DOCSLIB.ORG

On the Equivalence Functional Relation Estimated for Vector-Valued Singular Integrals with Semigroups of Operators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Equivalence Functional Relation Estimated for Vector-Valued Singular Integrals with Semigroups of Operators Republic of the Sudan Ministry of High Education and Scientific Research University of Nile Valley Faculty of Postgraduate Studies On The Equivalence Functional Relation Estimated For Vector-Valued Singular Integrals with Semigroups of Operators. A Thesis Submitted in Partial Fulfillment for the Degree of M.sc in Mathematics Prepared by: Mohammed Salih Mukhtar Dahab Ahmed Supervised by: Dr. Adam Abedalla Abbakar Hassan December, 2009 اﻵية ق ال اهلل تعالي ِ ِ }يَْومَ نَطْوِي السََّماء َكطَ ِّي السِّجلِّ لْل ُكتُ ِب كََما بَدَأْنَ ا أَوَّلَ ِ ِِ خَْل ٍق نُّعيدُهُ َو ْعداً عَل َْينَ ا إِن َّا ُكن َّا ف َاعلي َن {اﻷنبياء401 وق ال اهلل تعالي أيضا ِ ِ }َوَما قَدَُروا اللَّوَ َح َّق قَ ْدرِه َواْﻷَْر ُض َجميعا ً قَْب َضتُوُ يَْومَ ِ ِ ِ ِ ِِ اْلقيَ اَمة َوالسَّماَوا ُت َمطْوِي َّا ٌت بيَمينو ُسْبَحانَوُ َوتََعالَى عَمَّا يُ ْشرِ ُكو َن {الزمر76 وق ال اهلل تعالي أيضا ِ ٍ ِ }َوالسََّماء بَنَْينَ ا َىا بأَْيد َوإِن َّا لَُموسُعو َن * َواْﻷَْر َض فََر ْشنَ ا َىا فَنِعم اْلما ِىدون {الذاريات16، 14 ْ َ َ ُ َ صدق اهلل العظيم Dedication TO My parents who survived for me to be i My great thanks after my God to my supervisor Dr. Adam Abedd- Alla Abbakar who suggested this title , for his wise advice and guidance. Thanks are Also due to the library staff of the Facluty of Mathematical Sciences U.of.K and especially for Mr. Mohammed Yasein the secretary. Also are extended to the teaching staff and administrators of the M.sc program at the Nile Valley University Omdurman branch. ii Abstract In this thesis we have present a mathematical techniques in estimation of vector – valued singular integral by using equivalence functional relations with semigroups of operators. This thesis contains four chapters. The first chapter is the function and the functional relations. The second chapter with the title vector – valued singular integrals. Chapter three under the title of semigroups and semigroups of operators. The last , chapter four, in which the estimation of the singular integragrls have been done by developing a generalized Littlewood-paley theory for semigroups acting on Lp -spaces of functions with values in uniformly convex or smooth Banach spaces, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calder´on-Zygmund singular integral operators. iii الخﻻصة قددت في ددًيهدد ي لبحددبيب ددتي لات ٌدديفي ل ٌيمددٌدي ل ددا ت دي ددًياتددتٌ ي لامددي في ل ف تةيأوي لشي ةيي في لتٌ ددي تاايهٌددي لادًيا دا تعي ل قديفي لت لٌددي د ي دثر في شبهي لز .يٌحاوييهد ي لبحدبيليدًيأ ب ددي دو ي,ي لف د ي وو يب دو اي لت لدديوي ل قيفي لت لٌدي,يو لف ي لري ًيٌ ا تي لفمديا في ل ااهدديو لامدي في ل فد تةي أوي لشددي ةيي يآفي لتٌ دددي تاايهٌدددي.ي لف دد ي لريلددبيلدداي ددثر فيشددبهي لز دد ي,يأ دديي لبيبي ل ب يو و ٌ ياعي ٌهياتتٌ يايكي لامدي فيلدايي ٌدطيايدوٌ ي ظ ٌدديلٌايدووتي –يبٌيى(Littlewood-Paley theory) يلشبهي لز ي لاًيادثر ي دًيتو ي مديا في لبٌ ددٌج)Lebeseque space)ي في لتٌ دددي ددًي بدديفيبي ددي (Banach space)ي له ت ٌدي ل اظ دي ل حتبد(convex)يأويي لافيميٌدي ل يل د)smooth(يلا دبحيهد هي ل ظ ٌدي في فسي ٌزةي ثر فيميلٌت واي–يز ٌت و تييليامي في ل ف تةيأوي لشدي ةيي في لتٌ دددي تاايهٌدددي Calderon-Zagmound vector-valued singular) (integrals ي ي vi v List of contents Dedication…………………………………………………………i Acknoweledgement…………………………………………….…ii Abstract…………………………………………………………...iii Abstract (Arabic)………………………………………………....iv List of contents……..……………………………….…………….v List of symbols and abbreviations……………………………….vii Introduction……………………………………………….………x Chapter one 1-Function and functional relations.….…………………….……..1 11 The function…………………………………………………………..1 12 The functional relations…………………………………………6 13 The equivalence functional relations……………………..18 Chapter two 2-Vector –valued singular integrals……………………………..27 21 Vector space……………………………………………....28 22 Topological vector space…………………………………29 23 Kernel mathematic………………………………………..32 24 Convolution………………………………………………………..40 25 Integral transform……………………………………………….42 26 Singular integrals………………………………………………..43 27 Hardy – Littlewood maximal function…………………..47 28 Littlewood-Paley theory……………………………………….50 29 Fefferman - Stein function and vector-valued operators...53 v Chapter three 3-Semigroups and semigroups of operators……………………55 31 The semigroups…………………………………………...56 32 Semigroups action………………………………….