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Borel summation and splitting of separatrices for the Hénon map - Gelfreich, Sauzin (ResearchIndex) Borel summation and splitting of separatrices for the Hénon map (1999) V.Gelfreich, D.Sauzin, Borel summation and the splitting of separatrices for the Henon map. @misc{ gelfreich99borel, author = "V. Gelfreich and D. Sauzin", title = "Borel summation and the splitting of separatrices for the Henon map", text = "V.Gelfreich, D.Sauzin, Borel summation and the splitting of separatrices for the Henon map. citeseer.ist.psu.edu /433574.html   (642 words)

Divergent series - Wikipedia, the free encyclopedia The subject of divergent series, as a domain of mathematical analysis, is primarily concerned with explicit and natural techniques such as Abel summation, Cesàro summation and Borel summation, and their relationships. Summation of divergent series is also related to extrapolation methods and sequence transformations as numerical techniques. The third condition is less important, and some significant methods such as Borel summation do not possess it. en.wikipedia.org /wiki/Divergent_series   (969 words)

NationMaster - Encyclopedia: Borel's lemma   (Site not responding. Last check: ) In mathematics, Borel's lemma is an important result about partial differential equations named after Émile Borel. In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. This article incorporates material from Borel lemma (http://planetmath.org/?op=getobjandfrom=objectsandid=6185) on PlanetMath, which is licensed under the GFDL. www.nationmaster.com /encyclopedia/Borel%27s-lemma   (388 words)

Borel - Qwika Borel Borel is the name of several persons: Émile Borel, French mathematician and physicist Eugène Borel, Swiss politician Jean Louis Borel, French general Pascal Borel, German football... Borel measure In mathematics, the Borel algebra is the smallest σ-algebra on... Borel summation In mathematics, a Borel summation is a generalisation of the usual... www.qwika.com /find/Borel   (385 words)

Borel summation and momentum-plane analyticity in perturbative QCD Borel summation and momentum-plane analyticity in perturbative QCD We derive a compact expression for the Borel sum of a QCD amplitude in terms of the inverse Mellin transform of the corresponding Borel function. An interesting connection between the asymptotic behaviour of the Borel transform and the Landau singularities in the momentum plane is established. stacks.iop.org /1126-6708/1999/i=03/a=007   (313 words)

Borel Summation   (Site not responding. Last check: ) 50) this.border=1; this.alt='Thumbnails by Thumbshots.org';">SLAC-PUB-8059 -- Borel summation and momentum-plane analyticity in... SLAC-PUB-8059 Borel summation and momentum-plane analyticity in perturbative QCD... Borel summation finds application in perturbation theory where physicists frequently require the sum of a series even though it is divergent. www.borelsummation.info   (356 words)

SLAC-PUB-8059 -- Borel summation and momentum-plane analyticity in perturbative QCD SLAC-PUB-8059 -- Borel summation and momentum-plane analyticity in perturbative QCD Borel summation and momentum-plane analyticity in perturbative QCD We derive a compact expression for the Borel sum of a QCD amplitude in terms of the inverse Mellin transform of the corresponding Borel function. www.slac.stanford.edu /pubs/slacpubs/8000/slac-pub-8059.html   (162 words)

Borel summation: Encyclopedia II - Borel summation - Definition Suppose that has a nonzero radius of convergence as a function of t can be analytically continued to a function on all of the positive real line grows at most exponentially along the positive real line Then the Borel sum of y is given by the Laplace transform of. Borel summation, Borel summation - Definition, Borel summation - Discussion, Borel summation - Applications Then the Borel sum of y is given by the Laplace transform of. www.experiencefestival.com /a/Borel_summation_-_Definition/id/618916   (343 words)

[No title]   (Site not responding. Last check: ) A12 (1997) 3625-3663 \\ Methods of summation of power series relevant to applications in quantum theory are reviewed, with particular attention to expansions in powers of the coupling constant and in inverse powers of an energy variable. Alternatives to the Borel summation method are considered and their relevance to different physical situations is discussed. Applications of the renormalon language to perturbation expansions (resummation of bubble chains) in various QCD processes are reported and the importance of observing the full renormalization-group invariance in predicting observables is emphasized. www.thphys.uni-heidelberg.de /cgi-bin/abstracts/hep-ph:9704351   (200 words)

Pakistan encyclopedia : Cultural Information , Maps, Pakistan politics and officials, Pakistan History. Travel to ... Félix Édouard Justin Émile Borel (January 7, 1871 – February 3, 1956) was a French mathematician and politician. The concept of a Borel set is named in his honor. One of his books on probability introduced the amusing thought experiment that entered popular culture under the name infinite monkey theorem or the like. www.pakistaneworld.com /wiki-%C3%89mile_Borel   (193 words)

