Probing the singularities of the Landau-gauge gluon and ghost propagators with rational approximants

We employ Pad\'e approximants in the study of the analytic structure of the four-dimensional $SU(2)$ Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, serve as fitting functions for the lattice data. We carefully propagate the uncertainties due to the fitting procedure, taking into account all possible correlations. For the gluon-propagator data, we confirm the presence of a pair of complex poles at $p_{\rm pole}^2 = \left[(-0.37 \,\pm\, 0.05_{\rm stat}\,\pm\, 0.08_{\rm sys}) \pm i\,(0.66\, \pm\, 0.03_{\rm stat}\, \pm\, 0.02_{\rm sys})\right]\, \mathrm{GeV}^2$, where the first error is statistical and the second systematic. The existence of this pair of complex poles, already hinted upon in previous works, is thus put onto a firmer basis, thanks to the model independence and to the careful error propagation of our analysis. For the ghost propagator, the Pad\'es indicate the existence of a single pole at $p^2 = 0$, as expected. In this case, our results also show evidence of a branch cut along the negative real axis of $p^2$. This is corroborated with another type of approximant, the D-Log Pad\'es, which are better suited to studying functions with a branch cut and are applied here for the first time in this context. Due to particular features and limited statistics of the gluon-propagator data, our analysis is inconclusive regarding the presence of a branch cut in the gluon case.Comment: 36 pages, 12 figures. Small changes in the text, references added, results unchanged. Accepted for publication in JHE

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 Ingeniería MatemáticaPresidente: Francisco José Marcellán Español; Vocal: Alexander Ivanovich Aptekarev; Secretario: Andrei Martínez Finkelshtei

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd

Convergence and Asymptotic of Multi-Level Hermite-Padé Polynomials

Mención Internacional en el título de doctorPrograma de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Francisco José Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Arnoldus Bernardus Jacobus Kuijla

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 para­meters 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

Uniformization and Constructive Analytic Continuation of Taylor Series

We analyze the general mathematical problem of global reconstruction of a function with least possible errors, based on partial information such as n terms of a Taylor series at a point, possibly also with coefficients of finite precision. We refer to this as the "inverse approximation theory problem, because we seek to reconstruct a function from a given approximation, rather than constructing an approximation for a given function. Within the class of functions analytic on a common Riemann surface Omega, and a common Maclaurin series, we prove an optimality result on their reconstruction at other points on Omega, and provide a method to attain it. The procedure uses the uniformization theorem, and the optimal reconstruction errors depend only on the distance to the origin. We provide explicit uniformization maps for some Riemann surfaces Omega of interest in applications. One such map is the covering of the Borel plane of the tritronquee solutions to the Painleve equations PI-PV. As an application we show that this uniformization map leads to dramatic improvement in the extrapolation of the PI tritronquee solution throughout its domain of analyticity and also into the pole sector. Given further information about the function, such as is available for the ubiquitous class of resurgent functions, significantly better approximations are possible and we construct them. In particular, any one of their singularities can be eliminated by specific linear operators, and the local structure at the chosen singularity can be obtained in fine detail. More generally, for functions of reasonable complexity, based on the nth order truncates alone we propose new efficient tools which are convergent as n to infty, which provide near-optimal approximations of functions globally, as well as in their most interesting regions, near singularities or natural boundaries.Comment: 39 pages, 9 figures; v2 some clarifications adde