value problem (1)-(2).The similar problems are investigated, for example, in [1]-[3]. Keywords: Asymptotic formula; Minimality and basicity; Uniform convergence. References: [1] N. Yu. Kapustin, E. I. Moiseev, “ Convergence of spectral expansions for functions of the Hölder Class for two problems with a spectral parameter in the boundary condition”, Differential Equations, 36(2000), 1182-1188. [2] N. Yu. Kapustin, E. I. Moiseev, “A remark on the convergence problem for spectral expansions corresponding to a classical problem with spectral parameter in the boundary condition”, Differential Equations, 37(2001), 1677-1683. [3] D. B. Marchenkov, “On the convergence of spectral expansions of functions for problems with a spectral parameter in a boundary condition”, Differential Equations, 41(2005), 1496-1500.
158
Benjamin-Bona-Mahony Equation with Variable Coefficients: Conservation Laws Chaudry Masood Khalique
International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Republic of South Africa
Abstract: In this talk we present conservation laws for Benjamin-Bona-Mahony equation with variable coefficients. The Benjamin-Bona-Mahony equation is also known as the regularized long-wave equation and is applicable to shallow water waves and to the study of drift waves in plasma or the Rossby waves in rotating fluids. Keywords: Benjamin-Bona-Mahony equation; Conservation laws; Exact solutions.
159
Excess of Retro Banach Frames
G.Khattar and L.K.Vashisht G. Khattar, Department of Mathematics, University of Delhi, Delhi-110007, India
L. K. Vashisht, Department of Mathematics, University of Delhi, Delhi-110007, India [email protected], [email protected]
Abstract: The excess of a frame is greatest integer n such that n elements can be deleted from the frame and still leave a complete set, or if there is no upper bound to the number of elements that can be removed. In this paper, we discuss excess of retro Banach frame for Banach spaces. A characterization of finite excess of retro Banach frames in Banach spaces is given. A construction of a retro Banach frame from the given retro Banach Frame with Infinite excess is given. Keywords: Frames; Banach frames; Reconstruction property; Perturbation. References: [1] R.Balan, P.G.Cassaza, C.Heil and Z. Landau, “Deficits and excesses of frames”, Adv. Comp. Math., 18 (2003), 93-116. [2] P.G.Cassaza and G.Kutynoik, Finite Frames, “Birkhauser”, 2012. [3] I .Daubechies, A. Grossman and Y. Meyer, “Painless non orthogonal expansions”, J. Math. Phys, 27 (1986), 1271-1283. [4] R.J.Duffin and A.C. Schaffer, “A class of non harmonic Fourier series”, Trans. Amer. Math. Soc., 72 (1952), 341-366 [5] D.Gabor, “Theory of communications”, J. Inst. Elec. Engg., 93 (1946), 429-457.
160
On Binomial Transformations of a Product of Rising and Falling Factorials Emrah Kilic and Halit Ozturk
TOBB University of Economics and Technology, Mathematics Department, Ankara, Turkey [email protected], [email protected]
Abstract: In this paper, we give some new results including binomial transformations of a product of rising and falling factorials and their alternating analogous. Keywords: Binomial transform; Rising factorial; Falling factorials; Sums; Recursion. References: [1]R.L.Graham, D.E. Knuth,O.Patashnik,“Concrete Mathematics”,Second ed.,Addison-Wesley, Reading, MA, 1994. [2]H. P. Haukkanen, “Formal Power Series for Binomial Sums of Sequences of Numbers”, The Fibonacci Quarterly 31.1 (1993), 28--31. [3] P. H. Prodinger, “Some information about the binomial transform”,The Fibonacci Quarterly 32.5(1994), 412-415. [4] SP. M. Spivey, “Combinatorial sums and finite differences”, Discrete Mathematics 307 (2007), 3130-3146.
161
An examination on the positions of Frenet ruled surfaces along the evolute-involute curves,
according to their normal vector fields in Euclidean 3-space. Şeyda Kılıçoğlu
Baskent University, Ankara, Turkey [email protected]
Abstract: Deriving curves based on the other curves is a subject in geometry. Involute-evolute curves, Bertrand curves are this kind of curves. First by using the similiar method we produce a new ruled surface, called “Involutive B-scroll”, based on the other ruled surface, in [2]. Also D-scroll , which is known as the rectifying developable surface in [1], of any curve α and the involute D-scroll of the curve α are already defined , in Euclidean 3-space in [3]. In this paper we consider four special four Frenet ruled surfaces along to the involute-evolute curves, α* and α respectively, with curvature k₁≠0 ,together. We find the excplit equation of Frenet ruled surfaces along the involute curve in terms of the Frenet apparatus of evolute curve α. Further normal vector fields of these Frenet ruled surfaces be calculated too. In the and the positions of Normal vector fields of four Frenet ruled surfaces be examined, in terms of Frenet apparatus of evolute curve α. Keywords: Involute-evolute curves, ruled surface, Darboux vector.
References: [1] S. Izumiya, N.Takeuchi, “Special curves and Ruled surfaces “. Beitr¨age zur Algebra und Geometrie Contributions to Algebra and Geometry, Volume 44 (2003), No. 1, 203-212. [2] W.K. Schief, “On the integrability of Bertrand curves and Razzaboni surfaces”. Journal of Geometry and Physics, Volume 45, Issues 1--2, (2003), Pages 130-150, February.. [3] Ş. Kiliçoğlu, “On the Involutive B-scrolls in the Euclidean Three-space E 3”. XIIIth. Geometry Integrability and Quantization, Varna, Bulgaria: Sofia , (2012), pp 205-214,
162
Iterative Schemes and Common Fixed Point Problems on a Complete Geodesic Space Yasunori Kimura
Department of Information Science, Toho University, Miyama, Funabashi, Japan [email protected]
Abstract: A common fixed point problem for a family of nonexpansive mappings is one of the most important problems in nonlinear analysis and we may apply it to many other important problems such as convex minimization problems, variational inequality problems, equilibrium problems, minimax problems, and others. This problem has been investigated by a large number of researchers as a generalization of fixed point problem for a single mapping. In this talk, we propose several types of iterative methods to approximate a com mon fixed point of mappings defined on a complete geodesic space with curvature bounded above. When generating an iterative sequence, we consider calculation errors and observe its convergence property.We also show some recent development related to this topic. Keywords: Geodesic space; Nonexpansive mapping; Shrinking projection method,Iiterative scheme; Metric projection; Calculation error. References: [1] M. R. Bridson, A. Haefliger, “Metric spaces of non-positive curvature”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999.
163
Weighted Lacunary Statistical Convergence in a Locally Convex Topological Vector Space Sukran Konca and Metin Basarir
Department of Mathematics, Bitlis Eren University, Bitlis, Turkey, Department of Mathematics, Sakarya University, Sakarya, Turkey.
[email protected] , [email protected]
Abstract. In this paper, we extend the concept of weighted lacunary statistical convergence to Hausdorff locally convex topological vector spaces and present some inclusion theorems. Keywords: Weighted Lacunary Statistical Convergence; Sequence Space; Locally Convex Topological Vector Space; Hausdorff Spaces. References: [1] H. Albayrak, S. Pehlivan, “Statistical convergence and statistical continuity on locally solid Riesz spaces”, Topology Appl. 159 (2012), 1887-1893. [2] M. Basarir, S. Konca, “Weighted lacunary statistical convergence in locally solid Riesz spaces”, Filomat. (2014) (accepted). [3] M. Basarir, S. Konca, “On some spaces of lacunary convergent sequences derived by Norlund-type mean and weighted lacunary statistical convergence”, Arab J. Math. Sci., 20 (2), (2014), 250263, DOI: 10.1016/j.ajmsc.2013.09.002. [4] H. Fast, “Statistical convergence”, Colloq. Math. 2 (1951), 241-244. [5] J. A. Fridy, C. Orhan,(1993), “Lacunary statistical convergence”, Paci_c J. Math. vol. 160, 43-51. [6] V. Karakaya, T. A. Chishti, “Weighted statistical convergence”, Iran. J. Sci. Technol. Trans. A Sci.,33:A3 (2009), 219-223.
164
Numerical Solution for a Telegraph Equation via Bernestein Polynomials A. Kouadri and D. Belakroum
Signals and Systems Laboratory, Institute of Electrical and Electronic Engineering, University of Boumerdes, Av. de l’independence, 35000-Algeria.
University of Constantine 1, 25000-Algeria, [email protected], [email protected]
Abstract : In this work, a numerical solution of a telegraph partial derivative equation throughout Bernestein polynomials is proposed. The telegraph equation, describes wave propagation of electric signals in a cable transmission line, is a one dimensional linear time-invariant hyperbolic partial derivative equation. We adopt Bernestein polynomials to solve the system of equations defined on the region of an unit cable transmission line length and a given time window of size T, . These partial derivative equations with respect to the variable x are derived from the application of Rothe’s time discretisation scheme for a telegraph equation. The system of one dimensional partial derivative equations is transformed in a large sized diagonal matrix which can be viewed as the system of linear equations after dispersing the variable. Each element of the large sized diagonal matrix represents a Bernestein operational matrix at a given sample time. A numerical solution, by solving the linear system of algebraic equations, is obtained and its accuracy and complexity are discussed. Numerical experimentations are conducted to demonstrate the viability of Bernestain polynomials technique. It is also shown the impact of Bernestein polynomials’ orders and relevant sample points on the efficiency and accuracy of the proposed method.
165
Probability Theory on a Topological Monoid of Strings and Its Application to Statistical Machine Learning
Hitoshi Koyano and Morihiro Hayashida Laboratory of Biostatistics and Bioinformatics, Graduate School of Medicine,
Laboratory of Mathematical Bioinformatics, Institute for Chemical Research, Kyoto University, Kyoto, Japan
Abstract: Mathematicians have conducted detailed examinations of a large number of objects throughout the long history of mathematics, but they have not studied strings in detail. A string is an object that computer scientists have addressed in depth. Stringology, a field of computer science, has thoroughly investigated algorithms and data structures for string processing. However, computer scientists have not studied strings using a mathematical approach; for example, functions, operators, and probabilities on a set of strings provided with topological and algebraic structures have not been investigated. Numbers and numerical vectors account for a large portion of data. However, recently, the amount of string data, such as text data on the Web and biological sequences in life sciences, has increased dramatically. Consequently, a random string that randomly generates strings based on a probability law appears to become necessary for string data analysis, as a random variable that randomly generates numbers and a stochastic process that randomly generates functions are essential in various fields. Statistical methods for numerical data were rigorously con structed based on probability theory. Similarly, the development and systematization of methods on the basis of probability theory on a set of strings will be required for text mining techniques and methods for analyzing biological sequences. From the viewpoint of mathematics, a set A∗ of strings on an alphabet A is a monoid with concatenation and a metric space with the Levenshtein distance and forms a noncommutative topological monoid. In this presentation, we will first talk about the attempt to develop probability theory on A∗ to treat randomly generated strings and then the attempt to construct the theory of statistical machine learning to classify string data based on this probability theory [1, 2, 3]. Keywords: Strings; Probability theory; Statistical machine learning; Bioinformatics. References: [1] H. Koyano, H. Kishino, Phys. Rev. E, 81(6), 061912, 2010. [2] H. Koyano, T. Tsubouchi, H. Kishino, T. Akutsu, J. Geophys. Res. Biogeosciences, 2014. [3] H. Koyano, M. Hayashida, T. Akutsu, arXiv:1406.0597 [stat.ML].
166
On generalized absolute Cesàro summability of orthogonal series Xhevat Z. Krasniqi
Faculty of Education, Department of Mathematics and Informatics University of Prishtina “Hasan Prishtina”, Avenue “Mother Theresa”5
10000 Prishtina, Republic of Kosovo [email protected]
Abstract: In this paper we have introduced a new definition of generalized absolute Cesàro summability. Employing that definition we give some sufficient conditions in terms of the coefficients of an orthogonal series under which such series is generalized absolute Cesàro summable almost everywhere.
Keywords: Orthogonal series, absolute Cesàro summability, Fourier series.
References:
[1] M. Tanaka, On generalized Nörlund methods of summability, Bull. Austral. Math. Soc., vol. 19, (1978), 381-402. [2] Xh. Z. Krasniqi, An application of q-mathematics on absolute summability of orthogonal series, Analysis (Munich) 33, 35-43 (2013)/ DOI 10.1524/anly.2013.1169. [3] Xh. Z. Krasniqi, On |A,δ|k-summability of orthogonal series, Math. Bohem., 137, (2012), 17-25. [4] Xh. Z. Krasniqi, On absolute weighted mean summability of orthogonal series, Selçuk J. Appl. Math., 12, (2011), 63-70. [5] Xh. Z. Krasniqi, On φ-|C,1| summability of orthogonal series, Bull. Allahabad Math. Soc., 12, (2011), 63-70. [6] I. P. Natanson, Theory of functions of a real variable, Frederick Ungar New York 1955, 1961, MR 16-804, 26 # 6309. [7] Y. Okuyama, On the absolute generalized Nörlund summability of orthogonal series, Tamkang J. Math., 3, (2002), 161-165. [8] Y. Okuyama and T. Tsuchikura, On the absolute Riesz summability of orthogonal series, Anal. Math., 7, (1981), 199-208. [9] P. L. Ul'yanov, Solved and unsolved problem in the theory of trigonometric and orthogonal series, Uspekhi Mat. Nauk, 19, (1964), 3-69. [10] H. Seyhan, Ph. D. Thesis, Erciyes, Kayseri, 1995. [11] G. H. Hardy, Divergent series, first edition, Oxford
167
q-Barnes Multiple Bernoulli Polynomials Veli Kurt
Department of Mathematics, Akdeniz University, Antalya, Turkey [email protected]
Abstract: In this paper, we define q- Barnes Multiple Bernoulli Polynomials B_n(x,y;q∣a₁,...,a_r) and q- Barnes Multiple Euler Polynomials E_n(x,y;q∣a₁,...,a_r) . We give some identities for these polynomials. We also prove some recurrence relation between for these polynomials. Furthermore, we give some relations between for these polynomials and two variable Bernoulli polynomials B_n,q(x,y) and two variables Euler polynomials E_n,q(x,y). Keywords: Bernoulli polynomials, Euler polynomials, Barnes Multiple Bernoulli polynomials, q-Barnes Multiple Bernoulli polynomials. References: [1] L. Jang and T. Kim, q-analogue of Euler-Barnes' numbers and polynomials, Bull. Korean Math. Soc. 42 (2005), 491--499. [2] L.C. Jang, T.Kim, Y.H.Kim and K.W.Hwang, Note on the q-extension of Barnes's type multiple Euler polynomials, J. of Inequalties and Applications, 2009(2009), 1-7. [3] T. Kim, On Euler-Barnes Multiple Zeta Function, Russ. J. Math. Phys. 10 no. 3 (2003), 261-267. [4] D. S. Kim, T. Kim, T. Komatsu, Barnes' multiple Bernoulli and poly-Bernoulli mixed-type polynomials. arxiv. 1312. 717. 2013. [5] D. S. Kim, T. Kim, T. Komatsu, H.I. Kwon, Barnes' multiple Bernoulli and generalized Barnes' multiple Frobenius-Euler mixed-type polynomials, Advances in Difference Equations 2014, 2014:238.
168
Multi-Spectral Image Classification by Nonparametric Regression Splines Semih Kuter, Zuhal Akyurek and Gerhard-Wilhelm Weber
Cankiri Karatekin University, Faculty of Forestry, Department of Forest Engineering, Cankiri, Turkey Middle East Technical University, Faculty of Engineering, Department of Civil Engineering, Ankara, Turkey
Middle East Technical University (METU), Institute of Applied and Natural Sciences, Department of Geodetic and Geographic Information Technologies, Ankara, Turkey
Middle East Technical University, Institute of Applied Mathematics, Ankara, Turkey
Abstract: Multi-spectral image classification is one of the most active research areas in remote sensing [1, 2]. In this talk, we represent the application of Multivariate Adaptive Regression Splines (MARS) [3] for the classification of Moderate Resolution Imaging Spectroradiometer (MODIS) [4] images. MARS models with different model building settings are generated and applied on predefined test areas of MODIS images. The performance of the models is assessed by using error matrices [5]. The results are also compared with those obtained by the traditional maximum likelihood classification approach. Keywords: Multivariate Adaptive Regression Splines; Image Classification; Remote Sensing. References: 1. D. Lu, Q. Weng, “A survey of image classification methods and techniques for improving classification Performance”, International Journal of Remote Sensing, 2007. 28: p. 823-870. 2. B. Tso, P.M. Mather, “Classification Methods for Remotely Sensed Data”, 2nd ed. 2009: CRC Press. 3. J. H. Friedman, “Multivariate adaptive regression splines”, The Annals of Statistics, 1991. 19: p. 1-67. 4. J. J. Qu, W. Gao, M. Kafatos, R.E. Murphy, V.V. Salomonson, “Earth science satellite remote sensing”, Volume 1: Science and Instruments. 2006, Beijing: Springer. 5. R.G. Congalton, “A Review of Assessing the Accuracy of Classifications of Remotely Sensed Data”, Remote Sensing of Environment, 1991. 37: p. 35-46.
169
Controlling of Period-2 Unstable Orbits in a Bouncer System Using Patterns of H-Ranks Mantas Landauskas and Minvydas Ragulskis
Research Group for Mathematical and Numerical Analysisof Dynamical Systems, Kaunas University of Technology, Kaunas, Lithuania
[email protected], [email protected]
Abstract: In this paper the unstable period-2 orbit of a bouncer system is explored. The use of techniques based on the patterns of H-ranks is employed to keep these orbits temporary stable i.e. to keep the iterative process bounded in the certain location. New results shows some important facts to be considered whenever controlling the dynamical systems. Keywords: Bouncer system; Rank; Unstable Manifold; Basin of Attraction. References: [1] M. Landauskas, M. Ragulskis, “A pseudo-stable structure in a completely invertible bouncer system”, Nonlinear Dyn DOI 10.1007/s11071-014-1546-3 (accepted, in press). [2] A. Hutt, “Additive noise may change the stability of nonlinear systems”, EPL 84.3 (2008), 34,003-34,008. [3] E. Fermi, “On the Origin of the Cosmic Radiation”, Phys. Rev. 75.8 (1949), 1169-1174. [4] Y. Guo, A.C.J. Luo “Analytical predication of complex motion of a ball in a periodically shaken horizontal impact pair”, J. Comput. Nonlinear Dyn. 7.2 (2011), 021,001–021,009. [5] S.K. Joseph, M.A.F. Sanjuán, “Effect of the phase on the dynamics of a perturbed bouncing ball system”, Commun. Nonlinear Sci. Numer. Simulat. 17.8 (2012), 3279-3286.
170
Approximation by q-Balazs-Szabados Operators Nazim Mahmudov
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, T.R. North Cyprus, via Mersin 10, Turkey
Abstract: This paper deals with approximating properties of the newly defined q-generalization of the Balazs-Szabados operators in the case q>1. Quantitative estimates of the convergence and the Voronovskaja's theorem are given. In particular, it is proved that the rate of approximation by the q- Balazs-Szabados operators (q>1) is of order q⁻ⁿ versus 1/n for the classical Balazs-Szabados operators. Keywords: Complex Approximation, q-polynomials, Rate of Convergence. References: [1] Balazs, K. (1975) Approximation by Bernstein type rational functions, Acta Math. Acad. Sci. Hungar., 26, 123-134. [2] Balazs, K. and Szabados, J. (1982) Approximation by Bernstein type rational functions, II, Acta Math. Acad. Sci. Hungar., 40(3-4)331-337. [3] Abel, U. and Della Vecchia B. (2000) Asymptotic approximation by the operators of K. Bal azs and Szabados, Acta Sci. Math.(Szeged), 66, No. 1-2, 137-145. [4] N. Ispir and Y. Özkan, Approximation properties of complex q-Balázs-Szabados operators in compact disks, Journal of Inequalities and Applications 2013, 2013:361
171
Existence of Solutions of Fractional Boundary Value Problems with p-Laplacian Operator Nazım I.Mahmudov and a Sinem Unul
Department of Mathematics, Eastern Mediterranean University, Famagusta, Mersin, Turkey
[email protected] [email protected]
Abstract: In this talk, existence of solutions of fractional boundary value problems with p-Laplacian operator will be
discussed:
( ( )) ( , ( ))0 0
(0) (1)1 1
' '(0) (1)1 2
(0) 00
(1) ( )0 0
D D u t f t u tP
u u
u u
D u
D u D u
b af
m s
m s
a
a an h
=+ +
+ =
+ =
=+
=+ +
with 1 2; 1 2a b< £ < £
Key words: Differential equations, p-Laplacian operator, existence and uniqueness.