Citebase - Pade Approximants and Borel Summation for QCD Perturbation Expansions   (Site not responding. Last check: ) Pade Approximants and Borel Summation for QCD Perturbation Expansions If the latter has poles on the positive semiaxis, the Borel integral does not exist, but we point out that the Cauchy pricipal value integral can exist and that it represents one of the possible "sums" of the original series, the one that is real on the positive semiaxis. The conformal mapping of the Borel plane can be utilized for the analytic continuation of the Borel transform to the entire positive real semi-axis and is thus helpful in the resummation of divergent perturbation series in quantum field theory. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/9903151   (1104 words)

Springer Online Reference Works In Tauberian theorems concerning such cases, conditions on a series (sequence) are established under which convergence follows from summability by a given method. Tauber [1], who was the first to prove two theorems of this type for the Abel summation method: Apart from ordinary summability, in the theory of summation Tauberian theorems are considered for special types of summability (absolute, strong, summability with a weight, etc.). eom.springer.de /T/t092280.htm   (743 words)

Gelfreich, Sauzin: Borel summation and splitting of separatrices for the Hénon map Borel summation and splitting of separatrices for the Hénon map The Borel transform of the formal series defines an analytic germ. Hakim and K. Mallick, “Exponentially small splittings of separatrices, matching in the complex plane and Borel summation”, Nonlinearity 6 (1993), p. aif.cedram.org /aif-bin/item?id=AIF_2001__51_2_513_0   (463 words)

On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE's - Costin (ResearchIndex) It is shown that the formal asymptotic exponential series solutions (transseries solutions: countable linear combinations of formal power series multiplied by small exponentials) are Borel summable in a generalized sense along any direction in which the exponentials decay. 2: trans-series and generalized Borel summation for analytic nonlinear rank one sys.. Costin, O. On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE's. citeseer.ist.psu.edu /494396.html   (533 words)

[No title] Borel summation is very useful in quantum mechanics and quantum field theory. For example, consider the anharmonic oscillator, which is the quantum mechanical system with Hamiltonian H = (p^2 + q^2)/2 + c q^4 When the coupling constant c is greater than zero, the ground state energy of this system is an analytic function of c. Fortunately, this series is asymptotic - and you can use Borel summation when c > 0 to sum it to the right answer. www.math.niu.edu /~rusin/known-math/00/borel_sum   (677 words)

Mellin - Barnes regularization, Borel summation and the Bender - Wu asymptotics for the anharmonic oscillator The technique is shown to be numerically equivalent to the corresponding results obtained by Borel summation. The MB regularized versions, equations (6) and (7), were evaluated by using the NIntegrate routine from Mathematica, which was also used to evaluate the integrals in the Borel summed forms, equations (8) and (10). In each instance the sum of the truncated series with the remainder evaluated by using MB regularization and Borel summation yields the corresponding value in table 1. www.iop.org /EJ/article/0305-4470/31/38/002/ja31038l2.html   (2926 words)

Divergent series The operator giving the sum of a convergent sequence is linear, and it follows from the Hahn-Banach theorem that it may be extended to a summation method summing any bounded sequence. A desirable property for two distinct summation methods A and B to share is consistency: A and B are consistent if for every sequence s to which both assign a value, A(s) = B(s). The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation. www.dejavu.org /cgi-bin/get.cgi?ver=93&url=http://articles.gourt.com/%22http%3A%2F%2Farticles.gourt.com%2F%3Farticle%3DDivergent   (1028 words)

[0708.2201] A Modified Borel Summation Technique Abstract: We compare and contrast three different perturbative expansions for the quartic anharmonic oscillator wavefunction and apply a modified Borel summation technique to determine the energy eigenvalues. In the first two expansions this provides the energy eigenvalues directly however in the third method we tune the wavefunctions to achieve the correct large x behaviour. We also refine the modified Borel summation technique to improve its accuracy. babbage.sissa.it /abs/0708.2201   (131 words)

LARGE ORDERS OF THE 1/<I>n</I>-EXPANSION IN ATOMIC PHYSICS The reason is the divergence of the expansion, which renders convential summation methods ineffective beyond the lowest orders. The factorial increase of the coefficients leads to the singularity in the Borel function at z=a-1. So, by taking into account the divergent large-order behaviour of the expansion one can localize the nearest singularity to the origin in the Borel function and take full advantage of the Pade - Borel summation technique. www.asergeev.com /files/confs/pres25.htm   (960 words)

