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Supported by ¨ ˙ TUBITAK (http://www.tubitak.gov.tr/) Uludag University (http://www.uludag.edu.tr/) K¨ ult¨ ur Okulları(http://www.kulturokullari.net/) Emek Ya˘g (http://www.emekyag.com.tr/giris.html) Kafkas (http://www.kafkas.com/)

Edited by Dr. Ahmet TEKCAN (http://matematik.uludag.edu.tr/AhmetTekcan.htm)

August 21, 2008

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The 20 th International Congress of The Jangjeon Mathematical Society

COMMITTEES

INTERNATIONAL SCIENTIFIC COMMITTEE • DR. V. K. AATRE (INDIA) • DR. R. P. AGARWAL (INDIA) • DR. G. E. ANDREWS (USA) • DR. K. AOMOTO (JAPAN) • DR. B. C. BERNDT (USA) • DR. H. BOR (TURKIYE) • DR. J. BORWEIN (CANADA) • DR. S. COOPER (NEW ZEALAND) • DR. J. C. CORTET (FRANCE) • DR. L. C. JANG (SOUTH KOREA) • DR. T. KIM (SOUTH KOREA) • DR. A. KRASSIMIN (BULGARIA) • DR. H. K. PAK (SOUTH KOREA) • DR. K. RAMACHANDRA (INDIA) • DR. S. H. RIM (SOUTH KOREA) • DR. I. SHIOKAWA (JAPAN) • DR. A. SHTERN (RUSSIA) • DR. Y. SIMSEK (TURKIYE) • DR. K. N. SRINIVASARAO (INDIA) • DR. H. M. SRIVASTAVA (CANADA) • DR. X. WANG (CANADA) LOCAL ORGANIZING COMMITTEE • DR. MUSTAFA YURTKURAN (RECTOR, ULUDAG UNIVERSITY) • DR. GOKAY KAYNAK (DEAN, FACULTY OF ARTS & SCIENCE, ULUDAG UNIVERSITY) • DR. VELI KURT (AKDENIZ UNIVERSITY) • DR. YILMAZ SIMSEK (AKDENIZ UNIVERSITY) • DR. ISMAIL NACI CANGUL (ULUDAG UNIVERSITY) • DR. OSMAN BIZIM (ULUDAG UNIVERSITY) • DR. METIN OZTURK (ULUDAG UNIVERSITY) • DR. SIBEL YALCIN (ULUDAG UNIVERSITY) • DR. A. SINAN CEVIK (BALIKESIR UNIVERSITY) • DR. RECEP SAHIN (BALIKESIR UNIVERSITY) • DR. AHMET TEKCAN (ULUDAG UNIVERSITY) • DR. MUSA DEMIRCI (ULUDAG UNIVERSITY) • DR. SEBAHATTIN IKIKARDES (BALIKESIR UNIVERSITY) • HACER OZDEN (ULUDAG UNIVERSITY) • ILKER INAM (ULUDAG UNIVERSITY) • BETUL GEZER (ULUDAG UNIVERSITY) • AYSUN YURTTAS (ULUDAG UNIVERSITY) Place: The conference will be held in Karinna Hotel (http://www.karinnahotel.com) at the famous ski resort Uludag.

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About The Jangjeon Mathematical Society (JMS) The Jangjeon Mathematical Society (JMS), born in historic Hapcheon, seeks to carry on Hapcheon’s proud tradition of excellent scholarship coupled with unquestionable moral fidelity. Loyal to its Hapcheon heritage, JMS strives to maintain individual excellence, faithfulness to responsibility, and development of talents and abilities while adhering to core values of contributing to world peace and prosperity. JMS was founded in 1996 by Doctor Taekyun Kim to fulfill the aforementioned values through free discussion and cooperation amongst voluntarily participating scholars motivated by a common concern for the general welfare of mankind. This ideal of free and open discussion is mirrored by society’s name, Jangjeon, which rendered in pure Korean, meaning “Geul-Baat”, the place of studies. With this significant symbolism in mind, Dr. Kim selected his birthplace, Jangjeon, as the title of this society. Since ancient times, Hapcheon has served as the training round of many scholars who carried on the teachings of the great Korean scholars, Namyoung Jo Sik, and were renowned for their utmost moral character and honor. The Hapcheon tradition of excellence may still be seen in its profound influence on many modern scholars. The geographical attributes of Hapcheon serve as fitting symbols of its metaphysical properties, The Hwang River, flowing serenely past suggests a steadfastness of virtue, unshaken by secular concerns, infusing energy into all living things. Nearby stands the towering Hwang-mae Mountain whose sheer slopes represent the unwavering fidelity of Hapcheon’s scholars. There are many great scholars who are the members of JMS such as Dr. Seog-Hoon Rim (Managing Editor, Kyungpook University), Dr. Hari M. Srivastava (Chief, Victoria University), Dr. Alexander Shtern (Assistant Chief-in-Editors, Moscow State University), Dr. Krassimir Atanassov (Editor-in-Chief, Bulgarian Academy), Dr. Lee Chae Jang (Assistant-Managing Editor, Kunkook University), Dr. Hongkyung Pak (Adjustor, Daegu Haany University), Dr. Taekyun Kim (Founding Editor) etc. The 20th Congress of The Jangjeon Mathematical Society will be held in Uludag University, BursaTURKIYE. In conclusion, we are really appreciative of participants in the 20th Congress of The Jangjeon Mathematical Society and do hope that all participants, understanding the meaning of Jangjeon, enjoy this conference and work together for the development of world. We hope that all participants have free and active Jangjeon meaning “Geul-Baat”(place of studies) and meaningful discussions with other participants throughout the conference.

Organizing Committee of the 20th Congress of The Jangjeon Mathematical Society.

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The 20 th International Congress of The Jangjeon Mathematical Society

About Bursa Bursa is a city in northwestern (called Marmara) of Turkiye and the seat of Bursa Province. The earliest known site at this location was Cius, which Philip V of Macedonia granted to the Bithynian king Prusias I in 202 BC, for his help against Pergamum and Heraclea Pontica (Karadeniz Ere˘gli). Prusias renamed the city after himself, as Prusa. It was later a major city, located on the westernmost end of the famous Silk Road, and was the capital of the Ottoman Empire following its capture from the shrinking Byzantine Empire in 1326. The capture of Edirne in 1365 brought that city to the fore as well, but Bursa remained an important administrative and commercial center even after it lost its status as the sole capital. Shortly after it was taken by the Ottomans they developed a school of theology at Bursa. This school attracted Muslim schoolers from throughout the Middle East and continued to function after the capital had been moved elsewhere. During the Ottoman rule, Bursa was the source of most royal silk products. Aside from the local production, it imported raw silk from Iran, and occasionally China, and was the factory for the kaftans, pillows, embroidery and other silk products for the royal palaces up through the 17th century. Another traditional occupation is knife making and, historically, horse carriage building. Nowadays one can still find hand-made knives as well as other products in rich variety produced by artisans, but instead of carriages, there are two big automobile industries FIAT and Renault. The city is frequently cited as “Ye¸sil Bursa”(meaning Green Bursa) in a reference to the beautiful parks and gardens located across its urban tissue, as well as to the vast forests in rich variety that extend in its surrounding region. The city is synonymous with the mountain Uludag (Olympus) which towers behind the city core and which is also a famous ski resort. The mausoleums of early Ottoman sultans are located in Bursa and the numerous edifices built throughout the Ottoman period constitute the city’s main landmarks. The surrounding fertile plain, its thermal baths, several interesting museums, notably a rich museum of archaeology, and a rather orderly urban growth are further principal elements that complete Bursa’s overall picture. At present, there is a population of approximately 2 Million and it is Turkiye’s fourth largest city, as well as one of the most industrialized and culturally charged metropolitan centers in the country. Karag¨ oz and Hacivat shadow play characters were historic personalities who lived and are buried in Bursa. Bursa is ˙ also home to some of the most famous Turkish dishes, especially candied chestnuts and Iskender kebap. Its ˙ peaches are also well-renowned. Among its depending district centers, Iznik (historic Nicaea), is especially notable for its long history and important edifices. Bursa is home to Uludag University which is one of the high-scale universities in Turkiye and in the international area. It has 3.000 Academic staff and 40.000 students at different levels, 26.000 of them are undergraduates, 50 programs at 11 Faculties. It has one of the highest numbers, around 800, of foreign students amongst 102 Turkish universities because of its ongoing relationships at international arena.

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Foreword Dear Participants of The 20th International Congress of Jangjeon Mathematical Society, We warmly wellcome you all to the lovely city of Bursa and Uluda˘g University. As the members of The Mathematics Department at Uluda˘ g University, we are proud of having the chance of organizing one of the Congresses of Jangjeon Mathematical Society. We would like to thank all members of the Society, Prof. Dr. Taekyun Kim and Prof. Dr. Seog-Hoon Rim in particular, for giving us the opportunity of bringing together such a nice group of people. The first words about Bursa which comes to ones mind are green, peaches, towels, candied chestnuts, Iskender (D¨ oner) Kebap, history and industry. Being the first capital city of the Ottoman Empire, Bursa has its roots in the third century BC. But the recent findings proves that the real roots go back to 6000 BC. Having such a rich historical background, Bursa has a large number of archeological sites, together with ruins and monuments of many old cultures. Alongside this historical richness which makes it a touristical city, Bursa is one of the main industrial centers of T¨ urkiye. Being at the one end of the famous silk road, there are thousands of Textile factories, three car factories - Fiat, Renault and Peugeot, and one can find a lot of other kinds of industrial production. Being placed on a large plateau, Bursa is also famous with agricultural products. Uluda˘ g University hosting the congress is a middle aged University in Turkish standards. Being 33 years old, it has almost completed the infrastructure and human resources. It has concentrated on education and research in the last decade. As a result of these reforms, the number of publications has had increased five times. All the educational programs have been benchmarked with good universities in USA and Europe, which enabled the University to participate in many international projects and programmes. There are over 200 partner universities all over Europe with which Uluda˘g University is exchanging student and staff as well as academical knowledge. Mathematics Department is one of the oldest departments of the University which has been giving undergraduate and postgraduate education since 1983. Main research areas of the staff are Complex Analysis, Algebra, Discrete Group Theory, Number Theory, Elliptic Curves, p-adic Analysis, Differential Geometry, Projective Geometry, Mathematical Physics, Applications of Mathematics and Differential Equations. There is a total of 30 academic staff at the department which handles a high number of lectures in Mathematics, Physics, Chemistry, Biology Departments as well as all Engineering, Education and Agriculture Departments and 14 Vocational schools, and also carries research projects and other academical work. There is a high number of international partners of the Department and every year there are over 40 exchange students and a high number of visiting staff. Among these international partners, we are proud of having members of Jangjeon Mathematical Society in recent years which gave the chance to widen the research capacity of the Department members and the chance of organizing this memorable congress. We all hope that you enjoy the short period of time you will spend here and will try to create other chances to make your way to Uluda˘ g University and Bursa again.

Organizing Committee.

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The 20 th International Congress of The Jangjeon Mathematical Society

Gratitudes

There are many people who spent a lot of time and effort to make this Congress possible. I would like to thank especially to the following young colleagues who had contributed to the success of this Congress in different ways: • Dr. Ahmet TEKCAN (Editor) ˙ ˙ • Res.Asst. Ilker INAM ¨ • Res.Asst. Hacer OZDEN • Res.Asst. Elif YAS ¸ AR • Res.Asst. Aysun YURTTAS ¸

Without their help, this Congress would be just a dream.

The 20 th International Congress of The Jangjeon Mathematical Society

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Contents 1 On Sums of Squares Chandrashekar Adiga

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2 Some Abstract Convex Functions and Hermit-Hadamard Type Inequalities Gabil Adilov and Serap Kemali

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3 Neighborhoods of Multivalent Analytic Functions Osman Altınta¸ s

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4 Determination of Limit Cycles by Homotopy Perturbation Method for Nonlinear Oscillators Bahar Arslan and Ahmet Yıldırım 17 5 Geometric Approximations to Minimality of Monoids Firat Ate¸ s and Ahmet Sinan ¸ Cevik

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p(x)

6 On Some Properties of the Spaces Aw (Rn ) Ismail Aydın and A.Turan G¨ urkanlı

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7 New Complete Monotonicity Properties of the Gamma Function Necdet Batır

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8 On Values of Jacobi Forms and Shifted Elliptic Dedekind Sums Abdelmejid Bayad

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9 A Modular Transformation for a Generalized Theta Function with Multiple Parameters S.Bhargava, M.S.Mahadeva Naika and M.C.Maheshkumar 22 10 On Meromorphic Harmonic Starlike Functions with Missing Coefficients Hakan Bostanci and Metin ¨ Ozt¨ urk

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11 The Effect of Optically Thick Limit and Buoyancy Forces on the Stability of MHD Ekman Layer Mabrouk Bragdi and Mahdi Fadel Mosa 24 12 A Note on the Operator-Valued Poisson Kernel Serap Bulut

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13 A Generalization of Zakrzewski Morphisms M˘ ad˘ alina Roxana Buneci

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14 Some Closed Type Formulas for Bernoulli and Related Numbers Mehmet Cenkci

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15 On Neighborhood Number and its Related Parameters in Graphs B.Chaluvaraju

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16 The Thermal Properties of the Deformation Potential Materials in Circularly Oscillating Fields J.Y.Choi, J.Y.Sug, S.H.Lee, S.C.Park and Sa-Gong Geon 29

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The 20 th International Congress of The Jangjeon Mathematical Society

17 Permutation Polynomials on Finite Fields Mihai Cipu

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18 On the Eigenvalues of a Schr¨ odinger Operator with Matrix Potential Didem Cos kan and Sedef Karakılı¸ c .

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19 Weak Solutions in Asymmetric Elasticity Ion Al.Cr˘ aciun

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20 Tauberian Theorems for Abel Limitability Method Ibrahim C ¸anak and ¨ Umit Totur

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21 On (1 − u2 )-cyclic Codes over F2k + uF2k + u2 F2k Yasemin C ¸engellenmis

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22 (L, M )-intuitionistic Fuzzy Filters Vildan C ¸etkin, Banu Pazar, Halis Ayg¨ un

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23 Determination of Unknown Boundary Condition in a Quasi-linear Parabolic Equation Ali Demir and Ebru ¨ Ozbilge 36 24 A Quasilinear Elliptic System with Integral Boundary Conditions Mohammed Derhab

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25 On the q-Trigonometric, q-Hyperbolic Functions Ayhan Dil and Veli Kurt

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26 A Fixed Point Theorem H¨ ulya Duru

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27 On The Vague DeMorgan Complemented Partially Ordered Sets and Lattices Zeynep Eken

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28 Application of He’s Semi-Inverse Method to the Nonlinear Wave Equations Meryem Erdal and Ahmet Yıldırım

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29 A System of ODEs for Nonlinear Programming Problems with Smooth Penalty Function Fırat Evirgen and Necati ¨ Ozdemir 42 30 B(X , X ∗ )-Valued Kernels and B(X )-Modules P˘ astorel Ga¸ spar and Dimitru Ga¸ spar

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31 Singular Curves and Singular Elliptic Divisibility Sequences over Finite Fields Bet¨ ul Gezer, Osman Bizim and Ahmet Tekcan

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32 τc -Topology on Hypergroup Algebras Ali Ghaffari

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33 Application of Statistical Shape Analysis to the Classification of Renal Tumours Appearing in Early Childhood Stefan Markus Giebel 46

The 20 th International Congress of The Jangjeon Mathematical Society

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34 Kac-Moody-Virasoro Algebras as Symmetries of 2+1-dimensional Nonlinear Partial Differential Equations Faruk G¨ ung¨ or 47 35 Trigonometric Approximation of Functions in Weighted Lp Spaces Ali G¨ uven

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36 Approximation by Means of Fourier Trigonometric Series in Weighted Orlicz Spaces Ali G¨ uven and Daniyal M.Israfilov

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37 Finite Derivation Type for Graph Products of Monoids Eylem G¨ uzel

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38 On the Rational Operator Pencils in Banach Space Elman Hasanov

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39 Note On Generalized M*-Groups Sebahattin Ikikardes and Recep S ¸ahin

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40 Periodic Solutions for Singular Perturbation Problem of 2−Dimensional Dynamical System Under Matching Conditions Mohammed Jahanshahi 53 41 On The Weighted Composition Operators Khadijeh Jahedi and Sedigheh Jahedi

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42 Morita Equivalence and Outer Conjugacy of Dynamical Systems Maria Joita

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43 Entanglement Dynamics in Stochastic Atom-Field Interactions H¨ unkar Kayhan

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44 Elementary Abelian Coverings of Regular Hypermaps of Types {5, 5, 5} and {5, 2, 10} of Genus 2 Mustafa Kazaz 57 45 Some Subordination Results for Certain Analytic Functions of Complex Order Involving Carlson-Shaffer Operator ¨ Oznur ¨ Ozkan Kılı¸ c 58 46 Arrays with the Window Property and their Generalization Sang-Mok Kim

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47 p-Adic q-Integration on Zp Taekyun Kim

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48 On a Inverse and Direct Problems of Scattering Theory for a Class of Sturm-Liouville Operator with Discountinous Coefficient Nida Palamut Ko¸ sar and Khanlar R.Mamedov 61

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The 20 th International Congress of The Jangjeon Mathematical Society

49 Surfaces with Negative Gauss Curvature; Classification According to the Singularities of Attached Monogenous Functions Lidia Elena Kozma 62 50 Integral and Difference Inequalities in Several Independent Variables and their Discrete Analogues Emine Mısırlı Kurpınar and ¨ Ozlem Mo˘ gol 63 51 Rothe’s Method for Semilinear Parabolic Integrodifferential Equation with Integral Condition A.Guezane-Lakoud and Abderrezak Chaoui 64 52 The Magnetic Field Dependence of the Quantum Transition Properties of Si in the Linearly Polarized Oscillating Field S.H.Lee, J.Y.Sug, G.H.Rue, Sa-Gong Geon and J.Y.Choi 65 53 Gnan Mean and its Dual in Several Arguents Veerabhadraiah Lokesha

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54 Non-linear Multi-objective Transportation Problem: A Fuzzy Goal Programming Approach Hamid Reza Maleki and Sara Khodaparasti 67 55 Recursive Relations on the Coefficients of Some p-Adic Differential Equations Hamza Menken and Abdulkadir A¸ san

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56 On N (k)-Mixed Quasi Einstein Manifolds H.G.Nagaraja

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57 Some New Explicit Values for Ramanujan Class Invariants M.S.Mahadeva Naika

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58 The Necessarily Efficient Point Method for Interval MOLP Problems Hassan Mishmast Nehi and Marzieh Alineghad

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59 Application of Variational Iteration Method for Solving Some Partial Differential Equations Volkan Oban and Ahmet Yıldırım 72 60 The Limit q-Bernstein Operator Sofiya Ostrovska

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61 Verification of the Unknown Diffusion Coefficient by Semigroup Method Ebru ¨ Ozbilge and Ali Demir

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62 Fractional Optimal Control Problem in Cylindrical Coordinates Necati ¨ Ozdemir, Derya Karadeniz and Beyza B.Iskender

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63 Remarks on Interpolation Functions of q-Bernoulli Numbers Hacer ¨ Ozden, Ismail Naci Cang¨ ul and Yilmaz Simsek

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64 Non-local Gas Dynamics Equation and Invariant Solutions Teoman ¨ Ozer

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65 A Note on Multiplers of Lp (G, A) Serap ¨ Oztop

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66 On the Numerical Solutions of Bitsadze Samarskii Type Elliptic Equation with Nonlocal Boundary and Mixed Conditions Elif ¨ Ozt¨ urk and Allaberen Ashyralyev 79 67 Notes on Multiplicative Calculus Ali ¨ Ozyapıcı and Emine Mısırlı Kurpınar

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68 Interval-valued L-fuzzy Topological Groups Banu Pazar, Vildan C ¸etkin and Halis Ayg¨ un

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69 Binomial Thue Equations and their Applications Akos Pinter, Michael Bennett, Kalman Gyory, Lajos Hajdu and Istvan Pink

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70 A Distributional Approach to Classical Electromagnetism I: The Mathematical Tools Burak Polat 83 71 A Distributional Approach to Classical Electromagnetism II: The Physical Evidences Burak Polat 84 72 Numerical Study of non-Darcy Forced Convective Heat Transfer in a Power Law Fluid over a Stretching Sheet K.V.Prasad and V.Rajappa 85 73 Note on Genocchi Numbers and Polynomials Seog-Hoon Rim, Kyoung Ho Park, Yong Do Lim and Eun Jung Moon

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74 Generalized Sobolev-Shubin Spaces Ay¸ se Sandık¸ cı and A.Turan G¨ urkanlı

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75 On Tripotency and Idempotency of Some Linear Combinations of Two Commuting Quadripotent Matrices Murat Sarduvan and Halim ¨ Ozdemir 88 76 On Double Lacunary Statistical σ-Convergence of Fuzzy Numbers Ekrem Sava¸ s

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77 An Excursion into the World of Elliptic Hypergeometric Series Michael J.Schlosser

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78 Some Properties of Vague Rings Sevda Sezer

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79 Beta-Semigroup and Riesz Potentials Sinem Sezer and Ilham A.Aliev

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The 20 th International Congress of The Jangjeon Mathematical Society

80 Principle of Local Conservation of Energy-Momentum Garret Sobczyk and Tolga Yarman

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81 Concerning Fundamental Mathematical and Physical Defects in the General Theory of Relativity Garret Sobczyk, Stephen J.Crothers and Tolga Yarman 94 82 The Diophantine Equation x2 + 11m = y n G¨ okhan Soydan, Musa Demirci and Ismail Naci Cang¨ ul

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83 Fuzzy Triangular Inequality G¨ ultekin Soylu

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84 Bounds for Classical Orthogonal Polynomials and Related Special Functions H.M.Srivastava

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85 Some Glimpses of Hindu (or Vedic) Mathematics and Srinivasa Ramanujan (1887– 1920) Rekha Srivastava 98 86 On p-adic q-Dedekind Sums Yılmaz S ¸im¸ sek 87 On Quadratic Ideals and Indefinite Quadratic Forms Ahmet Tekcan, Osman Bizim and Bet¨ ul Gezer

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88 Improved Direct and Inverse Theorems of Approximation Theory in the MorreySmirnov Classes Defined on the Complex Plane N.Pınar Tozman and Daniyal M.Israfilov 101 89 On q−Laplace Type Integral Operators and their Applications Faruk U¸ car and Durmu¸ s Albayrak

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90 q-Laplace Transforms Burcu Vula¸ s and G¨ ulsen K¨ urem

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91 Relative Defect and Multiple Common Roots of Two Meromorphic Functions Harina P.Waghamore

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92 The Quantum Mechanical Mechanism Behind the end Results of the GTR: Matter is Built on the Lorentz Invariant Framework Energy x Mass x Length2 ∼ h2 Tolga Yarman 105 93 Some Applications of He’s Variational Approaches Ahmet Yıldırım

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94 An Efficient Method for Solving Singular Two-Point Initial Value Problems Ahmet Yıldırım and Deniz A˘ gırseven

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95 Holditch Theorem for the Closed Space Curves in Lorentzian 3-space Handan Yıldırım, Salim Y¨ uce and Nuri Kuruo˘ glu

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The 20 th International Congress of The Jangjeon Mathematical Society

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96 On the Numerical Solutions of Hyperbolic Equations with Nonlocal Boundary and Neumann Conditions ¨ Ozg¨ ur Yıldırım and and Allaberen Ashyralyev 109 97 The Estimation of Mean Modulus of Smoothness in Lpw Yunus Emre Yildirir and Daniyal Israfilov

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98 Differential Transform Method (DTM) for Solving Sine-Gordon Type Equations Eda Y¨ ul¨ ukl¨ u and Turgut ¨ Ozi¸ s

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99 Hardy Littlewood and Polya Inequalities and their Applications to Various Integral Transforms Osman Y¨ urekli 112 100 On Sum Degree Energy of a Graph R.K.Zaferani, C.Adiga and H.B.Walıkar

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101 A Goal Programming Method for Finding Common Weights in DEA Majid Zohrehbandian, Ahmad Makui and Alireza Alinezhad

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The 20 th International Congress of The Jangjeon Mathematical Society

On Sums of Squares Chandrashekar Adiga

Historically one of the problems receiving a good deal of attention is the representing integers as sums of squares. In this talk, we first briefly review some of the advances in this area and then we describe how Ramanujan’s continued fractions are useful in deriving Jacobi’s two-square and two-triangular theorems. Finally, we present a general relationbetween sums of squares and sums of triangular numbers.

References [1] Adiga C. and Vasuki K.R. On sums of triangular numbers. The Mathematics Student 70(2001), 185190. [2] Adiga C. Cooper S. and Han J.H. A general relation between sums of squares and sums of triangular numbers. Int. Jour. Number Theory 1(2)(2005), 175-182. [3] Barrucand P. Cooper S. and Hirschhorn M. Relations between squares and triangles. Discrete.Math. 248(1-3)(2002), 245-247. [4] Berndt B.C. Ramanujan’s Notebooks. Part III, Springer-Verlag,New York, 1991.

Address: Department of Studies in Mathematics, University of Mysore, Manasagangotri, Myore-570006-INDIA e-mail: adiga [email protected]

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Some Abstract Convex Functions and Hermit-Hadamard Type Inequalities Gabil Adilov and Serap Kemali

Studying Hermite-Hadamard type inequalities for some function classifications have been very important in recent years. These inequalities, which are known in convex functions, are also found in different function classifications ([4-5], [7]). One of these functions classifications is abstract convex functions. The problem of finding Hermite-Hadamard type inequalities for increasing positively homogeneous (IPH) functions, increasing radiant (InR) functions , increasing co-radiant (ICR) functions and increasing convex-alongrays (ICAR) functions, which are significant classifications of abstract convex functions, is investigated by different authors and the concrete results are found ([1, 2, 3, 6, 8]). In this article, the problem of calculating Hermit-Hadamard type inequalities is considered totally, older results are summarized, new results of some classification are achieved and the results of some other 2 , all the results defined for that specific classification are generalized. By considering a concrete area in R++ area.

References [1] Adilov G.R. and Kemali S. Hermite-Hadamard type inequalities for increasing positively homogeneous functions. Journal of Inequalities and Applications, Volume 2007, Article ID 21430, 10pp, dor 101155/2007/21430. [2] Adilov G.R. Increasing co-radiant functions and Hermite-Hadamard type inequalities. Mathematical Inequalities and Applications (Submitted). [3] Dragomir S.S., Dutta J. and Rubinov A.M. Hermite-Hadamard type inequalities for increasing convex along rays functions. Analysis (Munich) 24(2), 171–181. [4] Dragomir S.S. and Pearce C.E.E. Quasi-convex functions and Hadamard’s inequality. Bull. Australian Math. Soc. 57(1998), 377-385. [5] Pearce C.E.M. and Rubinov A.M. P-functions, quasiconvex functions and Hadamard-type inequalities. Journal of Mathematical Analysis and Applications 240(1999), 92-104. [6] Rubinov A.M. Abstract convexity and global optimization. Kluwer Academic Publishers, Dordrecht, 2000. [7] Rubinov A.M. and Dutta J. Hadamard type inequality for quasiconvex functions in higher dimensions. (Preprint) RGMIA Res. Rep. Coll., 4 (1) (2001), Article 9. [ONLINE: http:// rgmia.vu.edu.au/v4n1.htm1] [8] Sharikov E.V. Hermite-Hadamard type inequalities for increasing radiant functions. Journal of Inequalities in Pure and Applied Mathematics 4(2)(2003), Article 47.

