ISSN: XXXX-XXX - 2017.Vol.1/1 

ReviewsMath Journal of Mathematics

Editors;

Florentin SMARANDACHE,The University of New Mexico, Department of Mathematics, USA, smarand@unm.edu Nasruddin HASSAN, Universiy of Kebangsaan Malaysia, Science and Technology, MALAYSIA, nas@ukm.edu.my Said BROUMI,Université Hassan II de Casablanca, Science and Technology, MOROCCO, broumisaid78@gmail.com İsmet YILDIZ,University of Düzce, Department of Mathematics, TURKEY, ismetyildiz@duzce.edu.tr Adil KILIC,University of Gaziantep, Department of Mathematics,TURKEY, adilkilic@gantep.edu.tr Mustafa BAYRAM,Gelişim University , Department of Computer Engineering, TURKEY, mbayram@gelisim.edu.tr Memet ŞAHİN, University of Gaziantep, Department of Mathematics,TURKEY, mesahin@gantep.edu.t Necati OLGUN, University of Gaziantep, Department of Mathematics, TURKEY, olgun@gantep.edu.tr Anjan BISWAS, Alabama A&M University, Department of Physics, Chemistry and Mathematics ,USA, biswas.anjan@gmail.com Adem KILICMAN, University Putra Malaysia, Department of Mathematics, MALAYSIA, kilicman@yahoo.com Sunil KUMAR, Department of Mathematics National Institute of Technology, INDIA, skiitbhu28@gmail.com Aydin SECER, Yildiz Technical University, Department of Mathematics, TURKEY, asecer@yildiz.edu.tr Vishnu Narayan MISHRA, S.V. National Institute of Technology, Department of Mathematics, INDIA, vishnunarayanmishra@gmail.com Hari M. SRIVASTAVA, University Victoria, Department of Mathematics, CANADA, harimsri@math.uvic.ca Nuran GUZEL, Yildiz Technical University, Department of Mathematics, TURKEY, nguzel@yildiz.edu.tr Selcuk KUTLUAY, Inonu University, Department of Mathematics, TURKEY, selcuk.kutluay@inonu.edu.tr Coşkun ÇETİN, California State University, USA, cetin@csus.edu Mehmet Ali COBAN, University of Gaziantep, Department of Mathematics, TURKEY, coban@gantep.edu.tr

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www.ReviewsMath.com – editor@reviewsmath.com ISSN - - 2017Vol. 1/ 1 ..

    

1  

PARTIAL NORMED SPACES

Olgun, N.1, Şahin, M.2, Kargın, A.3, Çoban, M.A.4

ASBRACT

In this study, we define a partial normed space. At the same time, the relations between partial normed space and normed space have been investigated. Furthermore the transitions between partial metric space and partial normed space have been explained.

.

Keywords: Metric Spaces, Normed Spaces, Partial Normed Space

                                                            1Department of Mathematics, Gaziantep University, Turkey, olgun@gantep.edu.tr 2Department of Mathematics, Gaziantep University, Turkey, mesahin@gantep.edu.tr 3Department of Mathematics, Gaziantep University, Turkey, abdullahkargin27@gmail.com 4Department of Mathematics, Gaziantep University, Turkey, coban@gantep.edu.tr

 

2  

1 Introduction

Normed spaces hold an important position in computational science and in calculus analysis. It conveys the concept of metric spaces onto vector spaces and has many vector spaces properties for example, in a vector space of n dimensions allows the length of the vector to be described. Many researchers have worked on this topic. Some of them are [1,2,3]. Furthermore goled [4] be normed spaces in fuzzy set theory, Das et al. [5] have tried to explain normed spaces in soft set theory. To complete in adequate positions of metric spaces Matthows [6] defined partial metric space and thus in computer science in the case where the auto correlation (self –distance) of an element of a sequence may be different from zero, , defined as nonzero and removed the contrast from the center of the metric space. It has been helpful to researches the partial metric space, in computer science and in fixed point theorem. In this study we have defined a partial normed space by carrying the partial metric space concept over vector spaces.

