On a shifted LR transformation derived from the discrete hungry Toda equation

The discrete hungry Toda (dhToda) equation is known as an integrable system which is derived from the study of the numbered box and ball system. In the authors’ paper (Fukuda et al., in Phys Lett A 375, 303–308, 2010), we associate the dhToda equation with a sequence of LR transformations for a totally nonnegative (TN) matrix, and then, in another paper (Fukuda et al. in Annal Math Pura Appl, 2011), based on the dhToda equation, we design an algorithm for computing the eigenvalues of the TN matrix. In this paper, in order to accelerate the convergence speed, we first introduce the shift of origin into the LR transformations associated with the dhToda equation, then derive a recursion formula for generating the shifted LR transformations.We next present a shift strategy for avoiding the breakdown of the shifted LR transformations. We finally clarify the asymptotic convergence and show that the sequence of TN matrices generated by the shifted LR transformations converges to a lower triangular matrix, exposing the eigenvalues of the original TN matrix. The asymptotic convergence is also verified through some numerical examples.     

Quasilinear sequence transformations

We give a new characterization of quasilinear sequence transformations; we prove that any quasilinear transformation can be represented by its kernel. This approach is new and allows one to give a general result of convergence acceleration and tools for the construction of new quasilinear transformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Optimal transformations for scalar sequences

Summary We consider transformations which accelerate convergence in some specified classes of convergent sequences. As an asymptotic measure of acceleration we introduce the order of transformation. We find a sharp upper bound on the order and show the explicit form of transformations of maximal order. We consider also the efficiency of transformations for fast convergent sequences. As a special case we find that the Germain-Bonne version of Richardson extropolation has maximal order for linearly convergent sequences.     

Effect of transformations of numerical parameters in automatic algorithm configuration

We study the impact of altering the sampling space of parameters in automatic algorithm configurators. We show that a proper transformation can strongly improve the convergence towards better configurations; at the same time, biases about good parameter values, possibly based on misleading prior knowledge, may lead to wrong choices in the transformations and be detrimental for the configuration process. To emphasize the impact of the transformations, we initially study their effect on configuration tasks with a single parameter in different experimental settings. We also propose a mechanism for how to adapt towards an appropriate transformation and give exemplary experimental results of that scheme.     

Spectral methods for weakly singular Volterra integral equations with pantograph delays

In this paper, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation with pantograph delays defined on the interval [-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. We provide a rigorous error analysis for the proposed method in the L^∞-norm and the weighted L²-norm. Numerical examples are presented to complement the theoretical convergence results.     

Pointwise convergence of ergodic averages along cubes

Let (X, B, μ, T) be a measure preserving system. We prove the pointwise convergence of ergodic averages along cubes of 2 ^k − 1 bounded and measurable functions for all k. We show that this result can be derived from estimates about bounded sequences of real numbers and apply these estimates to establish the pointwise convergence of some weighted ergodic averages and ergodic averages along cubes for not necessarily commuting measure preserving transformations.     

The Convergence of Infinite Element Method for the Non-Similar Case

We have considered the infinite element method for a class of elliptic systems with constant coefficients in [1]. This class can be characterized as: they have the invariance under similarity transformations of independent variables. For example, the Laplace equation and the system of plane elastic equations have this property. We have suggested a technique to solve these problems by applying this property and a self similar discretization, and proved the convergence. Not only the average convergence of the solutions has been discussed, but also term-by-term convergence for the expansions of the solutions. The second convergence manifests the advantage of the infinite element method, that is, the local singularity of the solutions can be calculated with high precision.     

Central limit theorem for a class of one-dimensional kinetic equations

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation??s solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ?? _??, then the limit is a scale mixture of ?? _??. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.     

Weighted Means, Strict Ergodicity, and Uniform Distributions

We strengthen the well-known Oxtoby theorem for strictly ergodic transformations by replacing the standard Cesaro convergence by the weaker Riesz or Voronoi convergence with monotonically increasing or decreasing weight coefficients. This general result allows, in particular, to strengthen the classical Weyl theorem on the uniform distribution of fractional parts of polynomials with irrational coefficients.     

FAST PARALLELIZABLE METHODS FOR COMPUTING INVARIANT SUBSPACES OF HERMITIAN MATRICES

We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big O notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.     