References:
[1] Jinhua Wang, Hongjun Xiang; “Upper and Lower solutions Method Fo a Class of Singular Fractional Boundary
Value Problems with P-Laplacian Operator” Abstract and Applied Analysis Vol.2010 ID 971824
[2] Xiping Liu, Mei Jia, Xiufen Xiang; “On the Solvability of a Fractional Differential Equation Model Involving
The p-Laplacian Operator”Computers and Mathematics with Applications 64(2012)3267-3275
[3] Bashir Ahmad; "Nonlinear Fractional Differential Equations with Anti-Periodic Type Fractional Boundary
Conditions" Differ Equ Dyn Syst (October 2013)21(4).387-401
172
Some Paranormed Difference Sequence Spaces of Order m-Derived by Generalized Means and Compact Operators
Amit Maji and P. D. Srivastava Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, India
Abstract: Kizmaz [1] first introduced and studied the difference sequence spaces using difference operator. In 2011, Polat et al. [2] introduced new sequence spaces combining both the weighted mean and the difference operator. Subsequently, a lot of developments have been made in this direction due to Başarir and Kara (see [3], [4]). Mursaleen and Noman [5] also developed a sequence space of generalized means. Quite recently, Maji et al. [6] introduced and studied mth-order difference sequence space by using generalized means. In this paper, we have introduced a paranormed sequence space l(r, s, t, p ; ∆(𝑚𝑚)) combining by using generalized means and difference operator of order m. We have shown that the space l(r, s, t, p ;∆(𝑚𝑚)) is complete under some suitable paranorm and it has Schauder basis. Furthermore, the 𝛼𝛼-, 𝛽𝛽-, 𝛾𝛾- duals of this space is computed and also obtained necessary and sufficient conditions for some matrix transformations from l(r, s, t, p ; ∆(𝑚𝑚)) to 𝑙𝑙∞ and 𝑙𝑙1. We have also shown that the space has uniform Opial property. Finally, we have obtained some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of some matrix operators on the BK space 𝑙𝑙𝑝𝑝(r, s, t; ∆(𝑚𝑚)) by applying the Hausdorff measure of noncompactness. Keywords: Difference operator; Generalized means; Matrix transformation; Hausdorff measure of noncompactness; Compact operators. References: [1] H. Kizmaz, “Certain sequence spaces”, Canad.Math. Bull. 24(2): 169-176, 1981. [2] H. Polat, V. Karakaya, N. Simsek, “Difference sequence spaces derived by using a generalized weighted mean”, Appl. Math. Lett. 24(5): 608-614, 2011. [3] M. Basarir, E. E. Kara, “On some difference sequence spaces of weighted means and compact operators”, Ann. Funct. Anal, 2(2): 114-129, 2011 [4] M. Basarir, E. E. Kara, “On the B-difference sequence space derived by generalized weighted mean and compact operators”, J. Math. Anal. Appl. 391: 67-81, 2012. [5] M. Mursaleen, A. K. Noman, “On generalized means and some related sequence spaces”, Comput. Math. Appl. 61(4): 988-999, 2011. [6] A. Maji, A. Manna, P.D. Srivastava, “Some mth-order difference sequence spaces of generalized means and compact operators”, Ann. Funct. Anal., to appear.
173
Fuzzy and Set-Valued Stochastic Differential Equations Marek T. Malinowski
Faculty of Mathematics, University of Zielona Góra, Poland [email protected], [email protected]
Abstract: Stochastic differential equations are often used in modeling dynamic phenomena subjected to uncertainty which originates from random forces and stochastic noises. Their analysis and applications require notions of the stochastic integrals. On the other hand, in practice, one encounters uncertainties which are not of stochastic type. For example, an impreciseness of measurement equipments is not qualified as a random factor. Also some human judgments on parameters of a considered dynamic system can be vague, ambiguous or fuzzy. To handle the dynamics of systems operating in random and vague/fuzzy environment, we propose to consider fuzzy and set-valued stochastic differential equations [1]-[10]. This constitutes a quite new branch of research in modeling uncertain phenomena. We examine problem of existence and uniqueness of solutions. Some stability properties of solutions are established as well. Several examples are studied to indicate applicability of the theory of fuzzy stochastic differential equations in modeling real phenomena. Keywords: Fuzzy stochastic differential equations; Fuzzy stochastic process; Fuzzy random variable. References: [1] Y. Feng, “Fuzzy stochastic differential systems”, Fuzzy Sets Syst., 115(2000), 351-363. [2] M. T. Malinowski, “On random fuzzy differential equations”, Fuzzy Sets Syst., 160(2009), 3152-3165. [3] M. T. Malinowski, “Existence theorems for solutions to random fuzzy differential equations”, Nonlinear Anal. TMA, 73(2010),1515-1532. [4] M. T. Malinowski, “Random fuzzy differential equations under generalized Lipschitz condition”, Nonlinear Anal. Real World Appl., 13(2012), 860-881. [5] M. T. Malinowski, “Strong solutions to stochastic fuzzy differential equations of Ito type”, Math. Comput. Modelling, 55(2012), 918-928. [6] M. T. Malinowski, “Ito type stochastic fuzzy differential equations with delay”, Systems Control Lett., 61(2012), 692-701. [7] M. T. Malinowski, “Some properties of strong solutions to stochastic fuzzy differential equations”, Inform. Sci., 252(2013), 62-80. [8] M. T. Malinowski, “Modeling with stochastic fuzzy differential equations”, In: Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems, S. Chakraverty (Eds.), IGI Global, Hershey Pennsylvania (2014), 150-172. [9] M. T. Malinowski, “Set-valued and fuzzy stochastic differtential equations in M-type 2 Banach spaces”, Tohoku Math. J., (2014), to appear. [10] J. Y. Park, J. U. Jeong, “On random fuzzy functional differential equations”, Fuzzy Sets Syst., 223(2013),89-99.
174
A New Curvaturelike Tensor Field in an Almost Contact Riemannian Manifold Koji Matsumoto
Yamagata University, Japan [email protected]
Abstract: In the paper of Prof. M. Prvanovi´c ([2]), we can find a curvaturelike tensor field (holomorphic curvature tensor field) in an almost Hermitian manifold. In this talk, using the above tensor field, we define a new curvaturelike tensor field, named contact holomorphic (briefly CHR) curvature tensor field in an almost contact Riemannian manifold. And we give some geometrical properties of this tensor field. For example, we show that the non-existence of flat C-holomorphic curvature tensor in a Sasakian manifold. References: [1] D. E. Blair, “Contact manifolds in a Riemannian Geometry”, Lecture Notes in Math. Springeverg, Berlin Heiderberg Newyork, (1976) [2] M. Prvanovi´c, “Conformally invariant tensors of an almost Hermitian manifold associated with the holomorphic curvature tensor”, J. Geom. 103(2012), 89–103. [3] K.Yano, “Differential geometry on complex and almost complex spaces”, A. Pergamon Press, (1965).
175
Probability-Theoretic Foundations of International Trade Models Andrei Matveenko and Taras Hrendash
CERGE-EI, Prague, Czech Republic [email protected], [email protected]
Abstract: The model of technological ideas flow can be applied to the cross-country models of endogenous growth and trade. As shown in [1], the choice of technology in individual countries is closely linked to the country’s comparative advantages, which drive the international trade. In [1], [2] and [3], only Pareto distribution is used to describe the process of ideas flow. We develop further an approach proposed in [4] and consider the implications of properties of different probability distributions on the assumptions and conclusions of the models of international trade. Acknowledgments: The authors acknowledge financial support of the Grant Agency of Charles University in Prague (Grant 259324). Andrei Matveenko acknowledges financial support of the Russian Foundation for Basic Research (Project 14-01-00448). Keywords: Ideas flow; Pareto distribution; International trade models. References: [1] J. Eaton, S. Kortum, “Technology, geography, and trade”, Econometrica, 70(2002), 1741-1780. [2] F. Alvarez, R. Lucas, “General equilibrium analysis of the Eaton-Kortum model of international trade”, J. of Monetary Economics, 54(2007), 1726-1768. [3] F. Alvarez, F. Buera, R. Lucas, “Idea Flows, Economic Growth, and Trade”, NBER Working Papers, 19667(2013). [4] A. Matveenko, E. Polyakova, “Statistical modeling of technological changes and changes of consumer’s preferences”, 8th International Symposium of Statistics: Abstract Book. Eskisehir, Turkey, Anadolu Universitesi Basimevi, 2012, 268-269.
176
Asymptotic Behaviour of Solutions to Initial Boundary Value Problems for Marine Riser Equations
Muge Meyvaci Department of Mathematics, Mimar Sinan Fine Arts University, Istanbul, Turkey
Abstract: In this talk we will find decay estimates for solutions of the problem
𝑢𝑢𝑡𝑡𝑡𝑡 + 𝑘𝑘𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 − [𝑎𝑎(𝑥𝑥)𝑢𝑢𝑥𝑥]𝑥𝑥 + 𝛾𝛾𝑢𝑢𝑡𝑡𝑥𝑥 + 𝑏𝑏(𝑡𝑡)𝑢𝑢𝑡𝑡|𝑢𝑢𝑡𝑡|𝑝𝑝 = 0, 𝑥𝑥 ∈ [0, 𝑙𝑙], 𝑡𝑡 > 0, 𝑢𝑢(0, 𝑡𝑡) = 𝑢𝑢𝑥𝑥𝑥𝑥(0, 𝑡𝑡) = 𝑢𝑢(𝑙𝑙, 𝑡𝑡) = 𝑢𝑢𝑥𝑥𝑥𝑥(𝑙𝑙, 𝑡𝑡) = 0, 𝑡𝑡 > 0,
where 𝑘𝑘, 𝑝𝑝, 𝛾𝛾 are given positive numbers, 𝑎𝑎(𝑡𝑡), 𝑏𝑏(𝑡𝑡) are given functions. Keywords: Asymptotic behavior; Energy decay; Marine riser equations. References: [1] V.K. Kalantarov, A. Kurt, The Long-time behavior of solutions of a nonlinear fourth order wave equation, describing the dynamics of marine risers, ZAMM ,3, 1997, 209-215. [2] F. Tahamtan, Asymptotic stability and decay rate of solutions for a nonlinear fourth order wave equation, Korean J. Comp and Appl. Math. 4, 1997, 47-61. [3] M. Köhl, An extended Lyapunov approach to the stability assessment of marine risers, Zangew. math. Mech. 73, 1993, 85-92.
177
Co-integration Analysis between Oil price and Algeria Inflation Zouaoui Chikr el Mezouar,
Laboratory of Study Economics and Local Development in South West of Algeria
University of Bechar
E-mail : [email protected]
Abstract: This paper seeks to investigate the causal relationship between oil price and Inflation
in Algeria. The empirical analysis starts by analyzing the time series properties of the data which
is followed by examining the nature of causality among the variables. This study analyzes that,
oil price effect the Inflation in Algeria.
In the empirical analysis involves testing the time series characteristics of the data series
(stationary) using ADF (unit roots test) and finding the number of cointegration equations
between the two variables using Johennson maximum likelihood test and running the pairwise
Granger causality test based on EViews Software. References:
[1] Abosedra, S., Baghestani, H. (1991) New Evidence on the Causal Relationship between United States Energy
Consumption and Gross national product. The Journal of Energy andDevelopment 14(2): 285-292.
[2] Altinay, G., Karagol, E. (2004) Structural Break, Unit Root, and the causality between EnergyConsumption and
GDP in Turkey. Energy Economics 26: 985-994.
[3] Amaira B. (2012) The Relationship of Oil Prices and Economic Growth in Tunisia: A Vector Error Correction
Model Analysis. The Romanian Economic Journal 43: 3-22.
[4 Granger, C. W. J. (1988) Some Recent Developments in a Concept of Causality. Journal of Econometrics 39: 199-
211.
178
New Shift-Compactness Results Harry I. Miller
International University of Sarajevo
Abstract: This presentation deals with shift-compactness (see Group action and shift-compactness, JMAA, 392 (2012), 23-39, by the presenter and A.J.Ostaszewski). Two theorems are presented. Theorem 1. There exists a set A concentrated on the rationals that is shift-compact. Theorem 2. There is a non-measurable set A that is shift-compact. These results will be part of a joint paper by H.I.Miller, L. Miller, and A.J.Ostaszewski. L. Miller is an assistant professor at the International University of Sarajevo and A.J.Ostaszewski is a professor at the London School of Economics.
179
Some Statistical Cluster Point Theorems
Leila Miller, Van Wieren and Harry I. Miller International University of Sarajevo, Sarajevo, Bosnia-Herzegovina
[email protected] Abstract: In this paper, we present results related to sets of statistical limit points and cluster points of sequences and their matrix transformations, single and double sequences and stretchings of sequences. R.C. Buck ( [1] , [2] , [3] ), in a series of articles considered questions concerning subsequential limit points of a given sequence. Statistical limit points and statistical cluster points were first considered by J. A. Friday in [4]. Our results are concerned with single sequences as well as with double sequences. Also we present a result about stretchings of sequences. Keywords: Sequences; Statistical limit points; Statistical cluster points; Matrix transformations; Stretchings. References: [1] R.C. Buck, “A note on subsequences”, Bull.Amer.Math.Soc., 49(1943), 898-899. [2] R.C. Buck, “Limit points of subsequences”, Bull.Amer.Math.Soc., 50(1944), 395-397. [3] R.C. Buck, H.Pollard, “Convergence and summability properties of subsequences”, Bull.Amer.Math.Soc., 49(1943), 924-931. [4] J. A. Fridy, “On statistical convergence”, J.Math.Anal. and Appl., Vol. 208, 2(1997), 520-527.
180
Homogenization of the Membrane Model of Perforated Shell Ait Yahia Mohamed and Mourad Lhannafi
Faculty of Mathematics University of Sciences and Technology Houari Boumedienne, Algeria [email protected]
Abstract: In our work, we apply the method of two-scale convergence to homogenizing the two membrane linear models defined on a thin shell which is, periodically perforated with giving the corresponding correctors results and some important remarks are established about the validity of this models in the case of perforated thin shell Keywords: Homogenization; Two-scale convergence; Thin shell; Perforated domains. References: [1] M. Bernadou, P.G. Ciarlet, B. Miara, “Existance theorems for two-dimensional linear shell theories, Journal of elasticity”, 34 pp 111-138, (1992) Kluwer acad.Pub. [2] P.G. Ciarlet, Elasticité tridimensionnelle, Masson Paris, (1986). [3] P.G. Ciarlet, V. Lods, “Analyse asymptotique des coques linéairement élastiques, I : coques membranaires”, C.R.Acd.Sci.,Paris, 318 série I (1994) pp 863-868. [4] P.G. Ciarlet, V. Lods, Asymptotic analysis of linearly shells, I: justification of membrane shell equations, publication du lab. d'analyse numérique Pierre et Marie-Curie. R 95021, (1995). [5] P.G. Ciarlet, V. Lods, “On the ellipticity of linear membrane shell equation”, pub du lab, d'analyse numérique Pierre et Marie-Curie. R95, (1994).
181
A New Approach to Differential Evolution Algorithm for Solving Stochastic Programming Problems
Ali Wagdy Mohamed Department of Statistics, Faculty of Sciences, King Abdulaziz University, Saudi Arabia
Operations Research Department, Institute of Statistical Studies and Research, Cairo University, Giza, Egypt [email protected]
Abstract: Stochastic or probabilistic programming (SP) deals with situations where some or all of the parameters of the optimization problem are described by random or probabilistic variables rather than by deterministic quentities [1].The mathematical models of these problems may follow any particular probability distribution for model coefficients [2]. The main objective is to find the optimal value for model parameters influenced by random event. The basic idea used in stochastic programming is to convert the stochastic problem into an equivelanet deterministic problem which can be solved by using appropriate classical and / or modern numerical techniques. Differential Evolution (DE) is a stochastic population-based search method, proposed by Storn and Price [3]. DE is considered the most recent EAs for solving real-parameter optimization problems [4]. This paper presents a new approach to Differential Evolution algorithm for solving stochastic programming problems, named (DESP). The proposed algorithm introduces a new triangular mutation rule based on the convex combination vector of the triangle and the difference vector between the best and the worst individuals among the three randomly selected vectors. The proposed novel approach to mutation operator is shown to enhance the global and local search capabilities and to increase the convergence speed of the new algorithm compared with conventional DE . DESP uses a constraint handling technique based on feasibility and the sum of constraints violations without any additional parameters. Besides, a new dynamic tolerance technique to handle equality constraints is also adopted[5].Two models of SP problems are considered: Stochastic Fractional Programming Problems and Multiobjective Stochastic Linear Programming Problems. The comparison results between the DESP and basic DE , basic particle swarm optimization (PSO), Genetic algorithm (GA) and the available results from where it is indicated that the proposed DESP algorithm is competitive with , and in some cases superior to, other algorithms in terms of final solution quality, efficiency and robustness of the considered problems in comparison with the quoted results in the literature. Keywords: Differential evolution; Stochastic programming; Fractional Programming; Multiobjective programming. References: [1] S.S. Rao, “Engineering Optimization : Theory and Practice”,John Wiley & Sons Inc.,Hoboken, New Jersey,2009. [2] R. Thangaraj, M. Pant , P. Bouvry, A. Abraham, “Solving stochastic programming problems using modified differential evolution algorithms”, Logic Jnl IGPL,20.4 (2012), 732-746. [3] R. Storn, K. Price “Differential evolution- a simple and efficient adaptive scheme for global optimization over continuous spaces”, Technical Report TR-95-012, ICSI (http://.icsi.berkeley.edu/~storn/litera.html,1995. [4] R. Storn, K. Price, “Differential Evolution- a simple and efficient heuristic for global optimization over continuous spaces”, Journal of Global Optimization, 11.4(1997), 341-359. [5] A. W. Mohamed, H. Z. Sabry, “ Constrainrd optimization based on modified differential evolution algorithm, 194(2012), Information Sciences.
182
On the Travelling Wave Solutions of a Generalized Zakharov-Kuznetsov Equation Dimpho Millicent Mothibi and Chaudry Masood Khalique
International Institute for Symmetry Analysis andMathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Republic of South Africa
Abstract: In this talk we present exact solutions of a generalized (2+1)-dimensional Zakharov-Kuznetsov equation. It is well-known that the Zakharov-Kuznetsov equation governs the behaviour of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Three types of solutions are obtained; hyperbolic function solutions, trigonometric function solutions and rational solutions. Keywords: A generalized (2+1)-dimensional Zakharov-Kuznetsov equation; Hyperbolic function solution; Trigonometric function solutions and rational solutions.