…….60 33 Transformation semigroup………………………………..63 34 Inverse semigroup………………………………………...64 35 Semigroups of operators………………………………….67 36 Hille–Yosida theorem……………………………………69 37 C0-semigroup……………………………………………72 38 Special integral operators…………………………………77 Chapter four 4-Estimation of vector-valued singular integrals with semigroups of operators………………………………………………...……87 41 Developing a generalized Littlewood-Paley theory for semigroups acting on Lp -spaces…………………………………89 42 One-sided vector-valued Littlewood-Paley-Stein inequalities for semigroups………………………………………94 43 Duality…………………………………………………..110 44 Poisson semigroup on Rn ……………………………..113 45 Continuation of Poisson semigroup on Rn ……………..121 46 Ornstein-Uhlenbeck semigroup…………………………126 47 Almost sure finiteness…………………………………..130 References…………………………….………………………..137 vi List of symbols and abbreviations f Function CC, 0 Set of continuous functions R \0 Domain of real numbers except zero X Equivalence class : Binary relation ,,,,,,:; Equivalence relations ConX Complete lattice ker Mathematical kernel Rn N-dimensional Euclidean space (CZO) Calderón-Zygmund Operator (UMD) unconditional martingale differences The norm Lp Lebesgue spaces W kp, Sobolev spaces vii P Poisson kernel H p Hardy space M Maximal function gf Littlewood-Paley g- function f # The sharp maximal function Q Cubes H Hilbert space B Banach space Ttt0 Diffusion semigroup W Weierstrass Operators Gradient operator The Laplace operator n Gaussian measure L Ornstein–Uhlenbeck operator BMO Bounded mean oscillator G The classical Littlewood-Paley g -function viii Pt t0 The subordinated Poisson semigroup A The infinitesimal generator of semigroup q g -function” associated to subordinated Poisson semigroup T Torus Gq n -dimensional generalized g-function n Ott0 The Ornstein-Uhlenbeck semigroup on R O t t0 Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on gq Littlewood-Paley g -function associated to En Monotone sequence of conditional Expectations n Pt Poisson kernels both on and on R dμ Lebesgue measure Pr Poisson kernels on ix Introduction In this thesis we have to present a mathematical technique in estimation of vector – valued singular integral by using equivalence functional relations with semigroups of operators. This thesis contains four chapters. the first chapter is the function and the functional relations, in which we have introduced basic results of the function its (definition, representation, notation, properties and continuity), the functional relations (injection, bijection, surjection, isometry and homomorphism) and the equivalence functional relations with the known notations ,,,,, that used as tools to compare between mathematical terms or structures and help us to estimate. In the second chapter with the title vector – valued singular integrals we start with the vector space and topological vector spaces and their properties examples are Banach spaces, Hilbert spaces and Sobolev spaces. Those spaces are underlying valued spaces for the functions define and bounded on them. After this we view the technique to achieve the singular integral by giving definition , meaning ,and properties of the kernels like (Calderon – Zagmund kernels , Poisson kernels), then we introduce the convolution , where the singular integral is an integral transform given by convolution of the function against the kernels , also the integrable function ( Lebesque measurable functions and Bochner integrals) , integral transform , Hilbert transform in harmonic analysis in the L2 space, from this view we have studied the Calderon – Zagmund singular operator with kernels in Rn . The rest of this chapter studies the development of the function to maximal function, Hrdy-Littlewood maximal functions, Littlewood – Paley theory and Fefferman Stein extension to the latter theory. That sharp maximal function f # is defined as regarding singular integrals. x In chapter three under the title