Alternating polynomial

Abstract: A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.

Abstract: Polynomial identities and PI-algebras $S_n$-representations Group gradings and group actions Codimension and colength growth Matrix invariants and central polynomials The PI-exponent of an algebra Polynomial growth and low PI-exponent Classifying minimal varieties Computing the exponent of a polynomial $G$-identities and $G\wr S_n$-action Superalgebras, *-algebras and codimension growth Lie algebras and nonassociative algebras The generalized-six-square theorem Bibliography Index.

Abstract: It is proved that the special linear group of degree not less than three over the polynomial ring over a field is generated by the elementary matrices. Other results are obtained that relate to the structure of the special linear group and stabilization of the general linear group over arbitrary polynomial and Laurent rings.Bibliography: 9 titles.

Abstract: This work gives apolynomial time algorithm for learning decision trees with respect to the uniform distribution (This algorithm uses membership queries) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (ie, summation of a subset of the input variables over GF(2)) This paper shows how to learn in polynomial time any function that can be approximated (in norm L2) by a polynomially sparse function (ie, a function with only polynomially many nonzero Fourier coefficients) The authors demonstrate that any functionf whose L -norm (ie, the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions Moreover, it is shown that the functions with polynomial L -norm can be learned deterministically The algorithm can also exactly identify a decision tree of depth d in time polynomial in 2 a and n This result implies that trees of logarithmic depth can be identified in polynomial time