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Although scientific computing is very often associated with numeric computations, the use of computer algebra methods in scientific computing has obtained considerable attention in the last two decades. Computer algebra methods are especially suitable for parametric analysis of the key properties of systems arising in scientific computing. The expression-based computational answers generally provided by these methods are very appealing as they directly relate properties to parameters and speed up testing and tuning of mathematical models through all their possible behaviors. This book contains 8 original research articles dealing with a broad range of topics, ranging from algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers over methods for certifying the isolated zeros of polynomial systems to computer algebra problems in quantum computing.

superposition --- SU(2) --- pseudo-remainder --- interval methods --- sparse polynomials --- element order --- Henneberg-type minimal surface --- timelike axis --- combinatorial decompositions --- sparse data structures --- mutually unbiased bases --- invariant surfaces --- projective special unitary group --- Minkowski 4-space --- free resolutions --- Dini-type helicoidal hypersurface --- linearity --- integrability --- Galois rings --- minimum point --- entanglement --- degree --- pseudo-division --- computational algebra --- polynomial arithmetic --- projective special linear group --- normal form --- Galois fields --- Gauss map --- implicit equation --- number of elements of the same order --- Weierstrass representation --- Lotka–Volterra system --- isolated zeros --- polynomial modules --- over-determined polynomial system --- simple Kn-group --- sum of squares --- four-dimensional space

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This book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties. These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of local class field theory. The local Langlands conjecture for GLn(K), where K is a p-adic field, asserts the existence of a correspondence, with certain formal properties, relating n-dimensional representations of the Galois group of K with the representation theory of the locally compact group GLn(K). This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. The local Langlands conjecture is obtained as a corollary. Certain techniques developed in this book should extend to more general Shimura varieties, providing new instances of the local Langlands conjecture. Moreover, the geometry of the special fibers is strictly analogous to that of Shimura curves and can be expected to have applications to a variety of questions in number theory.

Mathematics --- Shimura varieties. --- MATHEMATICS / Number Theory. --- Varieties, Shimura --- Arithmetical algebraic geometry --- Math --- Science --- Abelian variety. --- Absolute value. --- Algebraic group. --- Algebraically closed field. --- Artinian. --- Automorphic form. --- Base change. --- Bijection. --- Canonical map. --- Codimension. --- Coefficient. --- Cohomology. --- Compactification (mathematics). --- Conjecture. --- Corollary. --- Dimension (vector space). --- Dimension. --- Direct limit. --- Division algebra. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Embedding. --- Equivalence class. --- Equivalence of categories. --- Existence theorem. --- Field of fractions. --- Finite field. --- Function field. --- Functor. --- Galois cohomology. --- Galois group. --- Generic point. --- Geometry. --- Hasse invariant. --- Infinitesimal character. --- Integer. --- Inverse system. --- Isomorphism class. --- Lie algebra. --- Local class field theory. --- Maximal torus. --- Modular curve. --- Moduli space. --- Monic polynomial. --- P-adic number. --- Prime number. --- Profinite group. --- Residue field. --- Ring of integers. --- Separable extension. --- Sheaf (mathematics). --- Shimura variety. --- Simple group. --- Special case. --- Spectral sequence. --- Square root. --- Subset. --- Tate module. --- Theorem. --- Transcendence degree. --- Unitary group. --- Valuative criterion. --- Variable (mathematics). --- Vector space. --- Weil group. --- Weil pairing. --- Zariski topology.

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The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.

Dynamics. --- Ergodic theory. --- Harmonic analysis. --- Lattice theory. --- Lie groups. --- Ergodic theory --- Lie groups --- Lattice theory --- Harmonic analysis --- Dynamics --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Dynamical systems --- Kinetics --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Groups, Lie --- Ergodic transformations --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Banach algebras --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Lie algebras --- Symmetric spaces --- Topological groups --- Continuous groups --- Mathematical physics --- Measure theory --- Absolute continuity. --- Algebraic group. --- Amenable group. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Automorphism. --- Borel set. --- Bounded function. --- Bounded operator. --- Bounded set (topological vector space). --- Congruence subgroup. --- Continuous function. --- Convergence of random variables. --- Convolution. --- Coset. --- Counting problem (complexity). --- Counting. --- Differentiable function. --- Dimension (vector space). --- Diophantine approximation. --- Direct integral. --- Direct product. --- Discrete group. --- Embedding. --- Equidistribution theorem. --- Ergodicity. --- Estimation. --- Explicit formulae (L-function). --- Family of sets. --- Haar measure. --- Hilbert space. --- Hyperbolic space. --- Induced representation. --- Infimum and supremum. --- Initial condition. --- Interpolation theorem. --- Invariance principle (linguistics). --- Invariant measure. --- Irreducible representation. --- Isometry group. --- Iwasawa group. --- Lattice (group). --- Lie algebra. --- Linear algebraic group. --- Linear space (geometry). --- Lipschitz continuity. --- Mass distribution. --- Mathematical induction. --- Maximal compact subgroup. --- Maximal ergodic theorem. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Monotonic function. --- Neighbourhood (mathematics). --- Normal subgroup. --- Number theory. --- One-parameter group. --- Operator norm. --- Orthogonal complement. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Pointwise convergence. --- Pointwise. --- Principal homogeneous space. --- Principal series representation. --- Probability measure. --- Probability space. --- Probability. --- Rate of convergence. --- Regular representation. --- Representation theory. --- Resolution of singularities. --- Sobolev space. --- Special case. --- Spectral gap. --- Spectral method. --- Spectral theory. --- Square (algebra). --- Subgroup. --- Subsequence. --- Subset. --- Symmetric space. --- Tensor algebra. --- Tensor product. --- Theorem. --- Transfer principle. --- Unit sphere. --- Unit vector. --- Unitary group. --- Unitary representation. --- Upper and lower bounds. --- Variable (mathematics). --- Vector group. --- Vector space. --- Volume form. --- Word metric.