Research & reviews discrete mathematical structures (vol1, issue2)

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Discrete Mathematical Research & Reviews: Structures STM JOURNALS Scientific Technical Medical (RRDMS) May - August 2014 www.stmjournals.com

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Page 1: Research & reviews discrete mathematical structures (vol1, issue2)

Discrete Mathematical Research & Reviews:

Structures

STM JOURNALSScientific Technical Medical

(RRDMS) May - August 2014

www.stmjournals.com

Page 2: Research & reviews discrete mathematical structures (vol1, issue2)

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Mahmood BakhshiDepartment of Mathematics, University

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Kai-Long HsiaoTaiwan Shoufu University, Taiwan.

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of Mathematics, Bornova, IZMIR, Turkey.

Dr. Anil KumarProfessor

World Institute of Technology, Gurgaon.

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General Education CenterChienkuo Technology University,

Taiwan.

Dr. Reza Chavosh KhatamyIslamic Azad University Tabriz Branch,

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Prof. Palle JorgensenProf. Palle Jorgensen

The University of Iowa, USA.

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I take the privilege to present the print version for the [Volume 1 Issue (2)] of Research & Reviews:

Discrete Mathematical Structures. The intension of RRDMS is to create an atmosphere that

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Page 10: Research & reviews discrete mathematical structures (vol1, issue2)

1. On Symmetry Preserving Diffeomorphisms of Generalized Symmetric Finsler Spaces Reza Chavosh Khatamy, Dariush Latifi 1

2. A Study on the Sparing Number of the Corona of Certain Graphs K.P. Chithra, K.A. Germina, N. K. Sudev 5

3. On Some Triple Almost Lacunary Sequence Spaces Defined by Orlicz FunctionsAyhan Esi 16

4. On the Riemann Hypothesis in The Light of The Littlewood Criterion of EquivalenceLukasz Andrzej Glinka 26

5. Origin of Modern Mathematical Numeral – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9: the Hindu-Indian-Brahmagubta, The Islamo-Arabic or the West? Auwalu Musa 36

ContentsResearch & Reviews: Discrete Mathematical Structures

Page 11: Research & reviews discrete mathematical structures (vol1, issue2)

RRDMS (2014)© STM Journals 2014. All Rights Reserved

Research & Reviews: Discrete Mathematical Structures

Volume 1, Issue 2

www.stmjournals.com

On Symmetry Preserving Diffeomorphisms of Generalized

Symmetric Finsler Spaces

Reza Chavosh Khatamy1, Dariush Latifi

2* 1Department of Mathematics, College of Science Tabriz branch, Islamic Azad University, Tabriz, Iran

2Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran

Abstract In this paper, we study generalized symmetric Finsler spaces. We first study symmetry

preserving diffeomorphisms, then we show that the group of symmetry preserving

diffeomorphisms is a transitive Lie transformation group. Finally we give some existence

theorems. MSC: 53C60, 53C30

Keywords: Homogeneous finsler space, generalized symmetric finsler space,

symmetry preserving diffeomorphism

Page 12: Research & reviews discrete mathematical structures (vol1, issue2)

RRDMS (2014)© STM Journals 2014. All Rights Reserved

Research & Reviews: Discrete Mathematical Structures

Volume 1, Issue 2

www.stmjournals.com

A Study on the Sparing Number of the Corona

of Certain Graphs

K.P. Chithra1, K.A. Germina

2, N. K. Sudev

3*

1Naduvath Mana, Nandikkara P O Thrissur - 680301, Kerala, India

2Department of Mathematics, School of Mathematical & Physical Sciences,

Central University of Kerala, Kasaragod - 671316, Kerala, India 3Department of Mathematics, Vidya Academy of Science & Technology,

Thalakkottukara, Thrissur - 680501, Kerala, India

Abstract Let ℕ0 be the set of all non-negative integers and 𝒫(ℕ0) be its the power set. An integer

additive set-indexer (IASI) is defined as an injective function 𝑓: 𝑉(𝐺) → 𝒫(ℕ0) such that

the induced function 𝑓+: 𝐸(𝐺) → 𝒫(ℕ0) defined by 𝑓+(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣) is also

injective, where 𝑓(𝑢) + 𝑓(𝑣) is the sum set of 𝑓(𝑢) and 𝑓(𝑣). If 𝑓+(𝑢𝑣) = 𝑘 ∀ 𝑢𝑣 ∈𝐸(𝐺), then 𝑓 is said to be a 𝑘-uniform integer additive set-indexer. An integer additive

set-indexer 𝑓 is said to be a weak integer additive set-indexer if |𝑓+(𝑢𝑣)| =

𝑚𝑎𝑥 ( |𝑓(𝑢)|, |𝑓(𝑣)|) ∀ 𝑢𝑣 ∈ 𝐸(𝐺). We have some characteristics of the graphs which admit weak integer additive set-indexers. In this paper, we study about the sparing number

of the corona of two graphs.