Address: Mersin University, Faculty of Arts and Science, Department of Mathematics, Mersin-TURKIYE Akdeniz University, Faculty of Arts and Science, Department of Mathematics, Antalya-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

Neighborhoods of Multivalent Analytic Functions Osman Altınta¸ s

Let T (n, p) denote the class of functions f (z) which are analytic and multivalent in the unit disk U = {z : z ∈ C and |z| < 1}. We define the (n, ε) − neighborhood of a function f (q+δ) (z) when f ∈ T (n, p). q We also let Tn,p (λ, α, δ) denote the subclass of T (n, p) consisting of functions f (z) which satisfy the following inequality zf (1+q+δ) (z) + λz 2 f (2+q+δ) (z) Re > α, λzf (1+q+δ) (z) + (1 − λ)f (q+δ) (z) where 0 ≤ λ ≤ 1, 0 ≤ δ < 1, 0 ≤ α < p − q − δ, p > q, p ∈ N, q ∈ N0 = N ∪ {0}. Finally Kn (p, q, δ, α, λ, µ) denote the subclass of the general class T (n, p) consisting of functions f ∈ T (n, p) which satisfy the following non-homogenous Cauchy-Euler differential equation: z2

d2+q+δ w d1+q+δ w dq+δ w dq+δ g + 2 (1 + µ) z + µ (1 + µ) = (p − q − δ + µ) (p − q − δ + µ + 1) , dz 2+q+δ dz 1+q+δ dz q+δ dz q+δ

q where w = f ∈ T (n, p) , g ∈ Tn,p (λ, α, δ) and µ > q − p + δ. In the present investigation, several results concerning the (n, ε) − neighborhoods, coefficient bounds, q (λ, α, δ) and Kn (p, q, δ, α, λ, µ) are distortion inequalities for functions f ∈ T (n, p) in both classes Tn,p given.

References [1] Altınta¸s O. On a subclass of certain starlike functions with negative coefficients. Math. Japonica 36 (1991), 489-495. [2] Altınta¸s O., Irmak H. and Srivastava H.M. Fractional calculus and certain starlike functions with negative coefficients. Comput. Math. Appl. 30(2)(1995), 9-15. ¨ ¨ and Srivastava H.M. Neighborhoods of a certain family of multivalent functions [3] Altınta¸s O., Ozkan O. with negative coefficients. Comput. Math. Appl. 47(2004), 1667-1672. [4] Altınta¸s O. Neighborhoods of certain p-valently analytic functions with negative coefficients. Appl. Math. and Computation 187(2007), 47-53.

Address: Baskent University, Department of Mathematics Education, Baglıca, TR 06530, Ankara-TURKIYE e-mail: [email protected]

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Determination of Limit Cycles by Homotopy Perturbation Method for Nonlinear Oscillators Bahar Arslan and Ahmet Yıldırım

In this paper, Homotopy perturbation method is applied to certain nonlinear oscillators with strong nonlinearity. The method is of deceptively simplicity and the insightful solutions obtained are of high accuracy even for the first-order approximations.

References [1] He JH. Non-Perturbative Methods for Strongly Nonlinear Problems. Dissertation.de- Verlag im Internet GmbH, 2006 [2] He JH. Some asymptotic methods for strongly nonlinear equations. International Journal of modern Physics B 20(10)(2006), 1141-1199. [3] He JH.Limit cycle and bifurcation of nonlinear problems. Chaos Solitons & Fractals 26(3)(2005), 827-833. [4] He JH. Determination of limit cycles for strongly nonlinear oscillators. Physical Review Letters 90(17)(2003), Art. No. 174301. ¨ [5] Ozis T. and Yıldırım A. A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity. Chaos Solitons & Fractals 34(3)(), 989-991. ¨ [6] Ozis T. and Yıldırım A. Determination of limit cycles by a modified straightforward expansion for nonlinear oscillators. Chaos Solitons & Fractals 32(2007), 445-448.

Address: Ege University, Faculty of Science, Department of Mathematics, Bornova 35100, Izmir-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

Geometric Approximations to Minimality of Monoids Firat Ate¸ s and Ahmet Sinan ¸ Cevik

It was mainly considered a geometric configuration, namely pictures, of monoid presentations. In fact, by using pictures, we showed that a specific monoid presentation (that is, the presentation of the semidirect product of finite cyclic monoids) is minimal while it is inefficient. Let A and K be arbitrary two monoids. For any connecting monoid homomorphism θ : A −→ End(K), let M = K oθ A be the corresponding monoid semi-direct product. In [2], Cevik clarified necessary and sufficient conditions for the standard presentation ℘M of M to be p-Cockcroft for any prime p or 0. Moreover, as an application of this above result, it has been showed the efficiency for the presentation ℘∗M of the semidirect product of any two finite cyclic monoids (in a joint work [1]). As a main tool of this talk, it will be given sufficient conditions for ℘∗M to be minimal but not efficient. To do that the same method in [2] will be used. The fundamental material of arbitrary semigroups can be found in the famous book [3]. We also note that while the geometric thecniques of semigroups (and also of monoids) have been studied in [4, 5], the homological methods and applications have been specified in [3]. Finally an application of the geometric part has been given on the semidirect product of monoids in [7].

References [1] Ate¸s F. and C ¸ evik A.S. Minimal but inefficient presentations for semidirect product of finite cyclic monoids. Groups St. Andrews 2005, Vol 1, L.M.S. Lecture Note Series, 339(2006), 170-185. [2] C ¸ evik A.S. Minimal but inefficient presentations of the semi-direct products of some monoids. Semigroup Forum 66(2003), 1-17. [3] Howie J.M. Fundamentals of Semigroup Theory. Oxford University press, 1995. [4] Pride S.J. Geometric methods in combinatorial semigroup theory. Semigroups, Formal Languages and Groups, (J.Fountain editor), Kluwer Academic Publishers, 215-232 (1995). [5] Pride S.J. Low-dimensional homotopy theory for monoids. Int. J. Algebra and Comput. 5(6)(1995), 631-649. [6] Squier C.C. Word problems and a homological finiteness condition for monoids. Journal of Pure and Appl. Algebra 49(1987), 201-216. [7] Wang J. Finite derivation type for semi-direct products of monoids. Theoretical Computer Science, 191(1-2)(1998), 219-228.

Address: Balikesir University, Faculty of Science and Arts, Department of Mathematics, Cagis Campus, 10145, Balikesir-TURKIYE e-mails: [email protected], [email protected] URLL: http://w3.balikesir.edu.tr/ scevik/

The 20 th International Congress of The Jangjeon Mathematical Society

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p(x)

On Some Properties of the Spaces Aw (Rn ) Ismail Aydın and A.Turan G¨ urkanlı p(x)

In this work we define Aw (Rn ) to be the vector space of all complex-valued functions in L1w (Rn ) whose Fourier transforms fb belong to the generalized Lebesgue space Lp(x) (Rn ), where p (x) is a measurable

p function from Rn into [1, ∞). We endow it with the sum norm kf kw = kf k1,w + fb and study some p

p(x)

important properties of this space. Furthermore we show that Aw (Rn ) is a Sw (Rn ) space [1]. Later we p(x) investigate the ideals of the space Aw (Rn ). At the end of this work we discuss inclusions, embedding p(x) p(x) and compact embeddings between the spaces Aw (Rn ) and the multipliers of the spaces Aw (Rn ).

References [1] Cigler J. Normed ideals in L1 (G). Indag Math. 31(1969), 272-282.

Address: Sinop University, Faculty of Arts and Sciences, Department of Mathematics, 57000, Sinop-TURKIYE Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, 55139. Kurupelit, Samsun-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

New Complete Monotonicity Properties of the Gamma Function Necdet Batır

R∞ As it is well known the classical gamma function is defined by the integral Γ(z) = 0 uz−1 e−u du, for Rez > 0. In this talk, we prove some new complete monotonicity theorems for this important function.

References [1] Alzer H. Gamma function inequalities. Numer. Algor. DOI10.1007/511075-008-9160-4, 2008. [2] Alzer H. and Batir N. Monotonicty properties of the gamma function. Appl. Math. Lett. 20(2007), 778-781. [3] Alzer H. Inequalities fo the gamma function. Proc. Amer. Math. Soc. 128(1999), 141-147. [4] Alzer H. and Grinshpan A.Z. Inequalities for the gamma and q-gamma functions. J. Approx. Theory 144(2007), 67-83. [5] Alzer H. and Berg C. Some Classes of completeley monotonic functions II. The Ramanujan Journal 11(2)(2006), 225-248. [6] Alzer H. On Ramanujan’s Double inequality for the gamma function. Bull. London Math. Soc. 35(2003), 601-607. [7] Alzer H. On some inequalities for the gamma and psi functions. Math. Comp. 66(217)(1997), 373-389. [8] Anderson G.D., Barnard R.W., Richards K.C., Vamanamurthy M.K. and Vuorinen M. Inequalities for zero-balanced hypergeometric functions. Trans Amer. Math. Soc. 347(1995), 1713-1723. [9] Andrews G., Askey R., and Toy R. Special functions, Encyclopedia of Mathematics and its Applications. V.71, Cambridge U. Pres, 1999. [10] Batir N. Sharp inequalities for factorial n. Proyecciones 27(1)(2008), 97-102. [11] Batir N. On some properties of the gamma function. Expo. Math. 26(2008), 187-196. [12] Batir N. Some new inequalities for gamma and polygamma functions. JIPAM. J. Inequal. Pure Appl. 6(4)(2005), Article 103, 9 pp. [13] Burnside W. A rapidly convergent series for logN!. Messenger Math. 46(1917), 157-159.

Address: Y¨ uz¨ unc¨ u Yil University, Faculty of Sciences and Arts, Department of Mathematics, 65080, Van-TURKIYE e-mail: necdet [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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On Values of Jacobi Forms and Shifted Elliptic Dedekind Sums Abdelmejid Bayad

It’s well known that the Jacobi forms in one variable are a cross between elliptic functions and modular forms in one variable. They have several applications in differents areas in mathematics, especially in number theory and arithmetical geometry [6]. In this talk , we are interesting by the study of values of some Jacobi forms in two variables. Precisely, we introduce shifted elliptic Dedekind sums in terms of special values of Jacobi forms in two variables and we state and prove their reciprocity Laws. In our study, we show how to use our techniques to obtain a closed new reciprocity law for the so-called Shifted-elliptic Dedekind-Sczech Sums.

References [1] Atiyah M.F. The Logarithm of the Dedekind-Function. Math. Ann 278(1987), 335-380. [2] Barge J. and Ghys E. Cocycles d’Euler et de Maslov. Math. Ann. 294(2)(1992), 235-265. [3] Maria Immaculada Galvez Carrillo. Modular invariants for manifolds with Boundary. Thesis (2001), http://www.tdx.cesca.es/TDX-0806101-095056/ . [4] Hirzebruch F. The signature theorem: reminiscences and recreation. Prospects in Mathematics. Ann. of Math.Studies 70, 3-31, Princeton University Press, Princeton, 1971. [5] Hirzebruch F., Berger T. and Jung R. Manifolds and Modular forms. Aspects of Math.E. 20, Vieweg (1992). [6] Hirzebruch F. and Zagier D. The Atiyah-Singer Theorem and Elementary Number Theory. Math. Lecture Series 3, Publish or Perish Inc, 1974. [7] Kirby R. and Melvin P. Dedekind sums, µ-invariants and the signature cocycle. Math. Annalen 299(1994), 231-267. [8] Sczech R. Dedekindsummen mit elliptischen Funktionen. Invent.math 76(1984), 523-551. [9] Weselmann U. EisensteinKohomologie und Dedekindsummen fur GL2 uber imaginarquadratischen Zahlenkorpern. J. reine. angew. Math. 389(1988), 90-121.

Address: Departement de mathematiques, Universit e d’Evry Val d’Essone Bd. F. Mitterrand, 91025 EVRY CEDEX e-mail: [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

A Modular Transformation for a Generalized Theta Function with Multiple Parameters S.Bhargava, M.S.Mahadeva Naika and M.C.Maheshkumar In this talk, We obtain a modular transformation for the theta function ∞ X ∞ X

2

q a(m

+mn)+cn2 +λm+µn+ν

ζ Am+Bn z Cm+Dn .

(9.1)

−∞ −∞

We are thus able to unify and extend several modular transformations in literature. We also establish the relations between the above identity and Jacobian Theta-function when a|b. The objective here is to obtain a modular transformation for (9.1). It is possible to first treat the above equation(9.1) with λ = µ = ν = 0, A = B = 1 = C, D = −1 and then effect suitable transformations on ζ and z to obtain our main result. But we have preferred to present our main result and all the related lemmas directly and in detail in order to bring out the motivation and lucidity of the inter play between the various parameters all through, than would be the case in the abbreviated alternative approach.

References [1] Adiga C., Berndt B.C., Bhargava S. and Watson G.N. Chapter 16 of Ramanujan’s Second Notebook, Theta functions and q-series. Mem. Am. Math. Soc. 53, no. 315(1985), American Mathematical Society, Providence, 1985. [2] Adiga C., Mahadeva Naika M.S. and Han J.H. General Modular Transformations for Theta Functions. Indian J. Math. 49(2)(2007), 239-251. [3] Baker H.F. An Introduction to the Theory of Multiply Periodic Functions. Cambridge University Press, (1907). [4] Bellmen R. A Brief Introduction to the theta functions. Holt, Rinehart and Winston, (1961). [5] Bhargava S. Unification of the Cubic Analogues of Jacobian Theta Functions. J. Math. Anal. Appl. 193(1995), 543–558. [6] Bhargava S. and Anitha N. A Triple Product Identity for the three − parameter Cubic Theta Function. Indian Journal of Pure and Applied Math. 36(9)(2005), 471–479. [7] Bhargava S. and Fathima S.N. Unification of Modular Transformations for Cubic Theta Functions. New Zealand J. Mathematics 33(2004), 121–127. [8] Borwein J.M. and Borwein P.B. A Cubic Counterpart of Jacobi’s Identity and the AGM. Trans. Amer. Math. Soc. 323(1991), 691–701. [9] CooperS. Cubic Theta Functions. J. Computational and Applied Math. 160(2003), 77–94. [10] Hecke E. Mathematische Werks. G¨ ottingen: Vordenhoeck und Ruprecht, 1959.

Address: Department of studies in Mathematics, University of Mysore, Manasagangotri, Mysore-570 006, INDIA Department of Mathematics, Bangalore University, Central College Campus, Bangalore-560 001, INDIA Department of Mathematics, Bangalore University, Central College Campus, Bangalore-560 001, INDIA e-mails: [email protected], [email protected], [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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On Meromorphic Harmonic Starlike Functions with Missing Coefficients Hakan Bostanci and Metin ¨ Ozt¨ urk

In this paper, we introduce a new class of meromorphic harmonic starlike functions with missing coefficients in the punctured unit disk U ∗ = {z : 0 < |z| < 1}. We obtain coefficient inequalities, distortion theorem and closure theorem. In addition, we investigated some properties of this class.

References [1] Cluine J. and Sheil-Small T. Harmonic Univalent Functions. Ann. Acad. Sci. Fenn. Ser. Al Math. 9 (1984), 3-25. [2] Hengartner W. and Schober G. Univalent Harmonic Functions. Trans. Amer. Math. Soc. 299 (1987), 1-31. [3] Jahangari J.M. Coefficient bounds and univalence criteria for harmonic functions with negative coefficients. Ann. Univ. Mariae Currie-Sklodowska, Sec. A, 52(1998), 57-66. [4] Jahangari J.M. Harmonic Meromorphic Starlike Functions. Bull. Korean Math. Soc. 37(2)(2000), 291-301. [5] Auf M.K. and Hossen H.M. New criteria for meromorphic p-valent starlike funcitons. Tsukuba J. Math. 17(2)(1993), 181-186. [6] Darwish H.E. Meromorphic p-valent starlike functions with negative coefficients. Indian J. Pure Apll. Math. 33(7)(2002), 967-976. [7] Uraleggaddi B.A. and Somanatha C. New criteria for meromorphic starlike univalent functions. Bull. Austral. Math. Soc. 43(1991), 137-140. [8] Joshi S.B. and Sangle N.D. Meromorphic Starlike functions with negative and missing coefficients. Far East J. Math. Sci. (FMJS) 26(2)(2007), 289-301. [9] Jahangari J.M. and Silverman H. Meromorphic Univalent Harmonic Functions with Negative Coefficients. Bull. Korean Math. Soc. 36(1999), 763-770. [10] Murugusundaramoorthy G. Starlikeness of multivalent meromorphic harmonic functions. Bull. Korean Math. Soc. 40(4)(2003), 553-564. [11] Murugusundaramoorthy, G. Harmonic meromorphic convex functions with missing coefficients, J. Indones. Math. Soc. (MIHMI), 10(1)(2004), 15-22.

Address: Uludag University, Faculty of Science, Department of Mathematics, G¨ or¨ ukle 16059, Bursa-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

The Effect of Optically Thick Limit and Buoyancy Forces on the Stability of MHD Ekman Layer Mabrouk Bragdi and Mahdi Fadel Mosa

Let us consider a Cartesian co-ordinate system of rotating uniformly with angular velocity Ω about the z-axis, the basic equations of motion for such a configuration in a non-dimensional form, as well established (cf. [3]): ∂u ∂t

∂θ − S ∂η

2

∂2u ∂η 2 1 Fr −

∂v ∂ v + E(u − 1) − S ∂η = 3D − N v + ∂η Grθ 2 + 2 2 2 ∂2θ Ec ∂Q ∂u ∂v = 3D P1r ∂η + ∂η + N Ec (u − 1) + v 2 . 2 − Bo ∂η + Ec ∂η ∂v ∂t

∂θ ∂t

− Ev − S ∂u ∂η = 3D − N (u − 1) +

The boundary conditions on velocities and temperature can be written in the form: u = 3Dv = 3D0 at η = 3D0; u → 1, v → 0 as η → ∞. θ = 3D1 at η = 3D0 for suction; θ = 3D TT∞ at η = 3D0 for blowing; 0 as η → ∞. θ = 3D TT∞ 0 A radiative optically thick limit case has been introduce into the energy equation of MHD Ekman layer. A steady solution for the velocity and temperature distribution is obtained by using a finite difference method that implements the 3-stage lobatto IIIa formula, the resulting solution have shown that the radiation has a remarkable effect on the temerature distribution. Also the stability of this model is investigated. The differential equations governing such a stability problem are introduced. A computer program for computing the eigenvalues of the system for the measured of the stability has been organized using Matlab V.6, and the neutral stability curves have been achievement at difference cases of parameters. The result of the mathematical analysis show that the optically thick limit and buoyancy forces produced a new addition case for stability.

References [1] Helliwell J.B. On the stability of thermally radiative = magnetofluiddynamic channel flow. J. Eng. Math. 11(1977), 67-80. [2] Helliwell J.B. and Mosa M.F. Radiative heat transfer= in horizontal magnetohydrodynamic channel flow with buoyancy effects and an axial temperature gradient. Int. J. Heat Mass Transfer 22(1979), 657-668. [3] Mosa M.F. Radiative heat transfer in MHD Ekman layer on a porous plate. Dirasat: The University of Jordan, 7(2)(1985), 167. [4] Mosa M.F. and Manaa S.A. Effects of radiative heat=transfer in the MHD Ekman layer on a porous plate. Mu’tah Journal for Research and Studies, Natural and Applied Sciences Series, 7(1992), 286.

Address: University Center of Larbi Ben M’hidi, Department of Mathematics, route de Constantine, DZ 04000. Oum El Bouaghi-ALGERIA e-mail: [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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A Note on the Operator-Valued Poisson Kernel Serap Bulut In this talk, we give a different proof of the integral formula Z 2π 1 Kr,t (T )dt = I, 2π 0

where Kr,t (T ) is the operator-valued Poisson kernel.

References [1] Chalendar I. The operator-valued Poisson kernel and its applications. Ir. Math. Soc. Bull. 51(2003), 21–44. [2] Taylor A.E. A Note on the Poisson Kernel. Amer. Math. Monthly 57(1950), 478–479.

Address: Kocaeli University, Faculty of Arts and Science, Department of Mathematics, Kocaeli-TURKIYE e-mail: [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

A Generalization of Zakrzewski Morphisms M˘ ad˘ alina Roxana Buneci

We briefly recall various notions of groupoid morphisms and their applications. Then we introduce a new notion of groupoid morphisms (in algebraic setting as well as in topological setting), starting from the characterization obtained in [1] for the Zakrzewski morphisms [8]. The classes of (algebraic, respectively, topological) groupoids with these new introduced morphisms form categories. We also prove that the isomorphisms of the resulted categories can be identified with the groupoid isomorphisms in the usual sense. We analyze the relation between the proposed notion of groupoid morphisms and other notions such as correspondences [7], bibundles [3] and we show that these new morphisms are generalizations of the morphisms in the sense of [8, 2].

References [1] Buneci M. Groupoid categories. Perspectives in Operator Algebras and Mathematical Physics 27-40, Theta 2008. [2] Buneci M. and Stachura P. Morphisms of locally compact groupoids endowed with Haar systems. arXiv: math.OA/0511613. [3] Landsman N.P. Operator algebras and Poisson manifolds associated to groupoids. Comm. Math. Phys. 222(2001), 97–116. [4] Muhly P., Reanult J. and Williams D. Equivalence and isomorphism for groupoid C*-algebras. J. Operator Theory 17(1987), 3–22. [5] Muhly P. Coordinates in operator algebra. Book in preparation. [6] Renault J. A groupoid approach to C ∗ -algebras. Lecture Notes in Math. Springer-Verlag, 793, 1980. [7] Macho-Stadler M. and O’Uchi M. Correspondences and groupoids. Proceedings of the IX Fall Workshop on Geometry and Physics, Publicaciones de la RSME 3(2000), 233–238. [8] Zakrzewski S. Quantum and Classical pseudogroups I. Comm. Math. Phys. 134(1990), 347–370.

Address: Department of Mathematics, University Constantin Brncui, Bld. Republicii, Nr. 1, 210152 Trgu Jiu- ROMANIA e-mail: [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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Some Closed Type Formulas for Bernoulli and Related Numbers Mehmet Cenkci

The object of this talk is to give further applications of the theorem relating to potential polynomial and exponential Bell polynomial stated by Howard [Discrete Math. 39 (1982) 129-143]. This theorem provides a methodical approach to a number of formulas and identities involving Stirling numbers. In particular, we derive several closed form formulas for higher order Bernoulli, Eulerian, Genocchi, tangent, (z) Apostol-Bernoulli, Apostol-Euler, Bernoulli numbers of the second kind and the numbers An .

References [1] Apostol T.M. On the Lerch zeta function. Pacific J. Math. 1(1951), 161-167. [2] Carlitz L. Eulerian numbers and polynomials. Math. Mag. 33(1959), 247-260. [3] Carlitz L. A note on Bernoulli and Euler polynomials of the second kind. Scripta Math. 25(1961), 323-330. [4] Cenkci M. and Howard F.T. Notes on degenerate numbers. Discrete Math. 307(2007), 2359-2375. [5] Comtet L. Advanced Combinatorics. Riedel, Dordrech, Boston, 1974. [6] Gould H.W. Combinatorial Identites. Morgantown, 1972. [7] Howard F.T. A sequence of numbers related to the exponential function. Duke Math. J. 34(1967), 599-616. [8] Howard F.T. Numbers generated by the reciprocal of ex − x − 1. Math. Comput. 31(1977), 581-598. [9] Howard F.T. A special class of Bell polynomials. Math. Comput. 35(1980), 977-989. [10] Howard F.T. A theorem relating potential and Bell polynomials. Discrete Math. 39(1982), 129-143. [11] Jordan C. Calculus of Finite Differences. Chelsea, New York, 1950. [12] Luo Q.-M. and Srivastava H.M. Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 308(2005), 290-302. [13] Luo Q.-M. and Srivastava H.M. Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 51(2006), 631-642. [14] N¨ orlund N. Vorlesungen u ¨ber Differenzenrechnung. Chelsea, New York, 1954. [15] Srivastava H.M. and Todorov P.G. An explicit formula for the generalized Bernoulli polynomials. J. Math. Anal. Appl. 130(1988), 509-513. [16] Todorov P.G. Une formule simple explicite des nombres Bernoulli g´en´eralis´es. C. R. Acad. Sci. Paris S´er. I Math. 301(1985), 665-666.

Address: Akdeniz University, Department of Mathematics, 07058 Antalya-TURKIYE e-mail: [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

On Neighborhood Number and its Related Parameters in Graphs B.Chaluvaraju

Given a simple graph G = (V, E), a subset S of V is called a neighborhood set provided G is the union of the subgraphs induced by the closed neighborhoods of the vertices in S. The neighborhood number η(G) of G is the minimum cardinality of a neighborhood set of G. In this paper, we initiate a study on neighborhood set and its related parameters and also its graph theoretical relationships are explored.

References [1] Chaluvaraju B. k-neighborhood, k-connected neighborhood and k-co-connected neighborhood number of a graph. J.of Analysis and Computation 3(1)(2007), 9-12. [2] Harary F. Graph theory. Addison-Wesley, Reading Mass. 1962. [3] Haynes T.W., Hedetniemi S.T and Slater P.J. Fundamentals of domination in graphs. Marcel Dekker, Inc., New York, 1998. [4] Hedetniemi S.M., Hedetniemi S.T, Laskar R.C, Markus L. and Slater P.J. Disjoint dominating sets in graphs. Proce. of Int. Conf. of Disct. Maths., IISc, Bangalore, (2006), 88-101. [5] Jayaram S.R. The nomatic number of a graph. Nat. Acad. Sci. Lett. 19(1996), 159-161. [6] Kulli V.R. and Sigarkanti S.C. Further results on the neighbourhood number of a graph. Indian J. Pure and Appl. Math. 23(8)(1992), 575-577. [7] Kulli V.R. and Soner N.D. The independent neighbourhood number of a graph. Nat. Acad. Sci. Letts. 19(1996), 159-161. [8] Sampathkumar E. and Neeralagi P.S. The neighbourhood number of a graph. Indian J. Pure and Appl. Math. 16(2)(1985), 126-132. [9] Soner N.D., Chaluvaraju B. and Janakiram B. The maximal neighborhood number of a graph. Far East J. Appl. Math. 5(3)(2001), 301-307. [10] Soner N.D., Chaluvaraju B. and Janakiram B. Maximal edge neighborhood number in graphs. Indian Math. Soc. (IMS) 46(2-3)(2004), 283-291.

Address: Department of Mathematics, Central College Campus, Bangalore University, Bangalore-560 001-INDIA e-mail: [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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The Thermal Properties of the Deformation Potential Materials in Circularly Oscillating Fields J.Y.Choi, J.Y.Sug, S.H.Lee, S.C.Park and Sa-Gong Geon

We study the thermal properties of electron-deformation potential phonon interacting system based on quantum transport theory. We use the projected Liouville equation method with Equilibrium Average Projection Scheme (EAPS). We consider two systems - the one system is under the right circularly polarized oscillatory external fields and the other system is under the left circularly polarized oscillatory external field. The main purpose of this work is that comparisons of QTLS which show the absorption power and QTLW in both directional circularly polarized oscillatory external field. Our results indicate that the absorption power of the right circularly polarized oscillatory external field is larger than that of the left circularly polarized oscillatory external field, while the opposite result is obtained for the QTLW. we analyze the temperature of the QTLS and the QTLW in various cases. In order to analysis the quantum transition process of two circularly oscillatory external field, we compare the temperature dependence of the QTLW and the QTLS of two transition process, the intra Landau level transition process and the inter Landau level transition process.

