Approximation with polynomial kernels and SVM classifiers

This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels. Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by Durrmeyer operators in a weighted L¹ space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially fast with the sample size, which is a special feature of polynomial kernels. Dedicated to Charlie Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 68T05, 62J02. Ding-Xuan Zhou: The first author is supported partially by the Research Grants Council of Hong Kong (Project No. CityU 103704).     

THE WEINSTEIN CONJECTURE ON C~1-SMOOTH HYPERSURFACE OF CONTACT TYPE

In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved still holding for it. As a consequence some results on Weinstein conjecture are generalized to C1-smooth hypersurface of contact type.     

A new positive‐definite regularization of incompressible Navier–Stokes equations discretized with Q1/P0 finite element

A new regularization method is proposed for the Galerkin approximation of the incompressible Navier–Stokes equations with Q1/P0 element, by newly introducing a square‐type linear form into the variational divergence‐free constraint regularized with the global pressure jump (GPJ) method. The addition of the square‐type linear form is intended to eliminate the hydrostatic pressure mode appearing in confined flows, and to make the discretized matrix positive definite and then non‐singular without the pressure pegging trick. Effects of the free parameters for the regularization on the solutions are numerically examined with a 2‐D driven cavity flow problem. Furthermore, the convergences in the conjugate gradient iteration for the solution of the pressure Poisson equation are compared among the mixed method, the GPJ method and the present method for both leaky and non‐leaky 3‐D driven cavity flows. Finally, the non‐leaky 3‐D cavity flows at different Re numbers are solved to compare with the literature data and to demonstrate the accuracy of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

Tikhonov regularization for a general nonlinear constrained optimization problem

The goal of this study is to analyze the Tikhonov regularization method as applied to a general nonlinear optimization problem that has been previously reduced to an unconstrained optimization problem. The stability properties of the method are examined, and its convergence is proved. The text was submitted by the author in English.     

Symplectic Shifted Tableaux and Deformations of Weyl's Denominator Formula for sp(2n)

A determinantal expansion due to Okada is used to derive both a deformation of Weyl's denominator formula for the Lie algebra sp(2n) of the symplectic group and a further generalisation involving a product of the deformed denominator with a deformation of flagged characters of sp(2n). In each case the relevant expansion is expressed in terms of certain shifted sp(2n)-standard tableaux. It is then re-expressed, first in terms of monotone patterns and then in terms of alternating sign matrices.     

On difference between the spectra of the simple groups Bn(q) and Cn(q)

We prove that the nonisomorphic simple groups B _n (q) and C _n (q) have different sets of element orders. Original Russian Text Copyright ? 2007 Grechkoseeva M. A. __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 1, pp. 89–92, January–February, 2007.     

带复数核的第一类Fredholm积分方程的正则化方法及其应用

本文推广了Ｔｉｋｈｏｎｏｖ正则化方法，导出了带复数核的第一类Ｆｒｅｄｈｏｌｍ积分方程的正则解应满足的正则积分微分方程，并讨论了正则解的收敛性·作为这一方法的应用，数值求解了与二维摇板造波问题相应的一类逆问题，并给出了选择最佳正则参数的一个实用的方法     

Some Convergence Results for Evolution Hemivariational Inequalities

This paper is devoted to the regularization of a class of evolution hemivariational inequalities. The operator involved is taken to be non-coercive and the data are assumed to be known approximately. Under the assumption that the evolution hemivariational inequality be solvable, a weakly convergent approximation procedure is designed by means of the so-called Browder-Tikhonov regularization method.     

关于k-广义酉矩阵

本文研究了k-广义酉矩阵的性质及其与酉矩阵、辛矩阵、Householder矩阵之间的联系,取得了许多新的结果,推广了酉矩阵及Householder矩阵的相应结果,特别将正交矩阵的广义Cayley分解推广到了广义酉矩阵上;并将各类酉矩阵及辛矩阵统一了起来.     

Explicit Runge-Kutta-like schemes to solve certain quantum operator equations of motion

We propose a kind of explicit sympletic Runge-Kutta-like scheme to solve operator equations of motion in quantum mechanics and in a quantum scalar field theory.