130 research outputs found
The Resurgence of the Cusp Anomalous Dimension
This work addresses the resurgent properties of the cusp anomalous
dimension's strong coupling expansion, obtained from the integral
Beisert-Eden-Staudacher (BES) equation. This expansion is factorially
divergent, and its first nonperturbative corrections are related to the mass
gap of the -model. The factorial divergence can also be analysed
from a resurgence perspective. Building on the work of Basso and Korchemsky, a
transseries ansatz for the cusp anomalous dimension is proposed and the
corresponding expected large-order behaviour studied. One finds
non-perturbative phenomena in both the positive and negative real coupling
directions, which need to be included to address the analyticity conditions
coming from the BES equation. After checking the resurgence structure of the
proposed transseries, it is shown that it naturally leads to an unambiguous
resummation procedure, furthermore allowing for a strong/weak coupling
interpolation.Comment: 12 pages, 5 figure
On the convergence of type I Hermite-Pade approximants
Fade approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite-Pade approximants. The convergence properties of type II Hermite-Pade approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite-Pade approximants for Nikishin systems of functions.Both authors were partially supported by research grant MTM2012-36372-C03-01 of Ministerio de EconomÃa y Competitividad, Spain
Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics
International audienceThe fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles
Systems of Markov type functions: normality and convergence of Hermite-Padé approximants
This thesis deals with Hermite-Padé approximation of analytic and merophorphic
functions. As such it is embeded in the theory of vector rational approximation of
analytic functions which in turn is intimately connectd with the theory of multiple
orthogonal polynomials. All the basic concepts and results used in this thesis involving
complex analysis and measure theory may found in classical textbooks...........Programa Oficial de Doctorado en IngenierÃa MatemáticaPresidente: Francisco José Marcellán Español; Vocal: Alexander Ivanovich Aptekarev; Secretario: Andrei MartÃnez Finkelshtei
The approximation of functions with branch points
In recent years Pade approximants have proved to be one of the most useful computational tools in many areas of theoretical physics, most notably in statistical mechanics and strong interaction physics. The underlying reason for this is that very often the equations describing a physical process are so complicated that the simplest (if not the only) way of obtaining their solution is to perform a power series expansion in some parameters of the problem. Furthermore, the physical values of the paraÂmeters are often such that this perturbation expansion does not converge and is therefore only a formal solution to the problem; as such it cannot be used quantitatively. However, the relevant information is contained in the coefficients of the perturbation series and the Fade approximants provide a convenient mathematical technique for extracting this information in a convergent way. A major difficulty with these approximants is that their convergence is restricted to regions of the complex plane free from any branch cuts; for example, the (N/N+j) Pade approximants to a series of Stieltjes converge to an analytic function in the complex plane cut along the negative real axis. The central idea of the present work is to obtain convergence along these branch cuts by using approximants which themselves have branch points.
The ideas presented in this thesis are expected to be only the beginning of a large investigation into the use of multi-valued approximants as a practical method of approximation. In Chapter 1 we shall see that such approximants arise as natural generalisations of Pade approximants and possess many of the properties of Pade approximants; in particular, the very important property of homographic covariance. We term these approximants ’algebraic' approximants (since they satisfy an algebraic equation) and we are mainly concerned with the 'simplest' of these approximants, the quadratic approximants of Shafer. Chapter 2 considers some of the known convergence results for Pade approximants to indicate the type of results we nay reasonably expect to hold (and to be able to prove) for quadratic (and higher order) approximants. A discussion of various numerical examples is then given to illustrate the possible practical usefulness of these latter approximants. A major application of all these approximants is discussed in Chapter 3, where the problem of evaluating Feynman matrix elements in the physical region is considered; in this case, the physical region is along branch cuts. Several simple Feynman diagrams are considered to illustrate (a) the potential usefulness of the calculational scheme presented and (b) the relative merits of rational (Pade), quadratic and cubic approximation schemes. The success of these general approximation schemes in one variable (as exhibited by the results of Chapters 2 and 3) leads, in Chapter 4 to a consideration of the corresponding approximants in two variables. We shall see that the two variable scheme developed for rational approximants can be extended in a very natural way to define two variable "t-power" approximants. Numerical results are presented to indicate the usefulness of these schemes in practice. A final application to strong interaction physics is given in Chapter 5, where the analytic continuation of Legendre series is considered. Such series arise in partial wave expansions of the scattering amplitude. We shall see that the Pade Legendre approximants of Fleischer and Common can be generalised to produce corresponding quadratic Legendre approximants: various examples are considered to illustrate the relative merits of these schemes
- …