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Diffusive motion–displacement due to the cumulative effect of irregular fluctuations–has been a fundamental concept in mathematics and physics since Einstein’s work on Brownian motion. It is also relevant to understanding various aspects of quantum theory. This book explains diffusive motion and its relation to both nonrelativistic quantum theory and quantum field theory. It shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis.
Singularities arise naturally in a huge number of different areas of mathematics and science. As a consequence, singularity theory lies at the crossroads of paths that connect many of the most important areas of applications of mathematics with some of its most abstract regions.The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis, geometry, physics, or other science (functions, varieties, mappings, vector or tensor fields, differential equations, models, etc.)
In July 1996, a conference was organized by the editors of this volume at the Mathematische Forschungsinstitut Oberwolfach to honour Egbert Brieskorn on the occasion of his 60th birthday. Most of the mathematicians invited to the conference have been influenced in one way or another by Brieskorn’s work in singularity theory. It was the first time that so many people from the Russian school could be present at a conference in singularity theory outside Russia. This volume contains papers on singularity theory and its applications, written by participants of the conference.
The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sharing, frameproof codes, and broadcast encryption.Suitable for researchers and graduate students in mathematics and computer science, this self-contained book is one of the first to focus on many topics in cryptography involving algebraic curves.
Authors: Jones, Gareth A., Wolfart, JürgenProvides basic material about maps and hypermaps on Riemann surfacesPresents many elementary and less elementary examples of Galois actions on dessins and their algebraic curves Emphasises the role of group theory in the classification of regular maps, regular dessins, and quasiplatonic surfaces Explains the links between the theory of dessins and other areas of arithmetic and geometryThis volume provides an introduction to dessins d’enfants and embeddings of bipartite graphs in compact Riemann surfaces.
This edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory.
The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry, i.e., the study of big linear series on algebraic varieties. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties. The book includes such fundamental results as arithmetic Hilbert-Samuel formula, arithmetic Nakai-Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang-Bogomolov conjecture and so on.
This book collects the results of the workshops on Applications of Algebraic Curves and Applications of Finite Fieldsat the RICAMin 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world, addressing old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.
Bioceramics: Properties, Characterization, and Applications will be a general introduction to the uses of ceramics and glasses in the human body for the purposes of aiding, healing, correcting deformities, and restoring lost function. With over 30 years experience, the author developed the text as an outgrowth of an undergraduate course for senior students in biomedical engineering and will emphasize the fundamentals and applications in modern implant fabrication, and will also deal with tissue engineering scaffolds made of ceramics.
This volume is a collection of papers from the International Conference on Tropical and Idempotent Mathematics, held in Moscow, Russia in August 2007. This is a relatively new branch of mathematical sciences that has been rapidly developing and gaining popularity over the last decade. Tropical mathematics can be viewed as a result of the Maslov dequantization applied to ‘traditional’ mathematics over fields. Importantly, applications in econophysics and statistical mechanics lead to an explanation of the nature of financial crises.
This volume contains invited expository and research papers from the conference Topology of Algebraic Varieties, in honor of Anatoly Libgober’s 60th birthday, held June 22-26, 2009, in Jaca, Spain. The volume contains four parts corresponding to the four main focal points of the conference: algebraic geometry and fundamental groups, braids and knots, hyperplane arrangements, and singularities. Together, the papers provide an overview of the current status of a broad range of topological questions in Algebraic Geometry.
The study of derived categories is a subject that attracts increasingly many mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory, and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic geometry such as the duality theorem. In the 1970s, Beilinson, Gelfand, and Gelfand discovered that a derived category of an algebraic variety may be equivalent to that of a finite-dimensional non-commutative algebra, and Mukai found that there are non-isomorphic algebraic varieties that have equivalent derived categories.
There is a particular fascination when two apparently disjoint areas of mathematics turn out to have a meaningful connection to each other. The main goal of this book is to provide a largely self-contained, in-depth account of the linkage between nonassociative algebra and projective planes, with particular emphasis on octonion planes. There are several new results and many, if not most, of the proofs are new. The development should be accessible to most graduate students and should give them introductions to two areas which are often referenced but not often taught.O
This book introduces the study of knots, providing insights into recent applications in DNA research and graph theory. It sets forth fundamental facts such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials. It also covers more recent developments and special topics, such as chord diagrams and covering spaces.The author avoids advanced mathematical terminology and intricate techniques in algebraic topology and group theory. Numerous diagrams and exercises help readers understand and apply the theory.
This volume contains many of the lectures delivered at the AMS Summer Research Institute on Algebraic Geometry held at the University of California, Santa Cruz, in July 1995. The aim of the conference was to provide a comprehensive view of the development of algebraic geometry in the past decade and to lay special emphasis on emerging new directions. The focus of the papers in these volumes is on expository surveys of important areas rather than on technical presentations of new results. This book is intended for graduate students and research mathematicains interested in algebraic geometry and related areas.
Since their invention in the late seventies, public key cryptosystems have become an indispensable asset in establishing private and secure electronic communication, and this need, given the tremendous growth of the Internet, is likely to continue growing. Elliptic curve cryptosystems represent the state of the art for such systems. Elliptic Curves and Their Applications to Cryptography: An Introduction provides a comprehensive and self-contained introduction to elliptic curves and how they are employed to secure public key cryptosystems.
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