On the Order of the Lebesgue Constants for Interpolation by Algebraic Polynomials from Values at Uniform Nodes of a Simplex

For interpolation processes by algebraic polynomials of degree n from values at uniform nodes of an m-simplex, where m ≥ 2, we obtain the order of growth in n of the Lebesgue constants, which coincides with that in the one-dimensional case for which Turetskii obtained an asymptotics earlier.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 814–831.Original Russian Text Copyright ©2005 by N. V. Baidakova.     

Probing physics at extreme energies with cosmic ultra-high energy radiation

The highest energy cosmic rays observed possess macroscopic energies and their origin is likely to be associated with the most energetic processes in the universe. Their existence triggered a flurry of theoretical explanations ranging from conventional shock acceleration to particle physics beyond the standard model (SM) and processes taking place at the earliest moments of our universe. Furthermore, many new experimental activities promise a strong increase of statistics at the highest energies and a combination with γ-ray and neutrino astrophysics will put strong constraints on these theoretical models. We give an overview over this quickly evolving research field with focus on testing new particle physics.     

Dominated splitting versus small angles

In this paper, we give a partial answer to the problem proposed by Lan Wen. Roughly speaking, we prove that for a fixed i, f has C^1 persistently no small angles if and only if f has a dominated splitting of index i on the C^1 i-preperiodic set P＊^1（f）. To prove this, we mainly use some important conceptions and techniques developed by Christian Bonatti. In the last section, we also give a characterization of the finest dominated splitting for linear cocvcles.     

Perturbation models

A Markov chain (with a discrete state space and a continuous parameter) is perturbed by forcing a chain to return to “permissible” states whenever it happens to enter “forbidden” states, with returns governed by a replacement distribution.The compensation method is employed to obtain the distribution for the modified chain, in terms of the original chain and the perturbation mechanism.Emphasis is placed on ergodic chains, and interpretation of results in terms of perturbation theory of semi-groups and the ergodic potential theory (based on the fundamental matrix of a chain) is mentioned.     

The G-Ellipticity for the Sub-Laplacian on Two Step Stratified Lie Groups

This paper reveals that the sub-Laplacian L₀ on two step stratified Lie groups has a similar behavior like elliptic operators on the Euclidean space, that is, the sub - Laplacian satisfies a group-elliptic estimate, called the G- elliptic estimate (or the L^p regularity), and the general left Invariant operator L_λ has such a behavior if and only if λ is admissible.     

The simple layer potential method of fundamental solutions for certain biharmonic problems

A novel formulation of the method of fundamental solutions for the numerical solution of plane biharmonic problems, based on the simple layer potential representation of Fichera, is presented. The applicability and accuracy of the method are demonstrated by examining its performance on a set of practical problems arising in Stokes fluid flow.     

On a Matrix Equation AX＋XB=C over a Skew Field

In this paper we study a matrix equation AX BX=C （Ⅰ）over an arbitrary skew field,and give a consistency criterion of （Ⅰ） and an explicit expression of general solutions of （Ⅰ）.A convenient,simple and practical method of solving （Ⅰ） is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.     

Internal fundamental sequences and approximative retracts

We introduce the notion of internal fundamental sequence and prove that any shape morphism from an arbitrary compactum X to an internally movable compactum Y is induced by an internal fundamental sequence. We use this special kind of fundamental sequences to give characterizations and some properties of AANR_C-sets and AANR_N-sets. The paper ends with a section devoted to internal FANR's.     

The Hausmann-Weinberger 4-manifold invariant of abelian groups

The Hausmann-Weinberger invariant of a group is the minimal Euler characteristic of a closed orientable 4-manifold with fundamental group . We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.