Dependence of the Convergence Domain of Spectral Expansions on the Geometry of the Set of Discontinuity of the Function Being Expanded

For piecewise smooth functions, we indicate their domains of convergence and Riecz summarizability of their spectral expansions related to the Laplace operator in ? ⁿ , depending on the geometry of points of discontinuity of the function being expanded.     

Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise

A Wentzell–Freidlin type large deviation principle is established for the two-dimensional Navier–Stokes equations perturbed by a multiplicative noise in both bounded and unbounded domains. The large deviation principle is equivalent to the Laplace principle in our function space setting. Hence, the weak convergence approach is employed to obtain the Laplace principle for solutions of stochastic Navier–Stokes equations. The existence and uniqueness of a strong solution to (a) stochastic Navier–Stokes equations with a small multiplicative noise, and (b) Navier–Stokes equations with an additional Lipschitz continuous drift term are proved for unbounded domains which may be of independent interest.     

新的不动点定理及应用

本文获得了几个压缩型映射的不动点定理,改进了相应文献的结果．应用该定理,给出了一类时滞积分方程的正概周期解的存在性结果．推广了已有的结果．     

Band Theory in the Context of the Hamilton-Jacobi Formulation

In the one-dimensional periodic potential case, we formulate the condition of Bloch periodicity for the reduced action by using the relation between the wave function and the reduced action established in the context of the equivalence postulate of quantum mechanics. Then, without appealing to the wave function properties, we reproduce the well-known dispersion relations which predict the band structure for the energy spectrum in the Krönig-Penney model.     

STURM COMPARISON THEOREMS FOR SECOND ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS

In this paper, we first establish some differential inequalities and then some Sturm comparison theorems are derived for the second order neutral nonlinear differential equations. Our results generalize some classical Sturm comparison theorems.     

On the classification of complex vector bundles of stable rank

One describes, using a detailed analysis of Atiyah-Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constructive terms the Chern classes of a vector bundle. Since deceased.     

K-拟可加模糊数值积分的自连续性

本文在K-拟可加模糊测度空间上建立了K-拟可加模糊数值积分,利用其积分转换定理和诱导算子的性质,将这种积分整体看成可测空间上取值于模糊值和集函数,从而使得这种模糊积分不仅具有自连续性,而且也满足逆自连续性。这些特性能更好地描述模糊值可测函数列和K-拟可加模糊数值积分序列的收敛性。     

Improper Riemann Integral and Henstock Integral in ℝn

The Henstock integral in ℝⁿ and its relation to the n-dimensional improper Riemann integral are studied. A Hake-type theorem for the Henstock integral in ℝⁿ is proved.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 251–258.Original Russian Text Copyright © 2005 by P. Muldowney, V. A. Skvortsov.     

Rook numbers and the normal ordering problem

For an element w in the Weyl algebra generated by D and U with relation DU=UD+1, the normally ordered form is w=∑c_i,jUⁱD^j. We demonstrate that the normal order coefficients c_i,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c_i,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D+U)ⁿ in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.     

Does distribution theory contain means for extending Poincaré-Bendixson theory?

We use the theory of distributions to extend the Poincaré-Bendixson theorem and the Bendixson criterion to piecewise Lipschitz continuous system possessing unique and continuous solutions. We demonstrate the use of these extensions by several examples that have recently appeared in the literature.