183
Mixed d Semi Prefuzzy Topological Spaces and Some Results of Separation Axioms in Mixed d Semi Prefuzzy Topological Spaces
Anjan Mukherjee Department of Mathematics, Tripura University, Suryamaninagar, Tripura, India
Abstract: The study of mixed topology was first invented by Finchtenholz [6] of the Polish School of Mathematics in the year 1938. Mixed topology originated from the work of the polish Mathematician Alexiewicz and Semadini. Mixed topology is a technique of mixing two topologies on the same set in order to obtain a third topology. The foundation of mixed topology is based on the theory of strict topology of the space of continuous function on locally compact space. Mixed topology attracted many of the functional analysts like Bennet, Cooper, Kalton and others due to its use in summability, schauder decomposion and function spaces with strict topology. Wiweger [15] defined a mixed topology in a linear space equipped with two linear topologies. In the year 1971 J.B Cooper [4] showed that strict topology in a function space is mixed topology derived from two linear topologies on the function space. It is observed that strict topology of Buck comes out as a corollary of Coopers mixed topology. Fascinated by the work of many researchers in general mixed topology. N. R Das and P.C Baishya [5] have continued these works in fuzzy setting. They have constructed a fuzzy topology on a set X from two given fuzzy topologies on X with the help of closure of neighborhood of one topology with respect to the other topology. They called this new topology as mixed fuzzy topology. In this paper we construct the fuzzy d semi pre mixed topological space from two different fuzzy topologies using fuzzy d semi pre-q-nbd of a fuzzy point with respect to one topology and fuzzy d -semi pre closure of a fuzzy set with respect to the another topology, then satisfying the four conditions of a fuzzy topological space under certain conditions, some relations are established between two topologies used and their corresponding mixed topology. Also some results of separation axioms in mixed d semi pre fuzzy topological space. References: [1] K. K. Azad, “On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly continuity”, J. Math. Anal. Appl. 82, 14-32,(1981). [2] M. Caldas, S. Jafari, R.K. Saraf, “Fuzzy (δ,P)T1Topological Spaces”, J. Tri. Math. Soc. ,9,1-4,(2007). C. L. Chang, “Fuzzy Topological Spaces”, J. Math. Anal. Appl., 24, 182-190,(1968). [3] J. B. Cooper, “The Strict topology and space with mixed topologies”, Proc. Amr. Math. Soc., 30,583-592,(1971). [4] N.R. Das, P. C. Baishya, “Mixed fuzzy topological spaces”, Journal of Fuzzy Math.,3(2),777-784,(1995). [5] G.Fichtenholz, “Surles fonctionnelles lineaires continues au sens generalise”, Mat. Sbornik., 4,193-214,(1938). [6] S.Ganguly, S. Saha, “A note on δ-continuity and δ-connectedness in fuzzy setting”, Simon Stevin, 62, 127-141,(1988). [7] A. Mukherjee, S. Debnath, “On δ semi open sets in fuzzy setting”, J. Tri.Math.Soc., 8,51-54,(2006). [8] J. H. Park, B.Y. Lee, M. J. Son, “On δ semi open sets in topological space”, J. Indian Acad. Math., 19, 59-67,(1997). [9] S. Roy Choudhury, M. N. Mukherjee, “On δ almost continuity and δ pre open sets”, Bull. Inst. Math. Acad. Sinica, 21,357-366,(1993). [10] M. H. Stone, “Application of the theory of boolean ring to general topology”, Trans. Amer.Math.Soc.,41,375-381,(1937). [11] S.S Thakur, R. K. Khare, “Fuzzy semi δ pre-open sets and fuzzy semi δ pre continuous mappings”, Stud. Ceret. Stint. Scr. Mat. Univ. Bacau. (2004). [12] M. K. R. S. Veerakumar, “δ semi pre open sets”, Antartica J. Math., 3(1),101-102,(2006).
184
Second-Order Operator-Differential Equations with Transmission Conditions Oktay Sh.Mukhtarov, Kadriye Aydemir and Hayati Olgar
Department of Mathematics, Gaziosmanpasa University, Tokat, Turkey [email protected], [email protected], [email protected],
Abstract: The development of classical, Sturm-Liouville theory in the years after 1950 can be found in various sources; in particular in the texts of Atkinson [1], Coddington and Levinson [8], Levitan and Sargsjan [3] and Naimark [4]. In this paper we give a comprehensive treatment for a new class Sturm-Liouville problems together with transmission (impulsive, jump or interface) conditions at the one interior singular point. We investigate some spectral properties of the considered problem. Keywords: Sturm-Liouville problem; Transmission conditions. References: [1] F. V. Atkinson, “On bounds for the Titchmarsh-Weyl m-coefficients and for spectral functions for second-order differential equations”, Proc. Royal Soc. Edinburgh, 97(1984), 1-7. [2] E. A. Coddington, N. Levinson, “The integrability methods”, McGraw-Hill, New York, 1955. [3] B. M. Levitan, I. S. Sargsjan , “Sturm-Liouville and Dirac operators”, Mathematics and its Applications (Soviet Series), 59. Kluwer Academic Publishers Group, Dordrecht, 1991. [4] M. A. Naimark, “Linear diferential operators: II”, A(Ungar, New York: Translated from the second Russian edition), 1968.
185
General Stability in Memory-Type Thermoelasticity with Second Sound Muhammad I. Mustafa
Mathematics Department, King Fahd University of Petroleum and Mineral, Dhahran, Saudi Arabia [email protected]
Abstract: In classical thermoelasticity, the heat conduction is governed by the Fourier’s law, which predicts an infinite speed of heat propagation. To overcome this physical paradox, the theory of 2nd sound thermoelasticity suggests replacing Fourier’s law by so called Cattaneo’s law which models thermal disturbances as wave-like pulses traveling at finite speed, see [1, 2]. In the presence of viscoelastic damping, it was shown in [3] that the coupling via Cattaneo’s law causes loss of the exponential decay usually obtained in the case of coupling via Fourier’s law [4]. In this paper we consider an n-dimentional thermoelastic system of second sound with viscoelastic damping. We establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a larger class of relaxation functions and generalizes previous results existing in the literature. Keywords: Thermoelasticity with second sound; Viscoelastic damping; General decay; Convexity. References: [1] B.D. Coleman, W.J. Hrusa, D.R. Owen, “Stability of equilibrium for a nonlinear hyperbolic system describing heat propagation by second sound in solids”, Arch. Rational Mech. Anal. 94 (1986), 267-289. [2] D.S. Chandrasekharaiah, “Hyperbolic thermoelasticity: a review of recent literature”, Appl. Mech. Rev. 51 (1998), 705-729. [3] S.A. Messaoudi, A. Al-Shehri, “General Boundary Stabilization of Memory-Type Thermoelasticity with Second Sound”, Z. Anal. Anwend. 31 (2012), 441-461. [4] S.A. Messaoudi, A. Al-Shehri, General boundary stabilization of memory type thermoelasticity, J. Math. Phys. 51, 103514 (2010).
186
Time Series Forecasting Using Box-Jenkins Methodology Application on Census Data in Iraq
Qais Mustafa Department of Banking Management, Dohuk Politechnic University, Zakho, Dohuk, Iraq
Abstract: The applications of Autoregressive Moving Average (ARMA) models to time series forecasting problems have popularized and become widely known by George E. and Gwilym M.Jenkins in 1970[1].Many applications have been done in this area such as the recent works of Mutar and Ilias[2]and also Tuama[3].In [4] Ghafil used the Box-Jenkins models for forecasting the production of the electric power.In this paper we shall apply the Box-Jenkins methodology to forecast the census in Iraq.We also build models of the time series and then we choose the best one to forecast values for future census depending on some performance measures. Keywords: Box-Jenkins; ARMA models; Time series forecasting; Census. References: [1] G. E. P. Box, G. M. Jenkins, G. C. Reinsel, “Time series analysis:Forecasting and control”, WILEY. 4 (2008). [2] D. R. Mutar, I. I. Ilias,“Analysis and modeling time series of water flow into Mosul city:A comparativestudy”, Iraqi Journal of Statistical Sciences.10(2010), 1-32. [3] S. A. Tuama, “Using analysis of time series to forecast numbers of the patients Malignant Tumors in Anbar province”, Al-Anbar University Journal of Economics and Administration Sciences, 4(2012), 371-393. [4] A. A. Ghafil, “Using Box-Jenkins (ARIMA)models for forecasting the production of electric power”, Journal of Karbala University, 11(2013), 196-207.
187
Fixed Point Theorems for Weakly T-Chatterjea and Weakly T-Kannan Contractions in b-Metric Spaces
Zead Mustafa, Jamal Rezaei Roshan, Vahid Parvaneh and Zoran Kadelburg Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar. Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran.
Faculty of Mathematics, University of Belgrade, Beograd, Serbia. [email protected], [email protected], [email protected], [email protected]
Abstract: In this paper we obtain some fixed point results for generalized weakly T-Chatterjea-contractive and generalized weakly T-Kannan-contractive mappings in the framework of complete b-metric spaces. Examples are provided in order to distinguish these results from the known ones. Keywords: Fixed point; Complete metric space; b-metric space; Weak C-contraction; Altering distance function. References: [1] S. Banach, “Surles operateurs dans les ensembles abstraits et leur application aux equations integra les”, Fundam.Math.3, 133-181 (1922). [2] R. Kannan, “Some results on fixed points”,Bull.Calcutta Math.Soc. 60,71-76 (1968). [3] SK. ChatterJea, “Fixed point theorems”. C. R. Acad. Bulgare Sci.25, 727-730 (1972). [4] BS. Choudhury, “Unique fixed point theorem for weak C-contractive mappings”, Kathmandu Univ. J. Sci. Eng. TechnoI.5(1),6-13(2009). [5] A. Razani, V Parvaneh, “Some fixed point theorems for weakly T-ChatterJea and weakly T-Kannan-contractive mappings in complete metric spaces”, Russ. Math. (lzv. VUZ) 57(3),38-45 (2013)
188
Nonlinear three point boundary value problem Farid Nouioua
Univ Souk Ahras [email protected]
Abstract : In this study, we start by setting up sufficient conditions for the existence of solutions for a three point boundary value problem generated by a third order differential equation. Then, we provide sufficient conditions permitting us to obtain the existence of a nontrivial solution. Later, we prove the existence of at least one solution of the posed problem by using the Leray Schauder nonlinear alternative. Finally, and as an application, we illustrate our results with two examples.
Keywords : Fixed point theorem, three point boundary value problem, Leray Schauder nonlinear alternative
References:
[1] John R.Graef and J. R. L. Webb, Third order boundary value problems with nonlocal boundary conditions nonliner Anal,71(2009),no 5-6, 1542-1551.
[2] A. Guezane-Lakoud, and S. Kelaiaia Solvability of a three point nonlinear boundary value problem, Electron. J.Differential Equations, 2010 (139), 1-9.
[3] Y. Sun, Positive solutions for third-order three-point nonhomogenous boundary value problem, Appl. Math. Lett., 22(1) (2009) 45-51.
189
A Note on the Solutions of the Nonlinear Fractional Differential Equations Meryem Odabasi and Emine Misirli
Tire Kutsan Vocational School, Tire, Izmir, Turkey Department of Mathematics, Ege University, Bornova, Izmir, Turkey
[email protected] ,[email protected]
Abstract: Nonlinear fractional differential equations have many applications in physical sciences and dynamical systems. In this work, the nonlinear fractional partial differential equations have been defined by the modified Riemann-Liouville fractional derivative [1] and converted into nonlinear non-fractional ordinary differential equations by using this fractional derivative. By means of the improved form of the trial equation method [2], a modified trial equation method [3], we have successfully obtained exact solutions including soliton solutions, single-kink solution and rational function solutions of some nonlinear fractional differential equations. This method is convenient to obtain exact solutions of nonlinear equations through its beneficial and applicable structure. Keywords: The modified Riemann-Liouville derivative, The modified trial equation method, Nonlinear fractional differential equations. References: [1] G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of non differentiable functions further results”, Computers and Mathematics with Applications, 51(2006), 1367-1376. [2] C. S. Liu, “Trial equation method to nonlinear evolution equations with rank inhomogeneous: Mathematical discussions and its applications”, Communications in Theoretical Physics, 45(2006), 219-223. [3]H. Bulut, H. M. Baskonus and Y. Pandir, “The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation”, Abstract and Applied Analysis, Article ID 636802(2013), 8 pages.
190
Charge or Dipole Simulation Method for Approximations of Complex Analytic Functions Hidenori Ogata
Department of Communications Engineering and Informatics, University of Electro Communications, Chofu, Tokyo, Japan [email protected]
Abstract: The charge simulation method [1] is a numerical solver for potential problems, that is, problems of the Laplace equation. The method approximates the solution by a superposition of point charges, which are logarithmic potentials in two dimensional cases. Remarking that the harmonic functions are the real part of complex analytic functions, we optain an approximation of complex analytic function 𝑓𝑓(𝑧𝑧) of the form 𝑓𝑓(𝑧𝑧) ≈ ∑ 𝑄𝑄𝑗𝑗 log (𝑧𝑧 − 𝜍𝜍𝑗𝑗)𝑁𝑁
𝑗𝑗=1 where 𝑄𝑄𝑗𝑗 are real coefficients and 𝜍𝜍𝑗𝑗 points outside the domain where 𝑓𝑓(𝑧𝑧) is analytic Amano’s method of numerical conformal mappings [2] can be regarded as an approximation of mapping functions by the charge simulation method. We can also approximate the solution by a superposition of electric dipoles which is the dipole simulation method [3]. The approximation of complex analytic function 𝑓𝑓(𝑧𝑧) derived from the method is of the form 𝑓𝑓(𝑧𝑧) ≈∑ 𝑝𝑝𝑗𝑗 log (𝑧𝑧 − 𝜍𝜍𝑗𝑗)𝑁𝑁
𝑗𝑗=1 where 𝑝𝑝𝑗𝑗 are complex coefficients and 𝜍𝜍𝑗𝑗 are point located outside the domain where 𝑓𝑓(𝑧𝑧) is analytic. This approximation can be regarded as a discretization of Cauchy’s Integral Formula. In addition we give an approximation of periodic analytic function by the charge or dipole simulation method. In this approximation the periodic analytic functions are approximated using the potentials due to periodic arrays of point charges or dipoles. Figure 1 shows the two-dimensional electric with a periodic array of circular conductors computed by our dipole simulation method.
Keywords: Potential problem; Charge simulation method; Dipole simulation method; Periodic domain. References: [1] H. Singer, H. Steinbigler and P. Weiss, “A charge simulation method for the calculation of high voltage fields”, IEEE Trans. Power. Apparatus Systems PAS-93, 5(1974), 1660-1668. [2] K. Amano, “A charge simulation method for numerical conformal mapping onto circular and radial slit domains, SIAM J. Sci. Comput. 19.4(1998) 1169-1187. [3] H. Ogata, “Dipole simulation method for two-dimensional potential problems”, Nonlinear Theory and Its applications, IEICE, 5,1(2014),2-14.
191
On New Cesàro-Orlicz Double Difference Sequence Space
Oguz Ogur and Cenap Duyar Department of Mathematics, Ondokuz Mayis University, Atakum, Samsun, Turkey
[email protected], [email protected]
Abstract: The aim of this paper is to introduce the Cesàro-Orlicz double difference sequence space ( )(2) ,MCes pD . We study some topologic properties of this space and obtain some inclusion relations.
Keywords: Double sequences; Orlicz function; Paranorm; Cesaro sequence space. References: [1] Z. U. Ahmad, Mursaleen, “Köthe-Toeplitz Duals of Some New Sequence Spaces and Their Matrix Maps”, Pub. Inst. Math. Beograd, 42 (56) (1987), 57-61.
[2] F. Basar, Y. Sever, “The Space qL of Double Sequences”, Math. J. Okayaa Univ., 51 (2009), 149-157. [3] F. Basar, “Summability Theory an Its Applications, Bentham Science Publishers, e-books”, Monographs, İstanbul, 2012. [4] M. Et, M. Basarır, “On Some New Generalized Difference Sequence Spaces”, Periodica Math. Hung., 35 (3), 1997, 169-176. [5] M. Et, R. Colak, “On Some Generalized Difference Sequence Spaces”, Soochow J. Math.,21(4), (1995), 377-386. [6] Y. Cui, H. Hudzik, N. Petrot, S. Suantai, A. Szkymaszkiewicz, “Basic Topological and Geometric Properties of Cesaro-Orlicz Spaces”, Proc. Indian Acad. Sci. (Math. Sci.) 115 (4), 2005. [7] H. Kızmaz, “On Certain Sequence Spaces”, Canad. Math. Bull., 24 (2), 1981. [8] D. Kubiak, “A Note on Cesaro-Orlicz Sequence Spaces”, J. Math. Anal. Appl., 349 (2009), 291-296. [9] M. A. Kranoselskii, Ya. B. Rutickii, “Convex Functions and Orlicz Spaces”, (Gooningen: P. Nordhoff Ltd.), 1961. [10] P. Y. Lee, “Cesaro Sequence Space”, Math. Chronicle, 13 (1984), 29-45. [11] I. J. Maddox, “Elements of Functional Analysis”, Cambridge Univ. Press, 1970. [12] E. Malkowsky, E. D. Parashar, “Matrix Transformation in Spaces of Bounded and Convergent Difference Sequence of Order m”, Analysis, 17 (1997), 87-97. [13] N. Petrot, S. Suantai, “On Uniform Kadec-Klee Properties and Rodundity in Generalized Cesaro Sequence Spaces”, Internat. J. Math. Sci., 2 (2004), 91-97. [14] A. Pringsheim, “Zur Theorie de Zwefach Unendlichen Zahlenfolgen”, Math. Ann., 53 (1900), 298-321. [15] W. Sanhan, S. Suantai, “On k-nearly Uniform Convex Property in Generalized Cesaro Sequence Spaces”, Internat. J. Math. Sci., 57 (2003), 3599-3607. [16] J. S. Shiue, “On the Cesaro Sequence Spaces”, Tamkang J. Math., 1 (1970), 19-25.
192
Positive Solutions for a Singular Semipositone Dynamic System on Time Scales Arzu Denk Oguz and S. Gulsan Topal
Department of mathematics, Ege University, İzmir, Turkey
Abstract : This work is concerned with the existence of two positive solutions for the system of second order
singular semipositone m-point boundary value problem on time scales under suitable conditions by applying fixed
point theorem in cones. We emphasize that the nonlinear term may take a negative value and be singular. As an
application,we also give an example to illustrate our results.
193
G2-Congruence Classes of Curves in Purely Imaginary Octonions Misa Ohashi
Department of Mathematics, Nagoya Institute of Technology, Nagoya, Japan [email protected]
Abstract: In [1], we determined G2-invariants of curves in purely imaginary octonions. In this case, we fined that the special helix in purely imaginary octonions whose SO(7)- invariants are constant but its G2-invariants are not constant. We call a curve G2-helix if its G2-invariants are constant. Since exceptional Lie group G2 is the subgroup of Lie group SO(7), G2-invariants of curves determine SO(7)-invariants of the one. Therefore G2-helix is an SO(7)-helix, however the converse is not true. In this talk, we shall give the relationship of invariants of curves between G2 and SO(7) congruence classes, and their properties. Also, we shall give some special curves in purely imaginary octonions, and the relationship of invariants of their curves between G2 and SO(7) congruence classes, concretely. Keywords: Exceptional Lie group G2; congruent class; Frenet-Serre -formula. References: [1] M. Ohashi, “G2-congruence theorem fore curves in purely imaginary octonions and its application”, Geometriae Dedicata. 163(2013), 1-17. [2] M. Ohashi, “The relationships between G2-invariants and SO(7)-invariants of curves in ImC”, Prospects of Differential Geometry and Its Related Fields, World Sci. Publ., 2013, 145-153.
194
On the Riesz Basis Property of Eigenfunctions of One Boundary-Value Problem with Transmission Conditions
Hayati Olgar and Oktay Sh. Mukhtarov Department of Mathematics, Gaziosmanpasa University, Tokat, Turkey
[email protected], [email protected]
Abstract: In this study we shall investigate a one discontinuous Boundary Value problem on a finite interval which consist of a Sturm-Liouville equation together with eigendependent boundary conditions and two supplementary transmission conditions at the point of discontinuiy. It is shown that the eigenfunctions for the problems considered form a Riesz basis of the appropriate Hilbert space. Keywords:Sturm-Liouville problems; eigenfunctions; boundary and transmission conditions; Riesz basis. References: [1] O. A. Ladyzhenskaia, “The Boundary Value Problems of Mathematical Physics”, Springer-Verlag, New York 1985. [2] M. V. Keldys, “On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations”, Dokl Akad. Nauk SSR (in Russian) 77 (1951), 11. [3] A. G. Kostyuchenko and A. A. Shkalikov, “Self-adjoint quadratic operator pencils and eliptic problems”, Functional Anal. Appl. 17 (1983), 109. [4] A. S. Markus, “Introduction to the Spectral Theory of Polynomial Pencils”, Translation of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island 1988. [5] O. Sh. Mukhtarov, M. Kadakal and F. S. Muhtarov, “On Discontinuous Sturm-Liouville Problems with Transmission Conditions”, J. Math. Kyoto Univ., Vol. 44, Number 4, ( 2004) 779-798.