Thus we save new properties of the concept of vector space which holds an important place (position) in mathematic under the partial norm. For example we have shown that in a vector space of n dimensions the auto correlation (self-distance) of an element of a sequence can be different from zero. At the same time, we have studied the familiar relation between the normed space and the partial normed space and in same special cases, partial normed spaces may provide normed space conditions and we have shown that a partial normed space can be obtained from every normed space. Furthermore we explained basic concepts in partial normed spaces by defining partial normed metric defined by partial metric such as open ball, closed ball, convergence and completeness. Finally, we have associated convergent sequences in normed spaces with convergent sequences in partial normed space.

2 Preliminaries

Definition 1 [6]: A partial metric is a function : → 0,∞ satisfying the following conditions

0 , , ∀ , ∈

, , , then

, ,

, , , , .

Definition 2 [6]:Let be a partial metric space an the family of open -ball is

, : ∈ , 0 where

, ∈ : , , ,∀ ∈ .

Similarly, a closed - ball is defined as

, ∶ , , .

3  

Definition3 [6] A sequence in partial metric space , is called Cauchy when ever

lim, →

,

exists.

Definition4 [6] A partial metric space , is said to be complete if every Cauchy sequence in converges such that

lim, →

, ,

Definition5 [7] A sequence in a partial metric space , is called 0- Cauchy if

lim, →

, 0.

A partial metric space , is said to be 0- complete if every 0- Cauchy sequence converges such that

, 0.

3. Partial Normed Spaces

Definition6Let be a vector space and let be a field. Assume that : → 0,1 is a function such that ⊆ , 0, 0, , 0 0forany ∈ , ∈ .

A partial norm is a function ‖. ‖ : → 0,∞ satisfying the following conditions

i) 0 ‖0‖ ‖ ‖ for ∈ . ii) ‖0‖ ‖ ‖ ⟺ 0

iii) , ‖0‖ ‖ ‖ ‖ ‖

iv) ‖ ‖ ‖0‖ ‖ ‖ ‖ ‖

Example 1 Let ‖. ‖: → 0,∞ be a norm and let

: 0,1 → 0,1 ,| |. ‖ ‖‖ ‖

be a function then ‖ ‖ ‖ ‖ , ∈ 0,∞ is a partial normed.

i) We have 0 ‖0‖ ‖ ‖ since ‖0‖ 0 and ‖0‖ ∈ 0,∞

ii) Let ‖0‖ ‖ ‖ if ‖ ‖ 0 then 0.

Let 0 then ‖ ‖ ‖ ‖ ‖0‖ ‖0‖

iii) Let ,| |.‖ ‖

‖ ‖.Then

, ‖ ‖ ‖0‖| |. ‖ ‖‖ ‖

. ‖ ‖ ‖0‖

4  

| |‖ ‖ ‖0‖

‖ ‖ ‖0‖

‖ ‖ ‖ ‖

iv) We have ‖ ‖ ‖ ‖ ‖ ‖ by the definition of a norm ‖ ‖ ‖ ‖ ‖ ‖ ⟹ ‖ ‖ 2 ‖ ‖ ‖ ‖

⟹ ‖ ‖ ‖ ‖ ‖ ‖

⟹ ‖ ‖ ‖0‖ ‖ ‖ ‖ ‖

Theorem 1 Any partial normed space is a normed space if ‖0‖ 0.

Proof: Let be any partial normed space we can define , | |

i) 0 ‖0‖ ‖ ‖ ⟹ 0 ‖ ‖

ii) ‖ ‖ 0 ⟺ 0since‖0‖ 0

iii) | |‖ ‖ ‖ ‖ since , | | and ‖0‖ 0

iv) ‖ ‖ ‖ ‖ ‖ ‖ since ‖0‖ 0

Theorem 2 Any normed space a partial normed space.

Proof: It is omitted.

Theorem 3 Any normed space is a partial metric space.

Proof: Let ‖. ‖ : → 0,∞ be a partial normed space. Then : → 0,∞ ,‖ ‖ is a partial metric

i) , ‖ ‖ ‖0‖ ⟹ 0 ‖0‖ , ,

ii) , , ⟹ 0 ⟹

since , , ‖0‖ and ‖0‖ ‖ ‖ ⟺ 0

iii) We have 1,‖ ‖ ‖ ‖

‖ ‖ by the definition of the norm and for 1

‖ ‖ ‖0‖‖ ‖

. ‖ ‖ ‖0‖ ‖ ‖

by the third condition of the definition of the partial norm ‖ ‖ ‖0‖ ‖0‖ ‖ ‖ ⟹ ‖ ‖ ‖ ‖

Hence we obtain , ‖ ‖ ‖ ‖ ,

Definition 7Let be a partial metric and ‖. ‖ be a partial norm. Then the partial metric space

, ‖ ‖

is called a partial normed metric space.

By using the notion of the partial normed metric space we can define an open ball, a closed ball, a convergent at a sequence in the partial normed space

5  

Definition8 Let be a partial normed space. The familly of open - balls is , : ∈ ,

0 where

, ∈ :‖ ‖ ‖0‖ ∀ ∈ .

Similarly, a closed - ball is defined as

, ∈ :‖ ‖ ‖0‖ ∀ ∈ .

Definition 9Let be a normed space and be a sequence in . is called Cauchy sequence whenever for all 0 there exists N such that

‖ ‖ ‖0‖ , .

Definition 10 A partial normed space is said to be complete if every Cauchy sequence in converges such that

lim, →

‖ ‖ ‖0‖ .

Definition 11 A sequence in a partial normed space is called 0 -Cauchy if

lim, →

‖ ‖ 0.

A partial normed space is said to be 0- complete if every 0- Cauchy sequence converges such that

‖0‖ 0.

Example 2‖. ‖: → 0,∞ be a norm and let ‖. ‖ : → 0,1 ‖ ‖ ‖ ‖ be a partial

norm. Then any convergent sequence in the norm space converges in the partial normed space.

lim→‖ ‖ 0

⟹ lim→

‖ ‖

lim→‖ ‖

lim→

‖ ‖ lim→

‖0‖

therefore

lim→

‖ ‖ ‖0‖ .

Lemma 1 Let , ‖. ‖ be a partial norm space suppose that , ‖ ‖ is a partial

metric for , ∈ . Then , ,

6  

a) , , . , , ∈ (Absolute homogeneity)

Proof:

a) , ‖ . ‖ ‖ ‖ ,

b) , ‖ ‖ ‖ ‖

, . ‖ ‖ ‖0‖

, , ,

Corollary 1 Let , ‖. ‖ be a partial normed space and , be a partial metric space with

the definition of , ‖ ‖ , then each partial metric space is partial normed. But

the inverse is not always true. Only metric spaces that satisfy the above lemma conditions are partial normed space.

Proposition 1 Let , ‖. ‖ be a partial normed space. For all , ∈ and , , … , ∈

a) ‖ ‖ ‖0‖ ‖ ‖

b) ‖ . . . ‖ ‖0‖ ‖ ‖ ‖ ‖ . . . ‖ ‖ , 1

Proof:

a) ‖ ‖ ‖0‖ ‖0 ‖ ‖0‖ ‖ ‖ ‖ ‖

since ‖ ‖ ‖ ‖ ‖0‖ ‖ ‖ 1

‖ ‖ ‖0‖ ‖ ‖ ‖0‖

‖ ‖ ‖ ‖

since ‖ ‖ ‖ ‖ ‖0‖ ‖ ‖ 2

when 1,‖ ‖ ‖ ‖

‖ ‖ is taken inf function, ‖ ‖ =‖ ‖ since the (2) property is

‖ ‖ ⋯ ‖ ‖ ‖ ‖ ‖ ‖ ‖0‖ ‖ ‖ 3

The proof has done from (1) and (3).

