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High Quality Content by WIKIPEDIA articles! In mathematics, the Seifert?van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.

Editors Smilka Zdravkovska, Peter L. Duren

This volume contains articles on the history of Soviet mathematics, many of which are personal accounts by mathematicians who witnessed and contributed to the turbulent and glorious years of Moscow mathematics. The articles in the book focus on mathematical developments in that era, the personal lives of Russian mathematicians, and political events that shaped the course of scientific work in the Soviet Union. Important contributions include an article about Luzin and his school, based in part on documents that were released only after perestroika, and two articles on Kolmogorov. The volume concludes with annotated bibliographies in English and Russian for further reading. The revised edition is appended by an article of Tikhomirov, which provides an update and general overview of 20th-century Moscow mathematics, and it also includes an Index of Names. This book should appeal to mathematicians, historians, and anyone else interested in Soviet mathematical history. Co-published with...

High Quality Content by WIKIPEDIA articles! In cryptography, an S-Box (Substitution-box) is a basic component of symmetric key algorithms which performs substitution. In block ciphers, they are typically used to obscure the relationship between the key and the ciphertext ? Shannon's property of confusion. In many cases, the S-Boxes are carefully chosen to resist cryptanalysis. In general, an S-Box takes some number of input bits, m, and transforms them into some number of output bits, n: an m?n S-Box can be implemented as a lookup table with 2m words of n bits each. Fixed tables are normally used, as in the Data Encryption Standard (DES), but in some ciphers the tables are generated dynamically from the key; e.g. the Blowfish and the Twofish encryption algorithms.

Marcel Berger

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to...

George E. P. Box

The authority on building empirical models and the fitting of such surfaces to data—completely updated and revised Revising and updating a volume that represents the essential source on building empirical models, George Box and Norman Draper—renowned authorities in this field—continue to set the standard with the Second Edition of Response Surfaces, Mixtures, and Ridge Analyses, providing timely new techniques, new exercises, and expanded material. A comprehensive introduction to building empirical models, this book presents the general philosophy and computational details of a number of important topics, including factorial designs at two levels; fitting first and second–order models; adequacy of estimation and the use of transformation; and occurrence and elucidation of ridge systems. Substantially rewritten, the Second Edition reflects the emergence of ridge analysis of second–order response surfaces as a very practical tool that can be easily applied in a variety...

High Quality Content by WIKIPEDIA articles! A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction Pk of nodes in the network having k connections to other nodes goes for large values of k as Pk ~ k?? where ? is a constant whose value is typically in the range 2<?<3, although occasionally it may lie outside these bounds. Scale-free networks are noteworthy because many empirically observed networks appear to be scale-free, including the protein networks, citation networks, and some social networks.

Margaret L. Lial, John Hornsby, Terry McGinnis

Key Message: The Lial series has helped thousands of readers succeed in developmental mathematics through its approachable writing style, relevant real-world examples, extensive exercise sets, and complete supplements package.

High Quality Content by WIKIPEDIA articles! In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square (the result of multiplying the number by itself) is x.Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted with a radical sign as scriptstyle sqrt{x}. The square root can also be written in exponent notation, as x1/2. For example, the principal square root of 9 is 3, denoted scriptstyle sqrt{9} = 3, because 32 = 3 ? 3 = 9 and 3 is non-negative. The principal square root of a positive number, however, is only one of its two square roots.

High Quality Content by WIKIPEDIA articles! In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g (mod n). That is, if g is a primitive root (mod n), then for every integer a that has gcd(a, n) = 1, there is an integer k such that gk ? a (mod n). k is called the index of a. That is, g is a generator of the multiplicative group of integers modulo n.

High Quality Content by WIKIPEDIA articles! In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels. The subject was originally and simultaneously developed by Nachman Aronszajn (1907?1980) and Stefan Bergman (1895?1977) in 1950. In this article we assume that Hilbert spaces are complex. The main reason for this is that many of the examples of reproducing kernel Hilbert spaces are spaces of analytic functions, although some real Hilbert spaces also have reproducing kernels.