195
Cauchy Type Problem for Nonlinear Fractional Integro-Differential Equations with Multivariate Mittag-Leffler Function in the Kernel
Mehmet Ali Ozarslan Department of Mathematics, Eastern Mediterranean University,
Gazimagusa, TRNC, Mersin 10, Turkey [email protected]
Abstract: In this paper, the existence and uniqueness of the solution to Cauchy type problem for nonlinear fractional integro-differential equations of Volterra type, involving a multivariate Mittag-Leffler (M-L) function in the kernel is investigated. Furthermore, the solution of nonhomogeneous Cauchy-type problem for Volterra-type fractional differintegral equation with multivariate M-L function in the kernel is obtained in the space of summable functions. An illustative example is also given. We should notice that the above mentioned problems include a large variety of applications, since the multivariate M-L functions contains several special function such as generalized Lauricella confluent hypergeometric function, multivariate Laguerre polynomial, univariate M-L function etc. Keywords: Nonlinear fractional integro-differential equations of Volterra type, Cauchy problem, Mittag-Leffler function, Lauricella confluent hypergeometric function, Fixed point theorem, Laplace transform. References: [1] S.C. Cheng, J.N Wu, M.R. Tsai, W.F. Hsieh, Spontaneous emission near the band edge of a three-dimensional photonic crystal: a fractional calculus approach, J. Phys.: Condens. Matter 21 (2009) 015503. [2] D. Craiem, F.J. Rojo, J.M. Atienza, R.L. Armentano, G. Guinea, Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries, Phys. Med. Biol. 53 (2008) 4543. [3] Xiao-Li Ding, Yao-Lin Jiang, Semilinear fractional di¤erential equations based on a new integral operator approach, Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), 5143-5150.
196
On a Double Integral Equation Including a Set of Two Variables Polynomials Suggested by Laguerre Polynomials
Mehmet Ali Ozarslan and Cemaliye Kurt Department of Mathematics, Eastern Mediterranean University,
Gazimagusa, TRNC, Mersin 10, Turkey [email protected] and [email protected]
Abstract: In this paper, we introduce general class of bivariate Laguerre polynomials and Mittag-Leffler functions . We investigate double Laplace transforms, double fractional integrals and derivative properties of the above mentioned classes. We further obtain linear, multilinear and mixed multilateral generating functions for in terms of . Finally, we consider a general singular integral equation with in the kernel and obtain the solution in terms of . Keywords: Bivariate Laguerre polynomial, bivariate Mittag-Leffler function, double Laplace transform, double fractional integral, double fractional derivative, generating function, singular integral equation. References: [1] J.D.E. Konhauser, Some properties of biorthogonal polynomials, J. Math Anal. Appl. 11 (1965) 242-260. [2] J.D.E. Konhauser, Biorthogonal polynomials suggested by the Laguerre polynomials,Pacific J. Math 21 (1967) 303-314. [3] M.A. Ozarslan, A. Altın, Some families of generating functions for the multiple orthogonal polynomials associated with modified Bessel K-functions, J. Math. Anal. Appl. 297 (2004) 186-193. [4] M.A.Ozarslan, Some families of generating functions for the extended Srivastava polynomials, Appl. Math. Comput. 218 (2) (2011) 959-964. [5] M.A. Ozarslan, On a singular integral equation including a set of multivariate polynomials suggested by Laguerre polynomials, Appl. Math. Comput. 229 (2014) 350-358. [6] T.R. Praphakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971) 7-15. [7] T.R. Praphakar, On a set of polynomials suggested by Laguerre polynomials, Pasific J. Math. 35 (1) (1970) 213-219
197
On Some Fixed Point Results for Hybrid Rational a-Type Contractive Mappings Mahpeyker Ozturk
Department of Mathematics, Sakarya University, Sakarya, Turkey [email protected]
Abstract: In this study, some coincidence and fixed point results for mappings satisfying A-type contractivity via rational expressions have been obtained which are extensions and generalizations of existing results in the literature. Keywords:Coincidence point; a-Type Contraction; Rational Expressions. References: [1] M. Akram, A. A. Zafar, A. A. Siddiqui, “A general class of contractions: A-contractions”, Novi Sad J. Math, 38(1), 25-33 (2002) [2] A. Petrusel, G. Petruşel, I.A. Rus, “Fixed Point Theory”, Cluj Universİty Press, (2008) [3] G. Akinbo, “Fixed Point Theorems for a Class of Contractions in Metric Spaces”, Bull. Math. Anal. Appl. 3(4), 80-83, (2011) [4] V. Gupta, R. Kaur, “Some Common Fixed Point Theorems for a Class of A-Contractions on 2-Metric Spaces”, Inter. J. Pure Appl. Math. 78(6), 909-916, (2012) [5] F. Zabihi, A. Razani, “Fixed Point Theorems for Hybrid Rational Geraghty Contractive Mappings in Ordered b-Metric Spaces”, Journal of Applied Mathematics, 2014, Article ID:929821, 9 pages, (2014)
198
New Integral Operator for Solution of Differential Equations Ali Ozyapici
Department of Engineering, Cyprus International University, Lefkosa [email protected]
Abstract: One of the important method to solve differential equations, systems of differential equations, integral equations and system of integral equations is to use an integral operator (transform). In this study, new and effective integral operator (transform) is defined for the solution of differential equations. Consequently, the relationship between new operator and well known Laplace transform, Elzaki transform and Sumudu transform will be given respectively. According to the applications of this integral operator, it provides an easiest approach for the solution of differential equation. Moreover, this new integral operator can be used for the selection of suitable integral operator for the given differential equation. Keywords: Integral transform; Sumudu transform; Elzaki transform; Laplace transform. References: [1] A. Kilicman, H.E.Gadain, “An application of double Laplace transform and Sumudu transform”, Lobachevskii J.Math. 30 (3) (2009), pp.214-223. [2] A. Kilicman, H. Eltayeb, “A note on Integral transform and Partial Differential Equations”, Applied Mathematical Sciences, 4(3) (2010), pp.109-118. [3] F. B. M. Belgacem, "Sumudu Applications to Maxwell.s Equations", Progress In Electromagnetics Research Symposium,Beijing,China,(2009)March 23-27. [4] H. Eltayeh, A. Kilicman, “on Some Applications of a new Integral transform”, Int. Journal of Math. Analysis, Vol. 4, (2010), no.3, 123-132. [5] G. K. Watugala, “Sumudu transform- an integral transform to solve di¤erential equations and control engineering problems”, Int.J.Math.Educ.Sci.Technol. Vol. 24,(1993), pp.35.42. [6] R. Darzi,B. Mohammadzade,S. Mousavi,R. Beheshti,"Sumudu transform method for solving fractional differential equations and fractional Diffusion Wave equation",Journal of Mathematics and Computer Science,Vol.6, (2013), pp. 79-84. [7] T M. Elzaki, S.M.Elzaki, “On the relationship between Laplace transform and new integral transform", Tarig transform", Elixir Applied Mathematic, vol:36, (2011), pp. 3230.3233.
199
The Differential Geometry of Regular Curves on Regular Time-Like Surface Emin Ozyilmaz and Yusuf Yayli
Department of Mathematics, Ege University, Bornova, Izmir, Turkey Department of Mathematics, Ankara University, Ankara, Turkey
Abstract: In this study, we consider time-like regular surface in Minkowski space as 𝑦𝑦 = 𝑦𝑦(𝑢𝑢, 𝑏𝑏) and investigate Darboux vectors of the time-like curves ontime-like surface as (𝑐𝑐), (𝑐𝑐1) and (𝑐𝑐2) which are not intersect perpendicularly. Moreover, we give a relation between the Darboux vectors of these Darboux frames. By this relation we obtain general Liouville formula and general form Euler and O.Bonnet. Keywords: Classical Differential Geometry; Time-Like Curves; Time-Like Surface; Minkowski 3-space.
References: [1] H. Ugurlu, A. Caliskan, “The Space-like and Time-like Surface Geometry by the Darboux Vector”, The press of the Celal Bayar Univ., 2012. [2] S.Yilmaz, E. Ozyilmaz, M. Turgut, “On The Differential Geometry of the Curves in Minkowski Space-Time II”, Int.jour. of Comput Math Sci 3:2, 2009. [3] Y. Yayli, H. H. Hacisalihoglu, “Closed Curves in the Minkowski 3-space”, Hadronic J. 23 (2000), No:3 [4] S. Yilmaz, “Spherical Indicators of Curves and Characterizations of some special curves in four dimensional Lorentzian Space”, Ph.D. dissertation, Dept. Math. Dokuz Eylül Univ, ·Izmir,Turkey,2001. [5] F. Akbulut, “Bir Yüzey Üzerindeki Egrilerin Darboux Vektörleri”, E. Ü. Faculty of Science, ·Izmir, 1983. [6] S. Kiziltug, Y. Yayli, “Space-like Curve on Space-like Parallel Surface in Minkowski 3-space”, Int. J. Math. Comput. 19 (2013), no. 2, 94.105 [7] A.T. Ali, R. Lopez, M. Turgut, “k-type partially null and pseudo null slant helices in Minkowski 4-space”, Math.Commun. 17, (2012), no1, 93-103. [8] M. Turgut, S. Yilmaz, “On The Frenet Frame and A Characterization of space-like Involute-Evolute Curve Couple in Minkowski Space-time”, Int. Math. Forum Vol.3 No. 16 (2008), 793-801.
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Lacunary Invariant Statistical Convergence of Sequences of Sets with Respect to a Modulus Function
Nimet Pancaroglu and Fatih Nuray Department of Mathematics, Afyon Kocatepe University, Afyonkarahisar, Turkey
[email protected], [email protected]
Abstract: Convergence concept for sequences of set had been studied by Beer [3], Aubin and Frankowska[1], Baronti and Papini [2]. The concepts of Wijsman statistical convergence and Wijsman strong Cesὰro summability were introduced by Nuray and Rhoades [4] in 2012. Pancaroglu and Nuray [5] introduced Wijsman invariant statistical and Wijsman lacunary invariant statistical convergence of sequences of set. In this paper, we introduced and study the concept of lacunary invariant convergence for sequences of sets with respect to modulus function f and give some inclusion relations. Keywords: Statistical Convergence; Lacunary invariant statistical convergence; Modulus function; Sequences of sets. References: [1] J. P. Aubin, H. Frankowska, “Set Valued Analysis”, Birkhauser, (1990). [2] M. Baronti, P.L. Papini, “Convergence of sequences of sets”, Methods of Functional Analysis in Approximation Theory, (1986),133-155. [3] G. Beer, “Convergence of continuous linear functionals and their level sets”, Archiv.der Math. 52(1989), 482-491. [4] F. Nuray, B. E. Rhoades, “Statistical convergence of sequences of sets”, Fasciculi Mathematici, 49(2012),87 99. [5] N. Pancaroglu, F. Nuray, “On invariant statistically convergence and lacunary invariant statistically convergence of sequences of sets”, Progress in Applied Math., 5(2013),23-29.
201
Stochastic Modelling and Inference of Model Parameters in Complex Biochemical Systems
Vilda Purutcuoglu Department of Statistics, Middle East Technical University, Ankara, Turkey
Abstract: The stochastic activations in a cell are one of crucial sorts of the biological processes as certain cellular activations such as the transcription and the translation merely occur randomly. Hereby in order to explain these biological events, the stochastic modelling is the natural way. There are different approaches to model a biological system probabilistically. The diffusion approximation is one of these methods whose model parameters, i.e. stochastic reaction rate constants, are estimated via Bayesian methods [1]. In this study we perform the discretized version of the diffusion approximation, called the Euler-Maruyama approximation, in inference of a biological system. As the novelty we implement the inference in a realistically complex system. We use the particle filtering approach and the data augmentation technique in the estimation of the model parameters. In the calculation we apply a dataset which is simulated via the Gillespie algorithm [2]. In this dataset we assume that the observations are gathered for all genes in the systems at the given time points [3]. Then we add latent states in these generated time-course data in order to reduce the bias caused by the discretization of the diffusion modelling. Then we perform the particle filtering approach to simulateneous update the latent states and model parameters within the Bayesian framework [4]. Finally the estimates are evaluated in terms of the accuracy and the acceptance ratio which detects the mixing feature of the algorithm. As the future work we intent to extent this idea in a partially observed time-course dataset with sparse measurements. Acknowledgement: The author would like to thank the Scientific and Technological Research Council (TÜBİTAK Project No: TBAG-112T772) for their financial support. Keywords: Diffusion approximation; Particle filtering approach; Data augmentation; System biology; Bayesian inference. References: [1] D. Wilkinson, “Stochastic Modelling for System Biology”, Chapman and Hall/CRC (2006). [2] D. T. Gillespie, “Exact stochastic simulation of coupled chemical reactions”, Journal of Physical Chemistry, 81, 25(1977), 2340-2361. [3] B. Eraker, “MCMC analysis of diffusion models with application to finance”, Journal of Business and Economic Statistics, 19, 2(2001), 177-191. [4] A. Golightly and D. J. Wilkinson, “Bayesian sequential inference for nonlinear multivariate diffusions”, Statistics and Computing, 516(2006), 323-338.
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Nash Equilibrium for Binary Convexities Taras Radul
Institute of Mathematics, Kazimierz Wielki University, Bydgoszcz, Poland [email protected]
Abstract: The notion of convexity we consider is considerably broader then the classic one; specifically, it is not restricted to the context of linear spaces. Such convexities appeared in the process of studying different structures like partially ordered sets, semilattices, lattices, superextensions etc. We base our approach on the notion of topological convexity from [1] where the general convexity theory is covered from axioms to application in different areas. Particularly, there is proved Kakutani fixed point theorem for abstract convexity. The classical Nash equilibrium theory is based on fixed point theory and is developed in frames of linear convexities. The mixed strategies of a player are probability (additive) measures on a set of pure strategies. But an interest to Nash equilibria in more general frames is rapidly growing in last decades. Games in capacities (games where players are allowed to play their mixed non-additive strategies) were investigated in [2] and [3]. There exists a linear convex structure on the space of capacities which was used to prove existence of Nash equilibrium point. There are also results about Nash equilibrium for non-linear convexities. For instance, Ch.Horvath proved in [4] existence of Nash equilibrium point for B-convexity and MaxPlus convexity. Let us remark that MaxPlus convexity is related to idempotent (Maslov) measures in the same sense as linear convexity is related to probability measures. The constructions of the spaces of probability measures, idempotent measures and capacities are functorial and could be completed to monads. There was introduced in [5] a convexity structure on each F-algebra for any monad F in the category of compact Hausdorff spaces and continuous maps. Particularly, topological properties of monads with binary convexities were investigated. We prove counterparts of Kakutani and Nash theorems for any binary convexity. This results we use to obtain Nash theorem for algebras of any L- monad with binary convexity. Since capacity monad is an L-monad with binary convexity [6], we obtain as corollary the corresponding result from [3]. Keywords: Nash equilibrium; Abstract convexity; Monad; Capacity. References: [1] M.van de Vel, “Theory of convex strutures”, North-Holland, 1993. [2] D. Glycopantis, A. Muir, “ Continuity of the payoff function”, Econ. Theory 16 (2000) 239-244. [3] R. Kozhan, M. Zarichnyi, “Nash equilibria for games in capacities”, Econ. Theory, 35 (2008) 321-331. [4] W. Briec, Ch. Horvath. “Nash points, Ku Fan inequality and equilibria of abstract economies in Max-Plus and B-convexity”, J. Math. Anal. Appl. 341 (2008), 188-199.
203
Approximate Solutions to a Problem of Two Moving Boundaries Governed with Fractional Time Derivative in Drug Release Devices
Rajeev and M.S. Kushwaha Department of Mathematical Sciences,
Indian Institute of Technology (B. H. U.), Varanasi India-221005
[email protected] (Rajeev) Abstract: We present approximate solutions of a two moving boundaries problem governed with fractional time derivative in Caputo sense of order α ∈(0,1] by using Adomian decomposition method and Optimal homotopy asymptotic method. This problem occurs in controlled drug release from a polymeric matrix. The results thus obtained are compared with the existing analytical solutions for the particular cases which are in good agreement. Analyses for the dependence of moving boundaries on some parameters are also performed. Keywords: Adomian decomposition method, optimal homotopy asymptotic method, two moving boundaries problem, fractional derivative, anomalous diffusion.
204
Carleman Estimate for a One-Dimensional System of m-Coupled Parabolic PDEs with BV Diffusion Coefficients
Hichem Ramoul Department of Mathematics and Informatics, Abbes Laghrour University, Khenchela, Algeria
Abstract: This work is devoted to deriving global Carleman estimate for a one-dimensional linear coupled parabolic system of m equations with bounded variations (BV) diffusion coefficients. This kind of estimate is a generalization of the scalar result [1]. The key ingredient, in a first step, is to derive a global Carleman estimate for piecewise- C1 diffusion coefficients based on the construction of a suitable weight function. The Carleman estimate in the case (BV) diffusion coefficients is then obtained using the approach of (BV) diffusion coefficients by piecewise-constant coefficients. This Carleman estimate is used to show the observability inequality which yields the controllability result. Keywords: Carleman estimate, Parabolic equations, BV coefficients, Controllability.
References: [1] J. Le Rousseau, “Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients”, J. Differential Equations. 233(2007), 417-447. [2] H. Ramoul, “Carleman estimate for a one-dimensional system of m coupled parabolic PDEs with BV diffusion coefficients”, Accepted in Boundary Value Problems, SpringerOpen.
205
Waterloo Numbers and Their Relation to Pascal Triangle and Polygons Wajdi Mohamed Ratemi and Otman Basir
Faculty of Engineering, University of Tripoli, Tripoli, Libya Electrical and Computer Engineering Department, University of Waterloo, Canada
[email protected], [email protected]
Abstract: The Pascal triangle is a well known and famous triangle which is dated long before Pascal has introduced it and got his name attached to it. Chinese, Persian, and Greeks had all dealt with the structuring of the numbers which are now known as the Pascal triangle. Many Properties have been noticed in the structure of the numbers in the Pascal triangle, Such as the Fibonacci numbers, the triangular numbers, the Hockey stick, the Sierpinski’fractals [1,2]. In this paper, further exploration is made for the Pascal triangle. The horizontal elements making the triangle represent the coefficients of nth powered binomial expansion of the form , or , and it has been noticed that the set of the diagonal elements as well as the vertical elements of the right angled Pascal triangle present the numbers of expansion terms of monomials, binomials, and polynomials of the form in a consecutive order . The values of those coefficients are ones for monomials, the horizontal elements of Pascal triangle for binomials, and for the higher polynomials the values of the coefficients are determined by the Embedded Pascal Triangles (EPTs) expansion method [3,4] which is an extension from special mathematical expansion called The Guelph Expansion method [5]. Those set of numbers determining the number of coefficients of the r-nomials are named as the Waterloo numbers (W-numbers), whereas the values of those numbers are called the attached values to waterloo numbers . Furthermore, the paper presents a geometrical representation to those set of numbers in a similar manner as the geometrical representation of Polygonal numbers. Possible applications for the Waterloo number and its attached values in pass wording. Keywords: Pascal’s Triangl; Embedded Pascal Triangles (EPTs); Guelph Expansion, Polynomial Expansions. References [1] J. H. Conway, R. K. Guy, “The Book of Numbers”, Springer-Verlag, NY, 1996. [2] Nathanson, B. Melvyn, “Additive Number Theory The Classical Bases”, Berlin: Springer, (1996). [3] W. M. Ratemi, “Embedded Pascal’s triangles for polynomial expansion” Abstract-19th International Workshop on Matrices and Statistics. Shanghai Finance University, China, pp: 45-46, 2010.. [4] B. Serge, M. Provost and Wajdi Ratemi, “Polynomial expansions via embedded Pascal's Triangles”, Acta et Commentationes Universitatis Tartuensis de Mathematica, Volume 15, Number 1, 2011. [5] W. M. Ratemi, H. Abdulla, “The Guelph expansion: a mathematical expansion for polynomial expansion”, Abstract- International Conference on Mathematical Sciences, Istanbul, p.381, (2009).