References [1] R. Zwanzig, J. Chm. phys. 33, 1338(1960); J. Chm. phys. 60, 2717(1960); Phys. Rev. 124. 983(1961); Physica 30. 1109(1964); J. stat. phys. 9, 215, 73 ; [2] V. M. Kenkre, Phys.Rev. A4, 2327(1971); V. M. Kenkre, Phys. Rev. A6, 769 (1972); V. M. Kenkre, Phys. Rev. A7, 772 (1973) [3] Joung Young Sug and Sang Don Choi. Phys. Rev.E Vol 55, 314.(1997); [4] J. Y. Sug, S. G. Jo, Jangil kim and S. D. Choi. Phys. Rev. B 64, 235210(2001), references therein.

Address: Electronic and Electric Eng. School, Kyungpook National University, Daegu 702-701, KOREA (Sa-Gong Geon) Dep. of Electrical Eng., Donga Uni. Pusan City, KOREA e-mails: [email protected], [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

Permutation Polynomials on Finite Fields Mihai Cipu

One of the problems of certain interest for cryptographers is to know polynomials that induce a bijection of a finite field under substitution. Our talk starts with a discussion of what does it mean ‘to know’ in this context. Then we proceed by presenting various answers given to the problem. The second part of the talk is devoted to Dickson (in fact, Chebyshev) polynomials of the second kind fn . The permutation properties of members of this class are hard to describe. In his 1982 Ph. D. thesis, R.W. Matthews [5] pointed out a sufficient condition assuring that a Dickson polynomial fn permutes a finite field F Fpd . The necessity of Matthews’s condition for the prime field case has been essentially established by S. D. Cohen [3]. In [4] it is tempted a proof for a field of prime square order p2 . The lecturer [1] introduced a more efficient method to attack the problem in these cases, which implicitly eliminates some of the flaws in the previous proofs. Joint work of these authors [2] produced a correct proof for the characterization of Dickson polynomials of the second kind that permute F Fpd with p > 7 prime and d = 1, 2.

References [1] Cipu M. Dickson polynomials that are permutations. Serdica Math. J. 30(2004), 177–194. [2] Cipu M. and Cohen S.D. Dickson polynomial permutations. to appear. [3] Cohen S.D. Dickson polynomials of the second kind that are permutations. Canad. Math. Bull., 40 (1994), 225–238. [4] Cohen S.D. Dickson permutations, in Number-theoretic and algebraic methods in computer science (Moscow, 1993), 29–51, World Sci. Publishing, River Edge, NJ, 1995. [5] Matthews R.W. Permutation polynomials in one and several variables. Ph. D. Thesis, University of Tasmania, 1982.

Address: The Simion Stoilow Institute of Mathematics, Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest-ROMANIA e-mail: [email protected]

Acknowledgement: The participation to the 20th International Congress of Jangjeon Mathematical Society was possible thanks to support from Grant CNCSIS 1116/2007.

The 20 th International Congress of The Jangjeon Mathematical Society

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On the Eigenvalues of a Schr¨ odinger Operator with Matrix Potential Didem Cos c .kan and Sedef Karakılı¸

In this talk, we consider a Shr¨ odinger Operator L = −∆ + V with matrix potential V (x) and Neumann boundary condition ∂Φ | = 0 in L2 (Q)m , where Q = [0, a1 ] × [0, a2 ] × · · · × [0, ad ], m ≥ 1, d ≥ 2, −∆ ∂Q ∂n is diagonal m × m matrix, its diagonal elements being the scalar Laplace operators, V is the operator of multiplication by a real valued matrix V (x). Shr¨ odinger operator is a fundamental operator of quantum physics. It has different areas of application according to the properties of the potential V (x). We begin our talk with a brief history of the spectral theory of the Shr¨ odinger operator. We then give some information about the method of solution we used. We obtain asymptotic formulas for the eigenvalues of the Schr¨ odinger operator defined above.

References [1] Atılgan S ¸ ., Karakılı¸c S. and Veliev O.A. Asymptotic Formulas for the Eigenvalues of the Schr¨ odinger Operator. Turk J Math 26(2002), 215-227. [2] Feldman J., Knoerrer H. and Trubowitz E. The Perturbatively Stable Spectrum of the Periodic Schr¨ odinger Operator. Invent. Math. 100(1990), 259-300. [3] Feldman J., Knoerrer H. and Trubowitz E. The Perturbatively Unstable Spectrum of the Periodic Schr¨ odinger Operator. Comment. Math. Helvetica 66(1991), 557-579. [4] Friedlanger L. On the Spectrum for the Periodic Problem for the Schr¨ odinger Operator. Communications in Partial Differential Equations 15(1990), 1631-1647. [5] Karakılı¸c S., Atılgan S ¸ . and Veliev O.A. Asymptotic Formulas for the Eigenvalues of the Schr¨ odinger Operator with Dirichlet and Neumann Boundary Conditions. Reports on Mathematical Physics (ROMP) 55(2)(2005), 221-239. [6] Karpeshina Y. Perturbation series for the Schr¨ odinger Operator with a Periodic Potential near Planes of Diffraction. Communication in Analysis and Geometry 4(3)(1996), 339-413. [7] Karpeshina Y. On the Spectral Properties of Periodic Polyharmonic Matrix Operators. Indian Acad. Sci. (Math. Sci.) 112(1)(2002), 117-130. [8] Veliev O.A. Asimptotic Formulas for the Eigenvalues of the Periodic Schr¨ odinger Operator and the Bethe-Sommerfeld Conjecture. Functsional Anal. i Prilozhen 21(2)(1987), 1-15. [9] Veliev O.A. The Spectrum of Multidimensional Periodic Operators. Teor. Funktsional Anal. i Prilozhen 49(1988), 17-34.

Address: Dokuz Eyl¨ ul University, Faculty of Arts and Science, Department of Mathematics, Buca 35160, Izmir-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

19

Weak Solutions in Asymmetric Elasticity Ion Al.Cr˘ aciun

In this paper, we consider the equilibrium theory of the interior of a material right cylinder B with the generic cross–section S and the lateral boundary Π. Assume that the cylinder is occupied by a homogeneous and isotropic linearly elastic micropolar (Cosserat) material. The Cosserat, micropolar or asymmetric elasticity was established by Eringen in [1] and then it was developed in many papers and books (see e. g. [2], [3]). A rectangular Cartesian coordinate frame is chosen in such a way that the x3 −axis is parallel to the generators of B. The micropolar elastic plane strain parallel to the x1 , x2 −plane is characterized by the following fields of displacenments and rotations on S¯ = S ∪ ∂S uα = uα (x1 , x2 ) = uα (x),

u3 = 0,

ϕα = 0,

ϕ3 = ϕ = ϕ(x),

α = 1, 2,

(19.1)

Functions (19.1) represent the unknowns of the equilibrium equation −L(∂x )u(x) = f (x), x ∈ S,

(19.2)

in which u(x) = (u1 (x), u2 (x), ϕ(x))T , f is the load vector, and L(∂x ) = L(∂/∂x) is the linear differential operator of the plane asymmetric elasticity theory of a homogeneous and isotropic material occupying the cylinder B. The boundary value problems (BVPs) of the plane Cosserat elasticity represent the BVPs of equation (19.2). The regular (classical) solutions of these BVPs have been obtained by various authors in the form of integral potentials in L2 (S) space. However, in L2 (S) such solutions can be found only if the boundary L is sufficiently smooth. Guided by [4], we formulate the BVPs of plane Cosserat elasticity in a Sobolev space V and introduce the corresponding weak solutions for the domains with irregular boundaries. By using the dual space V ∗ of the Hilbert space V, and the Riesz representation theorem we establish existence, uniqueness, continuously dependence of the data of the BVPs, and stability of the weak solutions of the BVPs of plane Cosserat elasticity.

References [1] Eringen A.C. Linear Theory of Micropolar Elasticity. J. Math. Mech. 15(1966), 909-923. [2] Nowacki W. Theory of Asymmetric Elasticity. Polish Scientific Publishers, Warszaw, 1986. [3] Kupradze V.D. et al. Three–Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North–Holland, 1979. [4] Gockenbach M.S. Understanding and Implementing the Finite Element Method. SIAM, 2006.

Address: “Gh. Asachi” Technical University of Ia¸si, Department of Mathematics, 700050 Ia¸si IS-ROMANIA e-mail: ion [email protected]

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Tauberian Theorems for Abel Limitability Method Ibrahim C ¸anak and ¨ Umit Totur

This talk addresses conditions for Abel method of limitability to imply convergence and subsequential convergence. Namely, in this paper, we prove that Abel limitability of (un ) implies convergence of (un ) if (I − S)m+1 u is slowly oscillating. Next, we prove that Abel limitability of (un ) implies boundedness of (un ) if (I − S)m+1 u is moderately oscillating. (1) Linear operators S and T defined by (Su)n = σn (u) and (T u)n = n∆u on the vector spaces of all sequences u = (un ) satisfy ST = T S = I − S, where I is the identity operator. The general control modulo of the oscillatory behavior of integer order m ≥ 1 of a real sequence u = (un ) inductively defined by ω (m) (u) = (I − S)m−1 T u − (I − S)m u where ω (0) (u) = T u, was introduced by Stanojevi´c. Dik proved several Tauberian theorems for Abel limitability method for which Tauberian conditions were given in the terms of the conditions on (I −S)T u. Stanojevi´c and Stanojevi´c proved that if (I −S)2 T u is left onesidedly bounded with respect to some nonnegative sequence, then slow oscillation of u is recovered out of Abel limitability of a sequence related to (un ). Recently, C ¸ anak and Totur have given the two new identities for the general control modulo and then shown that the condition in the main theorem of Stanojevi´c and Stanojevi´c can be replaced by the condition that (I − S)m T u is left onesidedly bounded with respect to some nonnegative sequence.

References ˙ and Totur U. ¨ A Tauberian theorem with a generalized onesided condition. Abstr. Appl. Anal., [1] C ¸ anak I. Art. ID 60360, 12 pp., 2007. [2] Dik F. Tauberian theorems for convergence and subsequential convergence of sequences with controlled oscillatory behavior. Math. Morav. 5(2001), 19–56. [3] Dik M. Tauberian theorems for sequences with moderately oscillatory control moduli. Math. Morav. 5(2001), 57–94. ˇ [4] Stanojevi´c C.V. Analysis of Divergence: Control and Management of Divergent Process, Graduate ˙ C Research Seminar Lecture Notes. (Edited by I. ¸ anak), University of Missouri - Rolla, 1998. ˇ [5] Stanojevi´c C.V. and Stanojevi´c V.B. Tauberian retrieval theory. Publ. Inst. Math., Nouv. Ser. 71(85) (2002), 105-111.

Address: Adnan Menderes University, Faculty of Science and Letters, Department of Mathematics, Kepez, 09010. Aydın-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

On (1 − u2 )-cyclic Codes over F2k + uF2k + u2 F2k Yasemin C ¸engellenmis

It is extended a result of [3] to codes over the commutative ring F2k + uF2k + u2 F2k where k ∈ N and u = 0. A Gray map between codes over F2k + uF2k + u2 F2k and F2k is defined. It is proved that the Gray image of the linear (1 − u2 )-cyclic code over the commutative ring F2k + uF2k + u2 F2k of length n is a distance-invariant quasicyclic code of index 22k−1 and the length 22k n over F2k . 3

References [1] Maria Carmen V. Amarra, Fidel R.Nemenzo. On (1 − u)− cyclic codes over Fpk + uFpk . Applied Mathematics Letters, 2008. [2] Greferath M. and Schmith S.E. Gray isometries for finite chain rings and a nonlinear Ternary (36, 312 , 15) code. IEEE Trans. Inform. Theory 45(7)(1999), 2522-2524. [3] Jian-Fa Qian, Li-Na Zhang and Shi-Xin Zhu. Constacyclic and cyclic codes over F2 + uF2 + u2 F2 . IEICE Trans. Fundamentals E89-A(6), June 2006. [4] Jian-Fa Qian, Li-Na Zhang and Shi-Xin Zhu. (1 + u) constacyclic and cyclic codes over F2 + uF2 . Applied Mathematics Letters 19(2006), 820-823.

Address: Department of Mathematics, Faculty of Science and Arts, Trakya University, 22030 Edirne-TURKIYE e-mail: [email protected]

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(L, M )-intuitionistic Fuzzy Filters Vildan C ¸etkin, Banu Pazar, Halis Ayg¨ un

In this talk, we consider the notions of (L,M)-intuitionistic fuzzy topological spaces and (L,M)intuitionistic fuzzy filters where L and M are different strictly two-sided commutative quantales. It is well known that, the notion of a filter on a set is a basic concept in topology and plays an important role in it. In this work, first of all, we give a brief history of intuitionistic fuzzy topological spaces and fuzzy filters, then we give some preliminaries which we need in the main sections. In the main sections firstly, we introduce the notions of (L,M)-intuitionistic fuzzy topological spaces, (L,M)-intuitionistic fuzzy base and study relations between them. Then, we introduce the notions of (L,M)intuitionistic fuzzy filters, (L,M)-intuitionistic fuzzy filter base and get some relations between them. We also study (L,M)-intuitionistic fuzzy filter convergence with respect to (L,M)-intuitionistic fuzzy topological spaces.

References [1] Abdel-Hamied Hussein U.M. On Fuzzy Topological Spaces. Ph.D Thesis, Beni-Suef University Faculty of Science, Egypt, 2006. [2] Atanassov K.Intuitionistic fuzzy sets. Fuzzy Sets and Systems 1(1986), 87-96. [3] Atanassov K. New operators defined over the intuitionistic fuzzy sets. Fuzzy Sets Syst. 61(2)(1993), 131-142. [4] Burton M.H., Muraleetharan M. and Garcia J.G. Generalized filters 1. Fuzzy Sets and Systems 106(1999), 275-284. [5] Burton M.H., Muraleetharan M. and Garcia J.G. Generalized filters 2. Fuzzy Sets and Systems 106(1999), 393-400. [6] Chattopadhyay K.C., Hazra R.N. and Samanta S.K. Gradation of openness: fuzzy topology. Fuzzzy Sets Syst. 49(1992), 237-242. [7] G¨ ahler W. The general fuzzy filter approach to fuzzy topology, I. Fuzzy Sets and Systems 76(1995), 205-224. [8] G¨ ahler, W. The general fuzzy filter approach to fuzzy topology, II. Fuzzy Sets and Systems 76(1995), 225-246. ˇ [9] H¨ ohle and Sostak A.P. A general theory of fuzzy topological spaces. Fuzzy Sets Syst. 73(1995), 131-149. ˇ [10] H¨ ohle U. ansd Sostak A.P. Axiomatic foundations of fixed-basis fuzzy topology. The Handbooks of Fuzzy Sets Series, vol.3, Kluewer Academic Publishers, Dordrecht, 1999 (Chapter 3). [11] Kim Y.C. and Ko J.M. Images and preimages of L-filterbases. Fuzzy Sets and Systems 157(2006), 1913-1927.

Address: Kocaeli University, Department of Mathematics, Umuttepe Campus, 41380, Kocaeli-TURKIYE e-mails: [email protected], [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

Determination of Unknown Boundary Condition in a Quasilinear Parabolic Equation Ali Demir and Ebru ¨ Ozbilge

In this article, a semigroup approach is presented for the mathematical analysis of the inverse problems of identifying the unknown boundary condition u(1, t) = f (t) in the quasi-linear parabolic equation ut (x, t) = (k(u(x, t))ux (x, t))x , with Dirichlet boundary conditions u(0, t) = ψ0 , u(1, t) = f (t) by making use of the over measured data u(x0 , t) = ψ1 and ux (x0 , t) = ψ2 seperately. The purpose of this study is to identify the unknown boundary condition u(1, t) at x = 1 by using the over measured data u(x0 , t) = ψ1 and ux (x0 , t) = ψ2 . First by using over measured data as a boundary condition we define the problem on ΩT0 = {(x, t) ∈ R2 : 0 < x < x0 , 0 < t ≤ T }, then the integral representation of this problem via semigroup of linear operators is obtained. Finally extending the solution uniquely to the closed interval [0, 1] we reach the result. The main point here is the unique extensions of the solutions on [0, x0 ] to the closed interval [0, 1] which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition u(1, t) at x = 1.

References [1] Demir A. and Ozbilge E. Semigroup approach for identification of the unknown diffusion coefficient in a quasi-linear parabolic equation. Mathematical Methods in the Applied Sciences 30(11)(2007), 12831294. [2] Cannon J.R. The One-Dimensional Heat Equation. Encyclopedia of Mathematics and Its Applications 23(1984), Addison Wesley, Massachusets. [3] DuChateau P. Introduction to inverse problems in partial differential equations for engineers, physicists and mathematicians. in Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology, J. Gottlieb and P. DuChateau, eds., Kliver Academic Publishers, the Netherland, (1996), pp. 3-38. [4] DuChateau P. Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems. SIAM J. Math. Anal. 26(1995), 1473-1487. [5] DuChateau P., Thelwell R. and Butters G. Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient. Inverse Problems 20(2004), 601-625.

Address: Applied Mathematical Sciences Research Center and Department of Mathematics, Kocaeli University, Ataturk Bulvari, 41300, Izmit, Kocaeli-TURKIYE Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No.156, 35330, Balcova, Izmir-TURKIYE e-mails: [email protected], [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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A Quasilinear Elliptic System with Integral Boundary Conditions Mohammed Derhab

The purpose of this work is to study the existence of solutions of the following quasilinear elliptic system with integral boundary conditions: 0 − (ϕp (u0 )) = f (x, u, v) , x ∈ ]0, 1[ , 0 − (ϕq (v 0 )) = g (x, u, v) , x ∈ ]0, 1[ , R1 R1 (24.1) u (0) = g0 (s) u (s) ds, u (1) = g1 (s) u (s) ds, 0 0 R1 R1 v (0) = g2 (s) v (s) ds, v (1) = g3 (s) v (s) ds, 0

0

p−2

where ϕp (y) = |y| y, p > 1, q > 1, f : [0, 1] × R2 → R, g : [0, 1] × R2 → R and gi : [0, 1] → R+ (i = 0, · · · , 3) are a continuous functions. Existence of solutions of (24.1) is discussed by using upper and lower solutions method.

References [1] Boucherif A. Positive solutions of second order differential equations with integral boundary conditions. Discrete and Continuous Dynamical Systems Supplement (2007), 155-159. [2] Boucherif A. and Bouguima S. M. Nonlinear second order ordinary differential equations with nonlocal boundary conditions. Comm. in Applied Nonl. Anal. 5(1998), 73-85. [3] Capasso V. and Kunisch K. A reaction-diffusion system arising in modeling man-environment diseases. Q. Appl. Math. 46(1988), 431-449. [4] Cherpion M., De Coster C. and Habets P. Monotone iterative methods for boundary value problems. Differential and Integral Equations 12(1999), 309-338. [5] Manuel D. and Antonio S. Existence of solutions for elliptic systems with H¨ older continuous nonlinearities. Differential and Integral Equations 13(4-6)(2000), 453-477.

Address: Department of Mathematics, Faculty of Sciences, University Abou-Bekr Belkaid Tlemcen, B.P.119, Tlemcen, 13000, ALGERIA e-mail: [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

On the q-Trigonometric, q-Hyperbolic Functions Ayhan Dil and Veli Kurt

In this study, with the help of q-Gauss binomial formula, q-factorial and q-shifted factorial we obtain some properties of q-trigonometric and q-hyperbolic functions. These results correspond to usual trigonometric and hyperbolic functions for the limit case q = 1. On the other hand we obtain the equality eq2 −z 2 = eq (iz) eq (−iz).

References [1] De Sole A. and Kac V. On the integral representation of q-gamma and q-beta functions. arXiv: math. QA/0302032. [2] Kac V. and Cheung P. Quantum Calculus. Springer, New York, 2002. [3] Kim T. q-extension of the Euler formula and trigonometric functions. Russ. J. Math. Phys. 14(3)(2007), 275–278. [4] Koornwinder T.H. Special functions and q-commuting variables. eprint arXiv:q-alg/9608008. [5] Schork M. Ward’s calculus of sequences, q-calculus and the limit q → −1. Adv. Stud. Contemp. Math. (Kyungshang) 13(2)(2006), 131–141. [6] Suslov S.K. Addition theorems for some q-exponential and q-trigonometric functions. Methods Appl. Anal. 4(2000), 11–32. [7] Ward M. A calculus of sequences. Amer. J. Math. 58(1936) 255–266.

Address: Akdeniz University, Faculty of Art Science, Department of Mathematics, 07058-Antalya-TURKIYE e-mails: [email protected], [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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A Fixed Point Theorem H¨ ulya Duru

Let (H, <>) be a real Hilbert space and let K be a compact and connected subset of Hl. We show that every continuous mapping T : K → K satisfying a mild condition has a fixed point.

References [1] Schauder J. Der fixpunktsatz in funktionalraumen. Studia Math. 2(1930), 171-180. ¨ [2] Brouwer L.E. Uber abbildungen von mannigfaltigkeiten. Math.Ann. 71(912), 97-115. [3] Goebel K. and Kirk W.A. Topics in metric fixed point theory. Cambridge University press. [4] Kirk W.A and Sims B. Handbook of Metric Fixed point Theory. Kluwer Academic Publishers, Dordrecht, 2001. [5] Browder F.E. Fixed point theorems for noncompact mappings in Hilbert space. Proc. Nat. Acad. Sci. U.S.A. 53(1965), 1272-1276. [6] Browder F.E. Nonexpansive nonlinear operators in a Banach space. Proc.Nat.Sci.U.S.A 54(1965) 10411044. [7] G¨ ohde D. Zum prinzep der kontractiven abbildung. Math. Nachr. 30(1965), 251-258. [8] Kirk W.A. A fixed point theorem for mappings which do not increase distances. Amer.Math.Monthly 72(1965), 1004-1006.

˙ ˙ Address: Istanbul University, Faculty of Science, Department of Mathematics, Vezneciler-Istanbul-TURKIYE e-mail: [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

On The Vague DeMorgan Complemented Partially Ordered Sets and Lattices Zeynep Eken

Lattice theory is a basic topic of mathematics and many application areas. It is important and useful to fuzzily define the concepts in lattice theory. Therefore, this work contributes to mathematics in the following way: In this work, fuzzy partially ordered sets and fuzzy lattices that are many-valued logical counterparts to partially ordered sets and lattices in classical lattice theory have been studied on the basis of many-valued equivalence relations. Fuzzy DeMorgan complemented partially ordered sets and lattices are introduced by using many-valued equivalence relations. Furthermore, various non-trivial examples for these concepts are provided.

References [1] Bodenhofer U. Similarity-Based Generalization of Fuzzy Orderings Preserving the Classical Axioms. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8(3)(2000), 593-610. [2] Bodenhofer U. Representations and Constructions of Similarity-Based Fuzzy Orderings. Fuzzy Sets and Systems 137(2003), 113-136. [3] Demirci M. A Theory of Vague Lattices Based on Many-Valued Equivalence Relations-I: General Representation Results. Fuzzy Sets and Systems 151(3)(2005), 437-472. [4] Demirci M. and Eken Z. An Introduction to Vague Complemented Ordered Sets. Information Science 177(2007), 150-160. [5] Klement E.P., Mesiar R. and Pap E. Triangular Norms. pp 385, 2000.

Address: Akdeniz University, Faculty of Science and Arts, Department of Mathematics, 07058, Antalya-TURKIYE e-mail: [email protected]

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Application of He’s Semi-Inverse Method to the Nonlinear Wave Equations Meryem Erdal and Ahmet Yıldırım

A variational theory is established for Nonlinear Schr¨odinger Equation (NLS) and Boussinesq equation by He’s semi-inverse method. Based on the obtained variational principle, a solitary wave solution is obtained.

References [1] He JH. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3-4)(1999), 257-262. [2] He JH. Homotopy perturbation method for bifurcation of nonlinear problems. International Journal of Nonlinear Sciences and Numerical Simulation 6(2)(2005), 207-208. [3] He JH. Limit cycle and bifurcation of nonlinear problems. Chaos Solitons & Fractals 26(3)(2005), 827-833. ¨ [4] Ozis T. and Yıldırım A. A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity. Chaos Solitons & Fractals 34(3)(), 989-991. ¨ [5] Ozis T. and Yıldırım A. Determination of limit cycles by a modified straightforward expansion for nonlinear oscillators. Chaos Solitons & Fractals 32(2007), 445-448 ¨ [6] Yıldırım A. and Ozis T. Solutions of singular IVPs of Lane-Emden type by Homotopy Perturbation Method. Physics Letters A 369(2007), 70-76. ¨ [7] Ozis T. and Yıldırım A. An application of He’s Semi-Inverse Method to the Nonlinear Schr¨ odinger (NLS) Equation. Computer Mathematics with applications 54 (2007) 1039-1042.

Address: Ege University, Faculty of Science, Department of Mathematics, Bornova 35100, Izmir-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

A System of ODEs for Nonlinear Programming Problems with Smooth Penalty Function Fırat Evirgen and Necati ¨ Ozdemir

In this talk, we present a differential system based on smooth penalty function for solving nonlinear programming problems (NLP) minimize

f (x)

subject to hi (x)

=

0

i = 1, 2, ..., m

gj (x) ≤

0

j = 1, 2, ..., p,

where x ∈ Rn , f : Rn → R, hi : Rn → R (m ≤ n) and gj : Rn → R are continuously differentiable functions. The penalty function l 1 , where is non-smooth, is used frequently in the literature to obtain a solution to NLP problems. Firstly, we implement a smoothing method for non-smooth penalty function l 1 to construct a system of differential equations based on smoothing techniques in [4], [5] and [6]. Then, we put forward relations between optimal solutions of non-smooth penalty problem and smooth penalty problem. Moreover, we proposed a differential equation approach for solving nonlinear programming problems. This approach shows that the stable equilibrium point of the dynamic system coincide with the optimal solution of the corresponding smooth penalty problem. Finally, some practical examples are tested to illustrate the applicability of the proposed approach.

References [1] Fiacco A.V. and Mccormick G.P. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley, 1968. [2] Jin L., Zhang L-W. and Xiao X.T. Two Differential Equation Systems for Inequality Constrained Optimization. Appl. Math. Comput. 188(2)(2007), 1334-1343. [3] La Salle J. and Lefschetz S. Stability by Liapunov’s Direct Method with Application. Academic Press, 1961. [4] Meng Z., Dang C. and Yang X. On the Smoothing of the Square-root Exact Penalty Function for Inequality Constrained Optimization. Comput. Optim. Appl. 35(2006), 375-398. [5] Pinar M.C. and Zenios S.A. On Smoothing Exact Penalty Functions for Convex Constrained Optimization. SIAM J. Optim. 4(3)(1994), 486-511. [6] Zenios S.A., Pinar M.C. and Dembo R.S. A Smooth Penalty Function Algorithm for Network-Structured Problems. European J. Oper. Res. 83(1995), 220-236.

