On an inequality for spectral radius

In this note, we first extend and then give a related result to an inequality involing the spectral radius of nonnegative matrices that recently appcared in the literature.     

Covering Planar Graphs with a Fixed Number of Balls

In this note we prove that there exists a constant ρ such that any planar graph G of diameter ≤ 2R can be covered with at most ρ balls of radius R, a result conjectured by Gavoille, Peleg, Raspaud, and Sopena in 2001.     

Jung's Theorem for Alexandrov Spaces of Curvature Bounded Above

The classical Jung theorem gives an optimal upper estimate for the radius of a bounded subset of R ⁿ in terms of its diameter and the dimension. In this note we present an analogue of this result for metric spaces of curvature bounded above in the sense of Alexandrov.     

磁流体方程的时间解析半径

本文考查了二维周期边界条件下的磁流体方程 ,证明了当初值位于全局吸引子上和定常解足够接近时 ,解在t=0的时间解析半径可以任意大     

Corrigendum to “A Morse–Smale index theorem for indefinite elliptic systems and bifurcation” [J. Differential Equations 258 (5) (2015) 1715–1748]

We discussed in a previous paper elliptic systems of partial differential equations on star-shaped domains and introduced the notions of conjugate radius and bifurcation radius. We proved that every bifurcation radius is a conjugate radius, and believed to have shown by an example that on the other hand not every conjugate radius is a bifurcation radius. This note reveals that our previous example was wrong, but it also introduces an improved example that shows the assertion that we claimed before.     

多连通区域上的Bloch常数

<正> 本文可以看作[1]文的一个应用,在这里将提供一个关于多连通区域的 Bloch 常数的估计的普遍方法.并且特别以于二连通区域的 Bloch 常数进行了具体的估计.     

Note on the existence theory for evolution equations with pseudo-monotone operators

In this note we present a framework which allows to prove an abstract existence result for evolution equations with pseudo-monotone operators. The assumptions on the spaces and the operators can be easily verified in concrete examples.     

On the Optimum Radius of Robust Stability for Schur Polynomials

A recent paper (Ref. 1) established a new approach to estimate the robust stability radius of a Schur polynomial. This note points out that the approach given in Ref. 1 is not correct and also gives a counterexample to the main result of Ref. 1.     

A note on the existence of positive bounded solutions for an epidemic model

In this short note, we establish an existence and uniqueness theorem about a positive bounded solution for a nonlinear infinite delay integral equation, which arises in some epidemic problems. As one can see, our main result can deal with some cases, to which many previous results cannot be applied. In addition, we show that our main result can also be applied to a Lasota–Wazewska model.     

图的点覆盖数与其Laplace谱半径

本文讨论图的点覆盖数与图的 Laplace谱半径的关系 ,利用特征向量的技巧得到由图的 L aplace谱半径所确定的关于图的点覆盖数的紧的界     

Spectral Radius of Refinement and Subdivision Operators with Power Diagonal Dilations

Two types of estimate for the spectral radius of the multivariate refinement operator with power diagonal dilations are presented. One type contains multiplicator norm of number matrices generated by the symbol of the corresponding operator and by specific subsets of repeating fractions. These subsets are used together with the little Fermat theorem to establish estimates that comprise integrals over tori of various dimensions. Moreover, we note certain classes of symbols when the exact value of the spectral radius of refinement operator can be found. For the spectral radius of subdivision operators point value estimates are established. Submitted: April 25, 2007. Accepted: November 5, 2007.     

Explicit bounds for multidimensional linear recurrences with restricted coefficients

This note employs path counting techniques to extend recent results on bounds for odd order linear recurrences to higher dimensions. The results imply optimal zero-free polydisks for multivariable power series with 0, 1 coefficients. Among the applications is a result that states that the optimal zero-free polydisk has radius approximately 1/(v+1), for large dimensions v.     

On the spectral radius of the Jacobi iteration matrix for a rectangular region with two different media

The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method.The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carré's method, Kulstrud's method and the stationary iterative method using Frankel's theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions.As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula.     

Singularity of generalized grey Brownian motions with different parameters

In this note we prove that the probability measures generated by two generalized grey Brownian motions with different parameters are singular with respect to each other. This result can be interpreted as an extension of the Feldman–Hájek dichotomy of Gaussian measures to a family of non-Gaussian measures.     

Approximate convexity of average sums of sets in normed linear spaces

This note gives a sufficient condition under which the average sum of a large but finite number of sets in an arbitrary normed linear space is approximately convex. The result can be deduced as a simple corollary of an analogous result for a nonstandard universe. The latter has independent interest.     

WEIGHTED BOUNDEDNESS OF A ROUGH MAXIMAL OPERATOR

1 IntroductionIn 1990, L.K.Chen and H.Lin[1] gave the H-boundedness of a maximal opetator related toa class of singulax integrals:Theorem A Let n 2 2 and fi E C(S"--') with fs.--, n(() d(= 0 where fi is of homogeneous of degree zero on Re. Denotewhere the supremum is taked over the set {h: IlhllLB(R ,d./.) 5 1}. Then IITfll. 5 Crllffl. for1 5 s 5 2, sn/(sn -- 1) < p. This range of p is the best possible.In this note we shall give the weighted variant of the above result. Before stati…     

On a class of almost unbiased ratio estimators

Summary Murthy and Nanjamma [4] studied the problem of construction of almost unbiased ratio estimators for any sampling design using the technique of interpenetrating subsamples. Subsequently, Rao [7], [8] has given a general method of constructing unbiased ratio estimators by considering linear combinations of the two simple estimators based on the ratio of means and the mean of ratios. However, it is difficult to choose an optimum weight (Rao [9]) which minimizes the variance of the combined estimator since the weights are random in certain cases. In this note, we consider a different method of combining these estimators and obtain a general class of almost unbiased ratio estimators of which Murthy and Nanjamma's is a particular case and derive an optimum in this class. The case of simple random sampling where a similar class of almost unbiased ratio estimators can be developed is briefly discussed. The results are illustrated by means of simple numerical examples.     

A simplified proof for a theorem of Dwork and van der Poorten

The aim of this note is to give a new proof of a Dwork-van der Poorten theorem [6] that computes the radius of convergence near the generic point for algebraic functions.     

Several theorems of combinatorial geometry

A set is said to be H-convex if it can be represented by an intersection of a family of closed half-spaces whose outer normals belong to a given subset of the set H of the unit sphereS ⁿ⁻¹⊂R. On the basis of Helly’s theorem for H-convex sets recently obtained by us, we prove in this note certain extensions of Blaschke’s theorem (on the radius of an inscribed sphere) and of several other well-known theorems of combinatorial geometry. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 117–124, January, 1977.     

A note on a parameter bound for global stability in the 2D Coupled Ricker equation

In prior work on the coupled 2D Ricker Map, a conjecture was made concerning an upper bound on parameters that would guarantee injectivity of the map in a certain region of the plane. This injective property was used to show global asymptotic stability of the coexistent fixed point. In this short note, we find an explicit formula for the optimum upper bounds.