确界不等式的证明方法

本文将从确界的定义以及确界原理开始,归纳总结证明确界不等式的三种最常见的方法,希望这些方法能给予读者一些证明确界不等式的思路.     

不含三角形图的$A_\alpha$-谱半径的界

设$G$是简单无向图. 对于实数$\alpha \in [0,1]$, Nikiforov于2017年定义图的$A_\alpha$-矩阵为$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, 其中$A(G)$和$D(G)$分别为图$G$的邻接矩阵和度对角矩阵. 图的$A_\alpha$-矩阵可以看着是图的邻接矩阵和无符号拉普拉斯矩阵的共同推广, 其最大特征值称为图的$A_\alpha$- 谱半径. 对于$\alpha\in[0,1)$, 本文确定了不含三角形图的$A_\alpha$-谱半径的一个下界;对于$\alpha \in[1/2, 1)$, 本文确定了不含三角形$k$圈图的$A_\alpha$-谱半径的一个上界.     

具有常量红利界的带干扰项的风险模型及最优红利界

对于保险公司来说,如何确定其红利策略,使得投保人利益最大化是一个需要研究的课题.研究了具有常量红利界的带干扰项的经典风险模型下,索赔量为混合指数分布情形时的最优红利界的计算方法.     

Hermite矩阵特征值的新扰动界

本文研究了Hermite矩阵特征值的任意扰动,给出了新的绝对和相对扰动界．所给出的界改进了Hoffman-Wielandt和Kahan早期的结果．     

最小距离固定且优于Gilbert—Varshamov界的码

本文构造了一类GF(q)上的码,其中GF(q)为q个元素的有限域.这些码的冗余取到渐进界r(q,n,7) 4 m,此界优于Gilbert-Varshamov存在界r(q,n,7) 5m.     

凸集值映射的整体误差界

考虑了凸集值映射的整体误差界，推广Li和Singer(1998)的主要定理到无界情形并肯定地回答了该文的猜想．作为应用，给出了线性Hoffman误差界定理一个简单的新证明．     

水波的界带有限元

研究了水波计算的位移法.采用物质坐标,以位移为基本未知量,考虑小变形条件,引入流函数满足不可压缩条件.于是,分析力学的变分原理可以运用了,界带有限元、正则变换、保辛积分等有效手段可使数值求解方便得多.     

Riesz界与条件数

在多分辨分析与小波分析中,人们经常要估计Riesz基的上下界.在有限维空间中,这等价于计算Riesz基所对应Gramian矩阵的条件数.本文给出Riesz基与条件数的关系并且讨论了Riesz基加入元素后对Riesz界产生的影响.     

也谈Minc-Sathre不等式的上下界

H．Minc—L．Sathre在1964年证明了下面的不等式：对于一切自然数n，有     

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs $G$ and $H$. Let $H^k\left[G\right]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k\left[G\right]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k\left[G\right]$ and $H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10¹⁰⁰ ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.     

Spectra of graphs attached to the space of melodies

For any integer n greater than or equal to two, two intimately related graphs on the vertices of the n-dimensional cube are introduced. All of their eigenvalues are found to be integers, and the largest and the smallest ones are also determined. As a byproduct, certain kind of generating function for their spectra is introduced and shown to be quite effective to compute the eigenvalues of some broader class of adjacency matrices of graphs.     

The number of spanning forests of a graph

In this paper we study the number of spanning forests of a graph. Let G be a connected simple graph. (1) We give a lower bound for the number of spanning forests of G in terms of the edge connectivity of G. (2) We give an upper bound for the number of rooted spanning forests of G. (3) We describe the elementary symmetric functions of inverse positive Laplacian eigenvalues of a tree. (4) We determine all Laplacian integral graphs with prime number of spanning trees. (5) We give a simple proof of a theorem of K. Hashimoto on Ihara zeta function.     

(v,e)-图的正常λ-着色的最大数目的新界(英文)

设P（G，λ）表示图G的色多项式．给定正整数n，e和λ,记f（v，e，λ）=max{P（G，λ）：G是个v点e边的图}．本文得到f（v，e，λ）的新上界和f（v，e，3）的新下界．     

求参数函数置信限的一般性WCF方法及其应用

该文将求参数函数置信下限的WCF方法推广到参数的估计量之间具有相关性的一般情形; 利用推广了的公式, 给出了求单个威布尔型设备及含威布尔型设备的系统可靠度置信下限的方法, 并进行了模拟研究. 模拟结果表明, 该方法优于基于渐近正态性的近似方法, 尤其是在中小样本情况.     

Nonlinear Bound States on Weakly Homogeneous Spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F _λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.     

The Degree Spectra of Definable Relations on Boolean Algebras

We study some questions concerning the structure of the spectra of the sets of atoms and atomless elements in a computable Boolean algebra. We prove that if the spectrum of the set of atoms contains a 1-low degree then it contains a computable degree. We show also that in a computable Boolean algebra of characteristic (1, 1, 0) whose set of atoms is computable the spectrum of the atomless ideal consists of all Π ₀ ² degrees.Original Russian Text Copyright © 2005 Semukhin P. M.The author was supported by the Russian Foundation for Basic Research (Grant 02-01-00593), the Leading Scientific Schools of the Russian Federation (Grant NSh-2112.2003.1), and the Program “Universities of Russia” (Grant UR.04.01.013).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 928–941, July–August, 2005.     

A Berry-Esseen Bound for k-sample Symmetric Statistics

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多重交换算子的谱与数值值域

本研究了Banach空间上多重交换算子的谱和数值值域之间的关系,同时给出了非线性控制中一个公开问题的部分解答。     

一类高阶微分方程第二特征值的上界

本文研究形如（１．１）的第二特征上界问题,得到了定理１ 和定理２,其中定理１ 的估计系数与［ａ,ｂ］无关,定理２ 的结果在一定条件下比定理１ 好．