b) Let us do this proof by induction method For 2

‖ ‖ ‖0‖ ‖ ‖ ‖ ‖

is true by partial norm definition For

‖ ⋯ . ‖ ‖0‖ ‖ ‖ ⋯ ‖ ‖

Let ‖ ‖ ⋯ ‖ ‖ … be true for

Let is show the truth for 1 ‖ ⋯ . ‖ ‖0‖ ‖ ‖ ⋯ ‖ ‖ ‖ ‖

7  

‖ ⋯ . ‖ ‖ ‖ ⋯ ‖ ‖

‖ ⋯ . ‖ ‖0‖ ‖ ‖ ⋯ ‖ ‖ ‖ ‖

Lemma 2 Let , be a partial metric space and , ‖. ‖ be a partial normed space. For

‖ ‖ , 0 and having sh iftining and absolute homogenity. Every cauchy sequence in

the partial metric space is a Cauchy sequence in the normed space.

Proof: Let be a Cauchy sequence in , . Namely for any number 0, at leaft one ∈ such that for all , ∈ , .

p has the partial metric shifting homogeneity property

‖ ‖ , 0

,

,

‖ ‖

‖ ‖

since , is the Cauchy sequence in the , ‖. ‖ partial normed space. The inverse of

this lemma is not true.

8  

REFERENCES

1.Alsina, C., Schweizer, B., &Sklar, A. (1993). On thedefinition of a probabilistic normedspace. Aequationes Mathematicae, 46(1), 91-98.

2. Popiołek, J. (1991). Real normedspace. FormalizedMathematics, 2(1), 111-115. 3. Durier, R., &Michelot, C. (1986). Sets of efficientpoints in a normedspace. Journal of

Mathematical Analysis andapplications, 117(2), 506-528. 4.Golet, I. (2009). On generalizedfuzzynormedspaces. In Int. Math. Forum (Vol. 4, No.

25, pp. 1237-1242). 5.Das, S., Majumdar, P., &Samanta, S. K. (2013). On

softlinearspacesandsoftnormedlinearspaces. arXivpreprint arXiv:1308.1016. 6.Matthews, S. G. (1994). Partialmetrictopology. Annals of the New York Academy of

Sciences, 728(1), 183-197. 7.Romaguera, S. (2009). A Kirktypecharacterization of

completenessforpartialmetricspaces. Fixed Point Theoryand Applications, 2010(1), 493298.

www.ReviewsMath.com – editor@reviewsmath.com ISSN - - 2017Vol. 1/ 1 ..

      

9  

ON THE APPLICATIN TO THE HALF AND QUARTER TIMES OF THE ,  PERIOD PAIRS OF THE THETA FUNCTIONS

IN THE FIRST ORDER

Yıldız, I.1, Akyar, A.2, Merek, Ü.3, Çelen, M.4

ASBRACT

In this study, two theorems that first order theta functions provide according to associated periods of

2r

and

2r

and to 1,2,... for characteristics values of

1 1 0 0

, , , mod 21 0 1 0

were expressed and proved. In addition, multiplicative factors and transformation of theta functions were studied by these associated periods.

Keywords: Theta Function, Characteristic and Theta Period

                                                            1İsmet YILDIZ, Department of Mathematics, Düzce University, Turkey, ismetyildiz@duzce.edu.tr 2Alaattin AKYAR, Department of Mathematics, Düzce University, Turkey, alaattinakyar@duzce.edu.tr 3Ümran MEREK, Department of Mathematics, Düzce University, Turkey, umranmerek34@gmail.com 4Melike ÇELEN, Department of Mathematics, Düzce University, Turkey, melikecelen44@gmail.com

 

10  

Introduction

Definition 1: We defined the first order theta function with characteristic

, argument u

and theta period by

2

( , ) exp 22 2 2n

u n i i n u

where n ranges over all the integers to and

2

1

2

2

wR

w , Im 0 , 12w , 22 , , Z, 2 .w C

Definition 2: A period denoted

, is + where and are integers 1 .

