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Creative Telescoping for Holonomic Functions
"... Abstract The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of ..."
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Abstract The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts. 1
Hermite Reduction and Creative Telescoping for Hyperexponential Functions
, 2013
"... We present a new reduction algorithm that simultaneously extends Hermite’s reduction for rational functions and the Hermitelike reduction for hyperexponential functions. It yields a unique additive decomposition that allows to decide hyperexponential integrability. Based on this reduction algorithm ..."
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We present a new reduction algorithm that simultaneously extends Hermite’s reduction for rational functions and the Hermitelike reduction for hyperexponential functions. It yields a unique additive decomposition that allows to decide hyperexponential integrability. Based on this reduction algorithm, we design a new algorithm to compute minimal telescopers for bivariate hyperexponential functions. One of its main features is that it can avoid the costly computation of certificates. Its implementation outperforms Maple’s function DEtools[Zeilberger]. We also derive an order bound on minimal telescopers that is tighter than the known ones.
Desingularization Explains OrderDegree Curves for Ore Operators
"... Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An orderdegree curve for a given Ore operator is a curve in the (r, d)plane such that for all points (r, d) above this curve, the ..."
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Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An orderdegree curve for a given Ore operator is a curve in the (r, d)plane such that for all points (r, d) above this curve, there exists a left multiple of order r and degree d of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies orderdegree curves which are extremely accurate in examples. Categories and Subject Descriptors
Fast generalized
 DFT and DHT algorithms,” Signal Processing
, 1998
"... Adapting to rank address the the problem of insufficient domainspecific labeled training data in learning to rank. However, the initial study shows that adaptation is not always effective. In this paper, we investigate the relationship between the domain similarity and the effectiveness of domain ad ..."
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Adapting to rank address the the problem of insufficient domainspecific labeled training data in learning to rank. However, the initial study shows that adaptation is not always effective. In this paper, we investigate the relationship between the domain similarity and the effectiveness of domain adaptation with the help of two domain similarity measure: relevance correlation and sample distribution correlation. Categories and Subject Descriptors
The Holonomic Toolkit
"... This is an overview over standard techniques for holonomic functions, written for readers who are new to the subject. We state the definition for holonomy in a couple of different ways, including some concrete special cases as well as a more abstract and more general version. We give a collection ..."
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This is an overview over standard techniques for holonomic functions, written for readers who are new to the subject. We state the definition for holonomy in a couple of different ways, including some concrete special cases as well as a more abstract and more general version. We give a collection of standard examples and state several fundamental properties of holonomic objects. Two techniques which are most useful in applications are explained in some more detail: closure properties, which can be used to prove identities among holonomic functions, and guessing, which can be used to generate plausible conjectures for equations satisfied by a given function.
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.
Bounds for Dfinite closure properties
, 2014
"... We provide bounds on the size of operators obtained by algorithms for executing Dfinite closure properties. For operators of small order, we give bounds on the degree and on the height (bitsize). For higher order operators, we give degree bounds that are parameterized with respect to the order and ..."
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We provide bounds on the size of operators obtained by algorithms for executing Dfinite closure properties. For operators of small order, we give bounds on the degree and on the height (bitsize). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (orderdegree curves).