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128
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 59 (13 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
The Components of Content
 Philosophy of Mind: Classical and Contemporary Readings. Oxford and
, 2002
"... and 9 are similar to the old version, but the other sections are quite different. Because the old version has ..."
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Cited by 34 (6 self)
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and 9 are similar to the old version, but the other sections are quite different. Because the old version has
Contiguous relations, basic hypergeometric functions and orthogonal polynomials
 I, J. Math. Anal. Appl
, 1989
"... Abstract. Explicit solutions for the threeterm recurrence satisfied by associated continuous dual qHahn polynomials are obtained. A minimal solution is identified and an explicit expression for the related continued fraction is derived. The absolutely continuous component of the spectral measure i ..."
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Cited by 28 (5 self)
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Abstract. Explicit solutions for the threeterm recurrence satisfied by associated continuous dual qHahn polynomials are obtained. A minimal solution is identified and an explicit expression for the related continued fraction is derived. The absolutely continuous component of the spectral measure is obtained. Eleven limit cases are discussed in some detail. These include associated big qLaguerre, associated Wall, associated AlSalamChihara, associated AlSalamCarlitz I, and associated continuous qHermite polynomials.
The Unsymmetric Lanczos Algorithms And Their Relations To Padé Approximation, Continued Fractions, And The QD Algorithm
, 1990
"... . First, several algorithms based on the unsymmetric Lanczos process are surveyed: the biorthogonalization (BO) algorithm for constructing a tridiagonal matrix T similar to a given matrix A (whose extreme spectrum is sought typically); the "BOBC algorithm", which generates directly the LU ..."
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Cited by 25 (6 self)
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. First, several algorithms based on the unsymmetric Lanczos process are surveyed: the biorthogonalization (BO) algorithm for constructing a tridiagonal matrix T similar to a given matrix A (whose extreme spectrum is sought typically); the "BOBC algorithm", which generates directly the LU factors of T ; the Biores (Lanczos/Orthores), Biomin (Lanczos/Orthomin or biconjugate gradient (BCG)), and the Biodir (Lanczos/Orthodir) algorithms for solving a nonsymmetric system of linear equations. The possibilities of breakdown in these algorithms are discussed and brought into relation. Then the connections to formal orthogonal polynomials, Pad'e approximation, continued fractions, and the qd algorithm are reviewed. They allow us to deapen our understanding of breakdowns. Next, three types of (bi)conjugate gradient squared (CGS) algorithms are presented: Biores 2 , Biomin 2 (standard CGS), and Biodir 2 . Finally, fast Hankel solvers related to the Lanczos process are described. 1 Key ...
Wellpoised hypergeometric service for diophantine problems of zeta values
 J. Theorie Nombres Bordeaux
, 2003
"... It is explained how the classical concept of wellpoised hypergeometric series and integrals becomes crucial in studing arithmetic properties of the values of Riemann’s zeta function. By these wellpoised means we obtain: (1) a permutation group for linear forms in 1 and ζ(4) = π 4 /90 yielding a ..."
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Cited by 22 (8 self)
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It is explained how the classical concept of wellpoised hypergeometric series and integrals becomes crucial in studing arithmetic properties of the values of Riemann’s zeta function. By these wellpoised means we obtain: (1) a permutation group for linear forms in 1 and ζ(4) = π 4 /90 yielding a conditional upper bound for the irrationality measure of ζ(4); (2) a secondorder Apérylike recursion for ζ(4) and some loworder recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Eulertype multiple integrals that generalize socalled Beukers’ integrals for ζ(2) and ζ(3).
Shiftregister synthesis (modulo m)
 SIAM J. Computing
, 1985
"... The BerlekampMassey algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. In this paper we extend the algorithm to the case when the elements of the sequence are integers modulo m, where m is ..."
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Cited by 21 (0 self)
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The BerlekampMassey algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. In this paper we extend the algorithm to the case when the elements of the sequence are integers modulo m, where m is an arbitrary integer with known prime decomposition.
An Apérylike difference equation for Catalan's constant
 The Electronic Journal of Combinatorics
, 2003
"... Applying Zeilberger's algorithm of creative telescoping to a family of certain verywellpoised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a secondorder difference equation for these forms and their coefficients. As a consequen ..."
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Cited by 20 (4 self)
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Applying Zeilberger's algorithm of creative telescoping to a family of certain verywellpoised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a secondorder difference equation for these forms and their coefficients. As a consequence we derive a new way of fast calculation of Catalan's constant as well as a new continuedfraction expansion for it. Similar arguments are put forward to deduce a secondorder difference equation and a new continued fraction for &zeta;(4) = &pi;^4/90.