Преобразование Меллина и теорема Планшереля для дискретной группы Гейзенберга

Through basic research in mathematics, the NTT Institute for Fundamental Mathematics aims to enrich the "fountain of knowledge" that nourishes science and technology.In this article, we first provide an overview of the research being carried out at the Institute then introduce the Institute's core research areas: number theory, especially arithmetic dynamics; algebraic and arithmetic geometry; and representation theory and automorphic forms.

Рассматривается теорема о вырождении некоммутативного обобщения спектральной последовательности Ходжа-де Рама, доказанная ранее первым автором. Приводится улучшенная и упрощенная версия ее доказательства, которая в явном виде использует спектральную алгебраическую геометрию. Также объясняется, почему для доказательства существенно необходимы методы алгебраической топологии.

Изложена краткая биография Игоря Ростиславовича Шафаревича, и приведен обзор всех его основных опубликованных научных результатов.

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

The new edition of this celebrated and long-unavailable book preserves the original book's content and structure and its unrivalled presentation of a universal method for the resolution of a class of singularities in algebraic geometry. At the same time, the book has been completely re-typeset, errors have been eliminated, proofs have been streamlined, the notation has been made consistent and uniform, an index has been added, and a guide to recent literature has been added. The book brings together ideas from algebraic geometry, differential geometry, representation theory and number theory, and will continue to prove of value for researchers and graduate students in these areas.

This upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing, securing, retrieving and communicating information in the electronic world. It will serve as a textbook for an undergraduate course in algebra with a strong emphasis on applications. The book offers a brief introduction to elementary number theory as well as a fairly complete discussion of major algebraic systems (such as rings, fields, and groups) with a view of their use in bar coding, public key cryptosystems, error-correcting codes, counting techniques, and elliptic key cryptography. This is the only entry level text for algebraic systems that includes an extensive introduction to elliptic curves, a topic that has leaped to prominence due to its importance in the solution of Fermat’s Last Theorem and its incorporation into the rapidly expanding applications of elliptic curve cryptography in smart cards. Computer science students will appreciate the strong emphasis on the theory of polynomials, algebraic geometry and Groebner bases. The combination of a rigorous introduction to abstract algebra with a thorough coverage of its applications makes this book truly unique.

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

With the beginning of 2012, a new editorial structure for the journal Mathematische Nachrichten was established. The four Editors-in-Chief Ben Andrews (Canberra/Australia), Robert Denk (Konstanz/Germany), Klaus Hulek (Hannover/Germany), and Frédéric Klopp (Paris/France) took over the strategic and mathematical responsibility, while Reinhard Mennicken (Regensburg/Germany) is supporting us as Managing Editor. With these changes, we succeeded in broadening the spectrum of this journal by sharing responsibilities and shortening the times to publication while increasing the quality and attractiveness of our articles. The increasing number of submitted manuscripts in combination with natural page limitations leads to a restrictive review process, which again helps to keep the high quality standard of Mathematische Nachrichten. As a result, the backlog of papers was significantly reduced during 2012. The structural changes in the Editorial Team were complemented by a complete change of the online submission system ScholarOne Manuscript. As all papers have to be submitted via this online system, the handling of articles and reviews can be done more transparent and faster, and decisions can be communicated to the authors and the editors immediately. In the year 2012, a special issue of Mathematische Nachrichten appeared devoted to Eduard R. Tsekanovskiǐ (Niagara University, Lewiston, USA) on the occasion of his seventy-fifth birthday. Guest editors for this issue were Fritz Gesztesy, Heinz Langer, Mark Malamud, and Reinhard Mennicken. At the beginning of 2013, a special issue dedicated to Hans Triebel (Jena) on the occasion of his seventy-fifth birthday will appear, edited by Dorothee D. Haroske and Hans-Jürgen Schmeißer. These special issues give us the opportunity to gather high-quality papers on selected topics, which are a highlight for all researchers in this area, and we thank our guest editors for their help. We look forward to receiving high-quality manuscripts in all fields of mathematics in 2013. We will continue to aim at a wide spectrum of topics, including analysis, differential geometry, algebraic geometry and number theory, stochastics and all other fields of pure and applied mathematics. In this direction, we will also continue to re-structure the board of associate editors. As a new associate editor, we welcome Stefan Wewers (Ulm/Germany) as a well-known specialist in number theory. We also take the opportunity to thank Fritz Gesztesy (Columbia), Guido Schneider (Stuttgart/Germany), and Giorgio Talenti (Florence/Italy) who left the editorial board on their own request for their long-time support of Mathematische Nachrichten. The Editors-in-Chief, the Managing Editor, and Wiley-VCH express their gratitude to all who contributed to the success of Mathematische Nachrichten, i.e. to our associate editors, to the numerous referees from all over the world, and to our authors for their submissions.