206
A Note on Derivations in Rings and Banach Algebras Mohd Arif Raza
Department of Mathematics, Aligarh Muslim University, Aligarh, India [email protected]
Abstract: Let 𝑅𝑅 be a prime ring, Utumi quotient ring 𝑈𝑈, Martindale quotient ring 𝑄𝑄, 𝑑𝑑 derivation of 𝑅𝑅, and 𝑚𝑚, 𝑛𝑛 are fixed positive integers. In this paper, we generalized the results obtained in [1] and [2]. Moreprecisely, we study the case when one of the following holds: (𝑖𝑖) 𝑑𝑑(𝑥𝑥)𝜊𝜊𝑛𝑛 𝑑𝑑(𝑦𝑦) = 𝑥𝑥𝜊𝜊𝑚𝑚𝑦𝑦 (𝑖𝑖𝑖𝑖) 𝑑𝑑(𝑥𝑥)𝜊𝜊𝑚𝑚𝑑𝑑(𝑦𝑦) = 𝑑𝑑(𝑥𝑥𝜊𝜊𝑦𝑦)𝑛𝑛, for all 𝑥𝑥, 𝑦𝑦 in some appropriate subset of 𝑅𝑅. We also examine the case where 𝑅𝑅 is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on Banach algebras. Keywords: Prime and semiprime rings; Derivation; Martindale ring of quotients; Banach algebra; Radical. References: [1] N. Argac, H.G. Inceboz,“Derivation of prime and semiprime rings”, J.Korean Math. Soc 46 (5) (2009), 997 1005. [2] M. Ashraf, N. Rehman, “On commutativity of rings with derivations”, Results Math. 42 (1-2) (2002), 3-8.
207
On Global Solutions of Fractional Evolution System Samira Rihani and Amor Kessab
Department of Mathematics, University of Sciences and Technology Houari Boumedien, Bab-Ezzouar, Algiers
Abstract : The aim of this research is to study the nonexistence of global solutions to the non-local in space syste of evolution equations by using the dualityargument combined with a convexity inequality [1, 3]. Our results can be seen as the continuation of those obtained by [2, 4]. Key words: Fractional Laplace operatorr, Nonlinear initial value problems, fractional derivatives, Blow-up. References: [1] C. Cordoba, D. Cordoba: A pointwise estimate for fractionary deriva-tives with applications to partial differential equations, Proc Natl Acad, Sci U S A. Dec 23 100(26) (2003), 15316–5317. [2] A. Hakem, M. Berbiche: Blow up results for fractional differential equations and systems, publication de l’institut math´ematique. 93 (107)(2013), 173-186. [3] N. Ju: The maximum principle and the global attractor for the dissipative 2-D quasi-geostrophic equations, Comm. Pure Appl. Ana (2005)161–181. [4] Q.S. Zhang: Blow Up and Global Existence of Solutions to an Inhomogeneous Parabolic System, Journal of Differential Equations 147(1998), 155–183.
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Risk Modeling Insurance IARD by Decision Trees and V-TEST Khadidja Sadi, Nora Lounnici and Hanya Kherchi
ENSSEA Algiers [email protected]
Abstract: Today, insurance companies have engorged masses of important data. Indeed, the low costs of the machines in terms of storage and power encouraged the companies to accumulate always more information. According to insurance companies, the estimate of the quantity of data collected in the world doubles every 20 months, whereas tapped information with added value of these data only increases very little. These silos of data must be explored in order to understand the direction of them and to detect the hidden relations, the rules, the tendencies, the models explaining their behavior… etc Accordingly, the constitution of a data base, gathering, in a homogeneous and coherent form, all the data of the insurance company, offers new prospects to the users, in particular in terms of extraction of knowledge thanks to the tools of Data Mining . The recent development and the analysis of the actuarial tools raise problems to which it is trying to apply the methods and techniques of data mining. The insurance companies are interested in the invoices most influential on the products of insurance damage IARD (Fire, Accidents and Risk Various), such as the products multirisk dwelling and professionals and test for there decreasing their gravities. In order to meet this aim, we put pursuant to the methods of dated mining within the framework from the training supervised for the determination the factors likely to influence the decision according to the branch of insurance IARD. On this subject, we give results of our experiments for the description and the characterization of the branch in question. In this context, we chose two methods of modeling: the value test and technique of the decision trees. The two models are applied to the data received from the national company of insurance (SAA).
209
Boundedness of Superposition Operators on the Double Sequence Spaces of Maddox ( )L p
Birsen Sagir and Nihan Gungor
Department of Mathematics, Ondokuz Mayıs University, Kurupelit, Samsun, Turkey Department of Mathematical Engineering, Gümüşhane University, Gümüşhane, Turkey
[email protected], [email protected]
Abstract: Sama-ae [2] have characterized local boundedness and boundedness of the superposition operator acting from the Maddox sequence space ( )l p into 1l . Sağır & Güngör [1] have defined the superposition operator gP
where 2:g ´ ®¥ ¡ ¡ by ( )( ) ( ), ,g ks ksP x g k s x= for all real double sequences ( )ksx . In this study, we have
generalized gP acting from Maddox double sequence spaces, ( ) ( ), 1
kspks ks
k sp x x x
¥
=
ì üï ï= = < ¥í ýï ïî þ
åL into qL where
( )ksp p= is a bounded sequence of positive numbers and 1 q£ < ¥ . The main goal of this study is constructing the
necessary and sufficient conditions for the local boundedness and boundedness of ( ):g qP p ®L L where 1 q£ < ¥ . Keywords: Superposition Operators; Local Boundedness; Boundedness; Double Sequence Spaces. References: [1] B. Sagir, N. Gungor, “Continuity Of Superposition Operators On The Double Sequence Spaces pL ”, (2014) to accepted Filomat. [2] A. Sama-ae, “Boundedness Of Superposition Operators On The Sequence Spaces Of Maddox”, Master Thesis, Chiang Mai University, 1997.
210
Some Convergence Results for Nonexpansive Mappings in Uniformly Convex Hyperbolic Spaces
Aynur Sahin and Metin Basarir Department of Mathematics, Sakarya University, Sakarya, Turkey
[email protected], [email protected]
Abstract: In this study, we establish some strong and D-convergence theorems of an iteration process for approximating a common fixed point of three nonexpansive mappings in a uniformly convex hyperbolic space. The results presented here extend and improve some well known results in uniformly convex Banach spaces as well as CAT(0) spaces. Keywords:D-convergence; Common fixed point; Nonexpansive mapping; Iterative process; Strong convergence; Uniformly convex hyperbolic space. References: [1] U. Kohlenbach, “Some logical metatheorems with applications in functional analysis”, Trans. Amer. Math. Soc. 357(1), 89-128 (2004) [2] K. Goebel, S. Reich, “Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings”, Marcel Dekker, New York (1984) [3] M. Bridson, A. Haefliger, “Metric Spaces of Non-Positive Curvature”, Springer, Berlin (1999) [4] T. Shimizu, W. Takahashi, “Fixed points of multivalued mappings in certain convex metric spaces”, Topol. Methods Nonlinear Anal. 8, 197-203 (1996) [5] SH. Khan, YJ. Cho, M. Abbas, “Convergence to common fixed points by a modified iteration process” J. Appl. Math. Comput. 35, 607-616 (2011) [6] L. Leustean, “Nonexpansive iterations in uniformly convex W-hyperbolic spaces”, In: Leizarowitz, A, Mordukhovich, BS, Shafrir, I, Zaslavski, A (eds.), Nonlinear Analysis and Optimization I: Nonlinear Analysis, Contemp. Math., Am. Math. Soc. AMS 513, 193-209 (2010) [7] AR. Khan, H. Fukhar-ud-din, MAA. Khan, “An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces”, Fixed Point Theory Appl. 2012 Article ID 54 (2012) doi:10.1186/1687-1812-2012-54
211
A Multi-Criteria Neutrosophic Group Decision Making Metod Based TOPSIS for Supplier Selection
Ridvan Sahin and Muhammed Yigider Department of Mathematics, Faculty of Science, Ataturk University, Erzurum
Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum [email protected], [email protected]
Abstract: The main purpose of this paper is to develop a neutrosophic technique for order preference by similarity to ideal solution (TOPSIS) method to improve the quality of decision making for ranking alternatives. The TOPSIS method based neutrosophic set is proposed to select appropriate supplier in group decision making. In decision making process, the neutrosophic weighted aggregation operators (arithmetic and geometric average operators) are employed to aggregate individual opinions of decision makers for rating the importance of criteria and alternatives. Finally, a numerical example is given to illustrate application of developed neutrosophic TOPSIS method for supplier selection. Keywords: Neutrosophic set, Supplier selection, TOPSIS method, Group decision making. References: [1] C.C. Li, Y.P. Fun, J.S. Hung, “A new measure for supplier performance evaluation”, IIE Transactions on Operations Engineering, 29(1997), 753–758. [2] H. Wang, F. Smarandache, Y.Q. Zhang and R. Sunderraman, “Single valued neutrosophic sets”, Multispace and Multistructure 4(2010), 410–413. [3] M.Y. Bayrak, N. Çelebi, H. Taşkın, “A fuzzy approach method for supplier selection”, Production Planning and Control: The Management of Operations, 18(2007), 54–63. [4] Z. Hongyu, J. Qiang Wang and X. Chen, “Interval Neutrosophic Sets and its Application in Multi-criteria Decision Making Problems”, The Scientific World Journal, (2013), to appear.
212
Existence and Stability of a Damped Wave Equation With Two Delayed Terms in Boundary
Zitouni Salah and Amiar Rachida University of Souk Ahras and University of Annaba
Abstract: This paper considers a linear damped wave equation with dynamic bouandary conditions where two feedback terms have a delay. In bounded domain, we .rst establish the question of well-posedness and uniqueness of the solution for the initial-boundary value problem, using semigroup argument. Next, by introducing suitable Lyapunov functionals, exponential stability estimates are obtained under conditions on the delay terms. References: [1] C. Abdallah, P. Dorato, J. Benitez-Read, and R. Byrne, Delayed positive feedback can stabilize oscillatory system, in: Proceedings of the 1993 American Control Conference, pp. 3106.3107, San Francisco, CA, USA, 1993. [2] K. T. Andrews, K. L. Kuttler and M. Shillor. Second order evolution equations with dynamic boundary conditions. J. Math. Anal. Appl.,197(3):781.795, 1996. [3] G. Chen. Control and stabilization for the wave equation in a bounded domain I-II. SIAM J. Control Optim., 17:66.81, 1979; 19:114.122, 1981.
213
Lp Boundedness for Marcinkiewicz Integrals and Extrapolation
Salti Samareh Department of Mathematics and Statistics Jordan University of Science and Technology, Jordan
Abstract: In this article, we study the Lp mapping properties of Marcinkiewicz integral operators with rough kernels. Our results are essential improvements and extensions of some known results on Marcinkiewicz integrals.
214
Approximation Properties of Max-Product Operators Engin Sari
Department of Mathematics, TOBB Economics and Technology University, Sogutozu, Ankara, Turkey [email protected]
Abstract: In this talk, we study some approximation properties of nonlinear max-product operators introduced by Bede et.al. [1], [2] and discuss their statistical approximation behavior given by Duman [3]. Some new applications of these operators based on q-integers are also presented. Since the notion of statistical convergence introduced by Fast [4] is weaker method than the ordinary convergence, its use in the approximation theory enables us many advantages. Keywords: Statistical convergence; Max-product operators; Shepard operators; Statistical rates.
References: [1] B. Bede, H. Nobuhara, M. Danková, A. Di Nola, “Approximation by pseudo-linear operators”, Fuzzy Sets & Systems, 159 (2008), 804-820. [2] B. Bede, H. Nobuhara, J. Fodor, K. Hirota, “Max-product Shepard approximation operators”, J. Adv. Comput. Intelligence Intelligent Informatics 10 (2006), 494-497. [3] O. Duman, “Statistical convergence of max-product approximating operators”, Turkish J. Math. 34 (2010), 501-514. [4] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2 (1951), 241-244.
215
Some Inequalities Associated with the Hermite-Hadamard-Fejér Type for Convex Function Mehmet Zeki Sarikaya, Hatice Yaldiz and Samet Erden
Department of Mathematics, Duzce University, Duzce-Turkey [email protected], [email protected], [email protected]
Abstract: In this talk, we extend some estimates of the right hand side of a Hermite- Hadamard-Fejér type inequality for functions whose first derivatives absolute values are convex.The results presented here would provide extensions of those given in earlier works. Keywords: Hermite-Hadamard-Fejer inequality; Trapezoid inequality; Convex function; Hölder inequality. References: [1] S. S. Dragomir, R.P. Agarwal, “Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula”, Appl. Math. lett., 11(5) (1998), 91-95. [2] S. S. Dragomir, C. E. M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Applications”, RGMIA Monographs, Victoria University, 2000. [3] L. Fejer, “Über die Fourierreihen”, II. Math. Naturwiss. Anz Ungar. Akad. Wiss., 24 (1906), 369--390. (Hungarian). [4] J. Pečarić, F. Proschan, Y.L. Tong, “Convex functions, partial ordering and statistical applications”, Academic Press, New York, 1991. [5] M. Z. Sarikaya, “On new Hermite Hadamard Fejer Type integral inequalities”, Studia Universitatis Babes-Bolyai Mathematica., 57(2012), No. 3, 377-386. [6] M.Z.Sarikaya, S. Erden, “On the weighted integral inequalities for convex function”, RGMIA Research Report Collection, 17(2014), Article 70, 10 pp. [7] M.Z.Sarikaya, S. Erden, “On the Hermite- Hadamard-Fejér type integral inequality for convex function”, RGMIA Research Report Collection, 17(2014), Article 69, 12 pp. [8] K-L. Tseng, G-S. Yang, K-C. Hsu, “Some inequalities for differentiable mappings and applications to Fejer inequality and weighted trapozidal formula”, Taiwanese J. Math. 15(4), pp:1737-1747, 2011. [9] S-R. Hwang, K-L. Tseng, K-C. Hsu, “Hermite-Hadamard type and Fejér type inequalities for general weights (I)”, J. of Inequalities and Applications 2013, 2013:170. [10] C.-L. Wang, X.-H. Wang, “On an extension of Hadamard inequality for convex functions”, Chin. Ann. Math. 3 (1982) 567--570.
216
Equivalent Norms in Nikol’skij-Triebel-Morrey Spaces and Lizorkin-Triebel-Morrey Spaces
Merey Sautbekova Department of Higher Mathematic, Eurasian national University named after L.N. Gumilyov, Astana, Kazakhstan
Abstract: Function spaces take an important role in the theory of functions. Since at least 60 years it has been well-known that the Nikol’skij-Besov spaces represent a useful notion of regularity not only in the context of approximation theory, but also in other branches of mathematics. A lot of developments have been made in this area by the works of P.I. Lizorkin, O.V. Besov, V.P. Il’in, S.M. Nikol’skij [1], H. Triebel [2], W. Sickel, H.-J. Schmeisser [3], Y. Sawano, V.I. Burenkov, M.L. Gol’dman and many other autors. In this paper, we consider equivalent norms in term of periodic Lizorkin-Triebel-Morrey and Nikol’skij-Besov-Morrey spaces. Keywords: Periodic Lizorkin-Triebel-Morrey spaces; Nikol’skij-Besov-Morrey spaces; Equivalent norms; Decomposition of unity. References: [1] O.V. Besov, V.P. Il’in and S.M. Nikol’skij “Integral representations of functions and imbedding theorems”, translated from Russian, Scripta series in Mathematics, Vol. I, II, V, 1978. [2] H. Triebel “Theory of function spaces”, Birkhauser, 1983. [3] H.-J. Schmeisser and W. Sickel “On strong summability of multiple Fourier series and approximation of periodic functions”, Math. Nachr., 133 (1987), 211-236. [4] Y. Sawano “Wavelet characterization of Besov-Morrey and Triebel-Lizorkin-Morrey spaces”, Funct. Approx. Comment. Math., 38 (2008), 93-107 [5] M. Sautbekova and W. Sickel “Strong summability of Fourier series and Morrey spaces”, Analysis Mathematica, 40 (2014), 31-62.
217
Ideal Statistical Quasi Cauchy Sequences Ekrem Savas and Huseyin Cakalli
Department of Mathematics, Istanbul Commerce University, Istanbul-Turkey Faculty of Arts and Sciences, Maltepe University, Maltepe, Istanbul, Turkey
ekremsavas@@yahoo.com, [email protected]
Abstract: We introduce S(I)-statistically ward compactness of a subset of R, the set of real numbers, and S(I)-statistically ward continuity of a real function in the senses that a subset E of R is S(I)-statistically ward compact if any sequence of points in E has an S(I)-statistically quasi-Cauchy subsequence, and a real function is S(I)-statistically ward continuous if it preserves S(I)-statistically quasi-Cauchy sequences. We investigate S(I)-statistically ward continuity, and prove interesting theorems. Keywords: ideal convergence, quasi Cauchy sequence, continuity References: [1] D. Burton, J. Coleman, “Quasi-Cauchy sequence”, Amer. Math. Monthly, 117.4(2010) 328-333. [2] A. Caserta, and Lj.D.R. Kocinac, “On statistical exhaustiveness”, Appl. Math. Lett., 25.10 (2012) 1447-1451. [3] H. Cakalli, Forward continuity”, J.Comput. Anal. Appl., 13.2(2011) 225-230. [4]H.Cakalli, “Statistical ward continuity”Appl.Math.Lett.24.10(2011)1724-1728. [5] H. Cakalli: “Statistical-quasi-Cauchy sequences”, Math. Comput. Modeling, 54. 5-6 (2011) 1620-1624. [6] P. Das, E. Savas: “On I-statistically pre-Cauchy sequences, Taiwanese J. Math. 18.1(2014) 115-126. [7] P. Das, E Savas, S. Kr. Ghosal, “On generalizations of certain summability methods using ideals” Appl. Math. Lett. 24.9 (2011) 1509-1514. [8] P. Kostyrko, T. Salat, and W. Wilczyacutenski: “I-convergence”, Real Anal. Exchange 26.2 (2000-2001) 669-686. [9] F. Nuray, and W.H. Ruckle, Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl., \textbf245 2 (2000) 513-527. [10] E. Savas, P. Das, “A generalized statistical convergence via ideals”, Appl. Math. Lett. 24.6 (2011), 826--830.
218
SBT - Hausdorff Space Guzide Senel and Naim Cagman
Amasya University, Department of Mathematics, Amasya, Turkey Gaziosmanpasa University, Department of Mathematics, Tokat, Turkey
Abstract: The concept of soft sets was introduced by Molodtsov [1] in 1999 as a general mathematical tool for dealing with uncertain objects. In 1963, Kelly [2] was defined bitopological space as an original and fundamental work by using two different topologies. In this paper, we first define soft bitopological (SBT) space on a soft set. Furthermore, the notions of soft open set, soft closed set, soft neighborhood, soft limit point and soft Hausdorff space are studied and several related properties of SBT – Hausdorff space are investigated. Keywords: Soft sets; Soft bitopology; Soft open-closed set; Soft neighborhood; Soft Hausdorff space. References: [1] D. A. Molodtsov, “Soft set theory-first results”, Computers and Mathematics with Applications 37 (1999) 19-31. [2] J. C. Kelly, “Bitopological Spaces”, Proceedings of the London Mathematical Society 13/3 (1963) 71-89.
219
Rate of Convergence of Generalized Favard-Szàsz Type Operators Sevilay Kirci Serenbay
Department of Mathematics Education, Başkent University, Bağlıca, Ankara, Turkey [email protected]
Abstract: In this paper, we give an estimate of the rate of convergence for modified Favard-Szàsz operators introduced by N. İspir and Ç. Atakut [4] of functions of bounded variation on [0,∞). In recent years, the rate of approximation of certain operators acting on functions of bounded variation has been intensively studied. Now, we mention some of the recently study on this topic. V. Gupta and N. İspir [1] estimated the rate of convergence for function of bounded variation for the generalized Kantorovich type Balazs operators. V. Gupta and R.P. Pant [2] obtained an estimate of the rate of convergence of modified Szàsz type operators on functions of bounded variation.
Keywords: Rate of convergence; bounded variation; Favard-Szàsz type operators. References: [1] V. Gupta, N. İspir, On the Bezier variant of generalized Kantorovich type Balazs operators, Appl. Math. Letters, 18(5) (2005), 1053-1061. [2] V. Gupta, R.P. Pant, Rate of convergence for the modified Szàsz-Mirakyan operators on functions of bounded variation, J. Math. Anal. Appl., 233, (1999), 476—483. [3] V. Gupta, Rate of approximation by a new sequence of linear positive operators, Comput. & Math. with Appl., 45(12), (2003), 895-1904. [4] N. İspir, Ç. Atakut, Approximation by modified Szàsz--Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. (Math. Sci.), 112(4), (2002),571—578.
