The problem of evaluating matrix polynomial I + A + A2 + ... + A(N-1), has been considered in [1]-[3]. The proposed algorithms require at most 3 . right perpendicular log2 N left perpendicular and 2 . right perpendicu...
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The problem of evaluating matrix polynomial I + A + A2 + ... + A(N-1), has been considered in [1]-[3]. The proposed algorithms require at most 3 . right perpendicular log2 N left perpendicular and 2 . right perpendicular log2 N left perpendicular - 1 matrix multiplications, respectively. In [2] the authors have conjectured that if the binary representation of N is (i(t)i(t-1)...i1i0)2, then the number of the matrix multiplication for the evaluation of this polynomial is at least 2 . right perpendicular log2 N right perpendicular -2 + i(t-1). In the present communication we shall prove that for many values of N there exists an algorithm requiring a fewer number of matrix multiplications, thus disproving Lei-Nakamura's conjecture.
The elementary theory of real algebra, including multiplication, is decidable. More precisely, there is an algorithm to eliminate quantifiers which does not introduce new free variables or new constants other than rat...
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. . . both Gauss and lesser mathematicians may be justified in rejoic ing that there is one science [numbertheory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it ...
. . . both Gauss and lesser mathematicians may be justified in rejoic ing that there is one science [numbertheory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. - G. H. Hardy, A Mathematician's Apology, 1940 G. H. Hardy would have been surprised and probably displeased with the increasing interest in numbertheory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography (secret codes). Less than a half-century after Hardy wrote the words quoted above, it is no longer inconceivable (though it hasn't happened yet) that the N. S. A. (the agency for U. S. government work on cryptography) will demand prior review and clearance before publication of theoretical research papers on certain types of numbertheory. In part it is the dramatic increase in computer power and sophistica tion that has influenced some of the questions being studied by number theorists, giving rise to a new branch of the subject, called "computational numbertheory. " This book presumes almost no background in algebra or number the ory. Its purpose is to introduce the reader to arithmetic topics, both ancient and very modern, which have been at the center of interest in applications, especially in cryptography. For this reason we take an algorithmic approach, emphasizing estimates of the efficiency of the techniques that arise from the theory.
Macsyma, the first computeralgebra software, was developed in 1959 at Massachusetts Institute of Technology. There are currently a number of computeralgebra programs available, such as Mathematica, Maple, etc. This ...
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Macsyma, the first computeralgebra software, was developed in 1959 at Massachusetts Institute of Technology. There are currently a number of computeralgebra programs available, such as Mathematica, Maple, etc. This paper focuses on Maple software for engineering technology education applications because it is easy to learn, which allows students to spend more time on engineering problems and less time learning a new computer language and programming.
Because of their applications in so many diverse areas, finite fields continue to play increasingly important roles in various branches of modern mathematics, including numbertheory, algebra, and algebraic geometry, ...
ISBN:
(数字)9780821877593
Because of their applications in so many diverse areas, finite fields continue to play increasingly important roles in various branches of modern mathematics, including numbertheory, algebra, and algebraic geometry, as well as in computer science, information theory, statistics, and engineering. Computational and algorithmic aspects of finite field problems also continue to grow in importance. This volume contains the refereed proceedings of a conference entitled Finite Fields: theory, applications and Algorithms, held in August 1993 at the University of Nevada at Las Vegas. Among the topics treated are theoretical aspects of finite fields, coding theory, cryptology, combinatorial design theory, and algorithms related to finite fields. Also included is a list of open problems and conjectures. This volume is an excellent reference for applied and research mathematicians as well as specialists and graduate students in information theory, computer science, and electrical engineering.
We derive algorithms for the iterative reconstruction of discrete band-limited signals from their irregular samples. They converge at a geometric rate, and all constants are computable explicitly. We also treat the re...
We derive algorithms for the iterative reconstruction of discrete band-limited signals from their irregular samples. They converge at a geometric rate, and all constants are computable explicitly. We also treat the related problem of interpolation of trigonometric polynomials and give estimates for the condition number of certain positive definite Toeplitz matrices.
In this survey paper codes in finite and infinite polynomial metric spaces with given values of parameters are considered. We are especially interested in such parameters as the minimal distance and the maximal streng...
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We consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski's high school algebra problem. The part not related directly to Tarski's high school algebra p...
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We consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski's high school algebra problem. The part not related directly to Tarski's high school algebra problem. Let U be a connected complex-analytic manifold. Denote by F(U) the minimal field containing all functions meromorphic on U and closed under exponentiation f bar arrow pointing right e(f). Let f(j) is-an-element-of F(U), p(j) is-an-element-of M(U) - algebra for 1 less-than-or-equal-to j less-than-or-equal-to m, and g(k) is-an-element-of F(U), q(k) is-an-element-of M(U)-algebra for 1 less-than-or-equal-to k less-than-or-equal-to n (where M(U) is the field of functions meromorphic on U). Let f(i) - f(j) is-not-an-element-of K(U) for i not-equal j and g(k) - g(l) is-not-an-element-of K(U) for k not-equal l (where K(U) is the ring of functions holomorphic on U) . If all zeros and singularities of h = SIGMA(j=1)m p(j)e(fj)/SIGMA(k=1)n q(k)e(gk) are contained in an analytic subset of U then m = n and there exists a permutation sigma of {l , ... , m} such that h = (p(j)/q(sigma(j))) . e(fj-gsigma(j)) for 1 less-than-or-equal-to j less-than-or-equal-to m. When h is-an-element-of M(U), additionally f(j) - g(sigma(j)) is-an-element-of K(U) for all j. On Tarski's high school algebra problem. Consider L = {terms in variables and 1, +, ., up}, where T: a, b bar arrow pointing right a(b) for positive a, b . Each term t is-an-element-of L naturally determines a function tBAR : (R+)n --> R+, where n is the number of variables involved. For S subset-of L put SBAR = (tbar\t is-an-element-of S}. (i) We describe the algebraic structure of LAMBDABAR and LBAR, where LAMBDA = {t is-an-element-of L\ if u up v occurs as a subterm of t then either u is a variable or u contains no variables at all}, and L = {t is-an-element-of L\ if u up v occurs as a subterm of t then u is-an-element LAMBDA} . Of these, LAMBDABAR is a free semiring with respect to addition and multiplication but LBAR is
After some general comments about computer arithmetic consideration is given to a number of branches of mathematics to see what topics are relevant to courses in computer science and to applications of computers. The ...
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Let k be an infinite and perfect field, x1, ..., xn indeterminates over k and let f1, ..., fs be polynomials in k[x1, ..., xn] of degree bounded by a given number d, which satisfies d≥n. We prove an effective affine ...
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