This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed. Then a formula for Gauss sums in terms of the p-adic gamma function is proved using the cohomology of Fermat and Artin-Schreier curves. Graduate students and research workers in number theory, algebraic geometry and parts of algebra and analysis will welcome this account of current research.

In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

Elliptic curves have been intensively studied in algebraic geometry and number theory. Recently, they have been used in devising efficient algorithms for factoring integers [8, 9], for primality proving [2, 11, 12] and for the construction of pseudorAndom bit generators [4]. In Chapter 8 we study how elliptic curves can be used in constructing efficient and secure cryptosystems, while in Chapter 10 we will discuss how good error-correcting codes may be obtained by using elliptic curves.

Binary sequences with good autocorrelation have many applications in raging systems, spread spectrum communication systems, multi-terminal system identification, code division multiple access communications systems, and stream ciphers. Magma is a large, well-supported software package designed for computations in algebra, number theory, algebraic geometry and algebraic combinatory. We design the system to compare and analysis the binary pseudorandom sequences. It is easy to analysis the existing binary sequence and new construction. And it can compare the performance of different sequences and draw the figure. We implement it on the C# and SQL database system.

The Möbius function is an important arithmetic function in number theory. It’s also very important to the examiner of algebraic patterns and combinatorial mathematics. Further to conventional range principle, it may be utilized in algebraic geometry, organization precept, and combinatorial extensive variety idea, among different things. These examine looks at the many approaches the Möbius function is probably used in combinatorial contexts. It is mainly used to observe partition characteristics, listing lattice factors, and turn sums over divisor features. The Möbius characteristic helps us find strong inversion formulae and hyperlinks that provide brief combinatory computation. As an instance, the Möbius inversion component simplifies the extraction of values from summitry traits. We additionally show how the Möbius feature interacts with algebraic systems such earrings and fields to provide a better know-how of how combinatorial gadgets are assembled. Examining those connections these days no longer only we could us research more approximately combinatorial arithmetic; it also gives us fresh thoughts to cope with tough issues in quantity concept. The paper also discusses advanced cryptography applications, zeta features, and partition principle’s Möbius characteristic involvement.

Methods of computer algebra become more familiar to a wide audience of theoretical mathematicians and physicists. The environment of computer algebra system leads to a greater acceptance of computer instruments in the mathematical research. Methods of symbolic manipulation provided by computer algebra systems in combination with high-power number crunching abilities of traditional hardware and software open the way to truly large scale computations often needed by mathematicians and physicists. This chapter describes new low complexity (both operational and logical) methods and algorithms for computations of solutions of differential equations and their efficient evaluation, and of solution of algebraic equations. These computations are often basic in applied problems, but even more so in a variety of problems of number theory and algebraic geometry. Power series manipulations over various rings and fields are one of the most important features of advanced computer algebra systems.

This article reports on some recent progresses in Bessel moments, which represent a class of Feynman diagrams in 2-dimensional quantum field theory. Many challenging mathematical problems on these Bessel moments have been formulated as a vast set of conjectures, by David Broadhurst and collaborators, who work at the intersection of high energy physics, number theory and algebraic geometry. We present the main ideas behind our verifications of several such conjectures, which revolve around linear and non-linear sum rules of Bessel moments, as well as relations between individual Feynman diagrams and critical values of modular $L$-functions.