220
Sufficient Tauberian Conditions for the Weighted Mean Summability of Sequences of Fuzzy Numbers
Sefa Anil Sezer, Ibrahim Canak and Zerrin Onder Department of Mathematics, Ege University, Bornova, Izmir, Turkey
Department of Mathematics, Istanbul Medeniyet University, Kadikoy, Istanbul, Turkey [email protected]
Abstract: In recent years there has been an increasing interest focusing on the summability methods of sequences of fuzzy numbers. Firstly, Subrahmanyam [1] has defined the Cesaro summability method for sequences of fuzzy numbers and obtained fuzzy analogues of some classical Tauberian theorems. Later, Talo and Çakan[2], Talo and Başar[3] and Çanak[4] have established some Tauberian theorems for Cesaro summability method of sequences of fuzzy numbers. In addition, Tripathy and Baruah [5] have introduced Nörlund and Riesz summability methods for sequences of fuzzy numbers and proved a Tauberian theorem for the Riesz summability method. In this paper, a Tauberian theorem of slowly decreasing type is proved for the weighted mean method of summability of sequences of fuzzy numbers. Keywords: Tauberian theorems; Weighted means; Sequences of fuzzy numbers. References: [1] P. V. Subrahmanyam, “Cesaro summability for fuzzy real numbers”, J. Anal. 7(1999), 159-168. [2] O. Talo, C. Cakan, “On the Cesaro convergence of sequences of fuzzy numbers”, Appl. Math. Lett., 25.4(2012), 676-681. [3] O. Talo, F. Basar, “On the slowly decreasing sequences of fuzzy numbers”, Abstr. Appl. Anal., 7(2013), Art. ID 891986. [4] I. Canak, “Tauberian theorems for Cesaro summability of sequences of fuzzy numbers”, J. Intell. Fuzzy Systems, DOI:10.3233/IFS-131053. [5] B. C. Tripathy, A. Baruah, “Nörlund and Riesz mean of sequences of fuzzy real numbers”, Appl. Math. Lett., 23.5(2010), 651-655.
221
On Fine Spectra and Subspectrum (Approximate Point, Defect and Compression) of Operator with Periodic Coefficients
Necip Simsek and Vatan Karakaya Department of Mathematics, Istanbul Ticaret University, Uskudar, Istanbul, Turkey
Department of Mathematical Engineering, Yıldız Technical University, Esenler, Istanbul, Turkey [email protected], [email protected]
Abstract: The main purpose of this work is to determine of fine spectra and subspectra such as approximate point spectrum, defect spectrum and compression spectrum of the difference operator with periodic coefficients over the sequence spaces. Keywords: Fine spectra, difference operator, infinite matrices, sequence spaces. References: [1] Akhmedov, A.M., El-Shabrawy, S.R., On the fine spectrum of the operator over the sequence space c, Computers and Mathematics with Applications, 61(2011), 2994-3002. [2] H Akhmedov, A.M., Başar, F., On spectrum of the Cesaro operator, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 19(2003), 3-8. [3] Appell, J., Pascale, E., Vignoli, A., Nonlinear Spectral Theory, Walter de Gruyter Berlin, New York, 2004. [4] Akhmedov, A.M., Başar, F., On spectra of the difference operator over the sequence space (1≤p<∞), Demonstratio Math., 39(2006), 585-595. [5] Başar F. , Durna N. and Yildirim M. , Subdivision operator over certain sequence spaces, Thai J. Math., 9 (2011) 2, 285-295.
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222
Pretopologies for Structuring the Digital Plane Josef Slapal
Department of Mathematics, Brno University of Technology, Brno, Czech Republic [email protected]
Abstract: To be able to study and process digital images, we need to provide the digital plane Z2 with a convenient structure. Here, convenience means that such a structure satisfes, in analogies, some basic geometric properties of the Euclidean plane R2. First of all, it is required that an analogue of the Jordan curve theorem be valid. In the classical approach (see [2]), graph theoretic tools are used for structuring Z2, namely the well-known binary relations of 4-adjacency and 8-adjacency. Unfortunately, neither 4-adjacency nor 8-adjacency itself allows an analogue of the Jordan Curve Theorem, so that a combination of the two adjacencies has to be used. To eliminate this inconvenience, a new, purely topological approach to the problem was proposed in [1] which utilizes a so-called Khalimsky topology for structuring the digital plane. This approach was then developed by many authors. In [3], it was shown by the author of this note that it may be advantageous to use closure operators (more general than topologies) for structuring the digital plane. In the present note, we study a special type of such operators on Z2, namely the pretopologies. We introduce a certain natural graph on the vertex set Z2 whose cycles are eligible for Jordan curves in Z2 and we solve the problem of finding pretopologies on Z2 with respect to which these cycles are Jordan curves. Keywords: Digital plane; Jordan curve; Pretopology; Connectedness graph. References: [1] E.D. Khalimsky, R. Kopperman, P.R. Meyer, “Computer graphics and connected topologies on finite ordered sets”, Topology Appl. 36(1990), 1-17. [2] A. Rosenfeld, “Digital topology”, Amer. Math. Monthly 86(1979), 621-630. [3] J. Slapal, “Closure operations for digital topology”, Theor. Comp. Sci. 305 (2003), 457—471.
223
Linear Parabolic Boundary-Value Problems in Generalized Morrey Spaces Lubomira Softova
Department of Civil Engineering, Design, Construction and Environment, Second University of Naples, Italy [email protected]
Abstract: We prove continuity in generalized Morrey spaces of sub-linear operators generated by the Calderon-Zygmund singular integrals with parabolic type kernel and commutators of these integrals with BMO functions. As a consequence we obtain global regularity of the strong solutions to the Cauchy - Dirichlet problem and the oblique derivative problem for linear uniformly parabolic equations with VMO coefficients. Keywords: Calderon – Zygmund singular integrals; Commutators; A priori estimates; Generalized Morrey spaces. References: [1] V. Guliyev, L. Softova, “Generalized Morrey regularity for parabolic equations with discontinuous data”, Proceedings of the Edinburgh Mathematical Society, (to appear); [2] L. Softova, “Parabolic oblique derivative problem with discontinuous coefficients in generalized Morrey spaces”, Ricerche di Matematica, Vol. 62, 265-278, (2013).
224
Main Chaos-Based Image Encryption Algorithms Ibtissem Talbi
Department of Mathematics, Constantine 1 University, Constantine, Algeria [email protected]
Abstract: During the last decade, a variety of chaos-based cryptosystems have been investigated. Most of them are based on the structure of Fridrich, which is based on the traditional confusion-diffusion architecture proposed by Shannon. Compared with traditional cryptosystems (DES, 3DES, AES, etc.), the chaos-based cryptosystems are more flexible, more modular and easier to be implemented, which make them suitable for large scale-data encyption, such as images and videos. The heart of any chaos-based cryptosystem is the chaotic generator and so, a part of the efficiency (robustness, speed) of the system depends greatly on it. In this talk, we give an overview of the state of the art of chaos-based block ciphers and we describe some of our schemes already proposed. Also we will focus on the essential characteristics of the digital chaotic generator, The needed performance of a chaos-based block cipher in terms of security level and speed of calculus depends on the considered application. There is a compromise between the security and the speed of the calculation. The security of these block block ciphers will be analyzed. Keywords: Chaos-based cryptosystems; Chaotic generator; Security analysis. References: [1] C. Shannon, "Communication Theory of Secrecy Systems". System Technical Journal, vol. 28, no.4, pp. 656-715. [2] J. Fridrich, “Symmetric ciphers based on two-dimensional chaotic maps", Int. J. Bifurcation and chaos, vol. 8, no. 6, 1998, pp. 1259-1284. [3] G. Chen, Y. Mao, C. K. Chui, "A symmetric image encryption schemes based on 3D chaotic cat maps", Chaos Solutions and Fractals, vol. 21, 2004, pp. 749-761. [4] S. Lian, J. Sun, Z. Wang, "A bloc cipher based on a suitable use of the chaotic standard map". Chaos Solutions and Fractals, vol. 26, 2005, pp. 117-129. [5] K-W. Wong, B. S-H. Kwok, W-S. Law, "A fast image encryption scheme based on chaotic standard map". Physics Letters A, vol. 372, no. 15, 2008, pp. 2645-2652.
225
The Way-Below Soft Set Relation Bekir Tanay and Gozde Yaylali
Department of Mathematics, Mugla Sitki Kocman University, Mugla, Turkey [email protected] [email protected]
Abstract: Order is in everywhere in mathematics and related fields. We use order on the soft set which is introduced by Molodtsov [3]. Firsthly, Babitha [5] gave the definition of soft set relation, then Babitha and Sunil [8] introduced partially ordered soft set. We introduced directed soft sets in [13]. Moreover, definition of soft lattice, complete soft lattice were given in [12]. In this paper, way-below soft set relation, a new soft set relation, is introduced. Also, soft continuous directed complete soft sets and soft domains are defined. Moreover some important results are obtained and proved in this study. Keywords: Soft set; Soft Set Relation; Directed Soft Set; Soft Domain; Way-Below Soft Set Relation. References: [1] K. Hrbacek, T. Jech, “Introduction to Set Theory”, Marcel Dekker Inc., 1984. [2] G. Gierz, K.H Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott, “Continuous Lattices and Domains”, Encyclopedia of Mathematics and its Applications 93 [3] D. Molodtsov, “Soft Set Theory-First Results”, Comput. Math.Appl. 37 (1999) 19-31. [4] P. K. Maji, R. Biswas, A. R. Roy, “Soft set theory, Comput”. Math.Appl.45 (2003) 555-562. [5] K. V. Babitha, J. J. Sunil, “Soft Set Relations and Functions”, Comput. Math.Appl. 60 (2010) 1840-1849 [6] Fu Li, “Soft Lattices”, Global Journal of Science Frontier Research 56 Vol.10 Issue 4(Ver 1,0) (2010) 7] Y. B. Jun, K. J. Lee, A. Khan, “Soft Ordered Semigroups”, Math. Log Quart. 56, No.1, 42-50(2010) [8] K. V. Babitha, J. J. Sunil, “Transitive Closures and Ordering on Soft Sets”, Comput. Math.Appl. 62 (2011) 2235-2239 . [9] M. I. Ali, “Soft Ideals and Soft Filters of Soft Ordered Semigroups”, Comp. Math. Appl. 62(2011)3396-3403 [10] H. Yang, Z. Guo, “Kernels and Closures of Soft Set Relations”, and Soft Set Relation Mappings, Comp. Math. Appl. 61(2011)651-662 [11] J. H. Park, O. H. Kim, Y. C. Kwun, “Some properties of soft set relations”, Compt. Math. Appl. 63 (2012) 1079-1088 [12] B. Tanay, G. Yaylalı, “Some New Results on Orderings on Soft Sets”, 2.nd International eurasian conference on mathematical sciences and appli- cations, (August, 2013). [13] B. Tanay, G. Yaylalı , “New structures On Partially Ordered Soft Sets and Soft Scott Topology”, Ann. Fuzzy Math. Inform., 7 89-97 (2014).
226
A New Operator Splitting Method for Non-Linear Systems and Its Abstract Analysis Gamze Tanoglu and Sila Ovgu Korkut
Department of Mathematics, Izmir Institute of Technology, Izmir, Turkey Izmir Katip Celebi University, Science of Engineering, Izmir, Turkey
Abstract: The iterative splitting methods have been extensively applied to solve complicated systems of differential equations. In this process, we split the complex problem into several sub-problems, each of which can be solved sequentially. The purpose of this article is to construct a new scheme, which is based on the iterative splitting scheme, for solving non-linear systems. We also investigate its convergence properties in abstract framework by using the concepts of stability, consistency, and order. Several numerical examples are illustrated to confirm the theoretical results by comparing frequently used methods. We also show that the proposed method converges with the expected rates. Keywords: Nonlinear Magnus Integrator; Operator splitting method; Iterative splitting; Non-linear system. References: [1] V. C. Aguilera - Navarro, G. A. Estévez, R. Guardiola, “Variational and perturbative schemes for a spiked harmonic oscillator”, J. Math. Phys. 31, 99 (1990). [2] C. M. Bender, S. A. Orszag. “Advanced Mathematical Methods for Scientists and Engineers”, Springer-Verlag, New York, (1999) [3] H. Berland, B. Skaflestad. “Solving the nonlinear Schrödinger equation using exponential integrators”, Technical Report 3/05, The Norwegian Institute of Science and Technology, 2005. http://www.math.ntnu.no/preprint/. [4] S. Blanes, P. C. Moan. “Splitting Methods for Non-autonomous Hamiltonian Equations”, Journal of Computational Physics 170 (2001), pp. 205–230. [5] F. Casas, A. Iserles, “Explicit Magnus expansions for nonlinear Equations”, J. Phys. A: Math. Gen. 39 (2006), pp. 5445–5461. [6] S.A. Chin, P. Anisimov, “Gradient Symplectic Algorithms for Solving the Radial Schrödinger Equation” , J. Chem. Phys. 124, (2006), 054106. [7] S. Descombes, M. Thalhammer, “An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime”, BIT. Numerical Mathematics, 50, no. 4, pp.729–749 (2010). [8] J. Geiser, “Nonlinear iterative operator splitting methods and applications for nonlinear differential equations”, Proc. Appl. Math. Mech., 7 (2007), pp. 1041205-1041206, DOI 10.1002/pamm.200700613. [9] G. Goldstein, D. Baye, “Sixth-order factorization of the evolution operator for time-dependent potentials”, Phys. Rev E, 70 (2004), 056703. [10] M.Hochbruck, A. Ostermann, “Exponential Integrator”, Cambridge University Press, Acta Numerica,(2010), pp.209-286, DOI:10.1017/S0962492910000048.
227
A New Three Step Iteration and Convergence of Pseudocontractive Mappings Balwant Singh Thakur
School of Studies in Mathematics, Pt.Ravishankar Shukla University, Raipur, India [email protected]
Abstract: In this paper, we study a new three step iteration process for pseudocontractive mappings and prove convergence result for a countable family of pseudocontractive mappings. Our results improve and generalize most of the results that have been proved for this important class of nonlinear mappings. We also compare the performance of iteration by an example. Keywords: Strong convergence; Pseudocontractive mapping; Iteration process.
228
Contrastings on Textual Entailmentness and Algorithms of Syllogistic Logics Selcuk Topal and Tahsin Oner
Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected]
Abstract: First attempt of completeness of Aristotelian syllogistic was introduced by Łukasiewicz [1]. After RD and D systems which was introduced by Corcoran and their semantics had altered point of view of Aristotelian syllogistic [2], Westershål revised the classical square of opposition [3], [4]. Moss studied on completeness of syllogistic fragments [5] and Winter initiated to construct semantics of fragments of some English sentences [6]. In this paper, we shall study compatibilty between semantics of natural languages and syllogistic logics. We also introduce contrast amongs algorithms of classical syllogistics and modern ones . Keywords: Logic; Applied Logic; Natural Logic; Algorithms; Logic in Computer Science; Computational Linguistics. References: [1] J. Łukasiewicz, “Aristotle’s syllogistic from the standpoints of modern formal logic”, Oxford, Clarendon Press, (1951), 141pp. [2] E. J. Andrade, E. Becerra, “Corcoran’s Aristotelian syllogistic as subsystem of first order logic”, Revista Colombiana de Matemáticas, 41(2007), 67-80. [3] D. Westershål, “Aristotelian syllogism and generalized quantifiers”, Studia Logica, 48(4) (1989), 577-585. [4] D. Westershål, “On the Aristotelian square of opposition”, Kapten Mnemos Kolumbarium, en festskrift med anledning av Helge Malmgrens 60-ársdag (2005). [5] L. S. Moss,“Completeness theorems for syllogistic fragments”,Logics for linguistic structures, 29(2008),143-173. [6] Y. Winter, “Elements of formal semantics”, Edinburgh University Press, (2013).
229
Some Tauberian Conditions for the Weighted Mean Method of Summability Umit Totur and Ibrahim Canak
Department of Mathematics, Adnan Menderes University, Aydin, Turkey Department of Mathematics, Ege University, Bornova, Izmir, Turkey
[email protected], [email protected]
Abstract: Let be a sequence of nonnegative numbers and as Let the
weighted general control modulo of the oscillatory behavior of integer order of a sequence be denoted
by . We prove that if the weighted generator sequence of a sequence of real numbers is
summable to a finite number by the weighted mean method, is increasing, the conditions
and
are satisfied, and certain conditions on are hold, then is slowly oscillating. Keywords: Tauberian theorems; Weighted means; Weighted general control modulo; Slowly decreasing sequence. References: [1] G. H. Hardy, “Divergent series”, Clarendon Press, Oxford, 1949. [2] I. Canak, U. Totur, “Some Tauberian theorems for the weighted mean methods of summability”, Comput. Math. Appl., 62 (2011), 2609-2615. [3] U. Totur,I. Canak, “Some general Tauberian conditions for the weighted mean summability method”, Comput. Math. Appl., 63 (2012), 999-1006. [4] R. Schmidt, “Über divergente Folgen und lineare Mittelbildungen”, Math. Z., 22 (1925) , 89-152. [5] I. Canak, U. Totur, “An Extended Tauberian theorem for the weighted mean method of summability”, Ukrainian Math. J., 65 (2013), 1032-1041.
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230
A Tauberian Theorem for the Power Series Method of Summability Umit Totur and Ibrahim Canak
Department of Mathematics, Adnan Menderes University, Aydin, Turkey Department of Mathematics, Ege University, Izmir, Turkey
[email protected], [email protected]
Abstract: Let (𝑢𝑢𝑛𝑛) be a sequence of real numbers. Assume that p = (pn) be a sequence of nonnegative numbers with p0 > 0, that Pn: = ∑ n
k=0 pk → ∞ as n → ∞, and that p(x) = ∑ ∞k=0 pkxk < ∞ 𝑓𝑓𝐿𝐿𝑟𝑟 0 ≤ 𝑥𝑥 < 1. If ∑ ∞
k=0 pkukxk is convergent for 0 ≤ x < 1, and
limx→1−1
p(x)∑ ∞
k=0 pkukxk = s, then we say that (un) is summable to s by the power series method (J, p). In this work, we introduce a one-sided Tauberian condition in terms of the weighted general control modulo of integer order 𝑚𝑚 ≥ 1 for the power series method of summability. Keywords: Weighted means; Weighted general control modulo; Slowly decreasing sequence; Tauberian theorems; Power series method. References: [1] G. A. Mikhalin, “Theorem of Tauberian type for (𝑱𝑱, 𝒑𝒑𝒏𝒏) summation methods”, Ukrain. Mat. Zh. 29 (1977), 763–770. English translation: Ukrain. Math. J. 29 (1977), 564–569. [2] U. Totur, I. Canak, “Some general Tauberian conditions for the weighted mean summability method”, Comput. Math. Appl. 63 (5) (2012), 999–1006. [3] I. Canak, U. Totur, “Tauberian theorems for the (𝑱𝑱, 𝒑𝒑) summability method”, Appl. Math. Lett. 25 (10) (2012), 1430–1434. [4] G. H. Hardy, “Divergent series”, Clarendon Press, Oxford, 1949. [5] H. Tietz, “Schmidtsche Umkehrbedingungen für Potenzreihenverfahren”, Acta Sci.Math.,54 (3-4)(1990)355-365.
231
Some Tauberian Theorems for the Logarithmic Integrability Method Umit Totur and Muhammet Ali Okur
Department of Mathematics, Adnan Menderes University, Efeler, Aydin, Turkey [email protected]
Abstract: Let 𝑓𝑓 be a real valued function which is continuous on [1, ∞] and 𝑠𝑠(𝑥𝑥) = ∫ 𝑓𝑓(𝑡𝑡)𝑑𝑑𝑡𝑡.𝑥𝑥1 The logarithmic
mean of 𝑠𝑠(𝑥𝑥) is defined by 𝜎𝜎𝑙𝑙(𝑥𝑥) = 1log 𝑥𝑥 ∫ 𝑠𝑠(𝑡𝑡)
𝑡𝑡𝑥𝑥
1 𝑑𝑑𝑡𝑡. The integral ∫ 𝑓𝑓(𝑡𝑡)𝑑𝑑𝑡𝑡∞1 is said to be integrable by logarithmic
method to a finite number 𝑠𝑠 if lim𝑥𝑥→∞ 𝜎𝜎𝑙𝑙 (𝑥𝑥) = 𝑠𝑠. (1) If the integral ∫ 𝑓𝑓(𝑡𝑡)𝑑𝑑𝑡𝑡 = 𝑠𝑠∞
1 (2) exist, then limit (1) also exist. The converse is not always true. However, with some additional conditions it can be true. These conditions and theorems are called as Tauberian conditions and Tauberian theorems, respectively. In this paper, we give some Tauberian theorems for the logarithmic integrability of 𝑠𝑠(𝑥𝑥) with some Tauberian conditions. Keywords: Tauberian theorem; Tauberian condition; Logarithmic method; Slowly oscillating sequence; Slowly decreasing sequence; One-sided condition. References: [1] K. Ishiguro, “Tauberian theorems concerning the summability method of logarithmic type”, Proc. Japan. Acad. 39 1963 156-159. [2] B. Kwee, “A Tauberian theorem for the logarithmic method of summations”, Proc. Cambridge Philos. Soc. 63 1966 401-405 [3] B. Kwee, “Some Tauberian theorems for the logarithmic method of summability”, Canad. J. Math. 20 1968 1324-1331. [4] B. Kwee, “On generalized logarithmic methods of summation”, J. Math. Anal. Appl. 35 1971 83-89. [5] I. Canak, U. Totur, “Tauberian conditions for Cesàro summability of integrals”, Appl. Math. Lett. 24 (6) (2011) 891-896.
