Eigenvalues of the Laplacian: an Extension to Higher Dimensions

The trace function where are the eigenvalues of the three-dimensional Laplacian, is studied for a variety of domains. The dependenceof (t) on the connectivity of domains and the boundary conditionsis analysed. Particular attention is given for a spherical shell.     

The Wave Equation Approach to Robbin Inverse Problems for a Doubly-Connected Region: An Extension to Higher Dimensions

The spectral function $\hatμ(t)=\sum\limits_{j=1}^\infty e^{-itλ^{\frac{1}{2}}_j}$ where $\{λ_j\}^\infty_{j=1}$ are the eigenvalues of the three-dimensional Laplacian is studied for a variety of domains, where $- \infty＜t＜\infty$ and $i=\sqrt{-1}$. The dependence of $\hat{\mu}(t)$ on the connectivity of a domain and the impedance boundary condition (Robbin conditions) are analyzed. Particular attention is given to the spherical shell together with Robbin boundary conditions on its surface.     

Erratum to: On Ramsey Numbers for Arbitrary Sequences of Graphs

Doklady Mathematics - An Erratum to this paper has been published: https://doi.org/10.1134/S1064562422340014     

Cleft扩张的余挠维数

众所周知,如果扩张AB是cleft的,则B≌A#_σH.本文主要讨论B和A的余挠维数之间的关系,得出结论LCD(B)■LCD(A)+RD(H),从而推广了群环余挠维数的结论.作为主要结果的应用,还给出了几个使得LCD(B)=LCD(A)的条件.     

Euler数和高阶Euler数的推广

The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.     

Operators Associated to the Cauchy-Riemann Operator in Elliptic Complex Numbers

In this article we provide a generalized version of the result of L.H. Son and W. Tutschke [2] on the solvability of first order systems on the plane whose initial functions are arbitrary holomorphic functions. This is achieved by considering the more general concept of holomorphicity with respect to the structure polynomial X ²+?? X+??. It is shown that the Son-Tutschke lemma on the construction of complex linear operators associated to the Cauchy-Riemann operator remains valid when interpreted for a large class of real parameters ?? and ?? including the elliptic case but also cases that are not elliptic. For the elliptic case, first order interior estimates are obtained via the generalized version of the Cauchy representation theorem for elliptic numbers and thus the method of associated operators is applied to solve initial value problems with initial functions that are holomorphic in elliptic complex numbers.     

Fuzzy数的扩张运算及其性质

本文讨论了Ｆ数的扩张加法和扩张数乘运算的性质；并举反例说明这样的扩张运．算不能使Ｆ数全体Ｆ＊构成线性空间；最后对可列个Ｆ数的运算进行了讨论．     

Erratum to: On the Dirac Cohomology of Complex Lie Group Representations

We correct an error in the paper [D]. For Proposition 1.3 and Theorem 1.4 there to hold, we need to assume that λ _L + sλ _L is dominant for $ {\Delta^{+}}\left( {{{\mathfrak{g}}_0},{{\mathfrak{h}}_0}} \right) $ .     

A Graphical Goodness-of-Fit Test for Dependence Models in Higher Dimensions

This article introduces a graphical goodness-of-fit test for copulas in more than two dimensions. The test is based on pairs of variables and can thus be interpreted as a first-order approximation of the underlying dependence structure. The idea is to first transform pairs of data columns with the Rosenblatt transform to bivariate standard uniform distributions under the null hypothesis. This hypothesis can be graphically tested with a matrix of bivariate scatterplots, Q-Q plots, or other transformations. Furthermore, additional information can be encoded as background color, such as measures of association or (approximate) p-values of tests of independence. The proposed goodness-of-fit test is designed as a basic graphical tool for detecting deviations from a postulated, possibly high-dimensional, dependence model. Various examples are given and the methodology is applied to a financial dataset. An implementation is provided by the R package copula. Supplementary material for this article is available online, which provides the R package copula and reproduces all the graphical results of this article.     

Extension Problems Related to the Higher Order Fractional Laplacian

Caffarelli and Silvestre [Comm. Part. Diff. Eqs., 32, 1245–1260(2007)] characterized the fractional Laplacian(-Δ)~s as an operator maps Dirichlet boundary condition to Neumann condition via the harmonic extension problem to the upper half space for 0 s 1. In this paper, we extend this result to all s 0. We also give a new proof to the dissipative a priori estimate of quasi-geostrophic equations in the framework of L~p norm using the Caffarelli–Silvestre's extension technique.     

A Random Set Extension of a Strong Law of Large Numbers of Cantrell and Rosalsky

Abstract

A strong law of large numbers under conditions irrespective of the joint distribution of the sequence is extended to random sets. The extension is such that the role of events of the form {||V _n || ≤ b _n } (where V _n is a random element of a separable Banach space) is played by events of the form {X _n  ? B _n } (where X _n is a random closed bounded set).     

斜群代数与平凡扩张的表示维数

设A是域k上的一个有限维自入射代数,G是一个有限群,且G的阶在A中可逆,A*G是斜群代数,T(A)是平凡扩张代数,∧V是外代数.本文证明了A的稳定范畴与A*G的稳定范畴的三角维数相等,得到了∧V*G及T(∧V*G)的表示维数.     

高阶Bernoulli数的递推公式及其应用

给出了高阶Bernoulli数的一个递推公式和Nrlund数的一个计算公式,推广了Namias[4],Deeba和Rodriguez[5],Tuenter[6]的结果.     

Discrepancy of Arithmetic Progressions in Higher Dimensions

K. F. Roth (1964, Acta. Arith.9, 257-260) proved that the discrepancy of arithmetic progressions contained in [1, N]={1, 2, …, N} is at least cN^1/4, and later it was proved that this result is sharp. We consider the d-dimensional version of this problem. We give a lower estimate for the discrepancy of arithmetic progressions on [1, N]^d and prove that this result is nearly sharp. We use our results to give an upper estimate for the discrepancy of lines on an N×N lattice, and we also give an estimate for the discrepancy of a related random hypergraph.     

高阶Bernoulli数的递推公式

本文得到了高阶 Bernoulli数的若干递推公式 ,这些公式不仅结构精美 ,递推关系鲜明 ,而且便于应用