of semigroups of operators, we exhibits the semigroups with it’s definition, types, structure, action, and it’s properties. Then we introduce the semigroup operators its definition and generation, under the fundamental Hille–Yosida theorem we study the C0-semigroup and its infinitesimal operator, also known as a (strongly continuous) one-parameter semigroup, the transformation and the analytical semigroup are special type of the C0-semigroup Then we define the symmetric diffusion semigroup ,it is a collection of linear operators Ttt0 defined on Ldp , over a measure space ,d which is used in the solution of the partial differential equation and share the same application with the singular integral. At the end of this chapter we introduce some integral operators with semigroups properties, The Weierstrass operators, Poisson operator, and the Ornstein– Uhlenbeck operator used in the development of Littlewood-Paley generalized function for the semigroups in spaces. The last , chapter four, in which the estimation of the singular integragrls have been done by developing a generalized Littlewood-paley theory for semigroups acting on Lp -spaces of functions. It is well known that martingale inequalities involving square function are closely related to the corresponding inequalities
Load more

Recommended publications
  • Dirichlet Character Difference Graphs 1
    DIRICHLET CHARACTER DIFFERENCE GRAPHS M. BUDDEN, N. CALKINS, W. N. HACK, J. LAMBERT and K. THOMPSON Abstract. We define Dirichlet character difference graphs and describe their basic properties, includ- ing the enumeration of triangles. In the case where the modulus is an odd prime, we exploit the spectral properties of such graphs in order to provide meaningful upper bounds for their diameter. 1. Introduction Upon one's first encounter with abstract algebra, a theorem which resonates throughout the theory is Cayley's theorem, which states that every finite group is isomorphic to a subgroup of some symmetric group. The significance of such a result comes from giving all finite groups a common ground by allowing one to focus on groups of permutations. It comes as no surprise that algebraic graph theorists chose the name Cayley graphs to describe graphs which depict groups. More formally, if G is a group and S is a subset of G closed under taking inverses and not containing the identity, then the Cayley graph, Cay(G; S), is defined to be the graph with vertex set G and an edge occurring between the vertices g and h if hg−1 2 S [9]. Some of the first examples of JJ J I II Cayley graphs that are usually encountered include the class of circulant graphs. A circulant graph, denoted by Circ(m; S), uses Z=mZ as the group for a Cayley graph and the generating set S is Go back m chosen amongst the integers in the set f1; 2; · · · b 2 cg [2].
    [Show full text]
  • Introducing the Mini-DML Project Thierry Bouche
    Introducing the mini-DML project Thierry Bouche To cite this version: Thierry Bouche. Introducing the mini-DML project. ECM4 Satellite Conference EMANI/DML, Jun 2004, Stockholm, Sweden. 11 p.; ISBN 3-88127-107-4. hal-00347692 HAL Id: hal-00347692 https://hal.archives-ouvertes.fr/hal-00347692 Submitted on 16 Dec 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Introducing the mini-DML project Thierry Bouche Université Joseph Fourier (Grenoble) WDML workshop Stockholm June 27th 2004 Introduction At the Göttingen meeting of the Digital mathematical library project (DML), in May 2004, the issue was raised that discovery and seamless access to the available digitised litterature was still a task to be acomplished. The ambitious project of a comprehen- sive registry of all ongoing digitisation activities in the field of mathematical research litterature was agreed upon, as well as the further investigation of many linking op- tions to ease user’s life. However, given the scope of those projects, their benefits can’t be expected too soon. Between the hope of a comprehensive DML with many eYcient entry points and the actual dissemination of heterogeneous partial lists of available material, there is a path towards multiple distributed databases allowing integrated search, metadata exchange and powerful interlinking.
