Fujita phenomenon in inhomogeneous fast diffusion system

This paper deals with the Fujita phenomenon for the Cauchy problem of an inhomogeneous fast diffusion system. Both the critical exponent and the second exponent are obtained. We observe that the inhomogeneous terms in the system substantially contribute to the critical exponent, in that the blow-up exponent region is obviously enlarged, with keeping the second critical exponent unchanged for small inhomogeneous sources.     

Sub-harmonicity,monotonicity formula and finite Morse index solutions of an elliptic equation with negative exponent

A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with "negative exponent" is established.It is well known that such a monotonicity formula plays an essential role in the study of finite Morse index solutions of equations with "positive exponent".Unlike the positive exponent case,we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.     

A coupling approach to estimating the Lyapunov exponent of stochastic max-plus linear systems

This paper addresses the problem of approximately computing the Lyapunov exponent of stochastic max-plus linear systems. Our approach allows for an efficient simulation of bounds for the Lyapunov exponent. We provide sufficient conditions for the convergence of the bounds. In particular, a perfect sampling scheme for the Lyapunov exponent is established. We illustrate the effectiveness of our bounds with an application to (real-life) railway systems.     

A note on semilinear heat equation in hyperbolic space

Bandle et al. [1] obtained a quite interesting result about a semilinear heat equation that the Fujita exponent relative to the whole hyperbolic space is just the same as that relative to bounded domain in Euclidean space, and, in addition, the properties of solutions are different in the critical exponent case. Our purpose is to answer an open problem proposed by Bandle et al. for the critical exponent case, and it, together with the one obtained by them, shows that the critical exponent case does belong to the non-blow-up case, which is completely different from the case in Euclidean space.     

A Sharp Exponent Bound for McFarland Difference Sets withp=2

We show that under the self-conjugacy condition a McFarland difference set withp=2 andf2 in an abelian groupGcan only exist, if the exponent of the Sylow 2-subgroup does not exceed 4. The method also works for oddp(where the exponent bound ispand is necessary and sufficient), so that we obtain a unified proof of the exponent bounds for McFarland difference sets. We also correct a mistake in the proof of an exponent bound for (320, 88, 24)-difference sets in a previous paper.     

Fujita Exponent for Porous Medium Equation with Convection and Nonlinear Boundary Condition

This paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that determines the critical exponent.     

带跳的非线性随机微分方程的Lyapunov指数的估计

本文研究了带跳的非线性随机微分方程Lyapunov指数的估计,在适当的条件下,确定其Lyapunov指数q的值．对于给定的步长h,考虑此微分系统的Euler离散化模型,给出了的理论误差估计.     

Minimal varieties of algebras of exponential growth

The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of given exponent and of finite basic rank. As a consequence, we describe the corresponding T-ideals of the free algebra and we compute the asymptotics of the related codimension sequences, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand-Kirillov dimension of the corresponding relatively free algebras of finite rank.     

A note on stable exponents

Given an element γ in a group γ, the stable exponent p+(γ) of γ is defined as p+(γ) =lim sup_n→∞P(γⁿ) denotes the exponent of P(γⁿ) = sup{k/ ?γ_o ∈ γ such that γⁿ = γ^k _o We prove that if γ acts properly discontinuously by isometrics on a proper geodesic Gromov-hyperbolic metric space and γ ∈ γ is of hyperbolic type, then P+(γ) is an integer. This implies that the stable exponent of every element of infinite order in a word hyperbolic group is an integer. We also show that, in a translation discrete group, the stable exponent of every element of infinite order is finite.     

求一个特殊矩阵的n次幂的方法

由两个特殊的低阶方阵的n次幂,猜测出一种特殊形式的k阶方阵A的n次幂,然后通过数学归纳法证明了结果的正确性.本文提供了一种求特殊幂的方法.对求矩阵的幂给出了一种新思路.     