Theorem 1: The function ( , )u

satisfies the functional equations

(a) ( , ) 1 ( , )u u

(b) 2( , ) 1 ( , )i iuu e u

(c) 2( , ) 1 ( , )i iuu e u

11  

Proof:

(a)

2

( , ) exp 22 2 2n

u n i i n u

2

exp 2 22 2 2n

n i i n u i n i

1 ( , )u

(b)

2

( , ) exp 22 2 2n

u n i i n u

2

exp 22 2 2n

n i i n u

2

exp 2 22 2 2n

n i i n u iu i i

21 ( , )i iue u

(c)

2

( , ) exp 22 2 2n

u n i i n u

2

exp 2 22 2 2n

n i i n u i iu i i

21 ( , )i iue u

12  

Theorem 2: Let

be any characteristic and 1

2 2 2

a b a

b

any half period. Then

21 1

, exp ,2 4 2

a ai au au i a i b u

b b

.

Proof :

21

, exp 22 2 2 2 2 2n

a b au n i i n u

b

2

exp 22 2 2 2 2n

a i b in i i n u an i bn i

2

, exp 22 2 2n

a a a bu n i i n u

b

2

exp 22 2 2n

n i i n u

2

2 2 4 2 2

an i b i a i a i ab ian i au i bn i

.

on the other hand,

2 1

exp ,4 2

ai aau i a i b u

b

2

exp 22 2 2n

n i i n u

2 2

a i b ian i bn i

.

from the equalities above, we obtain

13  

21 1

, exp ,2 4 2

a ai au au i a i b u

b b

.

Hence the theorem is proved. According to the above theorem, we can arrive at the following result:

1. If 2a p and 2b q for , p q integers, then

22 21, exp 2 2 ,

2 22

p pu p i pu i p i pq i u

q q

21 exp 2 ,p

p i pu i u

,

where 2

mod 2 mod 22

p

q

for 2 .2 0 mod 2p q . Using the above

equality, we have

2

,

1 exp 221

,2 2

p

u

p i pu ip

uq

.

i) If 0 mod 2p , then it follows

2exp 21

21 ,,2 2

p i pu i

pu ru

q

ii) If 1 mod 2p , then it follows

2exp 21

21 ,,2 2

p i pu i

puu

q

We consider the ,u defined by the double series

2

, ,

1 exp 21,

21 ,,2 2

p

p q p q

p i pu iu

puu

q

14  

where ,

.p q p q

Since this ( , )u

does not depend on , p q and

21 exp 2p

p i pu i

does not contain ,q we obtain

2

,

1 exp 21

,21 ,,

2 2

p

p

p q

p i pu i

up

uuq

These show that:

i) If 0 mod 2p and ,n p then

,

0,

01,

21 ,,2 2

p q

u

up

uuq

.

ii) If 1 mod 2p and ,n p then

,

0,

01

21 ,,2 2

p q

u

puu

q

.

We can arrive at the following result:

21 exp 2

,

p

p

p i pu i

u

is a constant or rate of two theta functions according to characteristic values .

Thus, the

function ,u is an quasi elliptic or an elliptic function.

15  

2. If , a b are odd integers, then

21 1

, exp ,2 4 2

a ai au au i a i b u

b b

2 11

exp 1 ,14 2

i aau i a i u

where 1

(mod 2) (mod 2)1

a

b

as , 1 mod 2a b . Similarly, we obtain

2 1exp 1

4 2111 ,,12

i aau i a i

auu

b

Let the function ,u be defined by the double series

2

, ,

1exp 1

4 21,

11 ,,12

a b a b

i aau i a i

ua

uub

where ,

.a b a b

Since this 1

( , )1

u

does not depend on , a b and

2 1

exp 14 2

i aau i a i

does not contain b , we deduce

2

,

1exp 1

4 21,

11 ,,12

a

a b

i aau i a i

ua

uub

16  

i) If 1 0 mod 2 , then it follows

2

,

exp41

,11 ,,

02

a

a b

i aau i

ua

uub

Evidently, , 2,u u .

ii) If 1 1 mod 2 , then

2

,

exp4 21

,11 ,,

12

a

a b

i a a iau i

ua

uub

22

2 2exp exp 2

4 4i u r

a a

i ai aa i u r e a u i

2

2 exp ,4

i u r

a

a ie au i

(1)

where 2a is an integer.