    [Show full text]
  • 201.1.8001 Fall 2013-2014 Lecture Notes Updated: December 24, 2014
    INTRODUCTION TO PROBABILITY ARIEL YADIN Course: 201.1.8001 Fall 2013-2014 Lecture notes updated: December 24, 2014 Contents Lecture 1. 3 Lecture 2. 9 Lecture 3. 18 Lecture 4. 24 Lecture 5. 33 Lecture 6. 38 Lecture 7. 44 Lecture 8. 51 Lecture 9. 57 Lecture 10. 66 Lecture 11. 73 Lecture 12. 76 Lecture 13. 81 Lecture 14. 87 Lecture 15. 92 Lecture 16. 101 1 2 Lecture 17. 105 Lecture 18. 113 Lecture 19. 118 Lecture 20. 131 Lecture 21. 135 Lecture 22. 140 Lecture 23. 156 Lecture 24. 159 3 Introduction to Probability 201.1.8001 Ariel Yadin Lecture 1 1.1. Example: Bertrand's Paradox We begin with an example [this is known as Bertrand's paradox]. Joseph Louis Fran¸cois Question 1.1. Consider a circle of radius 1, and an equilateral triangle bounded in the Bertrand (1822{1900) circle, say ABC. (The side length of such a triangle is p3.) Let M be a randomly chosen chord in the circle. What is the probability that the length of M is greater than the length of a side of the triangle (i.e. p3)? Solution 1. How to chose a random chord M? One way is to choose a random angle, let r be the radius at that angle, and let M be the unique chord perpendicular to r. Let x be the intersection point of M with r. (See Figure 1, left.) Because of symmetry, we can rotate the triangle so that the chord AB is perpendicular to r. Since the sides of the triangle intersect the perpendicular radii at distance 1=2 from 0, M is longer than AB if and only if x is at distance at most 1=2 from 0.
    [Show full text]
  • A48 Integers 11 (2011) Enumeration of Triangles in Quartic Residue
    #A48 INTEGERS 11 (2011) ENUMERATION OF TRIANGLES IN QUARTIC RESIDUE GRAPHS Mark Budden Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC [email protected] Nicole Calkins Department of Mathematics, Armstrong Atlantic State University, Savannah, GA [email protected] William Nathan Hack Department of Mathematics, Armstrong Atlantic State University, Savannah, GA [email protected] Joshua Lambert Department of Mathematics, Armstrong Atlantic State University, Savannah, GA [email protected] Kimberly Thompson Department of Mathematics, Armstrong Atlantic State University, Savannah, GA [email protected] Received: 10/10/10, Accepted: 7/19/11, Published: 9/13/11 Abstract For a fixed prime p 1 (mod 4), we define the corresponding quartic residue graph and determine the n≡umber of triangles contained in such a graph. Our computation requires us to compute the number of pairs of consecutive quartic residues modulo p via the evaluation of certain quartic Jacobi sums. 1. Introduction Although Raymond Paley’s life passed by quickly, his impact resonated across mul- tiple mathematical disciplines such as complex analysis, combinatorics, number theory, and harmonic analysis [2, 8, 9, 10]. Perhaps one of Paley’s most notable contributions occurred when he brought the fields of combinatorics and number theory even closer together. One such mathematical achievement came about in INTEGERS: 11 (2011) 2 1933, when Paley [8] used the quadratic residues of the field with prime order p 3 ≡ (mod 4) to construct Hadamard matrices of order p + 1. While Paley passed away that same year, his results spurred a great deal of interest. One person that took notice of Paley’s achievements was Horst Sachs [12].
