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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 studies the interplay between the geometry and topology of locally symmetric spaces, and the arithmetic aspects of the special values of L-functions.The authors study the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel-Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin-Selberg L-functions for GL(n) x GL(m), where n + m = N. The authors carry through the entire program with an eye toward generalizations.This book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.
Shimura varieties. --- Cohomology operations. --- Number theory. --- Arithmetic groups. --- L-functions. --- Functions, L --- -Number theory --- Group theory --- Number study --- Numbers, Theory of --- Algebra --- Operations (Algebraic topology) --- Algebraic topology --- Varieties, Shimura --- Arithmetical algebraic geometry --- Addition. --- Adele ring. --- Algebraic group. --- Algebraic number theory. --- Arithmetic group. --- Automorphic form. --- Base change. --- Basis (linear algebra). --- Bearing (navigation). --- Borel subgroup. --- Calculation. --- Category of groups. --- Coefficient. --- Cohomology. --- Combination. --- Commutative ring. --- Compact group. --- Computation. --- Conjecture. --- Constant term. --- Corollary. --- Covering space. --- Critical value. --- Diagram (category theory). --- Dimension. --- Dirichlet character. --- Discrete series representation. --- Discrete spectrum. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elaboration. --- Embedding. --- Euler product. --- Field extension. --- Field of fractions. --- Free module. --- Freydoon Shahidi. --- Function field. --- Functor. --- Galois group. --- Ground field. --- Group (mathematics). --- Group scheme. --- Harish-Chandra. --- Hecke L-function. --- Hecke character. --- Hecke operator. --- Hereditary property. --- Induced representation. --- Irreducible representation. --- K0. --- L-function. --- Langlands dual group. --- Level structure. --- Lie algebra cohomology. --- Lie algebra. --- Lie group. --- Linear combination. --- Linear map. --- Local system. --- Maximal torus. --- Modular form. --- Modular symbol. --- Module (mathematics). --- Monograph. --- N0. --- National Science Foundation. --- Natural number. --- Natural transformation. --- Nilradical. --- Permutation. --- Prime number. --- Quantity. --- Rational number. --- Reductive group. --- Requirement. --- Ring of integers. --- Root of unity. --- SL2(R). --- Scalar (physics). --- Sheaf (mathematics). --- Special case. --- Spectral sequence. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Theorem. --- Theory. --- Triangular matrix. --- Triviality (mathematics). --- Two-dimensional space. --- Unitary group. --- Vector space. --- W0. --- Weyl group.
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