Keywords: Integer additive set-indexers, mono-indexed elements of a graph, weak

integer additive set-indexers, sparing number of a graph

Page 13: Research & reviews discrete mathematical structures (vol1, issue2)

RRDMS (2014)© STM Journals 2014. All Rights Reserved

Research & Reviews: Discrete Mathematical Structures

Volume 1, Issue 2

www.stmjournals.com

On Some Triple Almost Lacunary Sequence Spaces

Defined by Orlicz Functions

Ayhan Esi* Department of Mathematics, University of Adiyaman, Turkey

Abstract In this paper we introduce the concept of almost lacunary strong convergent triple

sequences with respect to an Orlicz function and examine some properties of these

sequence spaces. We also introduce and study almost lacunary statistical convergence for triple sequences and also present some inclusion relations.

Keywords: Preliminaries, real numbers, Pringsheim, nondecreasing

Page 14: Research & reviews discrete mathematical structures (vol1, issue2)

RRDMS (2014)© STM Journals 2014. All Rights Reserved

Research & Reviews: Discrete Mathematical Structures

Volume 1, Issue 2

www.stmjournals.com

On the Riemann Hypothesis in The Light of The

Littlewood Criterion of Equivalence

Lukasz Andrzej Glinka* Science Editor, BSc Physics. Full Member at the American Association

of International Researchers, American Institute for Policy Development, New York

Abstract In 1859, Riemann extended the Euler study onto the case of a complex variable according to then new Cauchy's complex analysis. According to his point of view, all non-trivial

zeros of the Riemann zeta function are located on the critical line s=1/2+it, where t is a

real number. In 1912, Littlewood presented the equivalence criterion for the Riemann Hypothesis based on the Mertens function. In this paper, the Riemann Hypothesis is

discussed and this is shown that the Littlewood criterion can be immediately applied for

proving the Riemann Hypothesis, according to the analytic number theory.

Keywords: Analytic number theory, Riemann zeta function, rie-mann hypothesis,

Littlewood criterion, Mellin transforms, Mertens function

Page 15: Research & reviews discrete mathematical structures (vol1, issue2)

RRDMS (2014) 36-55 © STM Journals 2014. All Rights Reserved Page 36

Research & Reviews: Discrete Mathematical Structures

Volume 1, Issue 2

www.stmjournals.com

Origin of Modern Mathematical Numeral – 0, 1, 2, 3, 4, 5,

6, 7, 8, 9: the Hindu-Indian-Brahmagubta,

The Islamo-Arabic or the West?

Auwalu Musa* Mubi North Education Authority, Mubi North Local Government Area,

Adamawa State – Nigeria

Abstract The aim of this paper is to examine, analyze and ascertain the root of the modern

mathematical numeral system between the Indian-Hindu Brahmagubta, Islamo-Arabic

and the Western numeral systems. The paper utilized secondary source of data. The methodology adopted by the paper is content analysis. The findings of the paper revealed

that the origin of the shapes of our ten modern numerals do not concern the Indian mathematics history, as Hindus do not have full-fledged mathematical numeral system

before the development of Arabic Numerals between 8th-15th Centuries. The character of

Hindu-Indian numerals in whatever form and categories it belongs- Brahmagubta or anyone else, does not have any genealogical affiliations to the present modern

mathematical numeral versions, as well the same thing applies to the Western

mathematical systems. The paper argued that, the root/origin of the modern Arabic numeral is traced to its original and independent Arabic character versions, which

transmuted from one stage to another with some state of galvanization - from Abjadi, Mashriki to Ghubari. The paper argued that apart from the non-reliability of the thesis of

disparity in characters or shapes of the Arabic numerals compare to that of the Hindu-

Indian Brahmagubta, there is also that of philosophical caricature fiction of genealogical linkage as well as the phonological misplacement of language or terms. The paper also

argued that during the Golden-Age of Islam between 7th and 15th

centuries, European arithmeticians especially in the eleventh, twelfth and part of thirteenth centuries, knew

nothing about Arabic numerals; it was incredibly absent, they knew only the use of

antiquated Roman numerals and Abacus in counting. In that regard, the Arabic numeral symbols (or system) were developed by Islamo-Arabic mathematicians, learned by

Indians and some Europeans who cares about knowledge. In the east, the numeral system

attracted the Indian mathematicians, while in Europe, the numeral system was first employed in Italy, then later France for practical purposes after the translation of the

work of Al-Khwarizmi Mohammed Ibn Musa (Latin Algorithm) in 12th

century into Latin by Gerard of Cremona, also known as Leonardo Fibonacci.

Keywords: Abjadi, Europe, Ghubari, Hindu-Indian-Brahmagubta, modern,

mathematics, islamo-Arabic-Numeral, Origin, Science, West.