    [Show full text]
  • Newsletter of the European Mathematical Society
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY S E European M M Mathematical E S Society June 2019 Issue 112 ISSN 1027-488X Feature Interviews Research Center Determinantal Point Processes Peter Scholze Isaac Newton Institute The Littlewood–Paley Theory Artur Avila for Mathematical Sciences Artur Avila (photo courtesy of Daryan Dornelles/Divulgação IMPA) Journals published by the ISSN print 1463-9963 Editors: ISSN online 1463-9971 José Francisco Rodrigues (Universidade de Lisboa, Portugal), Charles M. Elliott (University 2019. Vol. 21. 4 issues of Warwick, Coventry, UK), Harald Garcke (Universität Regensburg,Germany), Juan Luis Approx. 500 pages. 17 x 24 cm Vazquez (Universidad Autónoma de Madrid, Spain) Price of subscription: 390 ¤ online only Aims and Scope 430 ¤ print+online Interfaces and Free Boundaries is dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems in all areas where such phenomena are pertinent. The journal aims to be a forum where mathematical analysis, partial differential equations, modelling, scientific computing and the various applications which involve mathematical modelling meet. Submissions should, ideally, emphasize the combination of theory and application. ISSN print 0232-2064 A periodical edited by the University of Leipzig ISSN online 1661-4354 Managing Editors: 2019. Vol. 38. 4 issues J. Appell (Universität Würzburg, Germany), T. Eisner (Universität Leipzig, Germany), B. Approx. 500 pages. 17 x 24 cm Kirchheim (Universität Leipzig, Germany) Price of subscription: 190 ¤ online only Aims and Scope Journal for Analysis and its Applications aims at disseminating theoretical knowledge 230 ¤ print+online The in the field of analysis and, at the same time, cultivating and extending its applications.
    [Show full text]
  • Hardy Spaces Nikolaï Nikolski Frontmatter More Information
    Cambridge University Press 978-1-107-18454-1 — Hardy Spaces Nikolaï Nikolski Frontmatter More Information CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 179 Editorial Board B. BOLLOBAS,´ W. FULTON, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO HARDY SPACES The theory of Hardy spaces is a cornerstone of modern analysis. It combines techniques from functional analysis, the theory of analytic functions, and Lesbesgue integration to create a powerful tool for many applications, pure and applied, from signal processing and Fourier analysis to maximum modulus principles and the Riemann zeta function. This book, aimed at beginning graduate students, introduces and develops the classical results on Hardy spaces and applies them to fundamental concrete problems in analysis. The results are illustrated with numerous solved exercises which also introduce subsidiary topics and recent developments. The reader’s understanding of the current state of the eld, as well as its history, are further aided by engaging accounts of the key players and by the surveys of recent advances (with commented reference lists) that end each chapter. Such broad coverage makes this book the ideal source on Hardy spaces. Nikola¨ı Nikolski is Professor Emeritus at the Universite´ de Bordeaux working primarily in analysis and operator theory. He has been co-editor of four international journals and published numerous articles and research monographs. He has also supervised some 30 PhD students, including three Salem Prize winners. Professor Nikolski was elected Fellow of the AMS in 2013 and received the Prix Ampere` of the French Academy of Sciences in 2010. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-18454-1 — Hardy Spaces Nikolaï Nikolski Frontmatter More Information CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board B.
    [Show full text]
  • HARDY SPACES the Theory of Hardy Spaces Is a Cornerstone of Modern Analysis
    CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 179 Editorial Board B. BOLLOBAS,´ W. FULTON, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO HARDY SPACES The theory of Hardy spaces is a cornerstone of modern analysis. It combines techniques from functional analysis, the theory of analytic functions, and Lesbesgue integration to create a powerful tool for many applications, pure and applied, from signal processing and Fourier analysis to maximum modulus principles and the Riemann zeta function. This book, aimed at beginning graduate students, introduces and develops the classical results on Hardy spaces and applies them to fundamental concrete problems in analysis. The results are illustrated with numerous solved exercises which also introduce subsidiary topics and recent developments. The reader’s understanding of the current state of the field, as well as its history, are further aided by engaging accounts of the key players and by the surveys of recent advances (with commented reference lists) that end each chapter. Such broad coverage makes this book the ideal source on Hardy spaces. Nikola¨ı Nikolski is Professor Emeritus at the Universite´ de Bordeaux working primarily in analysis and operator theory. He has been co-editor of four international journals and published numerous articles and research monographs. He has also supervised some 30 PhD students, including three Salem Prize winners. Professor Nikolski was elected Fellow of the AMS in 2013 and received the Prix Ampere` of the French Academy of Sciences in 2010. CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board B. Bollobas,´ W. Fulton, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press.
