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Computing periods of rational integrals

Pierre Lairez
2015 Mathematics of Computation  
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]

Stavros Garoufalidis
2002 arXiv   pre-print
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

Christian Bogner, Stefan Weinzierl
2008 Nuclear Physics B - Proceedings Supplements  
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

J. A. G. Roberts, F. Vivaldi
2003 Physical Review Letters  
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]

Christian Haase, Tyrrell B. McAllister
2007 arXiv   pre-print
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

Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, Brendan M. Quine
2021 Applied Mathematics  
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  
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]

Nicole Berline
2006 arXiv   pre-print
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  
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

Evelyn Frank
1969 Mathematics of Computation  
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

Evelyn Frank
1969 Mathematics of Computation  
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

John P. Boyd
2011 Journal of Computational and Applied Mathematics  
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

J. E. Cremona, E. Whitley
1994 Mathematics of Computation  
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

J. E. Cremona, E. Whitley
1994 Mathematics of Computation  
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  
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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