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Abelian varieties can be classified via their moduli. In positive characteristic the structure of the p-torsion-structure is an additional, useful tool. For that structure supersingular abelian varieties can be considered the most special ones. They provide a starting point for the fine description of various structures. For low dimensions the moduli of supersingular abelian varieties is by now well understood. In this book we provide a description of the supersingular locus in all dimensions, in particular we compute the dimension of it: it turns out to be equal to Äg.g/4Ü, and we express the number of components as a class number, thus completing a long historical line where special cases were studied and general results were conjectured (Deuring, Hasse, Igusa, Oda-Oort, Katsura-Oort).

Group theory --- Abelian varieties --- Algebraic varieties --- Moduli theory --- Classification theory --- Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Abelse varieteiten --- Coefficiententheorie --- Theorie des coefficients --- Varieties Abelian --- Variétés abéliennes --- Algebraic geometry. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Algebraic varieties - Classification theory

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This monograph covers dynamical inverse problems, that is problems whose data are the values of wave fields. It deals with the problem of determination of one or more coefficients of a hyperbolic equation or a system of hyperbolic equations. The desired coefficients are functions of point. Most attention is given to the case where the required functions depend only on one coordinate. The first chapter of the book deals mainly with methods of solution of one-dimensional inverse problems. The second chapter focuses on scalar inverse problems of wave propagation in a layered medium. In the final chapter inverse problems for elasticity equations in stratified media and acoustic equations for moving media are given.

Wave-motion, Theory of. --- Undulatory theory --- Mechanics --- Acoustic Equations. --- Coefficients. --- Determination. --- Dynamical Inverse Problems. --- Elasticity Equations. --- Functions of Point. --- Hyperbolic Equations. --- One-dimensional Inverse Problems. --- Scalar Inverse Problems. --- Wave Fields. --- Wave Propagation.

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"Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number P can be computed in time bounded by a fixed power of the logarithm of P. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations"-- "This book represents a major step forward from explicit class field theory, and it could be described as the start of the 'explicit Langlands program'"--

Galois modules (Algebra) --- Class field theory. --- Algebraic number theory --- Galois module structure (Algebra) --- Galois's modules (Algebra) --- Modules (Algebra) --- Arakelov invariants. --- Arakelov theory. --- Fourier coefficients. --- Galois representation. --- Galois representations. --- Green functions. --- Hecke operators. --- Jacobians. --- Langlands program. --- Las Vegas algorithm. --- Lehmer. --- Peter Bruin. --- Ramanujan's tau function. --- Ramanujan's tau-function. --- Ramanujan's tau. --- Riemann surfaces. --- Schoof's algorithm. --- Turing machines. --- algorithms. --- arithmetic geometry. --- arithmetic surfaces. --- bounding heights. --- bounds. --- coefficients. --- complex roots. --- computation. --- computing algorithms. --- computing coefficients. --- cusp forms. --- cuspidal divisor. --- eigenforms. --- finite fields. --- height functions. --- inequality. --- lattices. --- minimal polynomial. --- modular curves. --- modular forms. --- modular representation. --- modular representations. --- modular symbols. --- nonvanishing conjecture. --- p-adic methods. --- plane curves. --- polynomial time algorithm. --- polynomial time algoriths. --- polynomial time. --- polynomials. --- power series. --- probabilistic polynomial time. --- random divisors. --- residual representation. --- square root. --- square-free levels. --- tale cohomology. --- torsion divisors. --- torsion.

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UNIFAC (Computer program) --- Vapor-liquid equilibrium --- Data processing --- 541.121 <083> --- Chemical statics in general. Equilibrium in general--Tabellen. Lijsten. Indices --(niet-bibliografische) --- 541.121 <083> Chemical statics in general. Equilibrium in general--Tabellen. Lijsten. Indices --(niet-bibliografische) --- Equilibrium, Vapor-liquid --- Liquid-vapor equilibrium --- Phase rule and equilibrium --- UNIFAC. --- UNIQUAC functional-group activity coefficients

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The various types of special functions have become essential tools for scientists and engineers. One of the important classes of special functions is of the hypergeometric type. It includes all classical hypergeometric functions such as the well-known Gaussian hypergeometric functions, the Bessel, Macdonald, Legendre, Whittaker, Kummer, Tricomi and Wright functions, the generalized hypergeometric functions ? Fq , Meijer's G -function, Fox's H -function, etc. Application of the new special functions allows one to increase considerably the number of problems whose solutions are found in a closed

Legendre's functions. --- Spherical harmonics. --- Functions, Potential --- Potential functions --- Harmonic analysis --- Harmonic functions --- Functions, Legendre's --- Legendre's coefficients --- Legendre's equation --- Spherical harmonics --- Legendre's functions --- 517.58 --- 517.58 Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials. --- Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials.