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I. Intzodu cUon Risch's landmarh paper [Ris89] presen ted the first decision proced~re for the integration of elementary functions. In that paper he required that the functions appearing in the integrand be algebraically independent.... more
I. Intzodu cUon Risch's landmarh paper [Ris89] presen ted the first decision proced~re for the integration of elementary functions. In that paper he required that the functions appearing in the integrand be algebraically independent. Shortly afterwards in [Risalg] and [RisTO] he relaxed that restriction and outlined a complete decision procedure for the integration of elemeniary functions in finite terms. Unfortunately his algorithms for dealing ~th algebraic functions required considerably more complex machinery than his earlier ones for purely transcendental functions. ~qeses' implementation of the earlier approach in ~%IACS~f]v!~ [MAC??] demonstrated its practicality, whereas the same has yet to be done for Risch's more recent approach.
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We study the problem of the computation of the square-free decomposition for polynomials over fields of positive characteristic. For fields which are explicitly finitely generated over perfect fields, we show how the classical algorithm... more
We study the problem of the computation of the square-free decomposition for polynomials over fields of positive characteristic. For fields which are explicitly finitely generated over perfect fields, we show how the classical algorithm for characteristic zero can be generalized using multiple derivations. For more general fields of positive characteristic one must make an additional constructive hypothesis in order for the problem to be decidable. We show that Seidenberg'sCondition P gives a necessary and sufficient condition on the fieldK for computing a complete square free decomposition of polynomials with coefficients in any finite algebraic extension ofK.
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We study the problem of the computation of the radical of an ideal of polynomials with coefficients over fields of arbitrary characteristic. We show how to use Seidenberg's condition P to solve this problem in the case of positive... more
We study the problem of the computation of the radical of an ideal of polynomials with coefficients over fields of arbitrary characteristic. We show how to use Seidenberg's condition P to solve this problem in the case of positive characteristic.
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Without Abstract
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This book and the Axiom software is licensed as follows: Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:- Redistributions of source code must... more
This book and the Axiom software is licensed as follows: Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:- Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.- Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.- Neither the name of The Numerical ALgorithms Group Ltd. nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
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0.1 Introduction to Axiom......................... 1 0.1.1 Symbolic Computation..................... 1 0.1.2 Numeric Computation..................... 2 0.1.3 Graphics............................ 3
Stochastic approximation algorithms are iterative procedures which are used to approximate a target value in an environment where the target is unknown and direct observations are corrupted by noise. These algorithms are useful, for... more
Stochastic approximation algorithms are iterative procedures which are used to approximate a target value in an environment where the target is unknown and direct observations are corrupted by noise. These algorithms are useful, for instance, for root-finding and function minimization when the target function or model is not directly known. Originally introduced in a 1951 paper by Robbins and Monro, the field of Stochastic approximation has grown enormously and has come to influence application domains from adaptive signal processing to artificial intelligence. As an example, the Stochastic Gradient Descent algorithm which is ubiquitous in various subdomains of Machine Learning is based on stochastic approximation theory. In this paper, we give a formal proof (in the Coq proof assistant) of a general convergence theorem due to Aryeh Dvoretzky, which implies the convergence of important classical methods such as the Robbins-Monro and the Kiefer-Wolfowitz algorithms. In the process, w...
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ABSTRACT We illustrate how employing Graphics Processing Units (GPU) can speed-up intensive image processing operations. In particular, we demonstrate the use of the NVIDIA CUDA architecture to implement a color digital binary halftoning... more
ABSTRACT We illustrate how employing Graphics Processing Units (GPU) can speed-up intensive image processing operations. In particular, we demonstrate the use of the NVIDIA CUDA architecture to implement a color digital binary halftoning algorithm based on Direct Binary Search (DBS). Halftoning a color image is more computationally expensive than the single color case as there is a need to minimize dot interaction between different color planes as well. We propose processing all color planes in parallel. In addition we employ processing several non-overlapping neighborhoods in parallel, by utilizing the GPU's parallel architecture, to further improve the computational efficiency. This parallel approach allows us to use a large neighborhood and filter size, to achieve the highest halftone quality, while having minimal impact on performance.
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We study the problem of the computation of the square-free decomposition for polynomials over fields of positive characteristic. For fields which are explicitly finitely generated over perfect fields, we show how the classical algorithm... more
We study the problem of the computation of the square-free decomposition for polynomials over fields of positive characteristic. For fields which are explicitly finitely generated over perfect fields, we show how the classical algorithm for characteristic zero can be generalized using multiple derivations. For more general fields of positive characteristic one must make an additional constructive hypothesis in order for the problem to be decidable. We show that Seidenberg'sCondition P gives a necessary and sufficient condition on the fieldK for computing a complete square free decomposition of polynomials with coefficients in any finite algebraic extension ofK.
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It is proved that, under the usual restrictions, the denominator of the integral of a purely logarithmic function is the expected one, that is, all factors of the denominator of the integrand have their multiplicity decreased by one.... more
It is proved that, under the usual restrictions, the denominator of the integral of a purely logarithmic function is the expected one, that is, all factors of the denominator of the integrand have their multiplicity decreased by one. Furthermore, it is determined which new logarithms may appear in the integration.
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We propose a definition of regularity of a linear differential system with coefficients in a monomial extension of a differential field, as well as a global and truly rational (i.e. factorisation–free) iteration that transforms a system... more
We propose a definition of regularity of a linear differential system with coefficients in a monomial extension of a differential field, as well as a global and truly rational (i.e. factorisation–free) iteration that transforms a system with regular finite singularities into an equivalent one with simple finite poles. We then apply our iteration to systems satisfied by bases of algebraic function fields, obtaining algorithms for computing the number of irreducible components and the genus of algebraic curves.
We propose a definition of regularity of a linear differential system with coefficients in a monomial extension of a differential field, as well as a global and truly rational (i.e. factorisation–free) iteration that transforms a system... more
We propose a definition of regularity of a linear differential system with coefficients in a monomial extension of a differential field, as well as a global and truly rational (i.e. factorisation–free) iteration that transforms a system with regular finite singularities into an equivalent one with simple finite poles. We then apply our iteration to systems satisfied by bases of algebraic function fields, obtaining algorithms for computing the number of irreducible components and the genus of algebraic curves. Introduction This paper is concerned with differential systems of the form y ′ 1 .. y′ n = A y1 .. yn (1) where the entries of the matrix A are in a differential field F . While the cyclic vector method [2, 5, 14] reduces (in theory) the study of such systems to the study of scalar differential equations, that method is well-known to be impractical except for very small n (see for example the timings in [2]), which motivates the study of direct algorithms that do no...