    [Show full text]
  • Quantum Algorithms for Weighing Matrices and Quadratic Residues
    Quantum Algorithms for Weighing Matrices and Quadratic Residues Wim van Dam∗ UC Berkeley [email protected] August 29, 2018 Abstract In this article we investigate how we can employ the structure of com- binatorial objects like Hadamard matrices and weighing matrices to device new quantum algorithms. We show how the properties of a weighing ma- trix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one. It is pointed out that this scheme captures both Bernstein & Vazirani’s inner-product pro- tocol, as well as Grover’s search algorithm. In the second part of the article we consider Paley’s construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol χ (which indicates if an element of a finite field Fq is a quadratic residue or not). It is shown how for a shifted Legendre function fs(i) = χ(i + s), the unknown s ∈ Fq can be obtained exactly with only two quantum calls to fs. This is in sharp contrast with the observation that 1−ε any classical, probabilistic procedure requires more than log q + log( 2 ) queries to solve the same problem. arXiv:quant-ph/0008059v3 19 Mar 2001 1 Introduction The theory of quantum computation investigates how we can use quantum me- chanical effects to solve computational problems more efficiently than we can by classical means. So far, the strongest evidence that there is indeed a real and significant difference between quantum and classical computation is provided by Peter Shor’s polynomial-time factoring algorithm[31].
    [Show full text]
  • Paley Graphs and Their Generalizations
    Paley Graphs and Their Generalizations By Ahmed Noubi Elsawy A thesis submitted to Heinrich Heine University D¨usseldorf, Germany for the Degree of Master of Science Supervisors Prof. Dr. F. Grunewald Prof. Dr. E. Klimenko arXiv:1203.1818v1 [math.CO] 7 Mar 2012 Department of Mathematics Heinrich Heine University D¨usseldorf 2009 MASTER OF SCIENCE THESIS OF Ahmed Elsawy APPROVED: THESIS COMMITTEE: MAJOR PROFESSOR DEAN OF THE GRADUATE SCHOOL HEINRICH HEINE UNIVERSITY DUSSELDORF¨ 2009 Acknowledgment I would like to offer my deepest gratitude and thankfulness to Prof. Dr. F. Grunewald for suggesting the topics of this thesis, for his kind supervision, and for his invaluable help during the progress of present work. I owe my deepest gratitude and special thankfulness to Prof. Dr. E. Klimenko who offered me a great help in understanding the topics and to get this document into its present shape. Many thanks for her stimulat- ing discussion, sincere advice, incisive comments, and careful reading of the manuscript. Naturally, all the mistakes that can still be found in this thesis are mine. If you ever find a word that is spelled correctly, it must have been Prof. Dr. E. Klimenko dropping me a line. I want also to thank my family for their support and for keeping me going. A. Elsawy Contents Introduction 7 1 Finite Fields 11 1.1 Basic definitions and properties . 11 1.2 Classification of the finite fields . 14 1.3 Constructionofthefinitefields . 18 2 Paley Graph 23 2.1 Basic definitions and properties . 23 2.2 Paleygraphs............................ 27 2.2.1 Definition and examples .
