Lower Bounds on the (Laplacian) Spectral Radius of Weighted Graphs

The weighted graphs, where the edge weights are positive numbers, are considered. The authors obtain some lower bounds on the spectral radius and the Laplacian spectral radius of weighted graphs, and characterize the graphs for which the bounds are attained. Moreover, some known lower bounds on the spectral radius and the Laplacian spectral radius of unweighted graphs can be deduced from the bounds.     

A new approach for the derivation of bounds for the Jensen difference

There are many useful applications of Jensen's inequality in several fields of science, and due to this reason, a lot of results are devoted to this inequality in the literature. The main theme of this article is to present a new method of finding estimates of the Jensen difference for differentiable functions. By applying definition of convex function, and integral Jensen's inequality for concave function in the identity pertaining the Jensen difference, we derive bounds for the Jensen difference. We present integral version of the bounds in Riemann sense as well. The sharpness of the proposed bounds through examples are discussed, and we conclude that the proposed bounds are better than some existing bounds even with weaker conditions. Also, we present some new variants of the Hermite–Hadamard and Hölder inequalities and some new inequalities for geometric, quasi-arithmetic, and power means. Finally, we give some applications in information theory.     

On eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

In this paper, we give some structured perturbation bounds for generalized saddle point matrices and Hermitian block tridiagonal matrices. Our bounds improve some existing ones. In particular, the proposed bounds reveal the sensitivity of the eigenvalues with respect to perturbations of different blocks. Numerical examples confirm the theoretical results.     

Bounds and Numerical Results for Homogenized Degenerated p-Poisson Equations

In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.     

Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method

We deal with the problem of estimating the volume of inclusions using a small number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq–Vogelius in the limit when the volume fraction tends to zero. The bounds of this article, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.     

Altermatic number of categorical product of graphs

In this paper, we prove some relaxations of Hedetniemi’s conjecture in terms of altermatic number and strong altermatic number of graphs, two combinatorial parameters introduced by the present authors Alishahi and Hajiabolhassan (2015) providing two sharp lower bounds for the chromatic number of graphs. In terms of these parameters, we also introduce some sharp lower bounds for the chromatic number of the categorical product of two graphs. Using these lower bounds, we present some new families of graphs supporting Hedetniemi’s conjecture.     

关于特征值与广义特征值的Bauer-Fike型相对扰动界

本文研究特征值与广义特征值的Bauer-Fike型相对扰动界.我们给出了一些新的结果.这些界从一定的意义上改进了以往相应的结论.     

Improved bounds on the magnitude of the derivative of rational Bézier curves

In this paper we derive some new derivative bounds of rational Bézier curves according to some existing identities and inequalities. The comparison of the new bounds with some existing ones is also presented.     

Lagrangian bounds in multiextremal polynomial and discrete optimization problems

Many polynomial and discrete optimization problems can be reduced to multiextremal quadratic type models of nonlinear programming. For solving these problems one may use Lagrangian bounds in combination with branch and bound techniques. The Lagrangian bounds may be improved for some important examples by adding in a model the so-called superfluous quadratic constraints which modify Lagrangian bounds. Problems of finding Lagrangian bounds as a rule can be reduced to minimization of nonsmooth convex functions and may be successively solved by modern methods of nondifferentiable optimization. This approach is illustrated by examples of solving polynomial-type problems and some discrete optimization problems on graphs.     

图的Laplace谱半径的几类上界

本文得到图的Laplace谱半径的几类上界.通过选取适当的对角矩阵,我们得到了在一定程度上优于其他界的上界.     

Tail bounds for the joint distribution of the surplus prior to and at ruin

For the classical risk model with Poisson arrivals, we study the (bivariate) tail of the joint distribution of the surplus prior to and at ruin. We obtain some exact expressions and new bounds for this tail, and we suggest three numerical methods that may yield upper and lower bounds for it. As a by-product of the analysis, we obtain new upper and lower bounds for the probability and severity of ruin. Many of the bounds in the present paper improve and generalise corresponding bounds that have appeared earlier. For the numerical bounds, their performance is also compared against bounds available in the literature.     

