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72
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 48 (10 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 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 factors of ..."
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Cited by 24 (5 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 ...
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 22 (4 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.
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 consequence we deri ..."
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Cited by 16 (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 ζ(4) = π^4/90.
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 15 (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.
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 14 (7 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).
The Formal Theory of BirthandDeath Processes, Lattice Path Combinatorics, and Continued Fractions
, 1999
"... Classic works of KarlinMcGregor and JonesMagnus have established a general correspondence between continuoustime birthanddeath processes and continued fractions of the StieltjesJacobi type together with their associated orthogonal polynomials. This fundamental correspondence is revisited here ..."
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Cited by 13 (0 self)
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Classic works of KarlinMcGregor and JonesMagnus have established a general correspondence between continuoustime birthanddeath processes and continued fractions of the StieltjesJacobi type together with their associated orthogonal polynomials. This fundamental correspondence is revisited here in the light of the basic relation between weighted lattice paths and continued fractions otherwise known from combinatorial theory. Given that trajectories of the embedded Markov chain of a birthanddeath process are lattice paths, Laplace transforms of a number of transientcharacteristics can be obtained systematically in terms of a fundamental continued fraction and its family of convergent polynomials. Applications include the analysis of evolutions in a strip, upcrossing and downcrossing times under flooring and ceiling conditions, as well as time, area, or n umber of transitions while a geometric condition is satisfied.
The generating function of ternary trees and continued fractions, Electron
 J. Combin
"... Abstract. �Michael � Somos conjectured a relation between Hankel determinants whose 1 3n entries 2n+1 n count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a cont ..."
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Cited by 10 (1 self)
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Abstract. �Michael � Somos conjectured a relation between Hankel determinants whose 1 3n entries 2n+1 n count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss’s continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos’s Hankel determinants to known determinants, and we obtain, up to a power of 3, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of rary trees.