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Computing periods of rational integrals
2015
Mathematics of Computation
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A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. ...
I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. ...
Find a linear dependency relation ofier K = C(t ):
Pierre Lairez TU Berlin, Germany COMPUTING PERIODS OF RATIONAL INTEGRALS OBJECTIVE ...
doi:10.1090/mcom/3054
fatcat:o6hrtlbvcfaajd6gscskjxuocu
Periodicity of Goussarov-Vassiliev knot invariants
[article]
2002
arXiv
pre-print
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The paper is a survey of known periodicity properties of finite type invariants of knots, and their applications. ...
This illustrates the periodicity of the 2-loop part of the Kontsevich integral. ...
What is periodicity for the Kontsevich integral? ...
arXiv:math/0201055v4
fatcat:u74mh3dlmzhlfjx2ffrz5zs54a
Blowing up Feynman integrals
2008
Nuclear Physics B - Proceedings Supplements
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2008 pre-print arXiv:0806.4307v1 fatcat:opniln5zkvb7dh6dexzhei43hi
We report on an open-source implementation of this algorithm to compute numerically the Laurent expansion of divergent multi-loop integrals. ...
We also show how this method can be used to prove a theorem which relates the coefficients of the Laurent series of dimensionally regulated multi-loop integrals to periods. ...
The defining integrals of periods have integrands, which are rational functions with rational coefficients. ...
doi:10.1016/j.nuclphysbps.2008.09.113
fatcat:viirflpfazhxfjir4rpsrpoi7a
Arithmetical Method to Detect Integrability in Maps
2003
Physical Review Letters
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We develop a method to detect the presence of integrals of the motion in symplectic rational maps, by representing these maps over finite fields and examining their orbit structure. ...
We find markedly different orbit statistics depending upon whether the map is integrable or not. ...
The corresponding continued fractions give several thousand rational The convergence of the first four is confirmed by repeating the computation on subsets of these primes. ...
doi:10.1103/physrevlett.90.034102
pmid:12570490
fatcat:obmiymp7wjazbhkzkz4m24vnxu
Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes
[article]
2007
arXiv
pre-print
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We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope. ...
Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. ...
Thus we avoid computing the Ehrhart quasi-polynomials of the individual pieces. This approach provides a unified framework for demonstrating quasi-period collapse of rational polytopes. ...
arXiv:0709.4070v1
fatcat:ip5r7pmxrrh7ph6nxegosilbte
A Rational Approximation of the Fourier Transform by Integration with Exponential Decay Multiplier
2021
Applied Mathematics
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Recently we have reported a new method of rational approximation of the sinc function obtained by sampling and the Fourier transforms. ...
A MATLAB code showing algorithmic implementation of the proposed method for rational approximation of the Fourier transform is presented. ...
., York University and Epic College of Technology. ...
doi:10.4236/am.2021.1211063
fatcat:5zc7p6omevgc3m5ozxh3hahb3u
Page 6486 of Mathematical Reviews Vol. , Issue 98J
[page]
1998
Mathematical Reviews
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Comput. 88 (1997), no. 2-3, 267-274.
Summary: “We develop a family of rational functions for comput- ing Dawson’s integral F(x). ...
The results can be considered as generalizations of results in the one-periodic case of multiplicities by Ghizzetti and Ossicini and in the two-periodic case by Dryanov. ...
Local Euler-Maclaurin formula for polytopes
[article]
2006
arXiv
pre-print
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Then for every convex rational polytope P in a rational euclidean space V and every polynomial function f (x) on V, the sum of the values of f(x) at the integral points of P is equal to the sum, for all ...
faces F of P, of the integral over F of the function D(N(F)).f, where we denote by N(F) the normal cone to P along F. ...
Therefore, it is a periodic function of t with period at most equal to q f , the smallest integer such that q f < f > contains integral points. ...
arXiv:math/0507256v3
fatcat:zlspbcbpifer7nrltjjqtda3wu
Page 2019 of Mathematical Reviews Vol. , Issue 2003C
[page]
2003
Mathematical Reviews
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Summary: “Integration of periodic functions on the real line with an even rational weight function is considered. ...