Address: Balıkesir University, Faculty of Science and Art, Department of Mathematics, Cagis Campus 10150. Balıkesir-TURKIYE e-mails: [email protected], [email protected]

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B(X , X ∗ )-Valued Kernels and B(X )-Modules P˘ astorel Ga¸ spar and Dimitru Ga¸ spar

The starting point of this research is to obtain a unitary treatment of the very different frames in which the theory of infinite dimensional (Banach space valued) stochastic processes was developed, the talk being based on several joint papers with Dumitru Ga¸spar ([1], [2], [3], [4]). Denoting by B(X , X ∗ ) the Banach space of all bounded linear operators from a Banach space X to its conjugate dual X ∗ and by D2 (X , X ∗ ) the ideal of 2-dominated operators from X into X ∗ , in the first section two types of positive definiteness for B(X , X ∗ )-valued, respectively D2 (X , X ∗ )-valued kernels on an arbitrary set Λ are introduced. These kinds of positive definiteness are characterized and their basic properties outlined. In the second section two different locally convex B(X )-modules are introduced. For the first one – called a Hilbert B(X )-modulethe inner product is B(X , X ∗ )-valued, while for the second one-called D2 -normal Hilbert B(X )-module-the inner product is D2 (X , X ∗ )-valued. The main results of this section are about functional models for each of them in terms of bounded linear operators from X to a suitable Hilbert space. Let’s mention that for the D2 -normal case the involved operators cover only the absolutely 2-summable operators. In the third section the reproducing kernel (D2 -normal) Hilbert B(X )-modules are introduced. The corresponding reproducing kernels are positive definite and B(X , X ∗ )-valued (D2 (X , X ∗ )-valued respectively). The basic properties of such reproducing kernels are given, as well as the characterizations of some inequalities between the positive definite kernels in terms of the embedding relations of the corresponding reproducing kernel B(X )-modules. In the last section applications to Y-valued generalized (strong second order) stationary stochastic processes are mentioned, the correlation function of such processes being a positive definite B(X , X ∗ )-(D2 (X , X ∗ ) respectively) valued kernel on the time parameter set, whereas the time module of the process is module isomorphic to the reproducing kernel (D2 -normal) Hilbert B(X )-module generated by the correlation function (here X is the dual of Y). Let us remark that if X = E, a Hilbert space, then B(X , X ∗ ) = B(E) = B(X ) and D2 (X , X ∗ ) is the ideal of trace class operators on E, this particular case being in essence described in the book [5]

References [1] Ga¸spar D. and Ga¸spar P. An operatorial model for Hilbert B(X )- modules. Anal. Univ. de Vest Timi¸soara XL(2)(2002), 15–30. [2] Ga¸spar D. and Ga¸spar P. On normal Hilbert B(X )-modules. Anal. Univ. de Vest Timi¸soara XLI (1)(2003), 49–64. [3] Ga¸spar D. and Ga¸spar P. Reproducing kernel Hilbert B(X )-modules. Anal. Univ. de Vest Timi¸soara XLIII(2)(2005), 2–4. [4] Ga¸spar P. Harmonic Analysis on Spaces of Random Variables. Ser. ”Monografii Matematice”, Ed. Univ. de Vest Timi¸soara, 2008, to appear. [5] Kakihara Yu. Multidimensional Second Order Stochastic Processes. World Scientific Publ. Comp., River Edge, N. I., 1997.

Address: West University Timi¸soara, Faculty of Mathematics and Computer Science, Department of Mathematics, Bul. V. Pˆ arvan nr. 4, Timi¸soara-ROMANIA e-mails: [email protected], [email protected]

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Singular Curves and Singular Elliptic Divisibility Sequences over Finite Fields Bet¨ ul Gezer, Osman Bizim and Ahmet Tekcan

Let F be a field (char(F) 6= 2, 3). An elliptic curve E defined over F is given by an equation y 2 = x3 +ax+b where a, b ∈ F and the cubic x3 + ax + b has distinct roots. By definition, this set always contains the point O so that E(F) = {O} ∪ {(x, y) ∈ F × F | y 2 = x3 + ax + b}.The points of E form an additive abelian group with O as the identity point, see [3, 4]. What happens if the cubic equation x3 + ax + b = 0 has multiple roots? Does the analysis of the composition law fail in this case? We discard the singular point and the resulting group then has a particularly simple structure. A divisibility sequence is a sequence (hn ) for n ∈ N of positive integers with the property that m|n if hm |hn . There are also divisibility sequences satisfying a nonlinear recurrence relation. These are the elliptic divisibility sequences and this relation comes from the recursion formula for elliptic division polynomials associated to an elliptic curve. An elliptic divisibility sequence (or EDS) is a sequence of integers (hn ) satisfying a non-linear recurrence relation hm+n hm−n = hm+1 hm−1 h2n − hn+1 hn−1 h2m and with the divisibility property that hm divides hn whenever m divides n for all m ≥ n ≥ 1. EDSs are generalisations of a class of integer divisibility sequences called Lucas sequences and these are special case of a type of EDS called a singular EDS. The discriminant of an elliptic divisibility sequence (hn ) is defined by the 3 12 2 10 3 7 3 5 6 4 2 3 2 4 formula: ∆(h2 , h3 , h4 ) = h4 h15 2 − h3 h2 + 3h4 h2 − 20h4 h3 h2 + 3h4 h2 + 16h3 h2 + 8h4 h3 h2 + h4 . An elliptic divisibility sequence (hn ) is said to be singular if and only if its discriminant ∆(h2 , h3 , h4 ) vanishes. For the arithmetic properties of EDSs, see [1, 2, 5, 6]. Shipsey [2] used EDSs to study some applications to cryptography and ECDLP. We will be interested in the singular curves and elliptic divisibility sequences, equivalence of these and singular elliptic divisibility sequences over Fp where p > 3 is a prime.

References [1] Everest G., Poorten A. van der., Shparlinski I., and Ward T. Recurrence sequences, Mathematical Surveys and Monographs 104, AMS, Providence, RI, 2003. [2] Shipsey R. Elliptic Divisibility Sequences. Ph. D. thesis, Goldsmith’s (University of London), 2000. [3] Silverman J.H. The Arithmetic of Elliptic Curves. Springer-Verlag, 1986. [4] Silverman J.H. and Tate J. Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer, 1992. [5] Swart C.S. Elliptic Curves and Related Sequences. Ph. D. thesis, Royal Holloway (University of London), 2003. [6] Ward M. Memoir on elliptic divisibility sequences. Amer. J. Math. 70(1948), 31-74.

Address: Uludag University, Faculty of Science, Department of Mathematics, G¨ or¨ ukle 16059. Bursa-TURKIYE e-mails: [email protected], [email protected], [email protected]

This work was supported by The Scientific and Technological Research Council of Turkey, project no: 107T311.

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τc -Topology on Hypergroup Algebras Ali Ghaffari

Throughout the paper, K denotes a locally compact hypergroup with a left Haar measure λ. Integration R with respect to λ will be denoted by ... dx. The spaces Lp (G) (1 ≤ p < ∞) and L∞ (G) are as defined in [4]. The theory of hypergroup algebras has been extensively studied in such papers as [1], [2] and [3]. If φ and f are measurable functions on K, then Z 1 φ ~ f (x) = f (¯ y ∗ x ∗ y)∆(y) p φ(y)dy, whenever this makes sense. It is easy to see that for φ ∈ L1 (K) and f ∈ Lp (K), φ ~ f ∈ Lp (K) and kφ ~ f kp ≤ kφk1 kf kp . Under this multiplication, L1 (K) is a Banach algebra. Then L1 (K)∗∗ is a Banach algebra with the first Arens product [3]. For each φ in L1 (K) define ρφ : L∞ (K) → [0, ∞) by ρφ (f ) = kφf k∞ . Then ρφ is a seminorm. Let τc be the topology on L∞ (K) that has as a subbase the sets {h ∈ L∞ (K); ρφ (h − f ) < }, where φ ∈ L1 (K), f ∈ L∞ (K), and > 0. The main results of our investigations are the following theorems: Theorem 1) Let φ be a nonzero element in L1 (K). The following properties are equivalent: 1) There exist an > 0 such that {f ∈ L∞ (K); kφf k∞ < } is a weak∗ -neighborhood of zero. 2) {A(x,x) φ; x ∈ K} is part of a finite-dimensional subspace of L1 (K), where A(x,x) φ(y) = φ(¯ x∗y∗ x)∆(x) (x, y ∈ K). 3) Given > 0, there exists x1 , ..., xn in K and δ > 0 such that for f ∈ L∞ (K), the inequality |hf, A(xi ,xi ) φi| < δ for i = 1, ..., n implies that kφf k∞ < . Theorem 2) If K is a compact hypergroup and B = {f ∈ L∞ (K); kf k∞ ≤ 1}, then τc -topology and weak∗ -topology coincide on B. Theorem 3) Every norm-closed ball in L∞ (K) is τc -complete. Theorem 4) Suppose K is a compact hypergroup. If {fn } is a sequence in L∞ (K) that converges to some f ∈ L∞ (K) in the weak∗ -topology, then fn → f in the τc -topology.

References [1] Skantharajah M. Amenable hypergroups. Ill. J. Math. 36(1992), 15-46. [2] Pavel L. Multipliers for the Lp -spaces of a hypergroup. Rocky Mountain J. Math. 37(3)(2007), 987-1000. [3] Ghaffari A. Characterization of operators on the dual of hypergroups which commute with translations and convolutions. Acta Math. Sinica, English Series 20(2)(2004), 201-208. [4] Bloom W.R. and Heyer H. Harmonic analysis of probability measures on hypergroups. de Gruyter, Berlin, 1995.

Address: Department of Mathematics, Semnan University, Semnan-IRAN e-mail: [email protected]

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Application of Statistical Shape Analysis to the Classification of Renal Tumours Appearing in Early Childhood Stefan Markus Giebel

Most of the renal tumours affecting little children are Wilms-tumours. It is important but very difficult to differentiate Wilms-tumours from other renal tumour types like neuroblastoma, renal cell carcinoma etc.. The correct therapy depends on the diagnosis. For the diagnosis the radiologist has only the magnetic resonance imaging. In many cases the histology by extraction of body tissues is needed additionally. We try to establish a way to obtain a differential diagnosis using statistical shape analysis on the basis of magnetic resonance imaging. On the basis of magnetic resonance imaging a three-dimensional model of renal tumour is constructed. Then the centre of mass is calculated and the two-dimensional image of the area around the centre of mass is used for shape analysis. Landmarks are computed as the intersection of the boundary of the tumour and a straight line in a pre-determined angle from the centre of mass. The landmarks describe the shape of the tumours. To get a “pre-shape” the figure has to be centred and standardised. This allows comparing tumours with different sizes situated in different locations. The “mean shape” is the expected “pre-shape” of a group of objects/tumours. The “mean shape” of a group of tumours with a same diagnosis should be able to separate them from tumours with another diagnosis. Tumours with another diagnosis should have a greater distance from the “mean shape” than the tumours whereby the “mean shape” is calculated. The applicability of the ”mean shape” for separating groups with different diagnosis is tested by the test of Ziezold (1994). Finally, we present the three-dimensional landmarks and tests for identifying relevant landmarks to separate the different types of tumours.

References [1] Bookstein F.L. Biometrics, biomathematics and the morphometric synthesis. Bulletin of Mathematical Biology 58, 313-365. [2] Dryden I.L. and Mardia K.V. Statistical Shape Analysis. Wiley, Chichester, 1998 [3] Giebel Stefan Markus. Statistische Analyse der Form bei Nierentumoren von Kleinkindern. Unver¨ offentlichte Diplomarbeit in der Mathematik an der Universit¨at Kassel, 2007 [4] Huckemann S. and Ziezold H. Principal Component Analysis for Riemannian Manifolds with an Application to Triangular Shape Spaces. Adv. Appl. Prob. (SGSA) 38, 299-319. [5] Kendall D.G. The diffusion of shape. Adv. Appl. Probab. 9(1977), 428-430. [6] Small C.G. The Statistical Theory of Shape. Springer-Verlag, New York, 1996. [7] Ziezold H. On expected figures and a strong law of large numbers for random elements in quasi- metric spaces. Trans. 7th Prague Conference Inf. Th., Statist. Dec. Funct., Random Processes, Vol. A. Reidel, Dordrecht, Prag 1974, 591 – 602. [8] Ziezold H. Mean Figures and Mean Shapes Applied to Biological Figure and Shape Distributions in the Plane. Biometrical Journal 36(1994), 491-510.

Address: Universit¨ at Luxemburg, KfN L¨ utzerodestr. 9 Hannover, Berlinerstr. 46 34308 Bad Emstal-LUXEMBOURG e-mail: [email protected]

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Kac-Moody-Virasoro Algebras as Symmetries of 2+1-dimensional Nonlinear Partial Differential Equations Faruk G¨ ung¨ or

The notion of symmetry proved to be useful for detecting the integrability of nonlinear partial differential equations in 2+1 dimensions. In this talk I will survey some physical models such as generalized Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations admitting Kac-Moody-Virasoro algebras as symmetries and discuss how their presence can serve as a preliminary test for integrability.

References ¨ The generalized Davey-Stewartson equations, its Kac-Moody-Virasoro sym[1] G¨ ung¨ or F. and Aykanat O. metry algebra and relation to Davey-Stewartson equations. J. Math. Phys. 47(2006), 013510. [2] G¨ ung¨ or F. and Winternitz P. Generalized Kadomtsev-Petviashvili Equation with an Infinite Dimensional Symmetry Algebra, J. Math. Anal. and Appl. 276(2002), 314–328. [3] G¨ ung¨ or F. and Winternitz P. Equivalence classes and symmerties of the variable coefficient KP equation. Nonlinear Dynamics 35(2004), 381–396. [4] Lou S. Y. and Tang X. Y. Equations of arbitrary order invariant under the Kadomtsev-Petviashvili symmetry group. J. Math. Phys. 43(2004), 1020–1030.

˙ Address: Istanbul Technical University, Faculty of Science and Letters, Department of Mathematics, 34469, Maslak, ˙ Istanbul TURKIYE e-mail: [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

Trigonometric Approximation of Functions in Weighted Lp Spaces Ali G¨ uven

Let Lpw be the w−weighted Lebesgue space on the interval [0, 2π] , where 1 < p < ∞ and w a weight function belongs to the Muckenhoupt class Ap . The modulus of continuity Ω (f, ·)p,w of a function f ∈ Lpw is defined by Ω (f, δ)p,w = sup k∆h (f )kp,w , δ > 0, |h|≤δ

where ∆h (f ) (x) =

1 h

Zh |f (x + t) − f (x)| dt. 0

We define the Lipschitz class Lip (α, p, w) for 0 < α ≤ 1 by n Lip (α, p, w) = f ∈ Lpw : Ω (f, δ)p,w = O (δ α ) ,

o δ>0 .

The approximation properties of some means of Fourier series of the functions f ∈ Lip (α, p, w) were investigated. These results were applied to obtain similar approximation properties of means of the p−Faber p (G) , where G ⊂ C is a domain bounded series of the functions belongs to the weighted Smirnov spaces Ew by a Carleson curve.

References [1] Chandra P. Trigonometric approximation of functions in Lp −norm. J. Math. Anal. Appl. 275(2002), 13-26. [2] Hunt R., Muckenhoupt B. and Wheeden R. Weighted Norm Inequalities for the Conjugate Function and Hilbert Transform. Trans. Amer. Math. Soc. 176(1973), 227-251. [3] Ky N.X. Moduli of Mean Smoothness and Approximation with Ap −weights. Annales Univ. Sci. Budapest 40(1997), 37-48. [4] Muckenhoupt B. Weighted Norm Inequalities for the Hardy Maximal Function.Trans. Amer. Math. Soc. 165(1972), 207-226. [5] Zygmund A. Trigonometric Series, Vol I. Cambridge Univ. Press, 2nd edition, (1959).

Address: Balıkesir University, Faculty of Arts and Science, Department of Mathematics, 10145 Balkesir-TURKIYE e-mail: ag [email protected]

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Approximation by Means of Fourier Trigonometric Series in Weighted Orlicz Spaces Ali G¨ uven and Daniyal M.Israfilov

Let T denote the interval [−π, π] or the unit circle eit : −π ≤ t < π in the complex plane C. Let M be a Young function and LM (T) be the Orlicz function space generated by M on T. Let ω be a weight function, i. e., a nonnegative measurable function on T ω : T → [0, ∞], such that the set ω −1 ({0, ∞}) has Lebesgue measure zero.We denote by LM (T, ω) the linear space of all measurable functions f such that f ω ∈ LM (T) and set kf kLM (T, ω) := kf ωkLM (T) . The normed space LM (T, ω) is called a weighted Orlicz space. In this talk we give some estimation relating to the approximation properties of Cesaro, Zygmund and Abel-Poisson means of Fourier trigonometric series, using the modulus of continuity in reflexive weighted Orlicz spaces with Muckenhoupt weights. Than we give some applications of these estimations. Namely, we apply these result to estimate the rate of approximation of Cesaro, Zygmund and Abel sums of Faber series in weighted Smirnov-Orlicz classes defined on simply connected domains of the complex plane.

References [1] Israfilov D.M. Approximation by p−Faber Polynomials in the Weighted Smirnov Class Ep (G,w) and the Bieberbach Polynomials. Constr. Approx. 17(2001), 335-351. [2] Israfilov D.M. and Guven A. Approximation in Weighted Smirnov Classes. East J. Approx. 11 (2005), 91-102. [3] Israfilov D.M. and Guven A. Approximation by Trigonometric Polynomials in Weighted Orlicz Spaces. Studia Mathematica 174(2)(2006), 147-168. [4] Kokilashvili V. On the Approximation of Periodic Functions. Proceedings of Math. Inst. of Tbilisi 34(1968), 51-81. [5] Ky N.X. Moduli of Mean Smoothness and Approximation with Ap −weights. Annales Univ. Sci. Budapest 40(1997), 37-48. [6] Mastroianni G and Totik V. Jackson type inequalities for doubling and Ap weights. Rend. Circ. Mat. Palermo Serie II Suppl. 52(1998), 83-99.

Address: Balikesir University, Faculty of Art and Science, Department of Mathematics, Cagi¸s 10145, Balikesir-TURKIYE e-mails: ag [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

Finite Derivation Type for Graph Products of Monoids Eylem G¨ uzel

In the last years string-rewriting systems have received a lot of attention, both from Mathematics and Theoretical Computer Science. In particularly, finite and complete (noetherian and confluent) string-rewriting systems are used to solve word problems among other algebraic decision problems such as conjugacy and isomorphism problems. This application reveals the importance of string-rewriting systems. Unfortunately, the property of having finite and complete string-rewriting system is not invarible under monoid presentations. For the above reasons, it would be important to characterize algebraically the finitely presented monoids with solvable word problem that admit a finite and complete string-rewriting system. An important step in that direction was given by Squier [3] who defined a new combinatorial property of string-rewriting systems called finite derivation type (FDT). Since FDT is an invariant property of monoid presentations, it becomes important to know which monoid constructions preserve the property of FDT. In this talk we summarize some works studied on FDT and then give the necessary condition for graph products of monoids to have FDT property.

References [1] Book R.V. and Otto F. String-Rewriting Systems. Springer, New York, 1993. [2] Costa A.V. Graph Products of Monoids. Semigroup Forum, 63(2001), 247-227. [3] Squier C. A finiteness Condition for Rewriting Systems. rRvision by F. Otto and T. Kobayashi. Theo. Comp. Sci. 131(1994), 271-294. [4] Wang J. Finite Derivation Type for Semidirect Products of Monoids. Theo. Comp. Sci., 191(1998), 219-228. [5] Wang J. Finite Complete Rewriting Systems and Finite Derivation Type for Small Extensions of Monoids. Journal of Algebra, 204(1998), 493-503.

Address: Balikesir University, Faculty of Art and Science, Department of Mathematics, Cagis Campus, 10145. Balikesir-TURKIYE e-mail: [email protected]

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On the Rational Operator Pencils in Banach Space Elman Hasanov In this paper we consider operator pencils in the form A (λ) =

mi n X X

Aij j

i=1 j=1 (λ − ai )

+

m0 X

λk A0k ,

(1)

k=1

rationally depending on spectral parameter, where Aij are compact operators acting in complex Banach space B . The main purpose of the paper is to prove the following theorems: THEOREM 1. Let m X Ai A (λ) = + B (λ) , i i=1 (λ − a) where Ai i = 1, m are finite dimensional operators in Banach space and B (λ) is an operator valued function analytically depending on the spectral parameter λ at vicinity of a. Additionally, we assume that εn (B (a)) = inf kB (a) − Bn k → 0, where infimum is taken over the set of all finite dimensional operators in B. Then there is a positive number ε > 0 such that for all λ satisfying 0 < |λ − a| < ε, the equation (I − A (λ)) x = 0 has the same number linearly independent solutions. THEOREM 2. Let D be a connected domain in the complex plane, λ ∈ D and A (λ) =

ma X

Ai

i=1 (λ

− a)

i

+ B (λ) ,

where Ai i = 1, m are finite dimensional operators in Banach space and B (λ) is an operator valued function analytically depending on the spectral parameter λ on D, and εn (B (λ)) → 0. Under these −1 conditions if (I − A (λ)) exists at least at one point λ0 ∈ D, then it exists for all points of D except discrete points which may have limit point λ = a and is meromorphic function.

References [1] Allahverdiyev C.E. and Hasanov E.E. Completeness theorems in Banach space. Functional Analysis and its Applications 8(4)(1974), Moskow, (in Russian). [2] Allahverdiyev C.E. and Hasanov E.E. Theorems on completeness of eigenvectors of operators, rationally depending on parameter. roceedings of Academy of Sciences of Azerbaijan 6(1974), 20-32. (in Russian).

Address: Isk University, Department of Mathematics Istanbul-TURKIYE e-mail: [email protected]

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Note On Generalized M*-Groups Sebahattin Ikikardes and Recep ¸ Sahin

The study of Riemann and Klein surfaces with maximal automorphism groups has a long history. We begin our talk with a brief history. It is well known that a compact Riemann surface of genus g ≥ 2 admits at most 84(g − 1) automorphisms. Automorphism groups of Riemann surfaces with this maximal number of elements are called Hurwitz groups. Survey article [4] provides an excellent reference for the work on Hurwitz groups. Let X be a compact bordered Klein surface of algebraic genus p ≥ 2. In [6] May proved that the automorphism group G of X is finite, and the order of G is not greater than 12(p − 1). When this maximal bound is attained by a surface, its group of automorphisms is called an M ∗ −group [7]. M ∗ −groups were studied by May [7] first. Then these groups were investigated intensively. For examples of these studies see [1, 2, 3, 5, 8, 9]. May proved [7] that there is a relationship between the extended modular group and M ∗ −groups. The relationship says that a finite group of order at least 12 is an M ∗ −group if and only if it is a homomorphic image of the extended modular group Γ. Then generalized M*-groups are defined in [10] and relationship between extended Hecke Groups and generalized M*-groups are given. In this talk, we consider the generalized M*-groups and extended Hecke Groups H(λq ). We give new examples of generalized M*-groups which are finite quotient froups of H(λq ).

References [1] Bujalance E., Cirre F.J. and Turbek P. Groups acting on bordered Klein surfaces with maximal symmetry. Proceedings of Groups St. Andrews 2001 in Oxford. Vol. I, 50–58, London Math. Soc. Lecture Note Ser., 304, Cambridge, U.K.Cambridge University Press. (2003). [2] Bujalance E., Cirre F.J. and Turbek P. Subgroups of M ∗ -groups. Q. J. Math. 54(1)(2003), 49–60. [3] Bujalance E., Cirre F.J. and Turbek P. Automorphism criteria for M ∗ -groups. Proc. Edinb. Math. Soc. 47(2)(2004), 339–351. [4] Conder M.D.E. Hurwitz groups : a brief survey. Bull. Amer. Math. Soc. 23(1990), 359-370. [5] Greenleaf N. and May C.L. Bordered Klein surfaces with maximal symmetry. Trans. Amer. Math. Soc. 274(1)(1982), 265–283. [6] May C.L. Automorphisms of compact Klein surfaces with boundary. Pacific J.Math. 59(1975), 199-210. [7] May C.L. Large automorphism groups of compact Klein surfaces with boundary. Glasgow Math. J. 18 (1977), 1-10. [8] May C.L. A family of M ∗ -groups. Canad. J. Math. 38(5)(1986), 1094–1109. [9] May C.L. Supersolvable M ∗ -groups. Glasgow Math. J. 30(1)(1988), 31–40. ˙ ¨ Generalized M ∗ -groups. Internat. J. Algebra Comput. [10] Sahin R., Ikikardes S. and Koruo˘glu O. 16(6)(2006), 1211–1219.

Address: Balikesir University, Faculty of Science, Department of Mathematics, C ¸ a˘ gi¸s 10145, Balikesir-TURKIYE e-mails: [email protected], [email protected]du.tr

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Periodic Solutions for Singular Perturbation Problem of 2−Dimensional Dynamical System Under Matching Conditions Mohammed Jahanshahi

In this paper, we consider a linear perturbation problem which involve a two dimensional dynamical system with periodic boundary conditions. At first this system has regular case, and for this case periodic solutions are calculated. Next, we consider this system for a singular case. In this case, we have a boundary layer problem. The inner and outer asymptotic expansions of solution are determined. Then by making use of periodic boundary conditions and matching conditions of inner and outer expansions, the arbitrary constants in general solution of differential equations are calculated. Finally, the uniform and approximate solution are constructed by inner and outer expansions.

References [1] Cole J.D. Perturbation Method in Applied Mathematics. Ginn and Company Boston, 1958. [2] Tikhonov MA.N. System of differential equations containing Small Parameter in front of derivative. Math. Sb. 31(3)(1952), [3] O’Malley R.E. Jr. Singular Perturbation Methods for O. D. Es. Springer Verlag, 1991. [4] Jahanshahi M. Inverstigation of Boundary Layers in a Singular Perturbation Problem. 31 th Annual Iranian Mathematical Conference, Tehran, Iran, September 2000. [5] Pashaev R.T. and Aliev N. Uniform Correctness of B. V. P for second order O. D. E. News of Baku State University, NI, 1995.

Address: Dept. of Math., Azarbaijan University of Tarbiat Moallem, 35 km, Tabriz- Maraghe Road, Tabriz-IRAN e-mail: jahan [email protected]

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On The Weighted Composition Operators Khadijeh Jahedi and Sedigheh Jahedi

Let {β(n)}n be a sequence of positive numbers with β(0) = 1 and let 1 < p < ∞. Let f = {fˆ(n)}∞ n=0 P ˆ p P∞ be such that kf kp = |f n| β(n)p < ∞ The notation f (z) = n=0 fˆ(n)z n shall be used weather or not the series converges for any value of z. The space of such formal power series is called the weighted Hardy space, which is denoted by H p (β).We consider the weighted composition operators on the weighted Hardy spaces. In this talk, we investigate some relations between the function theoretic of the composition map and the weight function with the operator theoretic of the weighted composition operators.

References [1] Conway J.B. and MacCluer B.D. Composition operators on the spaces of analytic functions. CRC Press,1995. [2] Jahedi K. and Yousefi B. Compactness of composition operators on some function spaces. Int. J. Appl. Math. 18(4)(2005), 447-453 [3] Seddighi K., Hedayatian K. and Yousefi B. Operator acting on certain Banach spaces of analytic functions. International J. Math. and Math. Sci. 18(1)(1995), 107-110. [4] Yousefi B. and Jahedi S. Composition operators on Banach spaces of formal power series. Bollettino U.M.I, 6-B, 8(2003), 481-487.