[No title]   (Site not responding. Last check: ) In the present paper we obtain the Minkowskian quantities by the analytic continuation of the all-orders Euclidean amplitudes expressed in terms of the inverse Mellin transform of the corresponding Borel functions [2]. Our result shows that if the principle of analytic continuation is preserved in Borel-type resummations, the Minkowskian quantities exhibit a divergent increase in the infrared regime, which contradicts the claim made in [1]. We discuss the arguments given in [1] and show that the special redefinition of Borel summation at low energies adopted there does not reproduce the lowest order result obtained by analytic continuation. www.thphys.uni-heidelberg.de /cgi-bin/abstracts/hep-ph:0505016   (120 words)

Publication : T97/040 We examine here the influence of these additional terms on the estimates of critical exponents of the N-vector model, using some new ideas in the context of the Borel summation techniques. More importantly, perhaps, summation errors are better determined. The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values. www-spht.cea.fr /articles/t97/040   (253 words)

Stavros Garoufalidis: Abstract [1] Using a mild resurgence hypothesis for the function to be summed, we give a Borel summable transseries expression for the remainder term, as well as a Laplace integral formula, with an explicit integrand which is a resurgent function itself. In particular, our summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval. And another concerns resurgence of 1-dimensional sums of quantum factorials, that are associated to knotted 3-dimensional objects. www.math.gatech.edu /~stavros/publications/resurgenceeulermaclaurin.html   (216 words)

[No title] A "summation method" is a way of defining a sum for a certain class of divergent series! People have proved, for example, that applying "Borel summation" to the perturbation expansion for the ground state of the quantum anharmonic oscillator gives the exact answer. With more work, people have shown that Borel summation can allow you to get exact answers for phi^4 theory in 1+1 and 2+1 dimensions. www.math.niu.edu /~rusin/known-math/00_incoming/pathint   (1800 words)

Abstract of NTZ 03/00 This information may be recovered by means of a summation method applied to the series. The Borel summation method provides a basis for such summation methods adjusted to the considered differential equation. The Hankel functions were chosen as a natural basis of solutions related to the infinity which is an irregular singular point for the equation. www.uni-leipzig.de /~ntz/abs/abs0300.htm   (281 words)

Re: Borel summation, constructive QCD, etc. I can see > from the web that he proved a theorem about Borel summability, > but not what that theorem says. As presented in that book (Theorem I.5.1), the theorem states that a *sufficient* condition for f to be Borel summable is that it admit an asymptotic expansion about the origin r-1 k f(z) = sum a z + R (z) k=0 k r such that the bound r r R (z) Sokal, An improvement of Watson's theorem on Borel Summability, J. www.lns.cornell.edu /spr/2001-07/msg0034036.html   (232 words)

Gelfreich, Sauzin: Borel summation and splitting of separatrices for the Hénon map Gelfreich, Sauzin: Borel summation and splitting of separatrices for the Hénon map Borel summation and splitting of separatrices for the Hénon map. We explore the Riemann surface and singularities of its analytic continuation. www.numdam.org /item?id=AIF_2001__51_2_513_0   (364 words)

SPAMS Email Archive - Richard Stone - Fourier Series and Divergent Series We'll begin by quickly mentioning two of these, due to Cesaro and Borel, and describing an extension of them using eigenvalues and eigenfunctions of operators. We'll describe a famous theorem of Fejer that shows that the convergence behaviour of Fourier series is much better when we use Cesaro summation than just ordinary summation. But we'll then outline a computation suggesting that in some ways Borel summation may be even better. www-math.mit.edu /spams/spring1999/email7.htm   (150 words)

Borel Summation However, there does exist a perfectly reasonable exact answer to this problem, provided the additional potential is sufficiently well behaved. In some cases the exact answer can be obtained from the divergent series by a Borel summation of this series. The Borel summation consists in interchanging the sum and the integral. www.nbi.dk /~polesen/borel/node7.html   (295 words)

Citebase - Borel Summation of the Derivative Expansion and Effective Actions Borel Summation of the Derivative Expansion and Effective Actions Authors: Dunne, Gerald V. Hall, Theodore M. We give an explicit demonstration that the derivative expansion of the QED effective action is a divergent but Borel summable asymptotic series, for a particular inhomogeneous background magnetic field. Resummations of leading Borel approximations exponentiate to give perturbative corrections to the exponents in the non-perturbative pair production rates. www.citebase.org /abstract?id=oai:arXiv.org:hep-th/9902064   (183 words)

Michel Weber We obtain a general sufficient condition for a given matrix $A$ to have the Borel property. We show that this result cannot be extended to the study of the Borel summation method on arbitrary dynamical systems. However, in the $L^p$-setting, we establish necessary conditions of the same kind by using Bourgain's entropy criterion. www.univie.ac.at /EMIS/journals/CMUC/cmuc9502/abs/weber.htm   (91 words)

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