Bounds for the reliability functions of coherent systems with heterogeneous components

The computation of the reliability function of a (complex) coherent system is a difficult task. Hence, sometimes, we should simply work with some bounds (approximations). The computation of these bounds has been widely studied in the case of coherent systems with independent and identically distributed (IID) components. However, few results have been obtained in the case of heterogeneous (non ID) components. In this paper, we derive explicit bounds for systems with heterogeneous (independent or dependent) components. Also some stochastic comparisons are obtained. Some illustrative examples are included where we compare the different bounds proposed in the paper.     

Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods

In this study, the bounds for eigenvalues of the Laplacian operator on an L-shaped domain are determined. By adopting some special functions in Goerisch method for lower bounds and in traditional Rayleigh–Ritz method for upper bounds, very accurate bounds to eigenvalues for the problem are obtained. Numerical results show that these functions can also be successfully used to solve the problem on the region with other reentrant angle.     

Error Bounds for the Difference of Two Convex Multifunctions

In this paper, we consider error bounds for DC multifunctions (difference of two convex multifunctions) with/without set constraints. We give some Robinson-Ursescu type results in Banach spaces. Using some techniques of convex analysis, we present some results on the existence of error bounds in terms of normal cone and coderivative.     

Analyse sur les groupes de Lie à croissance polynômiale

From Gaussian upper bounds on the heat kernel we deduce similar upper bounds on the first space derivatives of the heat kernel. Gaussian lower bounds on the heat kernel are deduced and some applications are given.     

Sharp bounds for spectral radius of nonnegative weakly irreducible tensors

We obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor. By using the technique of the representation associate matrix of a tensor and the associate directed graph of the matrix, the equality cases of the bounds are completely characterized by graph theory methods. Applying these bounds to a nonnegative irreducible matrix or a connected graph (digraph), we can improve the results of L. H. You, Y. J. Shu, and P. Z. Yuan [Linear Multilinear Algebra, 2017, 65(1): 113–128], and obtain some new or known results. Applying these bounds to a uniform hypergraph, we obtain some new results and improve some known results of X. Y. Yuan, M. Zhang, and M. Lu [Linear Algebra Appl., 2015, 484: 540–549]. Finally, we give a characterization of a strongly connected k-uniform directed hypergraph, and obtain some new results by applying these bounds to a uniform directed hypergraph.     

On the number of integer polynomials with multiplicatively dependent roots

We give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for various cases, although in general there is a logarithmic gap between lower and upper bounds.     

α次殆β型螺形映照齐次展开式的精细估计

从Loewner链的角度讨论C~n中单位球B~n上α次殆β型螺形映照齐次展开式的相关项的上界,并作为特殊情况得出β型螺形映照、星形映照和α次殆星形映照齐次展开式的相关项的上界估计,推广了螺形映照及星形映照齐次展开式的二次项系数的上界估计.     

Sharp upper and lower bounds for the Laplacian spectral radius and the spectral radius of graphs

In this paper, sharp upper bounds for the Laplacian spectral radius and the spectral radius of graphs are given, respectively. We show that some known bounds can be obtained from our bounds. For a bipartite graph G, we also present sharp lower bounds for the Laplacian spectral radius and the spectral radius, respectively.     

Error analysis for convex separable programs: Bounds on optimal and dual optimal solutions

Computable lower and upper bounds on the optimal and dual optimal solutions of a nonlinear, convex separable program are obtained from its piecewise linear approximation. They provide traditional error and sensitivity measures and are shown to be attainable for some problems. In addition, the bounds on the solution can be used to develop an efficient solution approach for such programs, and the dual bounds enable us to determine a subdivision interval which insures the objective function accuracy of a prespecified level. A generalization of the bounds to certain separable, but nonconvex, programs is given and some numerical examples are included.