On Some Spectral Characteristics of Multiparameter Polynomial Matrices

Definitions of certain spectral characteristics of polynomial matrices (such as the analytical (algebraic) and geometric multiplicities of a point of the spectrum, deflating subspaces, matrix solvents, and block eigenvalues and eigenvectors) are generalized to the multiparameter case, and properties of these characteristics are analyzed. Bibliogrhaphy: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 166–173.     

Modifications of the ΔW-q Factorization Method for Multiparameter Polynomial Matrices and Their Properties

Properties of the method of ΔW-q factorization of multiparameter polynomial matrices are analyzed. Modifications of the method, used in solving spectral and other multiparameter problems of algebra, are discussed. Bibliogrhaphy: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 154–165.     

Data Adaptive Ridging in Local Polynomial Regression

Abstract

When estimating a regression function or its derivatives, local polynomials are an attractive choice due to their flexibility and asymptotic performance. Seifert and Gasser proposed ridging of local polynomials to overcome problems with variance for random design while retaining their advantages. In this article we present a data-independent rule of thumb and a data-adaptive spatial choice of the ridge parameter in local linear regression. In a framework of penalized local least squares regression, the methods are generalized to higher order polynomials, to estimation of derivatives, and to multivariate designs. The main message is that ridging is a powerful tool for improving the performance of local polynomials. A rule of thumb offers drastic improvements; data-adaptive ridging brings further but modest gains in mean square error.     

Application of the Rank-Factorization Method to the Analysis of Spectral Characteristics of Multiparameter Polynomial Matrices

The method of rank factorization (the W-q method), previously suggested by the author as a method for solving algebraic problems for a multiparameter matrix F polynomially dependent on parameters, is applied to analyze the finite spectrum of F. Special attention is paid to the part of the spectrum [F] of the q-parameter matrix F whose points are independent of at least one of the spectral parameters. Bibliography: 6 titles.     

Multiparameter resolvents

We give a characterization for multiparameter resolvents based upon an asymptotic condition and an analog of the resolvent equation.     

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.     

Multiparameter Shimizu equation

An analog of the Shimizu equation is considered, and the set of solutions is described for the case of diagonal matrices. The result is applied to one characterization problem. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995. Part III     

Multiparameter Ramanujan-Type q-Beta Integrals

We consider general multiparameter Ramanujan-type q-beta integrals and derive integral analogs of Slater's transformation formulas for bilateral basic hypergeometric series. Evaluation of some Ramanujan-type q-beta integrals is discussed in a unified form.     

Multiparameter resolvents. II

We give a characterization of generalized multiparameter resolvents based upon a reproducing kernel and asymptotic behavior.     

Multiparameter prediction of polymer creep

An investigation of the thermocreep of low-density polyethylene (LDP) and the vibrocreep of porous polyurethane (PPU) in complex states of stress has shown that multiparameter creep prediction based on the combined application of the time-stress, time-temperature, and time-vibration superposition principles can be used for rapid analysis of the nonlinear viscoelasticity and thermovibrocreep of polymeric materials under complex loading.For communication 1 see [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 416–420, May–June, 1971.     

Multiparameter quantum groups and multiparameterR-matrices

There exists an ( ₂ ⁿ ) + 1 parameter quantum group deformation of GL_n which has been constructed independently by several (groups of) authors. In this note, I give an explicitR-matrix for this multiparameter family. This gives additional information on the nature of this family and facilitates some calculations. This explicitR-matrix satisfies the Yang-Baxter equation. The centre of the paper is Section 3 which describes all solutions of the YBE under the restriction r _cd ^ab =0 unlessa, b=c, d. One kind of the most general constituents of these solutions precisely corresponds to the ( ₂ ⁿ ) + 1 parameter quantum group mentioned above. I describe solutions which extend to an enhanced Yang-Baxter operator and, hence, define link invariants. The paper concludes with some preliminary results on these link invariants.     

Multiparameter Schemes for Evolutionary Equations

Multiparameter extensions (MP) of (linear and nonlinear) descent methods have been proposed for the solution of finite dimensional time independent problems; these new methods are based on a different treatment of several blocks of components of the solution, basically via the substitution of a scalar relaxation by a (suitable) matricial relaxation. Similarly, the Nonlinear Galerkin Method (NLG), that stems from the dynamical system theory, propose to apply distinct temporal integration schemes to different sets of data scales when solving dissipative PDEs. In this paper, the algebraic similarity of Richardson iteration and Forward-Euler time integration is extended to new grounds through the expansion of the realm of MP methods to the field of the numerical integration of dissipative PDEs. The separation of the structures is realized by the utilization of hierarchical preconditioners in finite differences, which are conjugated to a MP temporal integration steeming from NLG theory. Numerical examples of fluid dynamics problems show the improved temporal stability of these new methods as compared to the classical ones.     

Stabilizers of Functional Menger Systems

A functional Menger system is a set of n-place functions containing n projections and closed under the so-called Menger's composition of n-place functions. We give the abstract characterization for subsets of these functional systems which contain functions having one common fixed point.     

Stabilizers of direct composition series

Let R be a domain, V a left R-module, and ${\mathcal{L}}$ a composition series of direct summands of V. Our main results show that if U is a stabilizer group of ${\mathcal{L}}$ containing the McLain-group associated with ${\mathcal{L}}$ , then U determines the chain ( ${\mathcal{L}, \subseteq}$ ) uniquely up to isomorphism or anti-isomorphism.     

Multiparameter ratio ergodic theorems for semigroups

After one-parameter treatment of ratio ergodic theorems for semigroups, we formulate the Sucheston a.e. convergence principle of continuous parameter type. This principle plays an effective role in proving some multiparameter generalizations of Chacon?s type continuous ratio ergodic theorems for semigroups and of Jacobs? type continuous random ratio ergodic theorems for quasi-semigroups. In addition, a continuous analogue of the Brunel–Dunford–Schwartz ergodic theorem is given of sectorially restricted averages for a commutative family of semigroups. We also formulate a local a.e. convergence principle of Sucheston?s type. The local convergence principle is effective in proving multiparameter local ergodic theorems. In fact, a multiparameter generalization of Akcoglu–Chacon?s local ratio ergodic theorem for semigroups of positive linear contractions on _L1$L_1$ is proved. Moreover, some multiparameter martingale theorems are obtained as applications of convergence principles.