Address: Islamic Azad University - Shiraz Branch, Graduate school, Shiraz -IRAN Shiraz University of Technology, Dept. of Mathematics, College of Basic Sciences, P.O.Box 7155313, Shiraz-IRAN e-mails: [email protected], [email protected]

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Morita Equivalence and Outer Conjugacy of Dynamical Systems Maria Joita

Pro-C ∗ -algebras are generalizations of C ∗ -algebras. Instead of being given by a single C ∗ -norm, the topology on a pro-C ∗ -algebra is defined by a directed family of C ∗ -seminorms. Clearly, any C ∗ -algebra is a pro-C ∗ -algebra. The class of pro-C ∗ -algebras is more large rather than the class of C ∗ -algebras, for example, the ∗-algebra Ccc ([0, 1]) of all complex valued continuous functions on [0, 1] with the topology of uniform convergence on the countable compact subsets of [0, 1] is a pro-C ∗ -algebra which is not topologically isomorphic with any C ∗ -algebra [2]. A pro-C ∗ -dynamical system is a triple (G, A, α), where G is a locally compact group, A is a pro-C ∗ algebra and α is a continuous inverse limit action of G on A (this is, we can write A as inverse limit lim Aλ of C ∗ -algebras in such a way that there are continuous actions αλ of G on Aλ , λ ∈ Λ such that ←λ∈Λ

αt = lim αtλ for all t in G [4]). ←λ∈Λ

The study of the pro-C ∗ -dynamical systems is motivated by the following example. The ∗-algebra C(X) of all continuous complex valued functions on a Hausdorff countably compactly generated topological space X (that is, there is a countable family of compact spaces K1 ⊆ K2 ⊆ ... ⊆ Kn ⊆ ... such that X = lim Kn [4]) equipped with the topology defined by the family of C ∗ -seminorms {pKn }n , where n→

pKn (f ) = sup{|f (x)| , x ∈ Kn }, is a pro-C ∗ -algebra. Suppose that (G, X) with G a compact group is a transformation group. Then, for each t ∈ G, the map αt : C(X) → C(X) defined by αt (f ) (x) = f t−1 · x is an isomorphism of pro-C ∗ -algebras. Moreover, the map t 7→ αt (f ) from G to C(X) is continuous for each f ∈ C(X), and the map t 7→ αt is a continuous inverse limit action of G on C(X). In the present talk, we extend a result of Combes [1] which states that two unital C ∗ -dynamical systems are stably outer conjugate if and only if they are strongly Morita equivalent to the context of pro-C ∗ -dynamical systems.

References [1] Combes F. Crossed products and Morita equivalence. Proc. London Math. Soc. 49(1984), 289–306. [2] Fragoulopoulou M. Topological algebras with involution. Amsterdam: Elsevier, 2005, (North-Holland mathematics studies; 200). [3] Joita M. Crossed products of locally C ∗ -algebras and strong Morita equivalence. Mediterr. J. Math. 5(2008). [4] Phillips N.C. Representable K-theory for σC ∗ -algebras. K-Theory 3(5)(1989), 441-478.

Address: Department of Mathematics, University of Bucharest-ROMANIA e-mail: [email protected]

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Entanglement Dynamics in Stochastic Atom-Field Interactions H¨ unkar Kayhan

The entanglement dynamics in a system of the stochastic interaction of an atom with a single-mode cavity field is studied by the Jaynes-Cummings model. Random phase telegraph noise is considered as the noise in the interaction and an exact solution to the model under this noise is obtained. The obtained solution is used to investigate the entanglement dynamics of the atom-field interaction. The mutual entropy is adopted for the quantification of the entanglement in the interaction. It is found that the entanglement is a non monotonic function of the intensity of the noise. The degree of the entanglement decreases to a minimum value for an optimal intensity of the noise and then increases for a sufficiently large intensity.

References [1] Jaynes E. T. and Cummings F. W. Comparison of Quantum and Semiclassical Radiation Theory with Application to the Beam Maser. Proc. IEEE 51(1963), 89-109. [2] Joshi A. Effects of phase fluctuations in the atom-field coupling coefficient of the Jaynes-Cummings model. J. mod. Optics 42(1995), 2561-2569.

Address: Abant Izzet Baysal University, Department of Physics G¨ olk¨ oy, Bolu-TURKIYE e-mail: hunkar− [email protected]

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Elementary Abelian Coverings of Regular Hypermaps of Types {5, 5, 5} and {5, 2, 10} of Genus 2 Mustafa Kazaz

In this talk, we find elementary abelian coverings of the regular hypermaps H1 of type {5, 5, 5} and M1 of type {5, 2, 10} of genus 2 corresponding to the orientation-preserving automorphism groups C5 , the cyclic group of order 5 and C10 , the cyclic group of order 10, respectively. We also determine the reflexibility of these coverings.

References [1] Breda d’ Azevedo A.J. and Jones G.A. Rotary Hypermaps of Genus 2. Contributies to Algebra and Geometry 42(1)(2001), 39-58. [2] Cori R. and MachıA. Maps, Hypermaps and their Automorphisms: a Survey I, II, III. Expositiones Math. 10(1992), 403-437, 429-447, 449-467. [3] Corn D. and Singerman D. Regular Hypermaps. Europ. J. Combinatorics 9(1988), 337-351. [4] Greenberg L. ALectures on Algebraic Topology. Benjamin, New York, 1966. [5] Jones G.A. Graph Imbeddings, Groups, and Riemann Surfaces. Colloq. Math. Soc. Janos Bolyai 25 (1978), 297-311. [6] Jones G. A. and Singerman D. Theory of Maps on Orientable Surfaces. Proc. Lon. Math. Soc. 3(37) (1978), 273-307. [7] Jones G.A. and Singerman D. Maps, Hypermaps and Triangle Groups. London Math. Soc. Lecture Note Ser. 200(1994), 115-145. [8] Kazaz M. Homology Action on Regular Hypermaps of Genus 2. Communications, Serias A1: Mathematics and Statistics 51(1)(2002), 1-18. [9] Rosen K.H. Elementary Number Theory and its Applications. Addison-Wesley, Reading, 1984. [10] Singerman D. Finitely Maximal Fuchsian Groups. J. London Math. Soc 2(6)(1972), 29-38.

Address: Department of Mathematics, Faculty of Science, University of Celal Bayar, Muradiye Campus, 45047, Manisa-TURKIYE. e-mail: m [email protected]

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Some Subordination Results for Certain Analytic Functions of Complex Order Involving Carlson-Shaffer Operator ¨ Oznur ¨ Ozkan Kılı¸ c

In this paper, we give some subordination results for the analytic functions that satisfy the following condition: eiλ (1 − δ) zL 0 (a, c) f (z) + δzL 0 (a + 1, c) f (z) Re 1 + −1 >0 b cos λ (1 − δ) L (a, c) f (z) + δL (a + 1, c) f (z) for some real λ (−π/2 < λ < π/2), b 6= 0, b ∈ C, 0 ≤ δ ≤ 1 and for all z ∈ U. This class is denoted by λ Pa,c (b, δ). We note that we obtain λ-spirallike functions of complex order for a = c = 1, δ = 0 and λ-Robertson functions of complex order for a = c = 1, δ = 1.

References ¨ ¨ Starlike, convex and close-to-convex functions of complex order. Hacettepe [1] Altınta¸s O. and Ozkan O. bulletin of natural sciences and engineering, Series B, 28(1999), 37-46 [2] Aouf M.K., Al-Oboudi F.M. and Haidan M.M. On some results for λ-spirallike and λ- Robertson functions of complex order. Publications de l’institut mathematique, Tome 75(91)(2005), 93-98 [3] Kim Y.C. and Srivastava H.M. Some subordination properties for spirallike functions. (in press) [4] Miller S.S. and Mocanu P.T. Differential Subordination, Theory and Applications, Series on Monographs and textbooks in Pure and Applied Mathematics. Vol 225, Marcel Dekker, New York, 2000.

Address: Baskent University, Department of Statistics and Computer Sciences, Baglıca, TR 06530, Ankara-TURKIYE e-mail: [email protected]

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Arrays with the Window Property and their Generalization Sang-Mok Kim

An aperiodic perfect map(APM) is a c-ary m × n array such that for given u ≤ m and v ≤ n every c-ary u × v array occurs exactly once as a subarray(called a window). This is called the perfect window property. An analogous perfect window property is also applicable to a periodic array called a Periodic Perfect Map(PM). Many results on periodic sequences with the perfect window property(called De Bruijn sequences) and PM can be seen in [1], [2], [10] and [8], [9](due to K. Paterson), respectively. In contrast to the rich theory of PM, not many results on APM are known. S.-M. Kim gives the existence of APM with 2×2 windows([3], 2002), a construction method for an infinite family of APMs derived from given de Bruijn sequences([4], 2008), and a simple construction of multi-dimensional APMs([5], 2005) as a generalisation of [3] in 2002. In this talk, with the reframed terminologies for multi-dimensional arrays in Kim[5], we generalise the criterion shown in Kim[4] to n-dimensional case. That means, (n + 1)−dimensional APMs can be obtained from n−dimensional PMs as if two-dimensional APMs are obtained from De Bruijn sequences(1dimensional PM) as shown in Kim[4]. We finally give some infinite families of multi-dimensional APMs by constructing such sequences T satisfying the criterion.

References [1] Good I.J. Normally recurring decimals. J. London Math. Soc. 21(1946), 167-169. [2] Fredricksen H. A survey of full length nonlinear shift register cycle algorithms. SIAM J. Algebraic and Discrete Methods 1(1980), 107-113. [3] Kim S.-M. On the existence of aperiodic perfect maps for 2 × 2 windows. Ars Comb. 65(2002), 111-120. [4] Kim S.-M. Aperiodic Perfect Maps from de Bruijn Sequences. Ars Comb. 86(2008), 201-216. [5] Kim S.-M. On a generalized aperiodic perfect map. Comm. of the KMS 20(4)(2005), 685-693. [6] Menezes A.J. Ddrschot P.C. and Vanstone S.A. Handbook of Applied Cryptography. CRC Press, 1997. [7] Mitchell C.J. Aperiodic and semi-periodic perfect maps. IEEE trans. on Info. Theory 41(1995), 88-95. [8] Paterson K.G. New Classes of Perfect Maps I. J. of Combinatorial Theory (Series A) 73(1996), 302-334. [9] Paterson K.G. New Classes of Perfect Maps II. J. of Combinatorial Theory (Series A) 73(1996), 335345. [10] Rees D. Note on a paper by I. J. Good. J. London Math. Soc. 21(1946), 169-172.

Address: Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701-S.KOREA e-mail: [email protected]

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p-Adic q-Integration on Zp Taekyun Kim

The first purpose of this paper is to present a systemic study of some families of multiple q-Bernoulli numbers and polynomials by using multivariate q-Volkenborn integral (=p-adic q-integral) on Zp . From the studies of these q-Bernoulli numbers and polynomials of higher order we derive some interesting q-analogs of Stirling number identities.

References [1] Cangul I.N., Kurt V., Simsek Y., Pak H.K. and Rim S.-H. An invariant p-adic q-integral associated with q-Euler numbers and polynomials. J. Nonlinear Math. Phys. 14 (2007), 8–14. [2] Cenkci M., Can M. and Kurt V. p-adic interpolation functions and Kummer-type congruences for q-twisted Euler numbers, Advan. Stud. Contemp. Math. 9 (2004), 203–216. [3] Kim T. On p-adic q-l-functions and sums of powers, J. Math. Anal. Appl. 329 (2007), 1472–1481. [4] Kim T. q-Volkenborn integration. Russ. J. Math. Phys. 9 (2002), 288–299. [5] Kim T. A Note on p-Adic q-integral on Zp Associated with q-Euler Numbers. Adv. Stud. Contemp. Math. 15 (2007), 133–138. [6] Kim T. On p-adic interpolating function for q-Euler numbers and its derivatives. J. Math. Anal. Appl. 339 (2008), 598–608. [7] Kim T. Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials. Russ. J. Math. Phys. 10 (2003), 91–98. [8] Ozden H., Simsek Y., Rim S.-H. and Cangul I.N. A note on p-adic q-Euler measure. Adv. Stud. Contemp. Math. 14 (2007), 233–239. [9] Srivastava H.M., Kim T. and Simsek Y. q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russ. J. Math. Phys. 12 (2005), 241–268.

Address: EECS, Kyungpook National University, Taegu 702-701, S. KOREA e-mail: [email protected], [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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On a Inverse and Direct Problems of Scattering Theory for a Class of Sturm-Liouville Operator with Discountinous Coefficient Nida Palamut Ko¸ sar and Khanlar R.Mamedov

In this paper a boundary-value problemwith piecewise continous coefficient on the half line [0, +∞) are considered: y 00 + q (x) y = λ2 ρ (x) y (48.1) y 0 (0) − λ2 y (0) = 0, where λ is a spectral parameter, q (x) is a real-valued function satisfying the condition

(48.2) +∞ R

(1 + x) |q (x)| dx <

0

+∞ and ρ (x) is real-valued function which has finite number of discontinouty point. In this work scattring datas for (48.1), (48.2) boundary-value problem are defined, the spectrum is investigated and the formula for the expansion of the eigenfunctions is optained. The Gelfand-LevitanMarchenko equation which has important role for solving the inverse problem is found. The inverse and direct problem for scattering theory for equation (48.1) without spectral parameter in boundary condition (48.2) was researced in [1] , [2]. The uniqueness of the solution to the inverse problem with a spectral parameter in the case of ρ (x) was investigated in [3] .

References [1] Guseinov I.M. and Pa¸saev R.T. On a Inverse Problem for a Second Order Differantial Equation. Usp. Math. Nauk 57(3)(2002), 597-598. [2] Mamedov Kh.R. Uniqueness of the Solution of the Inverse Problem of Scattring Theory for SturmLiouville Operator with Discontinous Coefficient. Proceeding of IMM of NAS Azerbaijan 24(2006), 263,272. [3] Mamedov KhR. Uniqueness of the Solution to the Inverse Problem Scattring Theory for Sturm-Liouville Operator with a Spectral Parameter in The Boundary Condition. Math. Notes 74(1)(2003), 136-140. [4] Menken H. and Mamedov Kh.R. On the Inverse Problem Of The Scattring Theory For A BoundaryValue Problem, Geomety. Integrability and Quantization, Sofia, 226-236, 2005.

Address: Mersin University, Faculty of Science, Department of Mathematics, Mersin-TURKIYE e-mails: [email protected], [email protected]

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Surfaces with Negative Gauss Curvature; Classification According to the Singularities of Attached Monogenous Functions Lidia Elena Kozma

The monogenous surfaces attached to a holomorphic function offer diverse application possibilities as much in the general approximation of surfaces having the total Gauss curvature negative [1] as in extending the solutions of some boundary problems [2]. This article analyzes the values of the total Gauss curvature of the monogenous surface ¯ (S) : r¯ = ¯iy + ¯ju(x, y) + kv(x, y) in the vicinity of singular isolated points of the holomorphic function: w = f (z) = u(x, y) + iv(x, y),

(x, y) ∈ D ⊂ C

References [1] Lidia Elena Kozma. About the Dirichlet and Neumann boundary value problems expressed by means of monogenous quaternions. Carpathian Journal of Mathematics 20(2)(2004), 193-196. [2] Lidia Elena Kozma. On the approximation of surface with negative Gauss curvature with surfaces attached to the monogenous functions. carpathian Journal of Mathematics 23(1-2)(2007), 100-107. [3] Lidia Elena Kozma. Applications of Pompeiu areolary derivative in expressing of the Gauss total negative curvature. General Mathematics, Proceedings of International Symposium on Complex Analysis 15(2-3)(2007), 141-153.

Address: Nort University of Baia Mare, Faculty of Sciences, Department of Mathematics, M62/A Victor Babe¸s Street, 430083 Baia Mare-ROMANIA e-mail: kozma [email protected]

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Integral and Difference Inequalities in Several Independent Variables and their Discrete Analogues Emine Mısırlı Kurpınar and ¨ Ozlem Mo˘ gol

We present some new results on the theory of integral inequality with two dependent limits. Our results have some relationships with certain Gronwall’s type integral inequalities which can be used to show the boundedness of the solutions of nonlinear differential equations . The analysis used in the proofs are quite easy and elementary and the results established here provide new estimates for these types of inequalities. Some applications are also given for our results.

References [1] Pachpatte B.G. On the discrete generalizations of Gronwall’s inequality. J.Indian Math. Soc. 37(1973), 147-156. [2] Pachpatte B.G. some new finite difference inequalities. Comput. Math. Appl. 28(1994), 227-241. [3] Pachpatte B.G. ”Inequalities for Differential and Integral Equations” Academic Press, New York, 1998. [4] Kurpınar Mısırlı E.On inequalities in the theory of differential equations and their discrete analogues. PAMJ, 1999. [5] Ashyralyev M. Generalizations of Gronwall’s integral inequality and their discrete analogies. Report MAS-EO520 September 2005. [6] Mamedov Y. and Askirov S. A Volterre type integral equation. UMJ 40(4)(1988), 510-515. [7] Ashyralyev M.Integral inequalities with four variable limits, in: Modeling of processes of development of gas deposits and applied tasks theoretical gas dynamics, “Ylym”. Ashgabat, Turkmenistan, 1998, pp.170-184. [8] Gronwall T.N. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Of Math. II., Ser. 20(1919), 292-296. [9] Mitrinovic D.S. “Analytic Inequalities”. Springer-Verlag, Berlin/New York, 1970.

Address: Ege University Science Faculty Department of Mathematic 35100 Bornova, Izmir-TURKEY. e-mails: [email protected], [email protected]

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Rothe’s Method for Semilinear Parabolic Integrodifferential Equation with Integral Condition A.Guezane-Lakoud and Abderrezak Chaoui

In this talk, we consider a semilinear parabolic integrodifferential equation with nonlocal integral condition. We study in the rectangle D = ]a, b[ × [0, T ] the following problem: ∂2v ∂v (x, t) − (x, t) = ∂t ∂2x R T a (x, t − s) K (x, s, v (x, s)) ds + f (x, t) 0 v (a, t) = v (b, t) = 0 b R v (x, t) dx = ϕ (t) . a

We apply Rothe’s method which is based on a semidiscretization with respect to the time variable to prove the unique solvability of this problem under certain conditions on the the nonlinear function K. Kacur has applied this method to solve a semilinear hyperbolic equation then Bahaguna has generelized these results to more general problems with various type of nonlocal history conditions. Now we try to apply the same method to an integroddiferential equation with integral condition.

References [1] Bahuguna D. and Raghavendra V. Rothe’s method to parabolic initial boundary value problems via abstract parabolic equations. Appl. Anal. 33(1989), 153-167. [2] Kacur J. Application of Rothe’s method to perturbed linear hyperbolic equations and variational inequalities. Czech. Math. J. 34(109)(1984), 92-106.

Address: Badji Mokhtar University, Annaba and Guelma University-ALGERIA e-mail: [email protected]

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The Magnetic Field Dependence of the Quantum Transition Properties of Si in the Linearly Polarized Oscillating Field S.H.Lee, J.Y.Sug, G.H.Rue, Sa-Gong Geon and J.Y.Choi

We consider the one system is subject to the linearly polarized oscillatory external field. We study the optical quantum transition Line shapes(QTLS) which show the absorption power and the quantum transition line widths(QTLW) of electron-deformation potential phonon interacting system. We analyze the magnetic field dependence of the QTLS and the QTLW in various cases. In order to analysis the quantum transition, we compare the magnetic field dependence of the QTLW and the QTLS of two transition process, the intra-Landau level transition process and the inter-Landau level transition process. We use the projected Liouville equation method with Equilibrium Average Projection Scheme (EAPS).

References [1] J. R. Barker, J. Phys. C 6, 2633(1973); Solid State electron, 21, 261(1978) ; J. R. Barker and D. K. Ferry, ibid, 23, 531(1980). [2] C. S. Ting, S. C. Ying and J. J. Quinn Phys. Rev. B16, 5394 (1977) R. S. Fishman, Phys. Rev. B39, 2994 (1989) [3] C. S. Ting and S. C. Ying and J. J. Quima, Phys. Rev. Lett. 37, 315 (1976) G. Y. Hu and R. F. Oconnell, Phys. Rev. B36, 5798 (1987) [4] Wu Xiaoguang, F. M. Peeters and J. T. Devreese, Phys. Rev. B34, 8800 (1986) R. S. Fishman and G. D. Mahan, Phys. Rev. B39, 2990 (1989) [5] P. Grigoglini and G. P. Parravidini, Phys. Rev. Bl25, 5180 (1982) [6] R. Kubo, J. Phys. Soc. Jpn. 12, 570(1957) ; [7] H. Mori, Progr. Theor. Phys. 33, 423(1965) ; 34, 399(1966) ; M. Tokuyama and H. Mori, Progr. Theor. Phys. 55, 2(1975) [8] A. Kawabata, J. Phys. Soc. Jpn. 23, 999(1967); K. Nagano, T. Karasudani and H. Okamoto, Progr. Theor. Phys. 63, 1904(1980)

Address: Electronic and Electric Eng. School, Kyungpook National University, Daegu 702-701, KOREA (Sa-Gong Geon) Dep. of Electrical Eng., Donga Uni. Pusan City, KOREA e-mails: [email protected], [email protected], [email protected]

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Gnan Mean and its Dual in Several Arguents Veerabhadraiah Lokesha

In this talk, we present our recent efforts which describes the basic aspects of gnan mean and its dual in two variables along with some fruitful results. Further, we define and explore the results of gnan mean and its dual in n variables.

References [1] Kuang J.-Ch. Ch´ angy` ong B´ udˇengsh`ı. Applied Inequalities, 2nd ed. Hunan Education Press, Changsha, China, 1993. (Chinese) [2] Lokesha V. and Zhang Zh.-H. The weighted heron dual mean in n variables. Adv. Stud. Contemp. Math. (Kyungshang) 13(2)(2006), 165-170. [3] Lokesha V., Zhang Zh.-H. and Nagaraja K.M. The Gnan mean for two variables. Far East Journal of Applied Mathematics 31(2)(2008), 263–272. [4] Lokesha V., Zhang Zh.-H. and Wu Y.-D. Two weighted product type means and its monotonicities. RGMIA Research Report Collection 8(1)(2005), Article 17, http://rgmia.vu.edu.au/v8n1.html [5] Nagaraja K.M., Padamanabhan S., Lokesha V. and Zhang Zh.-H. Gnan mean and its dual in n variables. Int. J. Pure & Appl. Math., Bulgeria (2008)(appear). [6] Zhang Zh.-H., Lokesha V. and Xiao Zh.-G. The weighted heron mean in n variables. J. Anal. Comput. 1(1)(2005), 57-68. [7] Zhang Zh.-H. and Wu Y.-D. The generalized Heron mean and its dual form. Appl. Math. E-Notes 5(2005), 16-23. http://www.math.nthu.edu.tw/˜amen/

Address: Professor and Head, Department of Mathematics, Acharya Institute of Technology, Soldevna Halli, Bangalore, Katnataka-INDIA e-mail: [email protected]

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Non-linear Multi-objective Transportation Problem: A Fuzzy Goal Programming Approach Hamid Reza Maleki and Sara Khodaparasti

A non-linear multi-objective transportation problem (NMOTP) refers to a special class of non-linear multi-objective problems. In this paper we review goal programming (GP) and fuzzy programming (FP) as two approaches for solving (NMOTP). Meanwhile, by extending Mohamed’s idea about fuzzy goal programming (FGP), we propose a fuzzy goal programming approach to solve the non-linear multi-objective transportation problem. To this end, we use a special type of non-linear (hyperbolic) membership functions. Then, by proving a theorem, the relationship between (FP) and (FGP) will be stated. At the end, a numerical example is given to illustrate the efficiency of the proposed approach.

References [1] Biswal M.P. and Verma R. Fuzzy Programming Technique to Solve A Non-Linear Transportation Problem. Fuzzy Mathematics, 1999. [2] A.K.Bit A.K., Biswal M.P and Alam S.S Fuzzy programming approach to multi-criteria decision making transportation problem. Fuzzy Sets and System, 1992. [3] Mohamed R.H. The relationship between goal programming and fuzzy programming. Chapman& Hall/CRC, 2001. [4] Chanas S. and Kuchta D. A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets and Systems., 1996. [5] Zangiabadi M. and Maleki H.R. Fuzzy Goal Programming for Multi-objective Transportation Problems. to be printed in Rocky J. Maths.

Address: Shiraz University of Technology, Department of Basic Sciences, P.O.Box 71555 - 31. Shira-IRAN e-mails: [email protected], [email protected]

This paper is supported in part by Fuzzy Systems and Applications Center of Excellence, Shahid Bahonar University of Kermen, I.R.of IRAN

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55

Recursive Relations on the Coefficients of Some p-Adic Differential Equations Hamza Menken and Abdulkadir A¸ san

In this talk, we consider some modified hypergeometric series of the form Fk (x) =

∞ P

n!Pk (n)xn where

n=0

Pk (n) = nk + Ck−1 nk−1 + · · · + C0 is a polynomial in n with Ci ∈ Q (or Ci ∈ Cp ). Such series are often encountered in p-adic analysis [1] and its applications and mathematical physics [2], [3]. 1 These power series are divergent for every 0 6= x ∈ R, but they are convergent p-adically in |x|p < p p−1 where p is any prime number and ||p is the p-adic norm. General theory of the p-adic hypergeometric series is given in [4]. It is well known [5] that there exist a first- and a second-order p-adic differential equation with the ∞ P analytic solution of the form Fk (x) = n!Pk (n)xn . For some special classes of Pk (n) the corresponding n=0

p-adic linear differential equations can be constructed. We construct new forms of the corresponding differential equations for some special classes of Pk (n). Then we obtain some recurrence relations for the coefficients of these differential equations.

References [1] Schikhof W.H. Ultrametric Calculus. Cambridge University Press, 2006. [2] Vladimirov V.S., Volovich I.V. and Zelenov E.I. p-adic Analysis and Mathematical Physics. World Scientific, 1998. [3] Khrennikov A. p-Adic Valued Distributions in Mathematical Physics. Dordrecht: Kluwer Academic Publishers, 1994. [4] Dwork B. Lecture on p-Adic Differential Equations. Springer-Verlag, 1982. [5] Gosson de M., Dragovich B. and Krennikov A. Some p−adic Differential Equations. Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 2001.

Address: Mersin University, Faculty of Science and Arts, Department of Mathematics, 33343. Mersin-TURKIYE e-mails: [email protected], [email protected]

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On N (k)-Mixed Quasi Einstein Manifolds H.G.Nagaraja

In this talk we consider N (k)-Mixed Quasi Einstein Manifolds(N (k) − (M QE)n ) and we give the proof of existence of these manifolds. We prove that hyper surfaces of Euclidean spaces are N (k) − (M QE)n manifolds. We also consider semi symmetric, ricci symmetric and ricci recurrent N (k)−(M QE)n manifolds.

References [1] Chaki M.C. and R.K.Maity. On Quasi Einstein manifolds. Publ. math. Debrecen 57(2000) 297-306. [2] Chen B.Y. Geometry of Submanifolds. Marcel Dekker, Inc., New York, 1973. [3] Chen B.Y. and Yano K. Hypersurfaces of a conformally flat space. Tensor, N.S. 26(1972), 318-322. [4] De U.C. and Gopal Chandra Ghosh. On quasi Einstein manifolds. Periodica Mathematica Hungarica 48(1-2)(2004), 223-231. [5] De U.C. and Gopal Chandra Ghosh. On generalized quasi Einstein manifolds. Kyungpook Math. J. 44(2004), 607-615. [6] Mukut Mani tripathi and Jeong-Sik Kim. On N (k)-quasi Einstein manifolds. Commun. Korean Math. Soc. 22(3)(2007), 411-417. [7] Tanno S. Ricci curvatures of contact Riemannian manifolds. Tohoku Math. J. 40(1988), 441-448.

