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A Course in Computational Algebraic Number Theory
A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
Advanced Topics in Computational Number Theory
Written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Chapters one through five form a homogenous subject matter suitable for a six-month or year-long course in computational number theory. The subsequent chapters deal with more miscellaneous subjects.
Number Theory
This book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.
Handbook of Categorization in Cognitive Science
Handbook of Categorization in Cognitive Science, Second Edition presents the study of categories and the process of categorization as viewed through the lens of the founding disciplines of the cognitive sciences, and how the study of categorization has long been at the core of each of these disciplines. The literature on categorization reveals there is a plethora of definitions, theories, models and methods to apprehend this central object of study. The contributions in this handbook reflect this diversity. For example, the notion of category is not uniform across these contributions, and there are multiple definitions of the notion of concept. Furthermore, the study of category and categori...
Numerical Algorithms for Number Theory: Using Pari/GP
This book presents multiprecision algorithms used in number theory and elsewhere, such as extrapolation, numerical integration, numerical summation (including multiple zeta values and the Riemann-Siegel formula), evaluation and speed of convergence of continued fractions, Euler products and Euler sums, inverse Mellin transforms, and complex L L-functions. For each task, many algorithms are presented, such as Gaussian and doubly-exponential integration, Euler-MacLaurin, Abel-Plana, Lagrange, and Monien summation. Each algorithm is given in detail, together with a complete implementation in the free Pari/GP system. These implementations serve both to make even more precise the inner workings of the algorithms, and to gently introduce advanced features of the Pari/GP language. This book will be appreciated by anyone interested in number theory, specifically in practical implementations, computer experiments and numerical algorithms that can be scaled to produce thousands of digits of accuracy.
Modular Forms
The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and “fun” subject in itself and abounds with an amazing number of surprising identities. This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke ...
Handbook of Elliptic and Hyperelliptic Curve Cryptography
The discrete logarithm problem based on elliptic and hyperelliptic curves has gained a lot of popularity as a cryptographic primitive. The main reason is that no subexponential algorithm for computing discrete logarithms on small genus curves is currently available, except in very special cases. Therefore curve-based cryptosystems require much smaller key sizes than RSA to attain the same security level. This makes them particularly attractive for implementations on memory-restricted devices like smart cards and in high-security applications. The Handbook of Elliptic and Hyperelliptic Curve Cryptography introduces the theory and algorithms involved in curve-based cryptography. After a very d...
New Perspectives on the Origins of Language
The question of how language emerged is one of the most fascinating and difficult problems in science. In recent years, a strong resurgence of interest in the emergence of language from an evolutionary perspective has been helped by the convergence of approaches, methods, and ideas from several disciplines. The selection of contributions in this volume highlight scenarios of language origin and the prerequisites for a faculty of language based on biological, historical, social, cultural, and paleontological forays into the conditions that brought forth and favored language emergence, augmented by insights from sister disciplines. The chapters all reflect new speculation, discoveries and more refined research methods leading to a more focused understanding of the range of possibilities and how we might choose among them. There is much that we do not yet know, but the outlines of the path ahead are ever clearer.
Consciousness and Cognition
What were the circumstances that led to the development of our cognitive abilities from a primitive hominid to an essentially modern human? The answer to this question is of profound importance to understanding our present nature. Since the steep path of our cognitive development is the attribute that most distinguishes humans from other mammals, this is also a quest to determine human origins. This collection of outstanding scientific problems and the revelation of the many ways they can be addressed indicates the scope of the field to be explored and reveals some avenues along which research is advancing. Distinguished scientists and researchers who have advanced the discussion of the mind...
Dance Me to the End of Love
10 years ago, Welcome Books published the star of its Art & Poetry Series, Dance Me to the End of Love, a deliriously romantic song by Leonard Cohen that was brilliantly visualized through the sensual paintings of Henri Matisse. Now for its 10-year anniversary, Welcome is thrilled to present the entirely re-imagined and redesigned Dance Me to the End of Love. With the art of Matisse and the words of Cohen still at the heart of the book, the new look and feel of this Art & Poetry book is overwhelmingly beautiful. Cohen's song is a lyrical tribute to the miracle of love, the grace it bestows on us and its healing, restorative power. Originally recorded on his Various Positions album, and featured in Cohen's anthology, Stranger Music, this poetic song is gloriously married to the art works by Henri Matisse, perhaps the greatest artist of the twentieth century. "I had this dance within me for a long time," Matisse once said in describing one of his murals. Dance Me to the End of Love is the perfect book for art lovers, song lovers, and all other lovers as well.