232
Eigenvalues for Kaehler Graphs of Connected Product Type Yaermaimaiti Tuerxunmaimaiti
Division of Mathematics & Mathematical Science, Nagoya Institute of Technology, Japan [email protected]
Abstract: Graphs which consist of sets of vertices and sets of edges are considered as discrete models of Riemannian manifolds of nonpositive curvature, and paths on graphs are regards as geodesics. In [4] we introduced discrete models of Riemannian manifolds admitting magnetic fields. We call a graph Kaehler if its set of edges is divided into two disjoint subsets, the set of principal edges and the set of auxiliary edges. Just like motions of electric charged particles, paths on the principal graph is bended under the influence of magnetic fields. In order to show how paths are bended we use paths on the auxiliary graph. We defined Laplacians of a Kaehler graph by use of adjacency operators and transition operators of the principal and the auxiliary graphs. In this paper we give some ways to construct Kaehler graphs and study their eigenvalues. As an application we give some examples of isospectral pairs of Kaehler graphs. Keywords: Kaehler graphs; Principal graphs; Auxiliary graphs; Laplacians; Isospectrality; Regular graphs; Bipartite double; Product graphs. References: [1] A.E. Brouwer, W.H. Haemers, “Spectra of graphs”, Springer 2012. [2] H. Fujii, A. Katsuda, “Isospectral graphs and the isoperimetric constant”, Discrete Math. 207(1999), 33-52. [3] T. Sunada, “Riemannian covering and isospectral manifolds”, Ann. Math. 121(1985), 169-186. [4] T. Yaermaimaiti, T. Adachi, “Isospectral Kaehler graphs”, preprint.
233
Generalized Weighted Norlund Mean and Ideal Convergence Orhan Tug and Feyzi Basar
Department of Mathematical Education, Ishik University, Erbil, Iraq Department of Mathematics, Fatih University, Turkey
[email protected], [email protected] Abstract: In this work, we introduce the generalized weighted Nörlund mean as a new summability method. We give some properties of this method and obtain some results by using ideal convergence and Nörlund mean. References: [1] B.Altay, F.Başar, Some Euler sequence spaces of non-absolute type, Ukrainian Math. J. 57 (1) (2005), 1-17. [2] F.M. Mears, The inverse Nörlund mean, Ann. Math. 44 (3) (1943), 401-409. [3] C.S. Wang. On Nörlund sequence space, Tamkang J. Math. 9 (1978), 269-274. [4] F.M. Mears, Some multiplication theorems for the Nörlund mean, The Ggeorge Washington University, Decemberr,1935, 875-880. [5] B. Altay, F. Başar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, J. Math. Anal. Appl. 336 (1) (2007), 632-645. [6] A. Wilansky, Summability through Functional Analysis, North-Holland Mathematics Studies 85, Amsterdam ∙ Newyork ∙ Oxford, 1984. [7] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, New York, 2000. [8] F. Başar, Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, İstanbul, 2012. [9] E.A. Aljimi, V. Loku, Generalized weighted Nörlund-Euler statistically convergence, Int. J. Math. Anal. 8 (7) (2014), 345-354. [10] E.A. Aljimi, V. Loku, The rate of weighted Nörlund-Euler statistically convergence, Int. J. Math. Anal. 8 (26) (2014), 1297-1304.
234
On Convergence Methods of Functions Defined on Time Scales Ceylan Turan and Oktay Duman
Department of Mathematics, TOBB Economics and Technology University, Sogutozu, Ankara, Turkey [email protected], [email protected]
Abstract: In [4], [5], we introduced the concepts of statistical convergence and lacunary statistical convergence of a function defined on a time scale, which cover the classical ones examined in [1], [2], [3]. The main purpose of this presentation is to give some recent results on these convergence methods and to discuss their applications. Keywords: Time Scale; Statistical Convergence; Lacunary statistical convergence. References: [1] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2 (1951), 241-244. [2] J. A. Fridy, C. Orhan, “Lacunary statistical convergence”, Pacific J. Math. 160 (1993), 43-51. [3] F. Móricz, “Statistical limits of measurable functions”, Analysis 24 (2004), 207-219. [4] C. Turan, O. Duman, “Statistical convergence on time scales and its characterizations”, Springer Proc. Math. Stat. 41 (2013), 57-71. [5] C. Turan, O. Duman, “Convergence methods on time scales”, AIP Conf. Proc. 1558 (2013), 1120-1123.
235
Classification of Integral Curves of a Linear Vector Field on (2n+1) Dimensional Semi
Euclidean Space Tunahan Turhan and Nihat Ayyildiz
Seydişehir Vocational School, Necmettin Erbakan University, Seydisehir, Konya, Turkey Department of Mathematics, Suleyman Demirel University, Isparta, Turkey
[email protected], [email protected]
Abstract: In this work, we give some results on classification of integral curves or flow lines of a linear vector field in (2n+1)-dimensional semi-Euclidean space The skew symmetric matrix has been found depending on the number of timelike vectors are odd or even. Taking into consideration of the structure, we obtained the linear first order system of differential equations. Then, we give solution of these systems and some examples. 2n 1 E . Keywords: Integral curve, linear vector field, semi-Euclidean space, skew-symmetric matrix. References: [1] A., Acratalishian, On the Linear Vector Field in E²ⁿ⁺¹, Commun. Fac. Sci. Ank., 1989, 39, 21-35. [2] A., Galbis M., Maestre, Vector Analysis Versus Vector Calculus, Springer, London, 2012, 375 pp. [3] A., Karger, J., Novak, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, 1978, 422 pp. [4] A., Yücesan, A.C., Çöken, N., Ayyıldız, G.S., Manning, On the Relaxed Elastic Line on Pseudo-Hypersurfaces in Pseudo-Euclidean Spaces, Applied Mathematics and Computation (AMC), 2004, 155(2), 353-372. [5] B., O'Neill, Semi-Riemann Geometry: with Applictions to Relativity. Academic Pres, New York, 1983, 469 pp. [6] T., Yaylacı, Linear Vector Fields and Applications, MSc thesis, Ankara Üniversity, The Institute of Science, Ankara, 2006, 49 pp.
236
More Results on the Upper Solution Bounds of the Continuous Algebraic Riccati Matrix Equation
Zubeyde Ulukok Selçuk University, Science Faculty, Department of Mathematics, 42075, Konya, Turkey.
Abstract. The algebraic Riccati and Lyapunov matrix equations are widely used and play an important role in various of engineering such as control system design and analysis [7, 8, 11, 14], and signal processing [17]. The continuous algebraic Riccati and Lyapunov matrix equations that we generally encounter in the literature are defined as below: The continuous algebraic Riccati matrix equation (CARE) is T TPA A P PBB P Q+ - = - (1)
where n nA ´Î ¡ and n mB ´Î ¡ are constant matrices, n nQ ´Î ¡ is a given positive semidefinite matrix, and the
matrix n nP ´Î ¡ is the unique positive semidefinite solution of the CARE (1). When 0B = and A is stable matrix, the CARE (1) becomes the continuous algebraic Lyapunov matrix equation (CALE) TA P PA Q+ = - . (2)
It is well known that the unique positive semidefinite solution P to the CARE (1) exists if the pair ( ),A B is
controllable (stabilizable) and the pair ( )1 2,A Q is observable (detectable). The computing of these equations’s analytic solutions are rather complicated in applications when the dimensions of system matrices are high. Therefore, in order to save time and decrease the burden of computation, instead of the exact solution, only the bounds as an approximation of the exact solution are sometimes needed. For example, for some applications such as stability analysis [16], without the burden of hard calculations, bounds are needed only for solution matrices. Furthermore, the solution bounds of the CARE (1) can be used to treat many control problems [4]. In this study, by constructing different equivalent forms of the continuous algebraic Riccati matrix equation (CARE) and using some linear algebraic techniques, we present the upper matrix bounds which depend on any positive definite matrix for the solution of the CARE. Based on these bounds, we develope iterative algorithms to obtain more sharper solution bounds. Furthermore, we give numerical examples to demonstrate that the new bounds are tighter than previous results in many cases. Keywords: Continuous algebraic Riccati matrix equation, Continuous algebraic Lyapunov matrix equation, Matrix bound. References: [1] M. Basin, J. Rodriguez-Gonzaleza, and L. Fridman, “Optimal and Robust Control for Linear State-Delay Systems”, Journal of the Franklin Institute, 344, (2007), 830-845. [2] D. S. Bernstein, Matrix Mathematics: “Theory, Facts and Formulas with Application to Linear Systems Theory”, Princeton University Press, Princeton, NJ, USA, 2005. [3] C. Y. Chen, C. H. Lee, “Explicit Matrix Bounds of the Solution for the Continuous Riccati Equation”, ICIC Express Letters, 3, (2009), 147-152. [4] R. Davies, P. Shi, and R. Wiltshire, “New Upper Solution Bounds for Perturbed Continuous Algebraic Riccati Equations Applied to Automatic Control”, Chaos, Solutions and Fractals, 32, (2007), 487-495. [5] R. Davies, P. Shi, R. Wiltshire, “New Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation”, Internat. J. of Control, Automat. and Systems, 6,(2008), 776-784. [6] S.W. Kim, P.G. Park, “Upper Bounds of the Continuous ARE Solution”, IEICE Trans. Fundam. Electron. Comm.
237
Global Attractors for Quasilinear Parabolic-Hyperbolic Equations Governing Longitudinal Motions of Nonlinearly Viscoelastic Rods
Suleyman Ulusoy Department of Mathematics, Zirve University, Gaziantep, Turkey
Abstract: We prove the existence of a global attractor and estimate its dimension for a general family of third-order quasilinear parabolic-hyperbolic equations governing the longitudinal motion of nonlınearly viscoelastic rods subject to interesting body forces and end conditions. The simplest version of the equations has the form w_tt = n(w_x,w_xt)_x where n is defined on (0,\infty)\times\bb R and is a strictly increasing function of each of its arguments, with $n \to -\infty$ as its first argument goes to 0. This limit characterizes a total compression, a source of technical difficulty, which new delicate a priori estimates prevent. We determine how the dimension of the attractor varies with several parameters of the problem giving conditions ensuring that the dimension is small. These estimates of dimension illuminate asymptotic analyses of the governing equation as parameters approach certain limits. This is a joint work with Stuart S. Antman. This work has been supported by TÜBİTAK grant 112T237. Keywords: Nonlinearly viscoelastic rods; Dimension of attractors; Quasilinear parabolic-hyperbolic equation. References: [1] S. S. Antman and T. I. Seidman, “Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity”, J. Differential Equations, 124(1996), 132-185. [2] S. S. Antman and S. Ulusoy, “The asymptotics of heavily burdened viscoelastic rods”, Quarterly J. Appl. Math, 170(2012), 437-467. [3] S. S. Antman and S. Ulusoy, “Global Attractors for Quasilinear Parabolic-Hyperbolic Equations Governing Longitudinal Motions of Nonlinearly Viscoelastic Rods”, Physica D , to appear.
238
()-boundedness of localization operators involving watson transform associated to regular representations and wavelet multipliers
S.K.Upadhyay Department of Mathematical Sciences,
IIT (BHU) and CIMS-DST (BHU), Varanasi - 221 005 (India)
E-mail address: [email protected] Abstract : ()-boundedness of localization operators associated to left reg-ular representations of locally compact group is investigated and using the theory of Watson Transformation relation between wavelet multipliers and localization operators is found. References: [1] F.M. Cholewinski, A Hankel convolution complex inversion. Theory. Mem. Amer. Math.Soc.58 (1965). [2] Richard R.Goldberg, Watson transform on groups, Annals of Mathematics, Second Series, 71(3) (1960), 522-528. [3] D.T. Haimo, Integral equation associated with Hankel convolution. Trans. Amer. Math.Soc.116 (1965), 330-375. [4] Hirschman Jr. I.I. Variation diminishing Hankel Transform. J. Analyse Math. 8 (1960-1961),307-336.
239
A Maximum Modulus Estimate for the Steady Stokes Equations Werner Varnhorn
Institute of Mathematics, Faculty of Mathematics and Natural Sciences, Kassel University, Germany [email protected]
Abstract: In the theory of partial differential equations the classical maximum principle is well known. It states that any non-constant harmonic function u takes its maximum (and minimum) values always at the boundary ∂G of the corresponding domain G. For higher order differential equations as well as for systems of differential equations such a principle doesn’t hold in general. In these cases, however, there is some hope for a so-called maximum modulus estimate of the form
maxG
|u(x)| ≤ c max∂G
|u(x)|, where c denotes some constant. We prove the validity of such an estimate for the linear Stokes system via the method of the boundary integral equations. Here G ⊂ ℝn (n ≥ 2) is some bounded or unbounded open set having a compact boundary ∂G of class C1,α (0 < 𝛼𝛼 ≤ 1) Keywords: Stokes Equations; Maximum modulus estimates.
References: [1] W. Varnhorn, “Maximum Modulus Estimates for the Linear Steady Stokes System”, Proc. Appl. Math. Mech., 12: 589–590. doi: 10.1002/pamm.201210283, 2012.
240
Approximation Properties of Chlodowsky-Durrmeyer Type q-Bernstein-Schurer-Stancu Operators
Tuba Vedi and Mehmet Ali Ozarslan Department of Mathematics, Eastern Mediterranean University,
Gazimagusa, TRNC, Mersin 10, Turkey [email protected], [email protected]
Abstract: The well known q-Durrmeyer operators and their approximation properties was studied by Gupta [7]. In this paper, we inroduce the Chlodowsky-Durrmeyer type q-Bernstein-Schurer-Stancu operators. We prove Korovkin-type approximation theorems on the unbounded domain and compute the rate of convergence of the operators. Keywords: q-Durrmeyer operators, q-Chlodowsky operators, q-Beta integral, q-calculus. References: [1] Agrawal P. N., Gupta V. and Kumar A. Sathish, On q-analogue of Bernstein-Schurer-Stancu operators, Applied Mathematics and Computation, 219 (14), (2013), 7754-7764. [2] Agrawal P. N., Kumar A. Sathish and Sinha T. A. K., Stancu type generalization of modified Schurer operators based on q-integers, Applied Mathematics and Computation, 226, (2014), 765-776. [3] Büyükyazıcı İ. And Sharma H., Approximation properties of two-dimensional q-Bernstein-Chlodowsky-Durrmeyer operators, Numerical Functional Analysis and Optimization, 33 (12), (2012), 1351-1371. [4] Chlodowsky, I: Sur le development des fonctions defines dans un interval infini en series de polynomes de M. S. Bernstein, Compositio Math., 4, 380-393 (1937). [5] Derriennic, M. M., Modified Bernstein polynomials and Jacobi polynomials in q-calculus, Rend. Circ. Mat. Palermo, Serie II (Suppl. 76), (2005), 269-290. [6] DeVore, RA, Lorentz, GG: Constructive Approximation, Springer-Verlag, Berlin (1993).
241
On Relative Homology Groups of Khalimsky Spaces Tane Vergili and Ismet Karaca
Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected], [email protected]
Abstract: In digital topology, the invariants such as the digital fundamental group [1, 2], the digital nth homotopy groups [5], the digital simplicial homology [3] and cohomology groups [4] of digital images are tools for comparing the digital images. The digital singular homology theory for Khalimsky spaces are introduced in [6] to distinguish the Khalimsky spaces regarding the functorial property. In this work, we introduce the digital relative singular homology groups and discover whether the excision, dimension, additivity and exactness axioms are valid for the digital singular homology of digital Khalimsky spaces. Keywords: digital homology groups, relative homology, Khalimsky space References: [1] E.H. Spanier, “Algebraic Topology”, Springer-Verlag, New York, (1966). [2] A. Rosenfeld, “Continuous functions on digital pictures”, Pattern Recognition Letters 4(1986), 177-184. [3] T.Y. Kong, “A digital fundamental group”, Computers and Graphics, 13(1989), 159-166. [4] L. Boxer, “A classical construction for the digital fundamental group”, Journal of Mathematical Imaging and Vision, 10(1994), 833-839. [5] L. Boxer, I. Karaca and A. Oztel, “Topological invariants in digital images”, Journal of Mathematical Sciences: Advances and Applications, 11(2011) No 2, 109-140. [6] O. Ege and I. Karaca, “Cohomology theory for digital images”, Romanian Journal of Information Science and Technology, 16(2013) No 1, 10-28. [7] T. Vergili and I. Karaca, “Some properties of higher dimensional homotopy groups for digital images”, Applied Mathematics and Information Science, (2014)(Submitted). [8] T. Vergili and I. Karaca, “On homology groups of certain digital spaces”, Mathematical Science Letters, (2014)(Submitted).
242
Stabilization in a n-Species Chemotaxis System with a Logistic Source Wang Wenjia
Department of Mathematics, Southeast University, Nanjing, China [email protected]
Abstract: In this paper we consider the following system involving more than two competitive populations of biological species all of which are attracted by the same chemoattractant,
1 1 1 1 1 1 11
1 1 1 1 1 1 11
1 1 1 1 1 1 11
11
u ( ) (1 ), , 0,
u ( ) (1 ), , 0,
u ( ) (1 ), , 0,
, , 0,
nt i ii
nt i ii
nt i ii
n
i
u u w u a u x t
u u w u a u x t
u u w u a u x t
w w u x t
c m
c m
c m
l
=
=
=
=
ì = D - Ñ × Ñ + - ÎW >ïï = D - Ñ × Ñ + - ÎW >ïïíï
= D - Ñ × Ñ + - ÎW >ïï
-D + = ÎW >ïî
åå
åå
L L
under homogeneous Neumann boundary conditions in a bounded domain ( 1)N NW Ì ³¡ with smooth boundary.
We prove that if ,i im c and the following matrix
11 12 1
21 22 2
1 2
n
n
n n nn
a a aa a a
A
a a a
æ öç ÷ç ÷=ç ÷ç ÷è ø
L
L
M M O M
L
satisfy certain properties, then all solutions of this system will stabilize towards to a positive equilibrium *
1,...,i i nu
=
which is globally asymptotically stable. Keywords: Keller-Segel system; Chemotaxis; Logistic source; Steady-state. References: [1] T. Cieślak, P. Laurençot, “Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(2010) , 437-446. [2] T. Cieślak, C. Stinner, “Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions”, J. Differential Equations, 252(2012), 5832-5851. [3] T. Cieslak, M. Winkler, “Finite-time blow-up in a quasilinear system of chemotaxis”, Nonlinearity, 21(2008), 1057-1076.