    [Show full text]
  • The Numdam Project
    Introducing the mini-DML project Thierry B Université Joseph Fourier (Grenoble) WDML workshop Stockholm June 27th 2004 A metadata project from Cellule MathDoc I. Presentation 1. Context mini-DML 2. Catalogues & lists 3. Where is the DML ? 4. Are lists useful ? 5. Where are Cauchy seminal papers ? 6. The mini-DML project 7. Back to Cauchy II. Demonstration III. Riemann hypothesis : A case study WDML — Stockholm — June 27th, 2004 — 1 1. C – Scientific journals exist since 1665 (Journal des savants, Philosophical transactionsmini-DML) – Mathematical journals exist since 1810 (Annales de Gergonne) – Many digitisation projects are heavily working in order to provide that intellectual treasury to the digital age – Freely downloadable grey litterature is omnipresent WDML — Stockholm — June 27th, 2004 — 2 – We face an exponentially growing amount of available material while – most researchersmini-DML aren’t even aware of this, and. – those who know have to bookmark hundreds of websites : repositories, specialised search engines, home pages, individual journal pages, and. – they need to figure a specific strategy to localise an item in each of these sites ! WDML — Stockholm — June 27th, 2004 — 3 2. C & Facing a similar situation, the mathematicians of the end of 19th century choosed to build lists of important published resources. mini-DML – Valentin’s famous lost catalogue (1885-1910) – The Répertoire bibliographique des sciences mathématiques (1894-1912, covering 19th century) – The Jahrbuch über die Fortschritte der Mathematik (1868-1944) WDML — Stockholm — June 27th, 2004 — 4 And later : – Zentralblatt-Math (1931- ) – Mathematical reviewsmini-DML(1940- ) Except in the (conjectural) first case, these lists are non-comprehensive and require an important intellectual input (choice, classification, review).
    [Show full text]
  • Self-Similar Solutions of Navier-Stokes Type Equations
    Eötvös Loránd University Faculty of Sciences Self-similar solutions of Navier-Stokes type equations Master’s Thesis Krisztián Benyó Mathematics MSc Paris, 2015 Supervisor: Consultant: Marco Cannone Ádám Besenyei Professor, Assistant Professor, Université Paris-Est Eötvös Loránd University ACKNOWLEDGEMENT First and foremost, I would like to express my deepest gratitude to the members of the Bézout LabEx (Laboratory of Excellence) for accepting my application to the Bézout Excellence Track program hence making it possible for me to spend the second year of my master’s degree program at the University of Marne-la-Vallée (Université Paris-Est, Marne-la-Vallée). My sincere appreciation is extended to my supervisor, Marco Cannone, for his support and aid not only during the preparation of the present report but also throughout the past two semesters. His knowledge, assistance and careful guidance proved to be invaluable for my studies and research. I would also like to extend my gratitude to Ádám Besenyei, my consultant at Eötvös Loránd University, whose sharp remarks, inspiring thoughts and precise cor- rections allowed me to polish and perfect the present work. Furthermore I owe a special thanks François Bouchut, my supervisor for the master’s thesis prepared for the University of Marne-la-Vallée, for his help during the finalisation of the report and for the suggestions made on the second part of it. I CONTENTS Introduction 1 1 Preliminaries 4 1.1 The Navier-Stokes equations . .5 1.2 Notion of the solution . .7 1.3 Some useful aspects of the Fourier analysis . 10 1.3.1 Fourier transform .
    [Show full text]
  • 21 Dirichlet Character Difference Graphs 1
    Acta Math. Univ. Comenianae 21 Vol. LXXXII, 1 (2013), pp. 21{28 DIRICHLET CHARACTER DIFFERENCE GRAPHS M. BUDDEN, N. CALKINS, W. N. HACK, J. LAMBERT and K. THOMPSON Abstract. We define Dirichlet character difference graphs and describe their basic properties, including the enumeration of triangles. In the case where the modulus is an odd prime, we exploit the spectral properties of such graphs in order to provide meaningful upper bounds for their diameter. 1. Introduction Upon one's first encounter with abstract algebra, a theorem which resonates throughout the theory is Cayley's theorem, which states that every finite group is isomorphic to a subgroup of some symmetric group. The significance of such a result comes from giving all finite groups a common ground by allowing one to focus on groups of permutations. It comes as no surprise that algebraic graph theorists chose the name Cayley graphs to describe graphs which depict groups. More formally, if G is a group and S is a subset of G closed under taking inverses and not containing the identity, then the Cayley graph, Cay(G; S), is defined to be the graph with vertex set G and an edge occurring between the vertices g and h if hg−1 2 S [9]. Some of the first examples of Cayley graphs that are usually encountered include the class of circulant graphs. A circulant graph, denoted by Circ(m; S), uses Z=mZ as the group for a Cayley graph and the generating set S m is chosen amongst the integers in the set f1; 2; · · · b 2 cg [2].
    [Show full text]