(YU-NISEE; Nis) Weighted integration of periodic functions on the real line. (English summary) Orthogonal systems and applications. Appl. Math. Comput. 128 (2002), no. 2-3, 365-378. ...
Computer use in continued fraction expansions
1969
Mathematics of Computation
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In this study, the use of computers is demonstrated for the rapid expansion of a general regular continued fraction with rational elements for \/C + L, where C and L are rational numbers, C positive. ...
But now the use of computers makes possible the study of a much greater variety of continued fraction expansions. | ...
for the unique values of the BJDn, is always periodic provided the PJRn in formulas (2.8) are integral. ...
doi:10.1090/s0025-5718-69-99645-8
fatcat:v533tmfonbbh7an74etjwyj73i
Computer Use in Continued Fraction Expansions
1969
Mathematics of Computation
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In this study, the use of computers is demonstrated for the rapid expansion of a general regular continued fraction with rational elements for \/C + L, where C and L are rational numbers, C positive. ...
But now the use of computers makes possible the study of a much greater variety of continued fraction expansions. | ...
for the unique values of the BJDn, is always periodic provided the PJRn in formulas (2.8) are integral. ...
doi:10.2307/2004440
fatcat:wbwsl52yindejhlxlwztr7xuzu
New series for the cosine lemniscate function and the polynomialization of the lemniscate integral
2011
Journal of Computational and Applied Mathematics
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For complex argument, we show that the lemniscate integral can be found to near machine precision (assumed as sixteen decimal digits) by computing the roots of a polynomial of degree thirteen. ...
We discuss the numerical computation of the cosine lemniscate function and its inverse, the lemniscate integral. These were previously studied by Bernoulli, Euler, Gauss, Abel, Jacobi and Ramanujan. ...
a b s t r a c t We discuss the numerical computation of the cosine lemniscate function and its inverse, the lemniscate integral. ...
doi:10.1016/j.cam.2010.09.020
fatcat:xy3r6tlhkjfhvo4u5dfkpbasr4
Periods of Cusp Forms and Elliptic Curves Over Imaginary Quadratic Fields
1994
Mathematics of Computation
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hand, rational newforms F of weight two for the congruence subgroups r0(n), where n is an ideal in the ring of integers R of K . ...
In each case we compute numerically the value of the L-series L(F, s) at s = 1 and compare with the value of L(E, 1 ) which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several ...
In (2.14), the "twisting cycle" y(X) is in the integral homology, since X is coprime to n ; hence we may express the integral ük L,X) F-ß as an integral multiple of the period Q(F). ...
doi:10.2307/2153419
fatcat:3bydqtpmujarzmygd2mp5ewraq
Periods of cusp forms and elliptic curves over imaginary quadratic fields
1994
Mathematics of Computation
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hand, rational newforms F of weight two for the congruence subgroups r0(n), where n is an ideal in the ring of integers R of K . ...
In each case we compute numerically the value of the L-series L(F, s) at s = 1 and compare with the value of L(E, 1 ) which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several ...
In (2.14), the "twisting cycle" y(X) is in the integral homology, since X is coprime to n ; hence we may express the integral ük L,X) F-ß as an integral multiple of the period Q(F). ...
doi:10.1090/s0025-5718-1994-1185241-6
fatcat:m5podd7gdngrdgucarqrqs33aa
Page 240 of Annals of Mathematics Vol. 21, Issue
[page]
1919
Annals of Mathematics
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The corresponding value of (1) is called a period. A cycle may be reducible by deformation to a line and yet vield a period. ...
INTEGRALS BELONGING TO ALGEBRAIC SURFACES. N i Doubli inte qrals and the iy re sidues.
19. Given a rational point function R(x, y, z) on F we may set up a double integral
(1) SS Roa, y, z\dxrdy. ...
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