General Galois Geometries

Author: James Hirschfeld,Joseph A. Thas

Publisher: Springer

ISBN: 1447167902

Category: Mathematics

Page: 409

View: 5579


This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of plane case (first volume) and three dimensions (second volume). This revised edition includes much updating and new material. It is a mostly self-contained study of classical varieties over a finite field, related incidence structures and particular point sets in finite n-dimensional projective spaces. General Galois Geometries is suitable for PhD students and researchers in combinatorics and geometry. The separate chapters can be used for courses at postgraduate level.

Arithmetic, Geometry, Cryptography and Coding Theory

Author: Alp Bassa,Alain Couvreur,David Kohel

Publisher: American Mathematical Soc.

ISBN: 1470428105

Category: Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Finite ground fields

Page: 199

View: 4479


This volume contains the proceedings of the 15th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT), held at the Centre International de Rencontres Mathématiques in Marseille, France, from May 18–22, 2015. Since the first meeting almost 30 years ago, the biennial AGCT meetings have been one of the main events bringing together researchers interested in explicit aspects of arithmetic geometry and applications to coding theory and cryptography. This volume contains original research articles reflecting recent developments in the field.

Codes, Cryptology and Curves with Computer Algebra:

Author: Ruud Pellikaan,Xin-Wen Wu,Stanislav Bulygin,Relinde Jurrius

Publisher: Cambridge University Press

ISBN: 1108547826

Category: Mathematics

Page: N.A

View: 3899


This well-balanced text touches on theoretical and applied aspects of protecting digital data. The reader is provided with the basic theory and is then shown deeper fascinating detail, including the current state of the art. Readers will soon become familiar with methods of protecting digital data while it is transmitted, as well as while the data is being stored. Both basic and advanced error-correcting codes are introduced together with numerous results on their parameters and properties. The authors explain how to apply these codes to symmetric and public key cryptosystems and secret sharing. Interesting approaches based on polynomial systems solving are applied to cryptography and decoding codes. Computer algebra systems are also used to provide an understanding of how objects introduced in the book are constructed, and how their properties can be examined. This book is designed for Masters-level students studying mathematics, computer science, electrical engineering or physics.

Foundations of Incidence Geometry

Projective and Polar Spaces

Author: Johannes Ueberberg

Publisher: Springer Science & Business Media

ISBN: 3642209726

Category: Mathematics

Page: 248

View: 9820


Incidence geometry is a central part of modern mathematics that has an impressive tradition. The main topics of incidence geometry are projective and affine geometry and, in more recent times, the theory of buildings and polar spaces. Embedded into the modern view of diagram geometry, projective and affine geometry including the fundamental theorems, polar geometry including the Theorem of Buekenhout-Shult and the classification of quadratic sets are presented in this volume. Incidence geometry is developed along the lines of the fascinating work of Jacques Tits and Francis Buekenhout. The book is a clear and comprehensible introduction into a wonderful piece of mathematics. More than 200 figures make even complicated proofs accessible to the reader.

Konstruktive Galoistheorie

Author: Bernd H. Matzat

Publisher: Springer-Verlag

ISBN: 3540479783

Category: Mathematics

Page: 294

View: 8161


This volume is based on a lecture course on constructive Galois Theory given in Karlsruhe by the author. The purpose of the course was to introduce students to the methods developed in the past few years for the realisation of finite groups as Galois groups over Q or over abelian number fields. Thus the book is addressed primarily to students with algebraic interests, as seminar material. Specialiists also will find in it a multitude of examples of polynomials with special Galois groups, which can of course also be used for the usual algebra courses.

Handbook of Finite Translation Planes

Author: Norman Johnson,Vikram Jha,Mauro Biliotti

Publisher: Chapman and Hall/CRC

ISBN: 9781584886051

Category: Mathematics

Page: 888

View: 2382


The Handbook of Finite Translation Planes provides a comprehensive listing of all translation planes derived from a fundamental construction technique, an explanation of the classes of translation planes using both descriptions and construction methods, and thorough sketches of the major relevant theorems. From the methods of André to coordinate and linear algebra, the book unifies the numerous diverse approaches for analyzing finite translation planes. It pays particular attention to the processes that are used to study translation planes, including ovoid and Klein quadric projection, multiple derivation, hyper-regulus replacement, subregular lifting, conical distortion, and Hermitian sequences. In addition, the book demonstrates how the collineation group can affect the structure of the plane and what information can be obtained by imposing group theoretic conditions on the plane. The authors also examine semifield and division ring planes and introduce the geometries of two-dimensional translation planes. As a compendium of examples, processes, construction techniques, and models, the Handbook of Finite Translation Planes equips readers with precise information for finding a particular plane. It presents the classification results for translation planes and the general outlines of their proofs, offers a full review of all recognized construction techniques for translation planes, and illustrates known examples.