Stability of Manifold-Valued Subdivision Schemes and Multiscale Transformations

Linear subdivision schemes can be adapted in various ways so as to operate in nonlinear geometries such as Lie groups or Riemannian manifolds. It is well known that along with a linear subdivision scheme a multiscale transformation is defined. Such transformations can also be defined in a nonlinear setting. We show the stability of such nonlinear multiscale transforms. To do this we introduce a new kind of proximity condition which bounds the difference of the differential of a nonlinear subdivision scheme and a linear one. It turns out that—unlike the generic nonlinear case and modulo some minor technical assumptions—in the manifold-valued setting, convergence implies stability of the nonlinear subdivision scheme and associated nonlinear multiscale transformations.     

On a generalization of the Chazy equation with a movable singular line

We generalize a third-order Chazy equation with a movable singular line, which has only negative resonances. For differential equations of order 2n+1 with resonances −1,−2, …, −(2n + 1), we study the convergence of the series representing their solutions, the existence of rational solutions, the invariance of these equations under certain transformations, and the existence of three-parameter solutions with a movable singular line.     

Normalized convergence in stochastic optimization

A new concept of (normalized) convergence of random variables is introduced. This convergence is preserved under Lipschitz transformations, follows from convergence in mean and itself implies convergence in probability. If a sequence of random variables satisfies a limit theorem then it is a normalized convergent sequence. The introduced concept is applied to the convergence rate study of a statistical approach in stochastic optimization.     

Stochastic Stability of Lyapunov Exponents and Oseledets Splittings for Semi‐invertible Matrix Cocycles

We establish (i) stability of Lyapunov exponents and (ii) convergence in probability of Oseledets spaces for semi‐invertible matrix cocycles subjected to small random perturbations. The first part extends results of Ledrappier and Young to the semi‐invertible setting. The second part relies on the study of evolution of subspaces in the Grassmannian; the analysis developed here is based on higher‐dimensional Möbius transformations and is likely to be of wider interest. © 2015 Wiley Periodicals, Inc.     

The pointwise convergence of p-adic Mbius maps

The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequences of Mbius maps acting on the projective line P1(C p),where C p is the completion of the algebraic closure of Q p.We show that if the set of pointwise convergence of a sequence of p-adic Mbius maps contains at least three points,the sequence of p-adic Mbius maps either converges to a p-adic Mbius map on the projective line P1(C p),or converges to a constant on the set of pointwise convergence with one unique exceptional point.This result generalizes the result of Piranian and Thron(1957)to the non-archimedean settings.     

On rates of acceleration of extrapolation methods for oscillatory infinite integrals

The purpose of this work is to complement and expand our knowledge of the convergence theory of some extrapolation methods for the accurate computation of oscillatory infinite integrals. Specifically, we analyze in detail the convergence properties of theW- and -transformations of the author as they are applied to three integrals, all with totally different behavior at infinity. The results of the analysis suggest different convergence and acceleration of convergence behavior for the above mentioned transformations on the different integrals, and they improve considerably those that can be obtained from the existing convergence theories.     

Convergence theorems for weakly almost periodic Markov operators

Convergence theorems of both pointwise and uniform type are obtained on certain subsets for weakly almost perodic Markov operators onC(X). Several examples are considered to examine the sharpness of these results. Finally these results are used to obtain necessary and sufficient conditions for Blum-Hanson convergence of point transformations and sufficient conditions for Blum-Hanson convergence of Markov operators.     

Multidimensional and Multivalued Ergodic Theorems for Measure-Preserving Transformations

The aim of this paper is to establish some multidimensional and multivalued Birkhoff’s ergodic theorems for measure preserving transformations. Two types of convergence will be considered: Mosco convergence and convergence in the gap topology. Some illustrative examples are provided.     

Composite sequence transformations

Summary The aim of this work is to introduce the new concept of composite sequence transformations and to show, by very simple examples and theorems, that it can be useful in accelerating the convergence of sequences. Generalizations of classical transformations and results are obtained.Work performed under the Nato Research Grant 027.81.Presented at the International Conference on Numerical Analysis, Munich, March 19–21, 1984     

Stability of a class of transformations of distribution-valued processes and stochastic evolution equations

Stability of a class of linear transformations of distribution-valued stochastic processes is studied. Two types of applications to convergence of solutions of stochastic evolution equations are given. One of them, for the case of continuous limits, simplifies the tightness problem considerably due to a recent result of Aldous.Centro de Investigación y de Estudios Avanzados.