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Fast Algorithms for Refined Parameterized Telescoping in Difference Fields
 in : Lecture Notes in Computer Science
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A Generalized ApagoduZeilberger Algorithm
"... The ApagoduZeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary ∂finite functions ..."
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The ApagoduZeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary ∂finite functions. In analogy to the hypergeometric case, we introduce the notion of proper ∂finite functions. We show that the algorithm always succeeds for these functions, and we give a tight a priori bound for the order of the output operator.
Parallel Telescoping and Parameterized Picard–Vessiot Theory
, 2014
"... Parallel telescoping is a natural generalization of differential creativetelescoping for single integrals to line integrals. It computes a linear ordinary differential operator L, called a parallel telescoper, for several multivariate functions, such that the applications of L to the functions yield ..."
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Parallel telescoping is a natural generalization of differential creativetelescoping for single integrals to line integrals. It computes a linear ordinary differential operator L, called a parallel telescoper, for several multivariate functions, such that the applications of L to the functions yield antiderivatives of a single function. We present a necessary and sufficient condition guaranteeing the existence of parallel telescopers for differentially finite functions, and develop an algorithm to compute minimal ones for compatible hyperexponential functions. Besides computing annihilators of parametric line integrals, we use the parallel telescoping for determining Galois groups of parameterized partial differential systems of first order.
FUNDAMENTAL LASER MODES IN PARAXIAL OPTICS: FROM COMPUTER ALGEBRA AND SIMULATIONS TO EXPERIMENTAL OBSERVATION
, 2015
"... We study multiparameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide, spiral light beams, and other important families of propagationinvariant laser modes in weakly varying media. A “s ..."
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We study multiparameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide, spiral light beams, and other important families of propagationinvariant laser modes in weakly varying media. A “smart ” lens design and a similar effect of superfocusing of particle beams in a thin monocrystal film are also discussed. In the supplementary electronic material, we provide a computer algebra verification of the results presented here, and of some related mathematical tools that were stated without proofs in the literature. We also demonstrate how computer algebra can be used to derive some of the presented formulas automatically, which is highly desirable as the corresponding hand calculations are very tedious. In numerical simulations, some of the new solutions reveal quite exotic properties which deserve further investigation including an experimental observation.
MULTIPARAMETER LASER MODES IN PARAXIAL OPTICS
, 2015
"... We study multiparameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide, spiral light beams, and other important families of propagationinvariant laser modes in weakly varying media. A “sm ..."
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We study multiparameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide, spiral light beams, and other important families of propagationinvariant laser modes in weakly varying media. A “smart ” lens design and a similar effect of superfocusing of particle beams in a thin monocrystal film are also discussed. In the supplementary electronic material, we provide a computer algebra verification of the results presented here, and of some related mathematical tools that were stated without proofs in the literature.