Theorie der Transformationsgruppen

Author: Sophus Lie,Friedrich Engel

Publisher: American Mathematical Soc.

ISBN: 9780828402323

Category: Mathematics

Page: 2043

View: 9851


Sophus Lie had a tremendous impact in several areas of mathematics. His work centered on understanding continuous transformation groups and showing how these groups supply an organizing principle for different areas of mathematics, including geometry and mechanics. This title presents a treatise on theory of transformation groups.

Brauer Groups, Hopf Algebras and Galois Theory

Author: Stefaan Caenepeel

Publisher: Springer Science & Business Media

ISBN: 9781402003462

Category: Mathematics

Page: 488

View: 2196


This volume is devoted to the Brauer group of a commutative ring and related invariants. Part I presents a new self-contained exposition of the Brauer group of a commutative ring. Included is a systematic development of the theory of Grothendieck topologies and �tale cohomology, and discussion of topics such as Gabber's theorem and the theory of Taylor's big Brauer group of algebras without a unit. Part II presents a systematic development of the Galois theory of Hopf algebras with special emphasis on the group of Galois objects of a cocommutative Hopf algebra. The development of the theory is carried out in such a way that the connection to the theory of the Brauer group in Part I is made clear. Recent developments are considered and examples are included. The Brauer-Long group of a Hopf algebra over a commutative ring is discussed in Part III. This provides a link between the first two parts of the volume and is the first time this topic has been discussed in a monograph. Audience: Researchers whose work involves group theory. The first two parts, in particular, can be recommended for supplementary, graduate course use.

Modern Projective Geometry

Author: Claude-Alain Faure,Alfred Frölicher

Publisher: Springer Science & Business Media

ISBN: 9401595909

Category: Mathematics

Page: 363

View: 6923


This monograph develops projective geometries and provides a systematic treatment of morphisms. It introduces a new fundamental theorem and its applications describing morphisms of projective geometries in homogeneous coordinates by semilinear maps. Other topics treated include three equivalent definitions of projective geometries and their correspondence with certain lattices; quotients of projective geometries and isomorphism theorems; and recent results in dimension theory.

Valued Fields

Author: Antonio J. Engler,Alexander Prestel

Publisher: Springer Science & Business Media

ISBN: 354030035X

Category: Mathematics

Page: 208

View: 2314


Absolute values and their completions – such as the p-adic number fields – play an important role in number theory. Krull's generalization of absolute values to valuations made possible applications in other branches of mathematics. In valuation theory, the notion of completion must be replaced by that of "Henselization". This book develops the theory of valuations as well as of Henselizations, based on the skills of a standard graduate course in algebra.

Mathematical Tools for Physicists

Author: Michael Grinfeld

Publisher: John Wiley & Sons

ISBN: 3527684271

Category: Science

Page: 632

View: 5705


The new edition is significantly updated and expanded. This unique collection of review articles, ranging from fundamental concepts up to latest applications, contains individual contributions written by renowned experts in the relevant fields. Much attention is paid to ensuring fast access to the information, with each carefully reviewed article featuring cross-referencing, references to the most relevant publications in the field, and suggestions for further reading, both introductory as well as more specialized. While the chapters on group theory, integral transforms, Monte Carlo methods, numerical analysis, perturbation theory, and special functions are thoroughly rewritten, completely new content includes sections on commutative algebra, computational algebraic topology, differential geometry, dynamical systems, functional analysis, graph and network theory, PDEs of mathematical physics, probability theory, stochastic differential equations, and variational methods.

Dessins d'Enfants on Riemann Surfaces

Author: Gareth A. Jones,Jürgen Wolfart

Publisher: Springer

ISBN: 3319247115

Category: Mathematics

Page: 259

View: 7167


This volume provides an introduction to dessins d'enfants and embeddings of bipartite graphs in compact Riemann surfaces. The first part of the book presents basic material, guiding the reader through the current field of research. A key point of the second part is the interplay between the automorphism groups of dessins and their Riemann surfaces, and the action of the absolute Galois group on dessins and their algebraic curves. It concludes by showing the links between the theory of dessins and other areas of arithmetic and geometry, such as the abc conjecture, complex multiplication and Beauville surfaces. Dessins d'Enfants on Riemann Surfaces will appeal to graduate students and all mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group actions, and arithmetic.