循环矩阵本原指数的上界估计

本文给出了循环矩阵本原指数上界的新的估计及一种由级数较低的循环矩阵的本原指数估计级数较高的循环矩阵的本原指数的方法，解决了一类循环矩阵本原指数的计算问题．     

Powers and Exponents

<正>In mathematics,a power is represented with a base number and an exponent.The base number tells what number is being multiplied.The exponent,a small number written above and to the right of the base number,tells how many times the base number is being multiplied.Today we will talk about     

Comparative analysis of bacterial essential and nonessential genes with Hurst exponent based on chaos game representation

Essential genes are indispensable for the survival of an organism. Investigating features associated with gene essentiality is fundamental to the prediction and identification of essential genes with computational techniques. We use fractal theory approach to make comparative analysis of essential and nonessential genes in bacteria. The Hurst exponents of essential genes and nonessential genes available in the DEG database for 27 bacteria are calculated based on their gene chaos game representations. It is found that for most analyzed bacteria, weak negative correlation exists between Hurst exponent and gene length. Moreover, essential genes generally differ from nonessential genes in their Hurst exponent. For genes of similar length, the average Hurst exponent of essential genes is smaller than that of nonessential genes. The results of our work reveal that gene Hurst exponent is very probably useful gene feature for the algorithm predicting essential genes.     

Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations

The purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones. In the random case, a notion of principal Lyapunov exponent serving as an analog of principal eigenvalue is introduced. It is shown that the principal Lyapunov exponent is deterministic and of simple multiplicity. It is also shown that there is a one-dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign, which serves as an analog of principal eigenfunction. In addition, monotonicity of the principal Lyapunov exponent with respect to the zero-order terms both in the equation and in the boundary condition is proved. When the second- and first-order terms are deterministic, it is proved that the principal Lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time-averaged equation. In the general nonautonomous case, the concepts of principal spectrum, which serves as an analog of principal eigenvalue, and principal Lyapunov exponents are introduced. As is known, the principal spectrum is a compact interval. It is proved in the paper that the principal spectrum contains all the principal Lyapunov exponents. When the second and first-order terms are time independent, a lower estimate of the infimum of the principal spectrum is given in terms of an associated time-averaged equation.     

牛顿(Newton)后向插指

利用牛顿后向插指算子将牛顿(Newton)插值推广到牛顿后向插指,并得到了牛顿后向插指算子与牛顿后向插值算子之间的关系.     

Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent

We give explicit examples of arbitrarily large analytic ergodic potentials for which the Schr?dinger equation has zero Lyapunov exponent for certain energies. For one of these energies there is an explicit solution. In the quasi-periodic case we prove that one can have positive Lyapunov exponent on certain regions of the spectrum and zero on other regions. We also show the existence of 1-dependent random potentials with zero Lyapunov exponent. Research partially supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), Institutional Grant 2002-2052. Received: February 2005; Accepted: May 2005     

The critical exponent for continuous conventional powers of doubly nonnegative matrices

We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than n-2 and we conjecture that it is always n-2 (as it is with Hadamard powering). We prove this conjecture when n<6 and in certain other special cases. We establish a quadratic bound for the critical exponent in general.     

Computable examples of the maximal Lyapunov exponent

Summary Some new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal Lyapunov exponent. The examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks. If the original sequence is the outcome of independent Bernoulli trials with success probability p, then the maximal Lyapunov exponent may be expressed in terms of power series in p, with explicit formulae for the coefficients. The convexity of the maximal Lyapunov exponent as a function of p is discussed, as is an application to branching processes in a random environment.     

The ?ojasiewicz exponent of nondegenerate surface singularities

We give some estimations of the ?ojasiewicz exponent of nondegenerate surface singularities in terms of their Newton diagrams. We also give an exact formula for the ?ojasiewicz exponent of such singularities in some special cases. The results are stronger than Fukui inequality?[8]. It is also a multidimensional generalization of the Lenarcik theorem?[13].     

动力系统实测数据的Lyapunov指数的矩阵算法

Lyapunov指数l是定量描述混沌吸引子的重要指标,自从1985年Wolf提出Lyapunov指数l的轨线算法以来,如何准确、快速地计算正的、最大的Lyapunov指数l_max便成为人们关注的问题,虽有不少成功计算的报导,但一般并不公开交流.在Zuo Bingwu理论算法的基础上,给出了Lyapunov指数l的具体的矩阵算法,并与Wolf的算法进行了比较,计算结果表明:算法能快速、准确地计算(主要是正的、最大的)Lyapunov指数l_max.并对Lyapunov指数l的大小所反应的吸引子的特性进行了分析,并得出了相应的结论.