Address: Department of Mathematics, Central College, Bangalore University, Bangalore-560 001, Karnataka-INDIA. e-mail: [email protected]

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Some New Explicit Values for Ramanujan Class Invariants M.S.Mahadeva Naika Ramanujan’s two class invariants Gn and gn are defined by 1

1

Gn = 2− 4 q − 24 χ(q)

and

where q = e−π

χ(q) = (−q; q 2 )∞ , and (a; q)∞ :=

∞ Y

1

1

gn = 2− 4 q − 24 χ(−q), √

n

,

(57.1)

n ∈ Q+

(1 − aq k ), |q| < 1.

k=0

√ √ 1 In the notation of H. Weber [5], Gn := 2 f ( −n) and gn := 2− 4 f1 ( −n). Weber and Ramanujan were calculated class invariants separately for different purposes. In [5], Weber was calculated 107 class invariants, or monic, irreducible polynomials satisfied by them to construct Hilbert class fields. On the other hand Ramanujan calculated them to approximate the value of π or to calculate the explicit evaluations of the ratios of theta-functions so that he could be able to find the evaluations of Ramanujan’s remarkable product of theta-function, cubic continued fractions etc. On page 291-299, Ramanujan [4] recorded table of 77 class invariants, or monic, irreducible polynomials satisfied by them. After coming to England, he came to know about Weber’s work, and therefore his table of 46 class invariants in [3] does not contain any that are found in Weber’s book [5]. G. N. Watson established 28 out of these 46 class invariants. The remaining were proved by B.C. Berndt, H.H. Chan and L.-C. Zhang [2] using modular equations, Kronecker’s limit formula and Watson’ empirical process. In [1] N.D. Baruah established G217 via modular equations of degrees 7 and 31. The main purpose of this paper is to establish some new values for Ramanujan’s class invariants using modular equations of degrees 19, 23 and 59. − 41

References [1] Baruah N.D. On some class invariants of Ramanujan. J. Indian Math. Soc. 68(2001), 113-131. [2] Berndt B.C., Chan H.H. and Zhang L.-C. Ramanujan’s class invariants, Kronecker’s limit formula and modular equations. Trans. Amer. Math. Soc. 349(6)(1997), 2125-2173. [3] Ramanujan S. Modular equations and approximation to π. Quart. J. Math. 45(1914), 350-372. [4] Ramanujan S. Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay, 1957. [5] Weber H. Lehrbuch der Algebra, dritter Band. Chelsea, New York, 1961.

Address: Department of Mathematics, Central College Campus, Bangalore University, 560 001, Bangalore-INDIA e-mails: [email protected], [email protected]

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The Necessarily Efficient Point Method for Interval MOLP Problems Hassan Mishmast Nehi and Marzieh Alineghad

In this talk, we gave a necessarily efficient point method for interval MOLP problems. In the most real world situations, an objective function is not satisfied the decision makers goals and reduce the efficiency of the models. Also the coefficients of decision variables are not exactly known. One way to illustrated the uncertainty is intervals. In this paper we consider multi objective linear programming with interval coefficients and solve it with respect to necessarily efficient points.

References [1] Bitran G.R. Linear multiple objective problems with interval coefficients. Management Science 26(1980), 694-706. [2] Chineck J.W and Ramadan K. Linear programming with interval coefficients. Journal of the operational Research Society 51(200), 209-220. [3] Hansen E. Global optimaization using interval analysis. New York Press, 2002. [4] Ida M. Necessary efficient test in interval multiobjective linear programming. In:Proceeding of 8TH international fuzzy system association world congress, 500-504, 2000. [5] Ida M. Efficient solution generation for multiple objective linear programming and uncertain coefficients. In:proceedings of 8th Bellman continuum, 132-136, 2000. [6] Ishibuchi H. and Tanaka H. Multiobjective programming in optimization of the interval objective Systems. Prentic-Hall,Englewood Cliffs, NJ, 1982. [7] Oliveira C. and Antunes C.H. Multiple objective linear programming modeles with interval coefficients -an illustrated overview. European journal of operational Research, 1-30, 2006. [8] Rohn J. System of interval linear equations and inequalities(rectangular case). Technical report no:875, 2002. [9] Steuer P.E.Multiple criteria optimization:theory,computation and application. John wiley & Sons, New york, 1986.

Address: Assistant Professor in Mathematics Department of Mathematics, Faculty of Science Sistan and Baluchestan University, Zahedan-IRAN. e-mails: [email protected], [email protected]

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Application of Variational Iteration Method for Solving Some Partial Differential Equations Volkan Oban and Ahmet Yıldırım

In this paper, He’s variational iteration method (VIM) is employed successfully for solving some partial differential equations. In this method, the solution is calculated in the form of a convergent series with an easily computable component. This approach does not need liberalization, weak nonlinearity assumptions or perturbation theory. The results show applicability, accuracy and efficiency of VIM in solving nonlinear differential equations with fully nonlinear dispersion term. It is predicted that VIM can be widely applied in science and engineering problem.

References [1] He JH. Variational iteration method – a kind of non-linear analytical technique: some Examples. International Journal of Non-linear Mechanics 34(1999), 699-708 [2] He JH. Variational iteration method for autonomous ordinary differential system. Applied Mathematics and Computation 114(2000), 115-123 [3] He JH and Wu XH. Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons and Fractals 29(2006), 108-113 ¨ [4] Ozis T. and Yıldırım A. A study of nonlinear oscillators with u1/3 force by He’s variational noindent iteration method. Journal of Sound and Vibration 306(2007), 372-376

Address: Ege University, Faculty of Science, Department of Mathematics, Bornova 35100, Izmir-TURKIYE e-mails: [email protected], [email protected]

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The Limit q-Bernstein Operator Sofiya Ostrovska

The limit q-Bernstein operator B∞,q emerges naturally as a q-version of the Sz´asz-Mirakyan operator related to the Euler distribution (cf. [4]). Alternatively (cf. [1]), the limit q-Bernstein operator arises as a limit for a sequence of the q-Bernstein polynomials in the case 0 < q < 1 or a sequence of q-Meyer-K¨ onig and Zeller operators (cf. [6]). The limit q-Bernstein operator has been widely studied lately (see [2, ?, 5]. It has been shown that B∞,q is a positive shape-preserving linear operator on C[0, 1] with kB∞,q k = 1. Its approximation properties, probabilistic interpretation, the behavior of iterates, eigenstructure, and the impact on the smoothness of a function have been examined. In this talk, we discuss the properties of the limit q-Bernstein operator. The talk contains new results as well as those known previously.

References [1] Il’inskii A. and Ostrovska S. Convergence of generalized Bernstein polynomials. J. Approx. Theory 116(2002), 100-112. [2] Ostrovska S. On the limit q-Bernstein operator. Mathematica Balkanica 18(2004), 165-172. [3] Ostrovska S. On the improvement of analytic properties under the limit q-Bernstein operator. J. Approx. Theory 138(2006), 37-53. [4] Ostrovska S. Positive linear operators generated by analytic functions. Proc. Indian Acad. Sci. (Math. Sci.) 117(4)(2007), 485-493. [5] Videnskii V.S. On some classes of q-parametric positive operators. Operator Theory, Advances and Applications 158(2005), 213-222. [6] Wang H. Properties of convergence for the q-Meyer-Konig and Zeller operators. J. Math. Anal. and Appl. 335(2)(2007), 1360-1373.

Address: Atilim University, Department of Mathematics, Incek, Ankara-TURKIYE e-mail: [email protected]

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Verification of the Unknown Diffusion Coefficient by Semigroup Method Ebru ¨ Ozbilge and Ali Demir

In this article a semigroup approach is presented for analysis of inverse coefficient problems.We want to identify the unknown coefficient k(u(x, t)) in the quasi-linear parabolic equation ut (x, t) = (k(u(x, t))ux (x, t))x , with Dirichlet boundary conditions u(0, t) = ψ0 , u(1, t) = ψ1 . Distinguishability of the input-output mappings Φ[·] : K → C 1 [0, T ], Ψ[·] : K → C 1 [0, T ] by semigroup theory is investigated.We showed that if the null space of the semigroup T (t) consists of only zero function, then the input-output mappings Φ[·] and Ψ[·] have the distinguishability property. With using the measured output data (boundary observations) f (t) := k(u(0, t))ux (0, t) or/and h(t) := k(u(1, t))ux (1, t), we can determine the values of k(ψ0 ) at (x, t) = (0, 0) and k(ψ1 ) at (x, t) = (1, 0) explicitly. Additionally, by using the input data, the values of the unknown diffusion coefficient k(u(x, t)), ku (/phi0 ) at (x, t) = (0, 0) and ku (ψ1 ) at (x, t) = (1, 0) are identified. Moreover, by an integral representation f (t) and h(t) are specified analytically.Consequently Φ[·] : K → C 1 [0, T ] and Ψ[·] : K → C 1 [0, T ] are given explicitly in terms of the semigroup.

References [1] Cannon J.R. The One-Dimensional Heat Equation. Encyclopedia of Mathematics and Its Applications 23(1984) Addison Wesley, Massachusets. [2] Renardy M. and Rogers R.C. An Introduction to Partial Differential Equations. Springer, New York, 2004. [3] DuChateau P. Introduction to inverse problems in partial differential equations for engineers, physicists and mathematicians. in Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology, J. Gottlieb and P. DuChateau, eds., Kliver Academic Publishers, the Netherland, (1996), pp. 3-38. [4] DuChateau P. Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems. SIAM J. Math. Anal. 26(1995), 1473-1487. [5] DuChateau P., Thelwell R. and Butters G. Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient. Inverse Problems 20(2004), 601-625. [6] Showalter R.E. Monotone Operators in Banach spaces and Nonlinear Partial Differential Equations. American Mathematical Society, Providence, 1997.

Address: Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No.156, 35330, Balcova, Izmir-TURKIYE Applied Mathematical Sciences Research Center and Department of Mathematics, Kocaeli University, Ataturk Bulvari, 41300, Izmit, Kocaeli-TURKIYE e-mails: [email protected], [email protected]

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Fractional Optimal Control Problem in Cylindrical Coordinates Necati ¨ Ozdemir, Derya Karadeniz and Beyza B.Iskender

In this work, we propose the solution of an axis-symmetric Fractional Optimal Control Problem (FOCP) defined in cylindrical coordinates. Find the control u(r, z, t) that minimizes the perfonmance index 1 J(u) = 2

Z1 ZL ZR

r Ax2 (r, z, t) + Bu2 (r, z, t) drdzdt

(62.1)

0 0 0

subject to the system dynamic constraints 2 ∂ x (r, z, t) 1 ∂x (r, z, t) ∂ 2 x (r, z, t) α + + u (r, z, t) , + 0 Dt x (r, z, t) = β ∂r2 r ∂r ∂z 2

(62.2)

initial condition x(r, z, 0) = x0 (r, z)

( 0 < r < R, 0 < z < L)

(62.3)

and the boundary conditions ∂x(0, z, t) ∂x(R, z, t) ∂x(r, 0, t) ∂x(r, L, t) = = = = 0, (62.4) ∂r ∂r ∂z ∂z where x(r, z, t) and u (r, z, t) are the state and the control functions depend on r, z which represent cylindrical coordinates and t is time variable. A and B are two arbitrary functions. R is radius and L is length of cylindrical membrane on which problem is defined. The upper limit for time t is taken as 1 for convenience. This limit can be any positive number. The fractional time derivative is described in the Riemann-Liouville sense. To find the solution of the problem, the method of separation of variables is used. For numerical purposes, the Gr¨ unwald-Letnikov approach is used. The comparison of analytical and numerical solutions is given using simulation results and also it can be seen that analytical and numerical results overlap. In addition, simulation results for different number of order of derivative, time discretizations and eigenfunctions are analyzed.

References [1] Podlubny I. Fractional Differential Equations. Academic Press, New York, 1999. [2] Podlubny I. Fractional-order systems and P I λ Dµ controllers. IEEE Transactions on Automatic Control. 44(1999), 208-214. [3] Agrawal O.P. A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics 38(2004), 323-337. ¨ ˙ [4] Ozdemir N., Agrawal O.P., Iskender B.B. and Karadeniz D. Fractional Optimal Control of a 2Dimensional Distributed System Using Eigenfunctions. Nonlinear Dynamics DOI:10.1007/s11071-0089360-4, (2008)

Address: Balıkesir University, Faculty of Science and Art, Department of Mathematics, Ca˘ g is Campus 10145 ¯ Balıkesir-TURKIYE e-mails: [email protected], fractional− [email protected], [email protected]

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Remarks on Interpolation Functions of q-Bernoulli Numbers Hacer ¨ Ozden, Ismail Naci Cang¨ ul and Yilmaz Simsek

In this talk, we consider q-Bernoulli numbers. By applying Mellin transformation to the generating function we obtain integral representation of new q-zeta function which interpolate q-Bernoulli numbers at non-positive integers. Some new relations related to q-Bernoulli numbers and interpolation functions are also given.

References [1] Carlitz L. q-Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc. 76(1954), 332–370. [2] Cenkci M., Kurt V., Rim S.H. and Simsek Y. On (i, q) Bernoulli and Euler numbers. Appl. Math. Letters 21(2008), 706-711. [3] Koblitz N. On Carlitz’s q-Bernoulli numbers. J. Number Theory 14(1982), 332-339. [4] Kim T. On explicit formulas of p-adic q-L-functions. Kyushu J. Math. 48(1994), 73–86. [5] Kim T. On a q-analogue of the p-adic log gamma function and related integrals. J. Number Theory 76(1999), 320–329. [6] Kim T. q-Volkenborn integation. Russ. J. Math. Phys. 9(2002), 288–299. [7] Kim T. Power series and asymptotic series associated with the q-analogue of two-variable p-adic Lfunction. Russ. J. Math Phys. 12(2)(2005), 186-196. [8] Kim T. On p-adic q-l-functions and sums of powers. J. Math. Anal. Appl. 329(2007), 1472–1481. [9] Kim T. A new approach to q-zeta functions. J. Comput. Anal. Appl. 9(4)(2007), 395–400. [10] Ozden H., Simsek Y., Rim S.H. and Cangul I.N. A note on p-adic q-Euler measure. Advan. Stud. Contemp. Math. 14(2)(2007), 233-239. [11] Ozden H., Simsek Y., Rim S.H. and Cangul I.N. Generating functions of the (h, q)-extension of Euler polynomials and numbers. Acta Math. Hungarica 120(3)(2008), 281-299. [12] Simsek Y. Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function. J. Math. Anal. Appl. 324(2006), 790–804. [13] Simsek Y. On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers. Russian J. Math. Phys. 13(3)(2006), 340–348. [14] Srivastava H.M., Kim T. and Simsek Y. q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russian J. Math. Phys. 12(2005), 241-268.

Address: Uludag University, Faculty of Science, Department of Mathematics, G¨ or¨ ukle 16059, Bursa-TURKIYE Akdeniz University, Faculty of Science, Department of Mathematics, Antalya-TURKIYE e-mails: [email protected], [email protected], [email protected]

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Non-local Gas Dynamics Equation and Invariant Solutions Teoman ¨ Ozer

In this study, as an application of Lie symmetry groups to integro-differential equations we analyze symmetry group properties of non-local gas dynamics equation, which is expressed in the system of the coupled nonlinear partial integro-differential equations and introduce the invariant criterion to analyze symmetry groups for integro-differential equations and then obtain the symmetries of the system of equations. Then, all possible similarity reduction forms and invariant solutions are discussed.

References [1] El Naschie MS. Notes on exceptional Lie symmetry groups hierarchy and possible implications for EInfinity high energy physics. Chaos, Solitons & Fractals, in press. [2] Marek-Crnjac L. On the unification of all fundamental forces in a fundamentally fuzzy Cantorian epsilon ((infinity)) manifold and high energy particle physics. Chaos, Solitons & Fractals 4(2004), 669-682. [3] Gibbons J. Collisionless Boltzmann equations and integrable moment equations. Pyhsica 3D(1981), 503-511. ¨ [4] Ozer T. Symmetry group classification of two-dimensional elastodynamics problem in nonlocal elasticity. Int J Engng Sci 41(2)(2003), 193-221. ¨ [5] Ozer T. Symmetry group analysis of Benney system and an application for shallow- water equation. Mech Res Comm 32(2005), 241-254. [6] Ovsiannikov LV. Group Analysis of Differential Equations. Nauka, Moscow, 1978. [7] Olver PJ. Application of Lie Groups to Differential Equations. Springer-Verlag, 1986. [8] Bluman GW and Kumei S. Symmetries and Differential Equations. Springer- Verlag, 1989. [9] Ibragimov NH. Ed.,CRC Handbook of Lie Group Analysis of Differential Equations Vol I, II, III. 1994. [10] Taranov VB. Symmetry of the one-dimensional high frequency motion of collisionless plasma. Zh Tekh Fiz 46(1976), 1271.

Address: Istanbul Technical University Faculty of Civil Engineering, Division of Mechanics, 34469 Maslak, Istanbul-TURKIYE e-mail: [email protected]

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A Note on Multiplers of Lp (G, A) Serap ¨ Oztop

Let G be a locally compact abelian group, 1 < p < ∞, and A be a commutative Banach algebra. In this talk, we are interested in the relationship between the multipliers L1 (G, A)−module and the multipliers on a certain Banach algebra.The multipliers of type (p, p) and multipliers of the group Lp −algebras were studied and developed by many authors. In these studies, a multiplier is defined to be an invariant operator, a bounded linear operator T commutes with translation. In the case of scalar function space on G, the multipliers are identified with the translation invariant operators. However, in the Banach valued function spaces, an invariant operator does not need to be a multiplier. In this paper a new approach is used for the generalization of the results of McKennon concerning multipliers of type (p, p)to the Banach-valued function spaces. This method is applied to obtain the multipliers on L1 (G, A) ∩ Lp (G, A).

References [1] Datry C. Multiplicateurs d’un L1 (G)-module de Banach consideres comme multiplicateurs d’une certaine alg`ebre de Banach. Group travail d’Analyse Harmonique, Institut Fourier, 1981. [2] Griffin J. and McKennon K. Multipliers and group Lp −algebra. Pasific J. Math. 49(1973), 365-370. [3] Lai H.C. Multipliers of Banach valued function spaces. J. Austral. Math. Soc.(Ser.A) 39(1985), 51-62. [4] McKennon K. Multipliers of type (p, p). Pasific j. Math. 43(1972), 429-436. [5] Wang J.K. Multipliers of commutative Banach algebras. Pasific j. Math. 11(1961), 1131-1149.

Address: Istanbul University, Faculty of Sciences, Department of Mathematics 34134 Vezneciler-Istanbul-TURKIYE e-mail: [email protected]

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On the Numerical Solutions of Bitsadze Samarskii Type Elliptic Equation with Nonlocal Boundary and Mixed Conditions Elif ¨ Ozt¨ urk and Allaberen Ashyralyev

In this talk, we are interested in studying the stable difference schemes for the numerical solution of the nonlocal boundary value problem with mixed conditions for the multidimensional elliptic equation . The first and second orders of accuracy difference schemes are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes for the two-dimensional elliptic differential equation. The method is illustrated by numerical examples.

References [1] Ashyralyev A. A note on the Bitsadze-Samarskii type nonlocal boundary value problem in Banach space. Journal of Mathematical Analysis And Applications 344(1)(2008), 557-573. [2] Bitsadze A.V. and Samarskii A.A. On some simplest generalizations of linear elliptic problems. Dokl. Akad. Nauk. SSSR 185, 1966 [3] Ashyralyev A. and Sobolevskii P.E. New Difference Schemes for Partial Differential Equations. Birkhauser- Verlag, Basel, 2004. [4] Kapanadze D.V. On the Bitsadze-Samarskii nonlocal boundary value problem. Dif. Equation 23 (1460)(1987), 543–545. [5] Berikelashvili G. On a nonlocal boundary value problem for a two-dimensional elliptic equation. Comput. Methods Appl. Math 3(1)(2003), 35-44.

˙ Address: Fatih University, Faculty of Science, Department of Mathematics, B¨ uy¨ uk¸cekmece 34500, Istanbul-TURKIYE e-mails: [email protected], [email protected]

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Notes on Multiplicative Calculus Ali ¨ Ozyapıcı and Emine Mısırlı Kurpınar

Multiplicative calculus is one of the infinitely many calculi that can be constructed. Multiplication and division play a central role in the multiplicative calculus whereas the same role is realized in ordinary calculus by addition and substraction. Thus, multiplicative calculus is quite different from the ordinary calculus Multiplicative calculus can be considered to define new scientific concepts, to state scientific laws, to formulate and solve new problems. Furthermore, it can provide alternative approaches for thinking about many problems and making decisions in a reasonable way.

References [1] Bashirov A.E., Mısırlı K.E. and Ozyapici A. Multiplicative calculus and its applications. Journal of Mathematical Analysis and Its Applications 337(1)(2008), 36–48. [2] Grossman M. and Katz R. Non-Newtonian Calculus. Pigeon Cove, Mass., Lee Press, 1972. [3] Grossman M. Bigeometric Calculus: A System with a Scale-Free Derivative. Archimedes Foundation, Rockport, Mass., 1983. [4] Stanley D. A Multiplicative Calculus. Primus, IX(4)(1999), 310-326.

Address: Department of Mathematics, Ege University, Izmir-TURKIYE e-mails: [email protected], [email protected]

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Interval-valued L-fuzzy Topological Groups Banu Pazar, Vildan C ¸etkin and Halis Ayg¨ un

In this paper, we introduce the concepts of interval-valued L-fuzzy topological space, interval-valued L-fuzzy subgroup and interval-valued L-fuzzy topological group. We first define interval-valued L-fuzzy set and interval-valued L-fuzzy topology. In the main section, we introduce interval-valued L-fuzzy subgroup and interval-valued L-fuzzy topological group and study some of their properties. Furthermore, we also give the theorems of the homomorphic image and preimage.

References [1] Biswas R. Rosenfeld’s fuzzy subgroup with interval-valued membership functions. Fuzzy Sets and Systems 63(1994), 87-90. [2] Foster D.H. Fuzzy topological groups. J. Math. Anal. Appl. 67(1979), 549-564. [3] Mondal T.K. and Samantha S.K. Topology of interval-valued fuzzy sets. Indian J. Pure Appl. Math. 30(1)(1999), 23-38. [4] Rosenfeld A. Fuzzy groups. J. Math and Appl. 35(1971), 512-517. [5] Xiapping L. and Gujin W. The SH interval-valued fuzzy subgroups. Fuzzy Sets and Systmes 112(2000), 319-325. [6] Zadeh L.A. The concept of a linguistic varible and its application to aproximate reasoning- I. I. Inf. and Control. 8(1975), 199-249.

Address: Kocaeli University, Department of Mathematics, Umuttepe Campus, 41380, Kocaeli-TURKIYE e-mails: [email protected], [email protected], [email protected], [email protected]

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Binomial Thue Equations and their Applications Akos Pinter, Michael Bennett, Kalman Gyory, Lajos Hajdu and Istvan Pink

In this survey talk we present some recent results concerning ternary equations and binomial Thue and Thue-Mahler equations and we give several applications of these theorems to the classical diophantine problems.

The 20 th International Congress of The Jangjeon Mathematical Society

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A Distributional Approach to Classical Electromagnetism I: The Mathematical Tools Burak Polat

In this talk we present the distributional derivatives in space (gradient, divergence, curl and Laplacian) and in time of generalized functions whose singular parts are concentrated on an arbitrary surface, an arbitrary space curve or a point in arbitrary motion. Such generalized functions are described in a SchwartzSobolev space setting and represent arbitrary source or field quantities in the equations of mathematical physics. Their regular component are locally integrable functions in the Lebesgue sense and their singular components are assumed to be constructed via the temporal and spatial (directional) derivatives of the Dirac delta distribution of every order.

References [1] Polat B. A Distributional Approach to Classical Electromagnetism. A series of papers to appear in the Electronic Journal of Generalized Functions. [2] Estrada R. and Kanwal R.P. Higher Order Fundamental Forms of a Surface and Their Applications to Wave Propagation and Distributional Derivatives. Rend.Cir.Mat.Palermo 36(1987), 27-62. [3] Estrada R. and Kanwal R.P. Distributional Analysis of Discontinuous Fields. J.Math.Anal.Appl. 105 (1985), 478-490. [4] Estrada R. and Kanwal R.P. Applications of Distributional Derivatives to Wave Propagation. J.Inst.Math.Appl. 26(1980), 39-63. [5] R˘ adulet¸ R. and Ciric I.R. Generalized Functions in the Theory of Fields. Revue Romaine des Sciences ´ ´ Techniques, S´erie Electrotechnique et Energ´ etique, 16, 4, 1971, pp. 565-591 (Editions de l’Academie de la Republique Socialiste de Roumanie, Bucarest).

Address: Uludag University, Faculty of Science, Department of Mathematics, G¨ or¨ ukle 16059. Bursa-TURKIYE e-mail: [email protected]

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A Distributional Approach to Classical Electromagnetism II: The Physical Evidences Burak Polat

In this continuation talk we illustrate the application of function theoretical techniques introduced in the first talk to the field equations of classical electrodynamics to illuminate the influence of differential geometric quantities on the field behavior on point, space curve and surface type singularity domains. The results cover initial, boundary/continuity, edge and tip conditions for concentrated sources in arbitrary motion and provide the basic tools for a postulation of Maxwell’s field theory.

References [1] Polat B. A Distributional Approach to Classical Electromagnetism. A series of papers to appear in the Electronic Journal of Generalized Functions. [2] Polat B. On Poynting’s Theorem and Reciprocity Relations for Discontinuous Fields. IEEE Antennas and Propagation Magazine, 49(4)(2007), 74-83. [3] Polat B. Remarks on the Fundamental Postulates on Field Singularities in Electromagnetic Theory. IEEE Antennas and Propagation Magazine 47(5)(2005), 47-54. [4] Polat B. A Note Regarding ”Remarks on the Fundamental Postulates on Field Singularities in Electromagnetic Theory”. IEEE Antennas and Propagation Magazine 48(3)(2006), 105. [5] Polat B. Spatiotemporal Coupling in Jump Conditions on a Planar Material Sheet. IVth International Workshop on Electromagnetic Wave Scattering September 18 - 22, 2006 Gebze, Kocaeli, TURKEY. ˙ [6] Idemen M. Universal Boundary Conditions and Cauchy Data for the Electromagnetic Field, in A. Lakhtakia (ed.), Essays on the Formal Aspect of Electromagnetic Theory. Singapore, World Scientific Pub. Co. Ltd, (1993), pp. 657-698. [7] R˘ adulet¸ R. and Ciric I.R. Generalized Functions in the Theory of Fields. Revue Romaine des Sciences ´ ´ Techniques, S´erie Electrotechnique et Energ´ etique, 16, 4, 1971, pp. 565-591 (Editions de l’Academie de la Republique Socialiste de Roumanie, Bucarest).