243
Hermite-Hadamard-Fejer Type Inequalities Hatice Yaldiz
Department of Mathematics, Duzce University, Duzce, Turkey [email protected]
Abstract: In this paper, we have established the left hand side of the Hermite-Hadamard-Fejer type inequalities for the class of functions whose derivatives in absolute value at certain powers are convex functions by using fractional integrals. Because of the wide application of Hermite-Hadamard type inequalities and fractional integrals, many researchers extend their studies to Hermite-Hadamard type inequalities involving fractional integrals not limited to integer integrals. Recently, more and more Hermite-Hadamard inequalities involving fractional integrals have been obtained for different classes of functions. Keywords: Convex function; Hermite-Hadamard inequality; Hermite-Hadamard-Fejer inequality; Riemann-Liouville fractional integral. References: [1] M. Z. Sarikaya, “On new Hermite Hadamard Fejér type integral inequalities”, Stud. Univ. Babeş-Bolyai Math. 57 (3) (2012), 377--386. [2] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, “Hermite -Hadamard's inequalities for fractional integrals and related fractional inequalities”, Mathematical and Computer Modelling, DOI:10.1016/j.mcm.2011.12.048, 57 (2013) 2403-2407. [3] I. Iscan, “Hermite-Hadamard-Fejer Type Inequalities For Convex Functions Via Fractional Integrals”, 2014, arXiv:1404.7722v1. [4] M. Z. Sarikaya, H. Ogunmez, “On new inequalities via Riemann-Liouville fractional integration”, Abstract an Applied Analysis, 2012 (2012) 10 pages, Article ID 428983. doi:10.1155/2012/428983. [5] L. Fejér, “Uberdie Fourierreihen”, II Math. Naturwise. Anz Ungar. Akad., Wiss, 24 (1906), 369-390, (in Hungarian).
244
A Note on a Functional Identity on Lie Ideals
Nihan Baydar Yarbil Department of Mathematics, Ege University, Bornova, Izmir, Turkey
Abstract: A functional identity can be defined as an identical relation including elements of a ring together with functions. The aim of the functional identity (FI) theory is to characterize the form of these functions. This theory was first introduced by M. Bresar. K. I. Beidar and M. A. Chebotar has given the foundations of FI theory. Let R be a ring. An additive map of R is said to be a Lie derivation if [ ] ( ) ( ), , ,F x y F x y x F y= +é ù é ùë û ë û for all ,x y RÎ . In this
talk, we will take two additive mappings F and D of R such that [ ] ( ) ( ), , ,F x y F x y x D y= +é ù é ùë û ë û for all ,x y LÎ
where L is a noncentral Lie ideal of R and determine the form of these mappings. Keywords: Prime ring, derivation , Lie derivation, functional identitity. References: [1] K. I. Beidar, M. A. Chebotar , “On functional identities and d-free subsets of Rings I”, Comm. Algebra 28 (2000), 3925-3951. [2] K. I. Beidar , On functional identities and commuting additive mappings, Comm. Algebra, 26 (6) (1998), 1819-1850. [3] M. Bresar, C. R. Miers, Commuting maps on Lie ideals, Comm. Algebra 23 (1995), 5339-5553. [4] M. Bresar, On generalized biderivations and related maps”, Jour. Algebra 172(3) (1995), 764-786. [5] M. Bresar, M. A. Chebotar, W. S. Martindale, “Functional identities”, Birkhauser Verlag, 2007, Basel, Switzerland.
245
On the Higher Order Gaussian Curvatures in Lorentz Space Ayse Yasar Yavuz and F. Nejat Ekmekci
Department of Mathematics Education, Necmettin Erbakan Konya University, Konya, Turkey Department of Mathematics, Ankara University, Ankara, Turkey
[email protected], [email protected]
Abstract: Let 𝑀𝑀 be hypersurfaces in 𝐸𝐸𝑛𝑛+1 and 𝑀𝑀 be a parallel hypersurfaces to 𝑀𝑀. The higher order Gaussian curvatures of 𝑀𝑀 and 𝑀𝑀 are known. We introduced higher order Gaussian curvatures of 𝑀𝑀 and 𝑀𝑀 in 𝐸𝐸1
𝑛𝑛+1 Lorentz Space our last studies [2] . And now in this paper we prove a new theorem in Lorentz space by using principal curvatures and higher order Gaussian curvatures in our last works. Keywords: Gaussian curvatures; Principal curvatures; Parallel hypersurfaces References: [1] B. O'Neill , “Semi Riemannıan Geometry”, Department of Mathematics University of California Los Angeles, California. 1983 [2] A. Yasar, “Higher Order Gaussian Curvatures of a Parallel Hypersurfaces in Lorentz Space”, Master Thesis Ankara University. 2010
246
On Soft Dual Space of Soft Normed Spaces Murat Ibrahim Yazar, Yilmaz Altun and Tunay Bilgin
Department of Mathematics, Kafkas University, Kars, Turkey Department of Mathematics, Artvin Coruh University, Artvin, Turkey
Department of Secondary Education Science and Mathematics Fields Teaching, Yuzuncu Yil University, Van, Turkey [email protected], [email protected], [email protected]
Abstract: The concept of Soft set theory introduced by Molodtsov in[1]. Soft real numbers and properties introduced in[2] and soft normed space is defined in [3]. In this study, firstly we obtain a soft normed space by defining a soft norm on ¡ (Real numbers) which is called soft normed real space. By using this normed space we define the soft linear functional and investigate some of its properties. Secondly, we introduce soft dual space and soft dual operator and investigate their properties. Finally, we state and prove the theorem about representation of soft linear functional by inner product in soft Hilbert space defined in [4]. Keywords: Soft normed spaces; Soft linear functionals; Soft dual spaces; Soft dual operators; Soft Hilbert spaces. References: [1] D. Molodtsov, “Soft set-theory-first results”, Computers and Mathematics with Applications. 37(1999), 19-31. [2] S. Das, S. K. Samanta, “Soft real sets, soft real numbers and their properties”,J. Fuzzy Math.20.3(2012), 551-576. [3] M.I. Yazar, T. Bilgin, S. Bayramov, C. Gunduz(Aras), “A new view on soft normed spaces”, International Mathematical Forum, accepted. [4] M.I. Yazar,“Soft metric and soft normed spaces”,PhD Thesis, Yüzüncü Yıl University,Institute of Sciences,2014.
247
On Some Properties of the B-Convex Functions Ilknur Yesilce and Gabil Adilov
Department of Mathematics, Mersin University, Mersin, Turkey Department of Mathematics, Akdeniz University, Antalya, Turkey
[email protected], [email protected]; [email protected] Abstract: A subset U of n
+¡ is B-convex if for all ,x y UÎ and all [ ]0,1l Î one has
( )1 1 2 2max , ,max , ,...,max ,n nx y x y x y x y Ul l l lÚ = Î . This definition is given by Briec and Horvath [1]. Also, B-convex sets are studied in [1-4]. Using B-convex set, the definition of B-convex function can be given as follows: Let nU Ì ¡ . A function :f U ® È ±¥¡ is called B-convex function if
( ) ( ) , : , ,epif x x U f xm m m= Î Î ³¡ is B-convex set. Furthermore, we can give a necessary and sufficent
condition for B-convex function: Let nU +Ì ¡ , :f U +® È +¥¡ . f is B-convex function if and only if U is B-
convex set and for all ,x y UÎ and all [ ]0,1l Î the following inequality holds:
( ) ( ) ( )f x y f x f yl lÚ £ Ú In this work, some properties of B-convex functions are analyzed. Keywords: B-convexity, B-convex Sets, B-convex Functions. References: [1] W. Briec and C. D. Horvath, “B-convexity”, Optimization, 53(2004), 103-127. [2] G. Adilov and A. Rubinov, “B-convex sets and functions”, Numerical Functional Analysis and Optimization, 27:3-4(2006), 237-257. [3] W. Briec, C. D. Horvath and A. Rubinov, “Seperation in B-convexity”, Pacific J. Optimiz., 1(2005), 13-30. [4] W. Briec and C. D. Horvath, “Halfspaces and HahnBanach like properties in B-convexity and Max-Plus convexity”, Pacific J. Optimiz., 4:2(2008), 293-317.
248
Statistical Convergence of Multiple Sequences on a Product Time Scale Emrah Yilmaz, Yavuz Altin and Hikmet Koyunbakan
Department of Mathematics, Firat University, Elazıg, Turkey [email protected], [email protected],[email protected]
Abstract: In this study, we extend the concepts and basic results on statistical convergence from multiple sequences to any product time scales. Moreover, various characterizations about these new notions are also obtained. Keywords: Statistical Convergence; Multiple Sequences; Product Time Scales. References: [1] F. Moricz, “Statistical convergence of multiple sequences”, Archiv der Mathematik, 81 (2003), 82-89. [2] C. Turan, O. Duman, “Statistical convergence on time scales and its characterizations”, Advances in Applied Mathematics and Approximation Theory, Springer Proceedings in Mathematics & Statistics, 41 (2013), 57-71. [3] M. S. Seyyidoglu, N. Ozkan Tan, “A note on statistical convergence on time scale”, Journal of Inequalities and Applications, (2012), 219-227. [4] S. Hilger, “Analysis on measure chains-A unified approach to continuous and discrete calculus”, Results in Mathematics, 18 (1990), 19-56. [5] M. Bohner, G. Sh. Guseinov, “Multiple integration on time scales”, Dynamic Systems an Applications, 14 (2005), 579-606. [6] B. Jackson, “Partial dynamic equations on time scales”, Journal of Computational and Applied Mathematics, 186 (2006), 391--415. [7] A. Zygmund, “Trigonometric Series”, Cambridge University Press, Cambridge, UK, 1979. [8] Y. Altin, H. Koyunbakan, E. Yilmaz, “Uniform Statistical Convergence on Time Scales”, Journal of Applied Mathematics, Volume 2014, Article ID 471437, 6 pages.
249
Optimality Conditions for Non-Lipschitz Optimization Nurullah Yılmaz and Ahmet Sahiner
Department of Mathematics, Suleyman Demirel University, Isparta, Turkey [email protected], [email protected]
Abstract: Most of nonsmooth optimization studies deal with the Lipschitz continuous objective functions. In recent years, some interesting studies arise on non-Lipschitz optimization, because of the wide application areas in image processing [1,2,3]. In this study, we introduce optimality conditions that are one of the corner stone of the optimization theory for non-lipschitz objective functions. Keywords: Nonsmooth optimization, lp minimization, optimzality conditions. References: [1] X. Chen, L. Niu and Y. Yuan, Optimality Conditions and smoothing trust region method for nonlipschitz optimization, SIAM J. Optim., 23 (3), (2013),1528-1552. [2] X. Chen, W. Zhou, “Convergence of the reweighted l1 minimization algorithm for l2 –lp minimization”, Comput. Optim. Appl. 59(2014), 47-61. [3] H. Wang, D.-H. Li, X.-J. Zhang and L. Wu, “Optimality Conditions for constrained Lp regularization”, Optimization, (2014),
Doi: 10.1080/02331934.2014.929678
250
Groups and Graphs Utku Yilmazturk
Department of Mathematics, Istanbul University, Fatih, Istanbul, Turkey [email protected]
Abstract: There are several graphs related to finite groups. One of these graphs, denoted by ( )GD , is the character degree graph. The definition of this graph is the following: Let G be a finite group and let ( )cd G denote the set of the irreducible complex character degrees of G . The vertex set of the character degree graph, denoted by ( )Gr , is the prime divisors of the elements of ( )cd G and there is an edge between two vertices p and q if and only if there exists an element of ( )cd G divisible by pq . In the literature, there are several studies on the relation between the invariants of the character degree graph of a finite group and the structure of that group. For example, Palfy [1] has proved that in the character degree graph of a finite solvable group any three vertices induce a non-empty graph. Lewis [2] has classified all solvable groups with disconnected character degree graph. Lewis and Meng [3] has shown that a solvable group whose character degree graph is square (the graph with four vertices and in which every vertex has degree 2) is a direct product of two subgroups. Zuccari and Paolo[4] has given a relation between the character degree graph of a finite group and fitting height of that group. In this talk we will focus on the relation between a given graph and the group admits that graph as the character degree graph. Keywords: Finite group; Character degree graph. References: [1] P. P. Palfy, “On the character degree graph of solvable groups. I. Three primes”, Period. Math. Hungar. 36 (1998), no. 1, 61-65 [2] M. L. Lewis, “Solvable groups whose degree graphs have two connected components”, J. Group Theory 4(2001), no. 3, 255-275. [3] M. L. Lewis, M. Qingyun, “Square character degree graphs yield direct products”, J. Algebra 349 (2012), 185-200 [4] M. Zuccari, C. Paolo, “Fitting height and diameter of character degree graphs” Comm. Algebra 41 (2013), no. 8, 2869-2878.
251
Center Coloring and the Other Colorings Zeynep Ors Yorgancioglu and Pinar Dundar
Department of Mathematics, Yasar University, Izmir, Turkey Department of Mathematics, Ege University, Izmir, Turkey [email protected], [email protected]
Abstract- For a nontrivial connected graph G , center coloring is a kind of coloring the vertices of a graph G in such a way that if vertices have different distance from the center, then they must receive different colors. Two adjacent vertices can receive the same color. The number of colors required of such a coloring is called center coloring number Cc ( G ) of G . [1] This coloring can be applied to hierarchy problems to find the number of structures, people, criteria and comparisons, etc. Moreover it can be applied to earthquake motion problems to find the number of settlements that are affected by an earthquake. In this paper, center coloring is compared with the other various colorings [2, 3, 4, 5, 6] and several bounds are established for the center coloring and the other colorings. Keywords: Center; Distance; Coloring; Center coloring. References: [1] Y. Immelman, “On The (Upper) Line-Distinguishing And (Upper) Harmonious Chromatic Numbers Of A Graph”, Dissertation, 2007. [2] Z. Yorgancıoglu, P. Dundar, M.E. Berberler, “Center Colorıng Of Graphs And Spanning Trees”, Ecco XXV European Chapter on Combinatorial Optimization, Antalya, Turkey, 2012. [3] X. Li, Y. Sun, “Rainbow Connections Of Graphs”, Springer, 2012. [4] M., Kubale, “Harmonious Coloring Of Graphs”, Graph Colorings, Contemporary Mathematics, 2002, ISSN 0271-4132; 352. [5] S. Isobe, X. Zhou, T. Nishizeki, “A Polynomial – Time Algorithm for Finding Total Colorings of Partial k-trees”, International Journal of Foundations of Computer Science 10, 100-113, 1999. [6] J. Vernold Vivin, M.M. Akbar Ali, “On Harmonious Coloring of Middle graph of , and ”, Note di Matematica, 201-211, 2009.
( )nC C 1,( )nC K ( )nC P
252
Dual Transformations and One-Paremeter Motions Gulsum Yuca and Yusuf Yayli
Aksaray Üniversitesi, Sabire Yazıcı Fen Edebiyat Fakültesi, Matematik Bölümü, 68000, Aksaray, Türkiye Ankara Üniversitesi, Fen Fakültesi, Matematik Bölümü, 06100, Ankara, Türkiye
[email protected], [email protected]
Abstract: In this study we examine one-parameter motions by a dual transformation. We give examples in Euclidian space and Lorentzian space. Geometric interpretations will be given for these examples. Keywords: Dual transformation, Axis of rotation, Euclidean space, Lorentzian space References: [1] Dohi, R., Maeda, Y., Mori, M., Yoshida, H. 2010. "A dual transformation between SO(n + 1) and SO(n, 1) and its geometric applications". Linear Algebra Appl., vol. 432; pp. 770-776. [2] Greub W.H., Linear Algebra, Springer-Verlag New York, 1967 [3] Karger, A., Novak, J. 1985. Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, Montreux. [4] Lopez, R. 2008. Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space. [5] O’Neill, B. Semi-Riemannian Geometry, with applications to relativity. Academic Press, New York, 1983
253
On Boundedness of Fractional Maximal Operator in the Weighted (.)(0,1)pL Space
Yusuf Zeren and Lutfi Akin Yildiz Technical University, Departmant of Mathematics, Turkey
[email protected], [email protected] Abstract: Investigations of variational problems with variable exponent started from the papers by V. Zhykov. Later M. Ruzicka studied the problems in the so called rheological and electrorheological fluids, which lead to spaces with variable exponent. Many mathematical models in fluid mechanics, elasticity theory, differential equations etc are naturally related to the problems with non- standard local growth.It should be stressed that historically, the boundedness of the Hardy–Littlewood maximal operator over bounded domain W was established by L. Diening. We prove a new sufficiency result for the two weighted boundedness of the fractional maximal operator in the variable exponent Lebesgue space (.)(0,1)pL . A complete analog of the Sawyer’s condition are obtained on the weight functions assuming a usual log-Holder continuity condition on the exponent function (.)p . Keywords: Fractional Maximal Operator; Variable exponent; Two weight inequality; Boundedness. References: [1] D. Cruz Uribe, A. Fiorenza. “Variable Lebesgue Spaces”, Foundations and Harmonic Analysis (Applied and Numerical Harmonic Analysis)" , Birkhauser,2013 [2] D. Cruz Uribe, A. Fiorenza. “Variable Lebesgue spaces: Foundations and Harmonic Analysis(Applied and numerical analysis)”, Birkhauser, 2013.
254
Global Existence and Asymptotic Properties of the Solution to a Two-Species Chemotaxis System
Qingshan Zhang Department of Mathematics, Southeast University, Nanjing, P. R. China
Abstract: This paper deals with the Cauchy problem for a two-species chemotactic Keller-Segel system 1
2
1 2
·( ),·( ),
t
t
t
u u u wv v v ww w w u v
cc
g a a
= D - Ñ Ñìï = D - Ñ Ñíï = D - + +î
in 2 [0, ),´ ¥¡ where 0,g ³ 1,c 2c and 1,a 2a are real numbers. We obtain the global existence of solutions if
0 1,u‖ ‖ 0 1,v‖ ‖ and 0 2wÑ‖ ‖ are small, and the asymptotic behavior of the small-data solution as follows: l If 0g = , the solution is asymptotic to a self-similar solution for large time; l If 0g > , the solution behaves like a multiple of the heat kernel as t ® ¥ .
Keywords: Two-species chemotaxis system, Global existence, Self-similar solution, Asymptotic profile.
255
One Dimensional Model of Biodegradable Elastic Curved Rods Bojan Zugec
Faculty of organization and informatics, University of Zagreb, Varazdin, Croatia [email protected]
Abstract: In this work we consider a curved elastic rod made of biodegradable material, situated in a liquid solvent. The rod is clamped in both ends. By asymptotic expansion we derive a one dimensional model of biodegradable elastic curved rod from the associated 3D problem. The model is described by two coupled equations describing behaviour of the linearized elastic curved rod and the evolution of the rod's material at time t. Two cases of biodegradation are being observed: the case where diffusion doesn't affect biodegradation, i.e. material loss, and the case where it does. In both of these cases the existence and uniqueness of the one dimensional model solution will be proved. Keywords: Biodegradation, 1D model, elasticity. References: [1] A. Göpferich. “Mechanisms of polymer degradation and erosion”. Biomaterials 17.2 (1996.), 103–114. [2] M. Jurak i J. Tambaca. “Derivation and justification of a curved rod model”. Mathematical Models and Methods in Applied Sciences 09.07 (1999.), 991–1014. [3] K. Rajagopal, A. Srinivasa i A. Wineman. “On the shear and bending of a degrading polymer beam”. International Journal of Plasticity 23 (2007.), 1618– 1636.
256
Convergence Analysis and Numerical Solution of Benjamin-Bona-Mahony Equation by Lie-Trotter Splitting
Fatma Zurnaci, Nurcan Gucuyenen, Muaz Seydaoglu and Gamze Tanoglu Department of Mathematics, Izmir Institute of Technology, Urla, Izmir, Turkey
Civil Engineering Department, Gediz University, Menemen, Izmir, Turkey Department of Mathematics, Ege University, Bornova, Izmir, Turkey
[email protected], [email protected], [email protected], [email protected]
Abstract: In this paper, an operator splitting method is used to analyse the nonlinear Benjamin-Bona-Mahony type equations. We split the equation as an unbounded linear part and a bounded nonlinear part and then Lie-Trotter splitting method is applied to the splitted equations. The local error bounds are obtained using the similiar approaches as in [1]; [2]. The approach bases on the differential theory of operators in Banach space and the quadrature errors estimated via Lie commutator bounds. The global error estimate is obtained via Lady Windermere’s fan argument. Finally, to confirm the expected convergence order, a numerical example is studied. Keywords: Lie-Trotter splitting; Convergence analysis; Benjamin-Bona-Mahony equation. References: [1] H. Holden, K. H. Karlsen, N. H. Risebro, “Operator splitting for the KdV Equation”, Math. Comp., 80(2011), 821-846. [2] H. Holden, C. Lubich, N. H. Risebro “Operator splitting for partial differential equations with Burgers nonlinearity”, Mathematics of Computation, 82(2013), 173-185. [3] T. B. Benjamin, J. L. Bona, J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems”, Philos. Trans. Roy. Soc. London, 272(1972), 47-78.