21 1 1 1( , ) exp 2

1 2 2 2n

u n i i n u

1 2 11 ,

1i u re u

2 1,

0i u re u

, (2)

17  

where 1 0 mod 2 .

We obtain , ,u u from the equations (1) and (2) since

2

exp4 2a

i a a ia i u

22 exp 2 2 24 2

i u r

a

i ie a a u i a

2

2 exp4 2

i u r

a

a i a ie au i

,

where 2a is an integer.

21 1 1 1( , ) exp 2

1 2 2 2n

u n i i n n

2

1 2 1 1 11 exp 1 2 1

2 2 2i u r

n

e n i i n u

2

1 2 1 1 11 exp 2

2 2 2i u r

n

e n i i n u

2 1,

1i u re u

,

where 1n is an integer and 1 1 mod 2 [3].

Theorem 3: The function ( , )u

satisfies functional equation where r N

21 12

2 2( , ) exp 22 2 2 2 2

r ri n i

r rn

u e e n i i n u

18  

Proof:

2

( , ) exp 22 2 2 2 2 2 2r r r r

n

u n i i n u

21 1 1 1

exp 22 2 2 2 2 2 2 2 2r r r r r r

n

n i i n u

21 1 1 1

22 2 2 2 2 2 2exp

2 2

2 2 2 2

r r r r

n

r r r r

n i i n u

n i n i i i

2

2exp 2 1 1

2 2 2 2 2r rn

n i in i i n u

21 12

2 2 exp 22 2 2

r ri n i

n

e e n i i n u

where 1 .

1. Transformations of theta functions associated with the periods 2r

and

2r

21 1 1 1

( , ) 1 exp 21 2 2 2 2 2 2 2

n

r r r rn

u i n i i n u

a.

1, , 2

0 2 21( , )

1 2 2 1, , 2 1

0 2 2

r r

r r

r r

i u n k Z

u

i u n k Z

20( , ) 1 exp 2

1 2 2 2 2n

r r r rn

u n i ni u

19  

b.

0, , 2

0 2 20( , )

1 2 2 0, , 2 1

0 2 2

r r

r r

r r

u n k Z

u

u n k Z

2. Multiplicative factors of theta functions associated with periods 4

and

4

220

( , ) exp 20 4 4 2

in

n

n iu e n i nui

a.

0, , 2 1

0 4

0 0( , ) , , 4

0 04 4 4

0, , 4 2

0 4

u n k Z

u u n k Z

u n k Z

22 40

( , ) exp 21 4 4 2 4 2

i in

n

n i i iu e e n i nui n i ui

b.

0, , 2 1

1 4

0 02( , ) 1 , , 4

1 14 4 2 4

0, , 4 2

1 4

u n k Z

u i u n k Z

u n k Z

22 41

( , ) exp 20 4 4 2 2

i in

n

n i iu e e n i n i niu ui

c.

1, , 2 1

0 4

1 12( , ) 1 , , 4

0 04 4 2 4

1, , 4 2

0 4

u n k Z

u i u n k Z

u n k Z

20  

22 41

( , ) exp 21 4 4 2 2 2

i in

n

n i i iu e e n i n i niu ui n i

d.

1, , 2 1

1 4

1 12( , ) 1 , , 4

1 14 4 2 4

1, , 4 2

1 4

u n k Z

u i u n k Z

u n k Z

with the help of this theorem 2 proved above, transformations among theta functions can be

found for characteristic values

according to all 1

2rmultiplies of the periods.

21  

REFERENCES

1. Harry E. Rouch: Elliptic Functions, Theta Functions and Riemann Surfaces, The Williams and Wilkins Company, Baltimore, Maryland, (1973), U.S.A. 75-79.

2. Patrick Du Val: Elliptic Functions and Elliptic Curves, London Mathematical Society Lecture Notes Series 9, Cambridge, At the University Press. (1973) 163-165.

3. İsmet Yıldız: The Application of the First Order Theta Functions to Additional Characteristic Values According to Half Periods, International Journal of Applied Mathematics, Volume 2 No:3 (2000), 311-317.