Address: Uludag University, Faculty of Science, Department of Mathematics, G¨ or¨ ukle 16059. Bursa-TURKIYE e-mail: [email protected]

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Numerical Study of non-Darcy Forced Convective Heat Transfer in a Power Law Fluid over a Stretching Sheet K.V.Prasad and V.Rajappa

An analysis has been carried out to study the non-Darcy flow behavior and heat transfer characteristics of a non-Newtonian power law fluid over a non-isothermal stretching sheet with viscous dissipation and internal heat generation/ absorption. Thermal conductivity is assumed to vary as a linear function of temperature. The partial differential equations governing the flow and heat transfer are converted into ordinary differential equations by a similarity transformation. The presence of non-Darcy forced convection and power law index leads to coupling and non-linearity in the boundary value problem. Because of the coupling and non-linearity, the problem has been solved numerically by Keller box method. The computed values of horizontal velocity and temperature, boundary layer thickness are shown graphically in tables and figures. Several reported works on the problem are obtained as limiting cases of the study. The results of the study have implications in extrusion processes and such other applications with porous media facilitating as a heat-retaining mechanism.

References [1] Andersson H.I. and Dandapat B.S. Flow of a power law fluid over a stretching sheet. Stability Appl.Anal.Continous Media, 1, 339 (1991). [2] Hassanien I.A., Abdullah A.A. and Gorla R.S.R. Flow and heat transfer in a power law fluid over a non-isothermal stretching sheet. Math. Comput. Model, 28,105 (1998). [3] Jadhav B.P. and Waghmode B.B. Heat transfer to non-Newtonian power law fluid past a continuously moving porous flat plate with heat flux. Warme-Und Stoffubertragung 25, 377 (1990). [4] Kambiz V. Hand book of porous media. Marcel Dekker, 2000.. [5] Prasad K.V., Abel M.S. and Khan S.K. Momentum and heat transfer in a visco-elastic fluid flow in a porous medium over a non-isothermal stretching sheet. Int. J. Numer Methods for heat and fluid flow, 10(8), 786 (2000). [6] Sakiadis B.C. Boundary layer behaviour on continuous solid surfaces. A.I.Ch.E.J. 7, 26 (1961). [7] Vajravelu K. Flow and heat transfer in a saturated porous medium over a stretching surface. ZAMM, 74, 605 (1994).

Address: Department of Mathematics, Central College Campus Bangalore University, Bangalore-560 001, INDIA Dean Academics, Cambridge Institute of Technology, K.R.Puram, Bangalore- 560 036-INDIA e-mails: [email protected], [email protected]

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Note on Genocchi Numbers and Polynomials Seog-Hoon Rim, Kyoung Ho Park, Yong Do Lim and Eun Jung Moon

The main purpose of this paper is to study the distribution of Genocchi polynomials. Finally, we construct the Genocchi zeta function which interpolates Genocchi polynomials at negative integers

References [1] Kim T. Euler numbers and polynomials associated with zeta functions. Abstract and Applied Analysis, 2008 (2008), Article ID 581582, 11pages. [2] Kim T. The modified q-Euler numbers and polynomials. Adv. Stud. Contemp. Math. 16(2008), 161170. [3] Cenkci M. and Kurt V. p-adic interpolation function and Kummer type congruence for q-twisted Euler numbers. Adv. Stud. Contemp. Math. 9 (2004), 203-216. [4] Kim T. A note on p-adic q-integral on Zp Associated with q-Euler numbers. Adv. Stud. Contemp. Math. 15(2007), 133-138. [5] Kim T. q-volkenborn Integration. Russ. J. Math. Phys. 9 (2002), 288-299. [6] Ozden H., Simsek Y. and Cangul I.N. A note on p-adic q-Euler measure. Adv. Stud. Contemp. Math. 14(2007), 233-239. [7] Simsek Y. On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers. Russ. J. Math. Phys. 13(2006), 340-348. [8] Schork M. Combinatorial aspects of normal ordering and its connection to q-calculus. Adv. Stud. Contemp. Math. 15(2007), 49-57. [9] Schork M. Ward’s “calculus of sequence”, q-calculus and the limit q → −1. Adv. Stud. Contemp. Math. 13(2006), 131-141. [10] Shiratani K. and YamamotoS. On a p-adic interpolating function for the Euler numbers and its derivatives. Mem. Fac. Sci. Kyushu Univ. Ser. A 39(1985), 113-125.

Address: Department of Mathematics Education, Kyungpook National University, Tagegu 702-701, S.KOREA Department of Mathematics, Kyungpook National University, Tagegu 702-701, S.KOREA Department of Mathematics, Kyungpook National University, Tagegu 702-701, S.KOREA Department of Mathematics, Kyungpook National University, Tagegu 702-701, S.KOREA e-mails: [email protected], [email protected], [email protected], [email protected]

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Generalized Sobolev-Shubin Spaces Ay¸ se Sandık¸ cı and A.Turan G¨ urkanlı

Let w be a Beurling weight function on R2d and 1 ≤ p, q ≤ ∞. In [5] a space M (p, q) Rd is defined and M (p,q) studied some properties of this space. In the present work we define a space Qg,w Rd as counter image of M (p, q) Rd under Toeplitz operator with symbol w. We endow this space with a suitable norm and M (p,q) study some properties. Afterwards we show that Qg,w Rd is a generalization of usual Sobolev-Shubin space Qs Rd . At the end of this work we discuss some of results on boundedness of Toeplitz operator.

References [1] Sandık¸cı A. and G¨ urkanlı A.T. The Space Ωpm Rd and Some Properties. Ukranian Mathematical Journal, 58(1)(2006), 139-145. [2] Reiter H. Classical Harmonic Analysis and Locally Compact Groups. Oxford University Press, Oxford, 1968. [3] Hunt R.A. On L (p, q) Spaces. Extrait de L’Enseignement Mathematique, T.XII, fasc.4(1966), 249-276. [4] Gr¨ ochenig K. Foundation of Time-Frequency Analysis. Birkh¨auser, Boston, 2001. [5] G¨ urkanli A.T. Time Frequency Analysis and Multipliers of the Spaces M (p, q) Rd and S (p, q) Rd . J. Math. Kyoto Univ., 46(3)(2006), 595-616. [6] Feichtinger H.G. and Strohmer T. Gabor Analysis and Algorithms Theory and Applications. Birkhauser, 1998. [7] Edmunds D.E. and Evans W. D. Hardy Operators Function Spaces and Embeddings. Springer Berlin Heidelberg New York, 2004. [8] Boggiatto P., Cordero E. and Gr¨ ochenig K. Generalized anti-weak operators with symbols in distributional Sobolev spaces. Integr. equ. oper. theory 48(2004), 427-442. [9] Boggiatto P. Localization operators with Lp symbols on the modulation spaces. Universita di Torino, Quederni del Dipartimento di Matematica, Quaderno N, 2003. [10] Boggiatto P. and Toft T. Embeddings and compactness for generalized Sobolev-Shubin spaces and modulation spaces. Applicable Analysis, 84(3)(2005), 269-282.

Address: Ondokuz Mayıs University, Faculty of Arts and Science, Department of Mathematics, 55139, Kurupelit Samsun-TURKIYE e-mails: [email protected], [email protected]

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On Tripotency and Idempotency of Some Linear Combinations of Two Commuting Quadripotent Matrices Murat Sarduvan and Halim ¨ Ozdemir

Let Q = c1 Q1 + c2 Q2 , where c1 , c2 are nonzero complex numbers and (Q1 , Q2 ) is a pair of two n × n nonzero commuting quadripotent matrices. Under the condition Q1 Q2 = η1 Q21 + η2 Q22 = Q2 Q1 with some scalars η1 ,η2 C, the problems of characterizing all situations in which the linear cobination matrix Q is a tripotent matrix or an idempotent matrix are considered.

References [1] Baksalary J.K. and Baksalary O.M. Idempotency of linear combinations of two idempotent matrices. Linear Algebra Appl. 321(2000), 3-7. ¨ ¨ [2] Ozdemir H. and Ozban A.Y. On idempotency of linear combinations of idempotent matrices. Appl. Math. Comput. 159(2004), 439-448. ¨ [3] Baksalary J.K., Baksalary O.M., and Ozdemir H. A note on linear combination of commuting tripotent matrices. Linear Algebra Appl. 388(2004), 45-51. ¨ [4] Sarduvan M. and Ozdemir H. On linear combinations of two tripotent, idempotent, and involutive matrices. Appl. Math. Comput. 200(2008), 401-406. ¨ ¨ [5] Ozdemir H., Sarduvan M., Ozban A.Y. and G¨ uler N. On idempotency and tripotency of linear combinations of two commuting tripotent matrices. submitted for publication. [6] Baksalary J.K., Baksalary O.M. and GroßJ. On some linear combinations of hypergeneralized projectors. Linear Algebra Appl. 413(2006), 264-273. [7] Baksalary O.M. and Ben´ıtez J. Idempotency of linear combinations of three idempotent matrices, two of which are commuting. Linear Algebra Appl. 424(2007), 320-337. [8] Horn R.A. and Johnson C.R. Matrix Analysis. Cambridge University Press, Cambridge, UK, 1985.

Address: Sakarya University, Faculty of Arts and Sciences, Department of Mathematics, TR54187. Sakarya-TURKIYE e-mails: [email protected], [email protected]

This work was supported by Sakarya University Scientific Research Projects Committee (No: 2007.50.02.021).

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On Double Lacunary Statistical σ-Convergence of Fuzzy Numbers Ekrem Sava¸ s

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An Excursion into the World of Elliptic Hypergeometric Series Michael J.Schlosser

In this talk, I will give a survey of the rather recent theory of elliptic hypergeometric series while focusing on selected results and aspects. Elliptic hypergeometric series first appeared as elliptic 6j coefficients, which are elliptic solutions of the Yang–Baxter equation, in work by Date, Jimbo, Kuniba, Miwa and Okado in 1987 [1]. Ten years later Frenkel and Turaev [2] discovered the (now called) 12 V11 transformation which came out as a consequence of the tetrahedral symmetry of the elliptic 6j coefficients. By specialization the 12 V11 transformation reduces to a summation formula, the 10 V9 summation, an identity which is fundamental to the theory of elliptic hypergeometric series. To illustrate some of the techniques applicable in the theory of elliptic hypergeometric series, I will present three different proofs(–my favourite ones!) of the 10 V9 summation. The first one uses induction and has been discovered independently by several authors (see [6, Ch. 11]). The second proof uses the combinatorics of lattice paths, which for our purposes are enumerated with respect to an elliptic weight function [4]. The third proof makes uses of an elliptic divided difference operator, the summation is established by means of elliptic Taylor series expansion [5].

References [1] Date E., Jimbo M., Kuniba A., Miwa T. and Okado M. Exactly solvable SOS models: local height probabilities and theta function identities. Nuclear Phys. B 290(1987), 231–273. [2] Frenkel I.B. and Turaev V.G. Elliptic solutions of the Yang–Baxter equation and modular hypergeometric functions. In V.I. Arnold et al. (eds.), The Arnold–Gelfand Mathematical Seminars, 171–204, Birkh¨ auser, Boston, 1997. [3] Gasper G. and Rahman M. Basic hypergeometric series, 2nd ed. Encyclopedia of Mathematics and Its Applications 96, Cambridge University Press, Cambridge, 2004. [4] Schlosser M.J. Elliptic enumeration of nonintersecting lattice paths. J. Combin. Theory Ser. A 114(3) (2007), 505–521. [5] Schlosser M.J. A Taylor expansion theorem for an elliptic extension of the Askey–Wilson operator. Contemp. Math., to appear; preprint arXiv:0803.2329.

Address: University of Vienna, Faculty of Mathematics, Nordbergstraße 15, A-1090 Vienna-AUSTRIA e-mail: [email protected]

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Some Properties of Vague Rings Sevda Sezer

The study of fuzzy algebraic structures was started with the introduction of the concept of fuzzy subgroups by Rosenfeld in 1971. Then, the concept of “vague group” was proposed by Demirci in 1999. Later on, the general theory of vague algebraic notions was established. In this work, using the definition of vague group and its some results, the concepts of vague ring and vague subring are defined, and fundamental properties of these concepts are obtained.

References [1] Demirci M. Vague Groups. Journal of Mathematical Analysis and Applications, 230(1999), 142-156. [2] Demirci M. Fundamentals of M-vague Algebra and M-vague Arithmetic Operations. Int. J. Uncertainty, Fuzziness and knowledge-Based Systems 10(2002), 25-75. [3] Demirci M. Foundations of Fuzzy Functions and Vague Algebra Based on Many-valued Equivalence Relations, Part II: Vague Algebraic Notions. Int. J. General Systems 32(2003), 157-175. [4] Demirci M. Foundations of Fuzzy Functions and Vague Algebra Based on Many-valued Equivalence Relations, Part III: Constructions of Vague Algebraic Notions and Vague Arithmetic Operations. Int. J. General Systems 32(2003), 177-201. [5] Demirci M. and Recasens J. Fuzzy Groups, Fuzzy Functions and Fuzzy equvalence Relation. Fuzzy Sets and Systems 144(2004), 441-458. [6] Rosenfeld A. Fuzzy Groups. Journal of Mathematical Analysis and Applications 35(1971), 512-517. [7] Sezer S. Vague Groups and Generalized Vague Subgroups on the Basis of ([0, 1], ≤, ∧). Information Sciences 174(2005), 123-142.

Address: Department of Mathematics, Faculty of Science, Akdeniz University, Antalya-TURKIYE e-mail: [email protected]

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Beta-Semigroup and Riesz Potentials Sinem Sezer and Ilham A.Aliev Denote β

−1 w(β) (y) = Fξ→y (e−|ξ| )(y) ≡ (2π)−n

Z

β

e−|ξ| eiy·ξ dξ,

Rn −1

where β > 0, y · ξ = y1 ξ1 + · · · + yn ξn , dξ = dξ1 · · · dξn and F is the inverse Fourier transform. We set (β) w(β) (y, t) = t−n/β w(β) (t−1/β y), (t > 0, y ∈ Rn ). The Beta-semigroup (Bt )t>0 generated by the kernel w(β) (y, t) is defined by Z (β) Bt f (x) = w(β) (y, t)f (x − y)dy, (f ∈ Lp (Rn )). Rn

By making use of this Beta-semigroup we introduce the following integral operator Iβα f (x)

1 = Γ(α/β)

Z∞

α

(β)

t β −1 Bt f (x)dt,

(Re α > 0).

(79.1)

0 (β)

For β = 1 and β = 2 Bt f coincides with the Poisson and Gauss-Weierstrass integrals, respectively, and (79.1) gives a semigroup representation of classical Riesz potentials. In this work we obtain explicit inversion formulas for integral operators Iβα f with the aid of some (β)

wavelet-like transform, associated with the Beta-semigroup (Bt )t>0 .

References [1] Aliev I.A., Rubin B., Sezer S. and Uyhan S.B. Composite Wavelet Transforms: Applications and Perspectives. Contemprary Mathematics: Radon Transforms, Geometry and Wavelets (to appear).

Address: Akdeniz University, Faculty of Education, 07058 Antalya-TURKIYE e-mails: [email protected], [email protected]

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Principle of Local Conservation of Energy-Momentum Garret Sobczyk and Tolga Yarman

Starting with Einstein’s theory of special relativity and the principle that whenever a celestial body or an elementary partical, subjected only to the fundamental forces of nature, undergoes a change in its kinetic energy then the mass-energy equivalent of that kinetic energy must be subtracted from the restmass of the body or particle, we derive explicit equations of motion for two falling bodies. In the resulting mathematical theory we find that there are no singularities and consequently no blackholes.

References [1] Einstein A., Lorentz H.A., Minkowski H. and Weyl H. On The Electro-dynamics of Moving Bodies, in The Principle of Relativity. Translated from “Zur Elektrodynamik bewegter K¨orper”, Annalen der Physik 17(1905), Dover Pablications, Inc. (1923). [2] Einstein A. The Meaning of Relativity. Princeton University Press, 1953. [3] Yarman T. The general equation of motion via special theory of relativity and quantum mechanics. Annales Fondation Louis de Broglie 29(3)(2004), 459-491. [4] Yarman T. The End Results of General Relativity Theory via just Energy Conservation and Quantum Mechanics. Foundations of Physics Letters 19(7)(2006), 675-694. [5] Sobezyk G. Geometry of Moving Planes. Submitted to American Mathematical Monthly, May 2008. (http://arxiv.org/PS cache/arxiv/pdf/0710/0710.0092v1.pdf) [6] Sobezyk G. Spacetime Vector Analysis. Physics Letters 84A(1981), 45-49. [7] Einstein A., Podolsky B. and Rosen N. Can quatum-mechanical description of physical reality be considered complete? Phys. Rev. 47(1935), 777.

Address: Universidad de Las Am´ ericas-Puebla, Departamento de Actuar´ıa, F´ısica y Matem´ aticas, 72820 Cholula, Puebla-MEXICO Okan University, Akfirat, Istanbul-TURKIYE e-mails: [email protected], [email protected]

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Concerning Fundamental Mathematical and Physical Defects in the General Theory of Relativity Garret Sobczyk, Stephen J.Crothers and Tolga Yarman

The physicists have misinterpreted the quantity ‘r’ appearing in the so-called “Schwarzschild solution” as it is neither a distance nor a geodesic radius but is infact the inverse square root of the Gaussian curvature of a spherically symmetric geodesic surface in the spatial section of the Schwarzschild manifold, and so it does not directly determine any distance at all in the Schwarzschild manifold - in other words, it determines the Gaussian curvature at any point in the spatial section of the manifold. The concept of the black hole is consequently invalid. It is also shown herein that the Theory of Relativity forbids the existence of point-mass singularities because they imply infinite energies (or equivalently, that a material body can acquire the speed of light in vacuo) and that Ric = Rµν = 0 violates Einstein’s ‘Principle of Equivalence’ and so does not describe Einstein’s gravitational field. It immediately follows that Einstein’s conceptions of the conservation and localisation of gravitational energy are invalid - the General Theory of Relativity violates the usual conservation of energy and momentum.

References [1] Levi-Civita T. The Absolute Differential Calculus. Dover Publications Inc., New York, 1977. [2] Tolman R.C. Relativity Thermodynamics and Cosmology. Dover Publications Inc., New York, 1987. [3] Abrams L.S. Black holes: the legacy of Hilbert’s error. Can. J. Phys. 67(1989), 919, arXiv:grqc/0102055, www.sjcrothers.plasmaresources.com/Abrams1989.pdf [4] Antoci S. David Hilbert and the origin of the “Schwarzschild” solution. 2001, arXiv: physics/0310104. [5] Loinger A. On black holes and gravitational waves. La Goliardica Paves, Pavia, 2002. [6] Mould R.A. Basic Relativity. Springer–Verlag New York Inc., New York, 1994. [7] Dodson C.T.J. and Poston T. Tensor Geometry-The Geometric Viewpoint and its Uses. 2nd Ed. Springer–Verlag, 1991. [8] Bruhn G.W. (public communication), www.sjcrothers.plasmaresources.com/BHLetters.html. [9] Wald R.M. General Relativity. The University of Chicago Press, Chicago, 1984. [10] Carroll B.W. and Ostile D.A. An Introduction to Modern Astrophysics. Addison–Wesley Publishing Company Inc., 1996.

Address: Universidad de Las Am´ ericas-Puebla, Departamento de Actuar´ıa, F´ısica y Matem´ aticas, 72820 Cholula, Puebla-MEXICO Okan University, Akfirat, Istanbul-TURKIYE Queensland-AUSTRALIA e-mails: [email protected], [email protected], [email protected]

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The Diophantine Equation x2 + 11m = y n G¨ okhan Soydan, Musa Demirci and Ismail Naci Cang¨ ul

The object of this talk is to give a new proof of the following Theorem. Let m be odd. Then the Diophantine equation x2 + 11m = y n , m > 1, n ≥ 3 has only one solution in positive integers x, y and the unique solution is given by m = 6M + 3, x = 9324.113M , y = 443.112M and n = 3.

References [1] Lebesgue V.A. Sur I’impossibilit´e en nombres entieres de I’l equation xm = y 2 + 1. Nouvelles Ann. des. Math. 9(1)(1850), 178-181. [2] Nagell T. Contributions to the theory of a category of diophantine equations of the second degree with two unknown. Nova Acta Reg. Soc. Upsal. Ser. 4(16)(1955), 1-38. [3] Cohn J.H.E.The Diophantine equation x2 + C = y n . Acta Arith. 65(4)(1993), 367-381. [4] Arif S.A. Muriefah F.S. Abu, The Diophantine equation x2 +3m = y n . Int. J. Math. Math. Sci. 21(1998), 619-620. [5] Arif S.A. and Muriefah F.S. Abu. The Diophantine equation x2 + 52k+1 = y n . Indian J. Pure Appl. Math. 30(1999), 229-231. [6] Arif S.A. and Muriefah F.S. Abu. On the Diophantine equation x2 + q 2k+1 = y n . J. Number Theory 95(2002), 95-100.

Address: Uludag University, Faculty of Arts and Science, Department of Mathematics, G¨ or¨ ukle 16059. Bursa-TURKIYE e-mails: [email protected], [email protected], [email protected]

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Fuzzy Triangular Inequality G¨ ultekin Soylu

Although there are several approaches to fuzzied metric definitions in the literature, they are missing the triangular inequality property in the sense that their defuzzifications do not give a metric. In this speech we introduce the definition of the Fuzzy Triangular Inequality and show that it is possible to derive E-vague Absolute Value functions which hold the fuzzy triangular inequality. These functions will play the key role in the construction of fuzzy metric functions possessing the triangular inequality property and therefore giving a metric when defuzzified.

References [1] Bodenhofer U. A Similarity-Based Generalization of Fuzzy Orderings Preserving the Classical Axioms. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8(3) (2000), 593-610. [2] Bodenhofer U. Representations and Constructions of Similarity-Based Fuzzy Orderings. Fuzzy Sets and Systems 137 (2003), 113-136. [3] George A. and Veeramani P.V. On some results in Fuzzy Metric Spaces. Fuzzy Sets and Systems 90 (1994), 395-399. [4] Kaleva O. and Seikkala S. On fuzzy metric spaces. Fuzzy Sets and Systems 12(1984), 215–229. [5] Klement E.P., Mesiar R. and Pap E. Triangular Norms. Volume 8 of Trends in Logic. Kluwer Academic Publishers, Dordrecht, 2000. [6] Koblitz N. A Course in Number Theory and Cryptography. Springer-Verlag, 1994. [7] Kramosil I. and Michalek J. Fuzzy metric and statistical metric spaces. Kybernetika 11(1975), 326–334. [8] Soylu G. E-Vague Positive Class. To be printed in Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems. [9] Soylu G. E-Vague Absolute Value. To be appear in Phd. Thesis. [10] Zadeh L. A. Similarity Relations and Fuzzy Orderings. Inform. Sci. 3(1971), 177-200.

Address: University of Akdeniz Faculty of Art and Science Department of Mathematics 07058 Antalya-TURKIYE e-mail: [email protected]

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Bounds for Classical Orthogonal Polynomials and Related Special Functions H.M.Srivastava

The main object of this talk is to present several bounding inequalities for the classical Jacobi function of the first kind. A number of closely-related inequalities for such other special functions as the classical Laguerre function are also considered.

Address: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W, 3R4-CANADA e-mail: [email protected] http://www.math.uvic.ca/faculty/harimsri/

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Some Glimpses of Hindu (or Vedic) Mathematics and Srinivasa Ramanujan (1887–1920) Rekha Srivastava

The main purpose of this talk is to present a brief introduction to ancient Hindu (or Vedic) mathematics. Owing mainly to the lack of authentic documentation and records, not much is widely known about the development of ancient Hindu (or Vedic) mathematics. Nevertheless, judging by the earliest history preserved in the 5000-year-old ruins of a city at Mohen-jo-Daro, there is suffient evidence to indicate the existence of a civilization which is at least as old as that found anywhere else in the ancient orient. These early peoples had their own system of writing, counting, weighing, and measuring, and they are known to have dug canals for irrigation purposes. All of these endeavors obviously required considerable knowledge of basic mathematics and engineering. This talk will also touch upon the life and mathematical contributions of the most spectacular Indian mathematician of modern times, Srinivasa Ramanujan (1887– 1920), an untrained genius who possessed the amazing ability to see quickly and deeply into intricate number relations.

Address: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V 8W 3R4-CANADA e-mail: [email protected] http://www.math.uvic.ca/other/rekhas/

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On p-adic q-Dedekind Sums Yılmaz S ¸im¸ sek

In this paper, we briefly give history of Dedekind sums. We also give some fundamental properties of this sums. By using p-adic q-Volkenborn integral, generating functions of Bernoulli numbers and Mellin transformation, we give q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function. In particular, by using p-adic q-Volkenborn integral, we give p-adic q-Dedekind type sums. Furthermore, p-adic interpolation function of the q-Dedekind type sums are studied. Finaly, we give some relations related to the Dedekind sums and the above functions.

References [1] Apostol T.M. Theorems on generalized Dedekind sums. Pacific J. Math. 2(1952), 1-9. [2] Bayad A. Sommes elliptiques multiples d’Apostol-Dedekind-Zagier (Multiple elliptic Apostol-DedekindZagier sums). C.R.Math. Acad. Sci. Paris 339(7)(2004), 457–462. [3] Berndt B.C. Generalized Dedekind eta functions and generalized Dedekind sums. Trans. Amer. Math. Soc., 178(1973), 495-508. [4] Can M., Cenkci M., Kurt V. and Simsek Y. Twisted Dedekind Type Sums Associated with Barnes, Type Multiple Frobenius-Euler l-Functions. arXiv:0711.0579v1 [math.NT] http://arxiv.org/abs/0711. 0579. [5] Kim T. On a q-Analogue of the p-adic log Gamma functions and related Integrals. J. Number Theory, 76(1999), 320-329. [6] Kim T. A note on p-adic q-Dedekind sums. C.R.Acad. Bulgare Sc., 54(10)(2001), 37-42. [7] Kurt V. On Dedekind Sums. Indian J. Pure Appl. Math., 21(10)(1990), 893-896. [8] Simsek Y. q-Dedekind type sums related to q-zeta function and basic L-series. J. Math. Anal. Appl. 318(1)(2006), 333-351. [9] Simsek Y. Generalized Dedekind sums associated with the Abel sum and the Eisenstein and Lambert series. Adv. Stud. Contemp. Math., 9(2)(2004), 125-137. [10] Srivastava H.M., Kim T. and Simsek Y. q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russ. J. Math Phys. 12(2)(2005), 241-268. [11] Srivastava H.M. and Choi J. Series Associated with the Zeta and Related Functions. Kluwer Acedemic Publishers, Dordrecht, Boston and London, 2001. [12] Srivastava H.M. and Pinter A. Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett. 17(4)(2004), 375-380. [13] Sitaramachandrarao R. Dedekind and Hardy Sums. Acta Arith., XLVIII(1978), 325-340. [14] Zhang W. On the mean values of Dedekind Sums. J. Theorie Nombres de Bordeaux, 8(1996), 429-442.

Address: University of Akdeniz Faculty of Art and Science Department of Mathematics 07058 Antalya-TURKIYE e-mail: [email protected]

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On Quadratic Ideals and Indefinite Quadratic Forms Ahmet Tekcan, Osman Bizim and Bet¨ ul Gezer

Let P and Q be two positive integers such that Q is odd and P is even, let D = P 2 + Q2 be √ a positive non-square integer. In this work, we consider some properties of quadratic irrationals γ = P +Q D , √ quadratic ideals Iγ = [Q, P + D] and indefinite quadratic forms Fγ (x, y) = Qx2 + 2P xy − Qy 2 of discriminant ∆ = 4D. We prove that Iγ is reduced and ambiguous, so is Fγ . We also prove that the cycle of Fγ can be derived by using the cycle of Iγ . Further we determine the order of the proper and improper automorphisms groups Aut(Fγ )+ and Aut(Fγ )− of Fγ , respectively.

