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.     

A generalization of local divergence measures

In this paper we propose a generalization of the concept of the local property for divergence measures. These new measures will be called g-local divergence measures, and we study some of their properties. Once this family is defined, a characterization based on Ling’s theorem is given. From this result, we obtain the general form of g-local divergence measures as a function of the divergence in each element of the reference set; this study is divided in three parts according to the cardinality of the reference set: finite, infinite countable or non-countable. Finally, we study the problem of componible divergence measures as a dual concept of g-local divergence measures.     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

不可数求和型Hahn-Schur定理

The classical countable summation type Hahn-Schur theorem is a famous result in summation theory and measure theory. An interesting problem is whether the theorem can be generalized to non-countable summation case? In this paper, we show that the answer is true.     

Approximation of local convoluted semigroups

We extend the Trotter-Kato theorem on C₀-semigroups to local convoluted semigroups on dual spaces and apply these results to the general Banach space setting. Compared to known results we obtain weaker convergence assumptions on the resolvent.     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.     

Simple Explanation of the No-Free-Lunch Theorem and Its Implications

The no-free-lunch theorem of optimization (NFLT) is an impossibility theorem telling us that a general-purpose, universal optimization strategy is impossible. The only way one strategy can outperform another is if it is specialized to the structure of the specific problem under consideration. Since optimization is a central human activity, an appreciation of the NFLT and its consequences is essential. In this paper, we present a framework for conceptualizing optimization that leads to a simple but rigorous explanation of the NFLT and its implications.     

A Functional Non-Central Limit Theorem for Jump-Diffusions with Periodic Coefficients Driven by Stable Lévy-Noise

We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Lévy-processes with stability index α>1. The limit process turns out to be an α-stable Lévy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity.     

广义Hojman定理

研究利用Lie对称的生成元τ(t,q,q·)和ξ_s(t,q,q·)来构造广义H ojman守恒 量，并讨论三种特殊情况，研究表明：Hojman守恒量是该广义守恒量的特例，且在Lie对称 的生成元的形式为τ(t,q)和ξ_s(t,q)时，该广义Hojman守恒量可以导出Lu tzky守恒 量，此外，还给出一个排除平凡守恒量的条件.最后，给出两个简单例子，作为所获得结果 的说明. 关键词： 动力学系统 广义Hojman定理 Lie对称 守恒量