References [1] Buchmann J. and Vollmer. U. Binary Quadratic Forms: An Algorithmic Approach. Springer-Verlag, Berlin, Heidelberg, 2007. [2] Buell. D.A. Binary Quadratic Forms, Clasical Theory and Modern Computations. Springer-Verlag, New York, 1989. [3] Flath D.E. Introduction to Number Theory. Wiley, 1989. [4] Mollin R.A. Quadratics. CRS Press, Boca Raton, New York, London, Tokyo, 1996. [5] Mollin R.A. and Cheng K. Palindromy and Ambiguous Ideals Revisited. Journal of Number Theory 74(1999), 98-110. [6] Tekcan A. and Bizim O. The Connection Between Quadratic Forms and the Extended Modular Group. Mathematica Bohemica 128(3)(2003), 225-236.

Address: Uludag University, Faculty of Science, Department of Mathematics, G¨ or¨ ukle 16059. Bursa-TURKIYE e-mails: [email protected], [email protected], [email protected] http://matematik.uludag.edu.tr/AhmetTekcan.htm

This work was supported by The Scientific and Technological Research Council of Turkey, project no: 107T311.

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Improved Direct and Inverse Theorems of Approximation Theory in the Morrey-Smirnov Classes Defined on the Complex Plane N.Pınar Tozman and Daniyal M.Israfilov

Let Γ be a rectifiable Jordan curve in the complex plane C. The Morrey spaces Lp,α (Γ), for a given 0 ≤ α ≤ 2 and p ≥ 1, we define as the set of functions f ∈ Lploc (Γ) such that

kf kLp,α (Γ) :=

sup B

Z

1 1− α 2

|B ∩ Γ|

B∩Γ

1/p p |f (z)| |dz| < ∞,

where the supremum is taken over all balls of C. Under this definition Lp,α (Γ) becomes a Banach space; for α = 2 coincides with Lp (Γ) and for α = 0 with L∞ (Γ). Denoting G := int Γ, we define the Morrey-Smirnov classes E p,α (G), 0 ≤ α ≤ 2 and p ≥ 1, of analytic functions in G as E p,α (G) := f ∈ E 1 (G) : f ∈ Lp,α (Γ) . In this talk, we discuss some improved direct and inverse problems of approximation theory in the Morrey-Smirnov classes defined on the complex plane C.

References [1] Duoandikoetxea J. Weights for maximal functions and singular integrals. NCTH 2005 Summer School on Harmonic Analysis in Taiwan. [2] Israfilov D.M. and Guven A. Improved Inverse Theorems in Weighted Lebesgue and Smirnov Spaces. Bull. Belg. Math. Soc. Simon Stevin 14(2007), 681-692.

Address: Balikesir University, Faculty of Art and Science, Department of Mathematics, 10145, Balikesir-TURKIYE e-mails: [email protected], [email protected]

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On q−Laplace Type Integral Operators and their Applications Faruk U¸ car and Durmu¸ s Albayrak

In this talk, we consider q−Laplace type integral transform. In the classical analysis, a ParsevalGoldstein type theorem involving the Laplace transform, the Fourier transform, the Stieltjes transform, the Glasser transform, the Mellin transform, the Hankel transform, the Widder-Potential transform and their applications are used widely in several branches of engineering and applied mathematics. Since the Laplace transform is useful for the differential equations solution, the Laplace transform has an important role among these transforms. Some integral transforms in the classical analysis have q-analogues in the theory of q-calculus. Some relationships between q-Laplace transform and q-Potential transform are given. Some examples are also given as illustrations of the results presented here.

References [1] Hahn W. Beitrage Zur Theorie der Heineschen Reihen, die 24 Integrale der hypergeometrischen qDiferenzengleichung, das q-Analog on der Laplace Transformation. Math. Nachr., 2(1949),340-379. [2] Abdi, W.H., On q-Laplace Transforms, Proc. Nat. Acad. Sci. India 29(1961), 389-408. [3] Thomae J. Beitrage zur Theorie der durch die Heinesche Reihe. J. Reine angew.Math 70(1869), 258– 281. [4] Jackson F.H. On q-definite Integrals. Quarterly J. Pure and Appl. Mathemathics 41(1910), 193-203. [5] Widder D.V. A transform related to the Poisson integral for a half-plane. Duke Math. J. 33(1966), 355–362. [6] Gasper G. and Rahman M. Generalized Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990). [7] Y¨ urekli O. Theorems on L2 -transforms and its applications. Complex Variables Theory Appl. 38 (1999), 95–107. [8] Y¨ urekli O. New identities involving the Laplace and the L2 -transforms and their applications. Appl. Math. Comput. 99(1999), 141–151. [9] Koornwinder T.H. Special functions and q-commuting variables, in Special functions, q-series and related topics. (Toronto,ON,1995), volume 14of Fields Inst. Commun., pp. 131–166, Amer.Math.Soc., Providence, RI, 1997. [10] Koelink H.T. and Koornwinder T.H. Q-special functions, a tutorial, in Deformation theory and quantum groups with applications to mathematical physics. (Amherst,MA,1990), volume134 of Contemp. Math., pp141–142, Amer.Math.Soc., Providence, RI, 1992. [11] Kac V.G. and Cheung P. Quantum Calculus. Universitext, Springer-Verlag, New York, 2002. [12] Kac V.G. and Alberto De Sole. On Integral representations of q-gamma and q-beta functions. Rend. Mat. Acc. Lincei 9(2005), 11-29. math.QA/0302032. [13] K¨ urem G. and Vula¸s B. Q-Laplace Transforms. (Manuscript). Address: Marmara University, Faculty of Arts and Sciences, Department of Mathematics, G¨ oztepe 34722, ˙ Kadık¨ oy-Istanbul-TURKIYE e-mails: [email protected], [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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q-Laplace Transforms Burcu Vula¸ s and G¨ ulsen K¨ urem

Hahn [4] defined two q-analogues of the well-known Laplace transform by the help of the Jackson integral, q Ls f

(x)

=

q Ls f

(x)

=

Z s−1 1 Eq (qsx) f (x) dq x 1−q 0 Z ∞ 1 eq (−sx) f (x) dq x. 1−q 0

In this paper, firstly convolutions of some functions are calculated and α−1 f (x) = sq Ls xα q Ls x

∞ X

q αk f q k x , α > 0

k=0

is proved via q-convolution theorem Hahn [4] proved q-convolution theorem for q Ls transform and gave some convolution examples. Abdi [2] studied some properties of q-Laplace transforms. q-Laplace transforms of some elementary functions are obtained by some different ways. Next, Z∞ α−1 1 Γq (α) (1 − q) α α−1 Ls f (x) = f (x) dq x q St f (x) =q Lt sq α 1−q (t + x)q 0

is defined and some properties are obtained. Finally, the derivation formulae n (n − 1) X m (m−j) f (0) s D q m n−m n (m) (n) m 2 sj−1 (q − 1) Dqs s ϕ − q Ls x Dq f (x) = (−1) q j−m qn (1 − q) j=1 is proved.

References [1] Abdi W.H. On Certain q-Difference Equations and q-Laplace Transform. Proc. Nat. Acad. Sci. India A, 1962, 1-15. [2] Abdi W.H. On q-Laplace Transforms. Proc. Nat. Acad. Sci. India, 29, 1961, 389-408. [3] Gasper G. Lecture Notes For An Introductory Minicourse on q-Series. 1995. [4] Hahn W. Beitr¨ age zur Theorie der Heineschen Reihen. Math. Nachr. 2(1949), 340-379. ˙ Address: Marmara University, Faculty of Science, Department of Mathematics, Kadık¨ oy, Istanbul-TURKIYE emails: b [email protected], [email protected]

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Relative Defect and Multiple Common Roots of Two Meromorphic Functions Harina P.Waghamore

The concept of absolute defect of ‘a’ with respect to the derivative f 0 was introduced by H. Milloux [2]. This definition was later extended by Xiong Qing-Lai [4]. He introduced the term. δr(k) (a, f ) = 1 − lim sup r→∞

N (r, 1/f (k) − a) T (r, f )

and called it as the relative defect of the value ‘a’ with respect to the derivative f (k) the suffix “r” in the left hand side has nothing to do with of the right hand side. It is just to denote the term “relative” and the usual defect of ‘a’ with respect to f (k) viz. δa(k) (a, f ) = 1 − lim sup r→∞

N (r, 1/f (k) − a) T (r, f (k) )

was called the absolute defect of the value ‘a’ with respect to f (k) .In this paper he found various relations between these two defects. Later A.P.Singh [3] defined the relative defects corresponding to distinct zeros and distinct poles and found various relations between these. In the present paper, we shall consider two different meromorphic functions having common roots and find some relations involving the relative defects.

References [1] Hayman W.K. Meromorphic functions. Oxford University Press, London, 1964. [2] Milloux H. Lee. d´ eriv´ ees des fonctions m´ eromorphes et la th´ eorie des d´ ef auts. Ecole Normale Superieure 63(3)(1946), 289-316. [3] Singh A.P. Relative defects of meromorphic functions. Journal of the Indian Math. Soc 44(1979), 191202. [4] Xiong Qing-Lai. A fundamental inequality in the theory of meromorphic functions. Chinese Mathematics 9(1)(1967), 146-167.

Address: Department of Mathematics, Central College Campus, Bangalore University, Bangalore-560001-INDIA e-mail: [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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The Quantum Mechanical Mechanism Behind the end Results of the GTR: Matter is Built on the Lorentz Invariant Framework Energy x Mass x Length2 ∼ h2 Tolga Yarman

In a previous article, we have provided a whole new approach toward the end results of the general theory of relativity (GTR), based on just the energy conservation law, in the broader sense of the concept of “energy” embodying the mass & energy equivalence of the special theory of relativity. Thus, our approach was solely based on this latter theory (excluding the necessity of assuming the principle of equivalence of Einstein). According to our approach, the rest mass of an object embedded in a gravitational field (in fact in any field the object interacts with) decreases as much as the binding energy coming into play. Thereby, based on a general quantum mechanical theorem we prove, its internal energy, weakens as much; thus the classical red shift and time dilation. This theorem (we did not have any room to provide a general proof of, previously), basically says that, if in a relativistic or non-relativistic quantum mechanical description, composed properly, the mass of the object in hand is multiplied by an arbitrary number γ, then the total energy of it, is multiplied by γ, and its size is divided by γ. This number however may very well not be arbitrary. For example, it would specify how much the rest mass of the object is altered when this is embedded in a gravitational field, leading via quantum mechanics, strikingly at once, to the end results of the GTR. This manipulation further yields the invariance of the quantity [energy x mass x size2]. We conclude that, it is this quantum mechanical invariance, necessarily strapped to the square of the Planck Constant, which constitutes a given framework regarding the matter architecture, and insures the end results of the GTR. Not only that our approach is incomparably simple as compared to the GTR, but it also avoids all incompatibilities (such as the breaking of the relativistic relationship E = mc2 ), or inconsistencies (such as the breaking of the energy conservation law, as well as the breaking of momentum conservation law), or blockades (such as the impossibility of the quantization of the gravitational field), thus opens a whole clean avenue toward a unification of fields, and understanding of the matter and the universe at all levels, with just the same set of tools. Since we do not have to use the principle of equivalence of the GTR, amongst others, we could show that, just like the gravitational field, the electric field too slows down the internal mechanism of a clock, had this interacted with the field. This result explains substantially, the retardation of the decay of the muon, bound to a nucleus.

References [1] Yarman T. and Rozanov V. The Mass Deficiency Correction To Classical And Quantum Mechanical Descriptions: Alike Metric Change And Quantization Nearby An Electric Charge, And A Celestial Body, Physical Interpretations of the Theory of Relativity. Bauman Moscow State Technical University, 4 – 7 July 2005. Also, Paper in Press in the International Journal of Computing Anticipatory Systems. [2] Yarman T. The General Equation of Motion via the Special Theory of Relativity and Quantum Mechanics. Annales de La Fondation Louis de Broglie 9(3)(2004), 459. http://www.ensmp.fr/aflb/AFLB293/aflb293m137.htm. [3] Yarman T. The End Results of General Theory of Relativity, Via Just Energy Conservation And Quantum Mechanics. Foundations of Physics Letters 19, December 2006. Address: Okan University, Akfirat, Istanbul-TURKIYE e-mail: [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

93

Some Applications of He’s Variational Approaches Ahmet Yıldırım

In this paper, He’s variational method is applied to certain nonlinear oscillators with strong nonlinearity. The method is of deceptively simplicity and the insightful solutions obtained are of high accuracy even for the first-order approximations.

References [1] He JH. Non-Perturbative Methods for Strongly Nonlinear Problems. Dissertation.de- Verlag im Internet GmbH, 2006. [2] He JH. Some asymptotic methods for strongly nonlinear equations. International Journal of modern Physics B 20(10)(2006), 1141-1199. [3] He JH. Limit cycle and bifurcation of nonlinear problems. Chaos Solitons & Fractals 26(3)(2005), 827-833. ¨ [4] Ozis T. and Yıldırım A. A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity. Chaos Solitons & Fractals 34(3)(), 989-991. ¨ [5] Ozis T. and Yıldırım A. Determination of limit cycles by a modified straightforward expansion for nonlinear oscillators. Chaos Solitons & Fractals 32(2007), 445-448.

Address: Ege University, Faculty of Science, Department of Mathematics, Bornova 35100, Izmir-TURKIYE e-mail: [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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An Efficient Method for Solving Singular Two-Point Initial Value Problems Ahmet Yıldırım and Deniz A˘ gırseven

In this paper, a class of nonlinear singular two-point initial value problems are solved by homotopy perturbation method. The approximate solution of this problem is calculated in the form of series with easily computable components. Finally, we give some numerical examples to show efficiency and simplicity of the method.

References ¨ s T. and Yıldırım A. A note on He’s homotopy perturbation method for van der Pol oscillator with [1] Ozi¸ very strong nonlinearity. Chaos,Solitons & Fractals 34(3)(2007), 989-991. ¨ s T. Solutions of Singular IVPs of Lane-Emden type by Homotopy Perturbation [2] Yıldırım A. and Ozi¸ Method. Physics Letters A 369(1-2)(2007), 2), 70-76. [3] Yıldırım A. Exact solutions of Nonlinear Differential-difference Equations by Homotopy Perturbation Method. Int. J. Nonlinear Sci. Num. Simulation 9(2)(2008), 111-114. ¨ s T. and Yıldırım A. Travelling Wave Solution of Korteweg-de Vries Equation using He’s Homotopy [4] Ozi¸ Perturbation Method. International Journal of Nonlinear Sciences and Numerical Simulation 8(2)(2007), 239-242. [5] He J.H. Homotopy Perturbation Method:A New Nonlinear analytical technique. Applied Mathematics and Computation 135(2003), 73-79. [6] He J.H. Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering 178(1999), 257-262. [7] He J.H. A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-linear Problems. International Journal of Non-linear Mechanics 35(2000), 37-43.

˙ Address: Department of Mathematics, Science Faculty, Ege University, 35100 Bornova, Izmir-TURKIYE Department of Mathematics, Science and Art Faculty, Trakya University, 22030 Edirne-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

Holditch Theorem for the Closed Space Curves in Lorentzian 3-space Handan Yıldırım, Salim Y¨ uce and Nuri Kuruo˘ glu

In this study, firstly we give the area of the closed projection curve of a closed space curve in Lorentzian space L3 . Secondly, during the 1-parameter closed motion in L3 , we obtain the Holditch Theorem for the closed space curves by using their orthogonal projection onto the Euclidean plane in the fixed space. Moreover, we generalize the Holditch Theorem for noncollinear three fixed points in L3 .

References [1] Greub W.H. Linear Algebra, 3rd ed. Springer-Verlag and Academic Press, New York 1967. [2] Dzan J.J. Trigonometric Laws on Lorentzian Sphere S12 . Journal of Geometry 24(1985), 6-13. [3] Holditch H.Geometrical Theorem. Q. J. Pure Appl. Math. 2(1858), 38. [4] M¨ uller H.R. Erweiterung des Satzes von Holditch f¨ ur geschlossene Raumkurven. Abh. Braunschw. Wiss. Ges. 31(1980), 129-135. [5] M¨ uller H.R. Ein Holditch-Satz f¨ ur Fl¨ achenst¨ ucke im R3 . Abh. Braunschw. Wiss. Ges. 39(1987), 37-42. [6] O’ Neill B.Semi-Riemannian Geometry. Academic Press, New York 1983.

˙ Address: University of Istanbul, Faculty of Science, Department of Mathematics, Vezneciler, 34134, Istanbul-TURKIYE ˙ Yıldız Technical University, Faculty of Arts and Science, Department of Mathematics, Esenler, 34210, Istanbul-TURKIYE University of Bah¸ce¸sehir, Faculty of Arts and Science, Department of Mathematics Computer Sciences, Be¸sikta¸s, 34100, ˙ Istanbul-TURKIYE e-mails: [email protected], [email protected], [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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On the Numerical Solutions of Hyperbolic Equations with Nonlocal Boundary and Neumann Conditions ¨ Ozg¨ ur Yıldırım and and Allaberen Ashyralyev

In this talk, the first and second order of accuracy difference schemes for the numerical solutions of the hyperbolic equations with nonlocal boundary and Neumann conditions are presented. The stability estimates for the solutions of the difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of a one-dimensional hyperbolic equations.

References [1] Ashyralyev A. and Aggez N. A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations. Numerical Functional Analysis and Optimization 25(5-6)(2004), 1-24. [2] Ashyralyev A., Martinez M., Paster J. and Piskarev S. Weak maximal regularity for abstract hyperbolic problems in function spaces. Abstracts of 6-th International ISAAC Congress, Ankara, Turkey, p. 90., 2007 [3] Ashyralyev A. and Yildirim O. On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations. Taiwanese Journal of Mathematics, 22p., accepted, 2008. [4] Ashyralyev A. and Sobolevskii P.E. A note on the difference schemes for hyperbolic equations. Abstract and Applied Analysis. 6(2)(2001), 63-70. [5] Fattorini H.O. Second Order Linear Differential Equations in Banach Space. North-Holland: Notas de Matematica, 1985. [6] Sobolevskii P.E. Difference Methods for the Approximate Solution of Differential Equations. Izdat. Voronezh. Gosud. Univ., Voronezh, (Russian), 1975.

˙ Address: Fatih University, Faculty of Science, Department of Mathematics, B¨ uy¨ uk¸cekmece 34500, Istanbul-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

The Estimation of Mean Modulus of Smoothness in Lpw Yunus Emre Yildirir and Daniyal Israfilov

In this talk, we obtain inverse type inequalities for 2π-periodic functions and their derivatives in terms of r-th mean modulus of smoothness. Our new results are the following. Theorem 1. Let f ∈ Lpw (T), 1 < p < ∞, and w ∈ Ap (T). Then for a given r ∈ Z+ , and γ = min {2, p} Ωr (f, π/(n + 1))Lpw

c ≤ (n + 1)r

n X

!1/γ (k +

1)rγ−1 Ekγ (f )Lpw

k=0

with a constant c independent of n and f. Corollary 1. Let f ∈ Lpw (T), 1 < p < ∞, and w ∈ Ap (T). If ∞ X

k αγ−1 Ekγ (f )Lpw < ∞

k=1 α and for n = 0, 1, 2, ... for some natural number α and γ = min {2, p} , then f ∈ Wp,w !1/γ !1/γ n ∞ X X c (α+r)γ−1 γ αγ−1 γ p p (k + 1) E (f ) + k E (f ) Ωr (f (α) , π/(n + 1))Lpw ≤ L L k k w w (n + 1)r k=0

k=n+1

with a constant c independent of n and f. Corollary 2. If En (f )

Lp w

=O

1 nr+α

then for γ = min {2, p} 1/γ

Ωr (f

(α)

, π/(n + 1))Lpw = O

(ln n) (n + 1)r

! .

References [1] Ky N.X. Moduli of mean smoothness and approximation with Ap -weights. Annales Univ. Sci. Budapest 40(1997), 37-48. [2] Kokilashvili V.M. and Yildirir Y.E. On the approximation in weighted Lebesgue spaces. Proceedings of A. Razmadze Math. Inst. 143(2007), 103-113.

Address: Balikesir University, Faculty of Education, Department of Mathematics, 10100 Balikesir-TURKIYE e-mails: [email protected], [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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Differential Transform Method (DTM) for Solving Sine-Gordon Type Equations Eda Y¨ ul¨ ukl¨ u and Turgut ¨ Ozi¸ s

In this paper, the differential transform method (DTM) is applied to solve various forms of Sine-Gordon type equations. The application of differential transform method is extended to derive approximate analytical solutions of Sine-Gordon type equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The results show that the approach is easy to implement and accurate when applied to Sine-Gordon type equations. The method introduces a promising tool for solving many linear and nonlinear differential equations.

References [1] Chen C.K. and Ho S.H. Solving partial differential equations by two dimensional differential transform. Appl. Math. Comput. 106(1999), 171-179. [2] Jang M.J., Chen C.L. and Liu Y.C. Two-dimensional differential transform for partial differential equations. Appl. Math. Comput. 121(2001), 261-270. [3] Wazwaz A.M. Exact solutions to nonlinear diffusion equations obtained by the decomposition method. Appl. Math. Comput. 123(2001), 109-122. [4] Chen C.K. and Ho S.H. Application of differential transformation to eigenvalue problems. Appl. Math. Comput. 147(1996), 173-188. [5] Bildik N., Konuralp N., Orakcı A.B.F. and K¨ u¸cu ¨karslan S. Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method. Appl. Math. Comput. 172(2006), 551-567. [6] Chen C.L. and Liu Y.C. Solution of two-boundary-value problems using the differential transformation method. J.Optim. Theory Appl. 99(1998), 23-35. [7] Abdel-Halim Hassan I.H. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos, Solitons and Fractals 36(2008), 53-65. [8] Ayaz F. On two-dimensional differential transform method. Appl. Math. Comput. 143(2003), 361-374. [9] El-Sayed S.M. The decomposition method for studying the Klein-Gordon equation. Chaos, Solitons & Fractals 18(2003), 1025-30. [10] Jang M.J., Chen C.L. and Liu Y.C. On solving the initial value problems using the differential transformation method. Appl. Math. Comput. 115(2000), 145-60.

˙ Address: Ege University, Science Faculty, Department of Mathematics, 35100 Bornova Izmir-TURKIYE e-mails: [email protected], [email protected]

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The 20 th International Congress of The Jangjeon Mathematical Society

Hardy Littlewood and Polya Inequalities and their Applications to Various Integral Transforms Osman Y¨ urekli

In this talk we present a theorem on a certain class of integral transforms of Hardy, Littlewood and Polya. We define the modified generalized Stieltjes transform and give a generalization of Hilbert’s inequality for the Stieltjes transform to the modified generalized Stieltjes transform. We also present inequalities of Hardy and Littlewood for the Riemann-Liouville and Weyl fractional integral and generalize them to the Erdelyi-Kober fractional integrals. Furthermore we present an inequality of Hardy for the classical Laplace transform. We generalize these inequalities to the L2 -transform and the K-transform. Applications of these results to various integral transforms including the Widder transform and Bessel transform will also be discussed.

References [1] Hardy G.H. and Littlewood J.E. Some properties of fractional integrals I. Mathematische Zeitschrift 27(1927), 565-606. [2] Hardy G.H., Littlewood J.E. and Polya G. Inequalities. Cambridge university Press, Cambridge, 1967. [3] Srivastava H.M. and Y¨ urekli O. A theorem on a Stieltjes-type integral transform and its applications. Complex Variables Theory Appl. 28(1995), 159-168. [4] Y¨ urekli O. A theorem on the generalized Stieltjes transform. J. Math. Anal. Appl. 168(1992), 63-71. [5] Y¨ urekli O. and Sadek I.A Parseval-Goldstein type theorem on the Widder potential transform and its applications. Internat. J. Math and Math. Sci. 14(1991), 517-524.

Address: Ithaca College, Humanities and Sciences, Department of Mathematics, Ithaca, NY-USA e-mail: [email protected]

The 20 th International Congress of The Jangjeon Mathematical Society

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113

On Sum Degree Energy of a Graph R.K.Zaferani, C.Adiga and H.B.Walıkar

In this talk, we introduce the sum degree matrix SD(G) of a simple graph G of order n and obtain a bound for eigenvalues of SD(G). The energy of G is defined as the sum of the absolute values of the eigenvalues of the graph G and is denoted by ESD (G). We prove that the energies of certain classes of graphs are less than that of ESD (Kn ). We also compute the energy of certain graphs.

References [1] Behzad M., Chartrand G. and Foster L.L. Graphs and digraphs. Wadsworth International Grop, Belmont, 1979. [2] Chung F.R.K. Spectral graph theory. CBMS 92, American Mathematical Society, Providence, 1997. [3] Cvetkovic D.M., Doob M., Gutman I. and Torgasev A. Recent results in the theory of graph spectra. North Holland, Amsterdam, 1988. [4] Cvetkovic D.M., Doob M. and Sachs H. Spectra of graphs. Academic Press, New York, 1980. [5] Gutman I. The Energy of a graph. Ber. Math. Stat. Sckt. Forschungzentrum Graz 103(1978), 1-22. [6] Harrary F. Graph theory. 7th Reprint, Narosa Publishing House, New Delhi, 1995.

Addresses: Department of Studies in Mathematics, University of Mysore, Manasagangothri, MYSORE- 570 006, INDIA Department of Mathematics, Karnatak University, DHARWAD -580003-INDIA e-mails: c [email protected], [email protected], [email protected]

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A Goal Programming Method for Finding Common Weights in DEA Majid Zohrehbandian, Ahmad Makui and Alireza Alinezhad

In this talk, we examine the application of goal programming approach for generating common set of weights under the DEA framework. A characteristic of data envelopment analysis (DEA) is to allow individual decision making units (DMUs) to select the factor weights that are the most advantageous for them in calculating their efficiency scores. This flexibility in selecting the weights, on the other hand, deters the comparison among DMUs on a common base. For dealing with this difficulty and assessment of all the DMUs on the same scale, this paper proposes to use a multiple objective linear programming (MOLP) approach for generating common set of weights under the DEA framework. This is an advantageous of the proposed approach against general approaches in the literature which are based on multiple objective nonlinear programming.

References [1] Charnes A. and Cooper W.W. Management Models and Industrial Applications of Linear Programming. John Wiley, New York, 1961. [2] Kao C. and Hung H.T. Data envelopment analysis with common weights: the compromise solution approach. Journal of the Operational Research Society 56(2005), 1196-1203. [3] Kornbluth J. Analysing policy effectiveness using cone restricted data envelopment analysis. Journal of the Operational Research Society 42(1991), 1097-1104. [4] Roll Y., Cook W.D. and Golany B. Controlling factor weights in data envelopment analysis. IIE Transactions 23(1)(1991), 2-9.

Address: Department of Mathematics, Islamic Azad University-Karaj Branch, P.O.Box 31485-313, Karaj-IRAN. e-mails: [email protected], [email protected], alinezhad [email protected]

THE 20 th INTERNATIONAL CONGRESS OF THE JANGJEON MATHEMATICAL SOCIETY August 21-23, 2008 Uludag University, Bursa-TURKIYE
Supported by ¨ ˙ TUBITAK (h...