恰含两个圈的二部图的能量

设λ1,λ2,…,λn是n阶图G的特征值,图G的能量是E(G)=|λ1| |λ2| … |λn|,设G(n)是n个顶点n 1条边的恰有两个圈的连通二部图的集合,Z(n;4,4)是G(n)中的一个图,它的两个长为4的圈恰有一个公共点,其余n-7个点都是悬挂点且均与这个公共点相邻．文中证明了Z(n;4,4)是G(n)中具有最小能量的图。     

关于正整数奇偶分拆数的计算问题

正整数n的分拆是指将正整数n表示成一个或多个正整数的无序和,设O(n,m)表示将正整数n分拆成m个奇数之和的分拆数;e(n,m)表示将正整数n分拆成m个偶数之和的分拆数.本文用初等方法给出了将O(n,m),e(n,m)分别化为有限个O(n,2),e(n,2)的和的计算公式,进而达到计算O(n,m),e(n,m)的值.同时,还讨论了将正整数n分拆成互不相同的奇数或偶数的分拆数的相应的递推计算方法.     

立方幂补数除数函数的均值

设 n是正整数 ,S(n)是 n的立方幂补数 ,τ(n)表示 n的除数函数 .本文的主要目的是探讨∑n xτ(S(n) )n 和 ∑n xτ(S(n) ) 的渐近性质 ,得到了两个渐近公式 ,进一步解决 F.Smarandache教授提出的第2 8个问题 .     

分空间最多区域数的新探索

张定强在文[1]中介绍了以下结论:n个不同的点可将直线分成n 1段;n条处于一般位置的直线将一个平面最多分成n(n 1)/2 1部分;n个处于一般位置的平面最多将空间分割成n(n2 5)/6 1部分.     

有限循环环的构造

给出了T(n)个互不同构的n阶循环环,两个n阶循环环同构的一个充要条件,进一步,得到了n阶循环环的个数是T(n)的一个新证明.     

F.Smarandache的一个问题

设 n是正整数 ,a( n)表示 n的三次方幂补数 .本文的主要目的是研究 a( n)和 a( n)n 的 k次均值性质 ,解决 F.Smarandache教授提出的第 2 8个问题 ,并用解析方法给出两个有趣的渐近公式     

大卫星恒等式的组合证明

以Np表示1到p的自然数集合,Fn(1,2,…,p)表示元素取自Np的有序n元组的集合,Fnn1n2…np1 2…p表示所有含n1个1,n2个2,…,np个p(n1 n2 … np=n)的有序n元组的集合.通过组合分析法可以证明这两类n元组集合的性质,进而给出"大卫星恒等式"及一些组合恒等式的组合性证明.     

一个组合恒等式的两种证法

在前不久 ,作业中出现了这样一道题 :证明 :(C0 n) 2 +(C1 n) 2 +(C2 n) 2 +… +(Cnn) 2 =Cn2n.我想了半天终于想出了常用的一种证法 ,如下 :设一袋中有n个白球 ,n个黑球 ,每次摸出n个 ,其中包含 0个白球 ,n个黑球 ,1个白球 (n - 1)个黑球 ,… ,直至n个白球与 0个黑球 ,它们恰好组成一个必然事件 ,故这个事件发生的概率总和为 1,即得到C0 n·CnnCn2n+C1 n·Cn -1 nCn2n+… +Cnn·C0 nCn2n=1.两边同乘以C1 2n,即得所证。经过几天的思考 ,我又想出了另一种比较新颖的证法 ,如下 :∵　 ( 1+x) n=C0 n+C1 nx +… +Cnnxn,　　 ( 1+1x) n=…     

连通图的最小Laplace谱半径的排序

首先找出了具有最小Laplace谱半径的第2个至第5个n阶单圈图和具有最小Laplace谱半径的n阶双圈图.然后结合有关n阶树的最小Laplace谱半径的排序,给出了所有n阶连通图中Laplace谱半径最小的14个图,当n为偶数时,它们达到了所有佗阶连通图中Laplace谱半径最小的9个值(其中有并列的),而当n为奇数时,它们则达到了Laplace谱半径最小的8个值(其中有并列的).     

立方幂补数倒数的均值

设 n为任意正整数 ,S( n) 表示 n的立方幂补数 .本文的主要目的是研究 ∑n x1S( n) 和 ∑n xnS( n) 的渐近性质 ,并用初等方法得到了两个渐近公式 ,进一步解决 F.Smarandache教授在文献 [1 ]中提出的第 2 8个问题 .     

Algorithms for approximation of invariant measures for IFS

We prove convergence of two algorithms approximating invariant measures to iterated function systems numerically. We consider IFSs with finitely many continuous and injective non-overlapping maps on the unit interval. The first algorithm is a version of the Ulam algorithm for IFSs introduced by Strichartz et al. [16]. We obtain convergence in the supremum metric for distribution functions of the approximating eigen-measures to a unique invariant measure for the IFS. We have to make some modifications of the usual way of treating the Ulam algorithm due to a problem concerning approximate eigenvalues, which is part of our more general situation with weights not necessarily being related to the maps of the IFS. The second algorithm is a new recursive algorithm which is an analogue of forward step algorithms in the approximation theory of ODEs. It produces a sequence of approximating measures that converges to a unique invariant measure with geometric rate in the supremum metric. The main advantage of the recursive algorithm is that it runs much faster on a computer (using Maple) than the Ulam algorithm.Mathematics Subject Classification (2000): 37A30, 37C30, 37M25, 47A58Acknowledgement I would like to express my deep gratitude to Andreas Strömbergsson and to the anonymous referee. The referee had several very enlightening comments, which Andreas helped me to deal with. Section 4 is essentially due to Andreas and he also came up with the new Proposition 3 and helped me to improve Lemma 1. Thanks also to Svante Janson, Anders Johansson, Sten Kaijser, Robert Strichartz and Hans Wallin.     

Equations for coefficients of tactical decomposition matrices for t-designs

Equations for coefficients of tactical decomposition matrices for 2-designs are well-known and they have been used for constructions of many examples of 2-designs. In this paper, we generalize these equations and propose an explicit equation system for coefficients of tactical decomposition matrices for \(t\text{-}(v,k,\lambda _t)\) designs, for any integer value of \(t.\)      

Rates of Convergence for a Class of Rank Tests for Independence

An almost optimal rate of convergence estimate is obtained for a large class of rank statistics for testing independence, including Gini's and Spearman's rank correlation coefficients as well as Spearman's footrule. Bibliography: 18 titles.     

Equivalence of conditions for convergence of iterative methods for singular equations

Equivalence is shown between different conditions for convergence of iterative methods for consistent singular systems of linear equations on Banach spaces. These systems appear in many applications, such as Markov chains and Markov processes. The conditions considered relate the range and null spaces of different operators.     

Extensions of ValEncIA-IVP for reduction of overestimation,for simulation of differential algebraic systems,and for dynamical optimization

Simulation techniques are commonly used to analyze the influence of uncertainties of initial conditions and systemparameters on the trajectories of the state variables of dynamical systems. In this context, interval arithmetic approaches are of interest. They are capable of determining guaranteed bounds of all reachable states if worst-case bounds of the above-mentioned uncertainties are known. Furthermore, interval algorithms ensure the correctness of numerical results in spite of rounding errors which inevitably arise if floating point operations are carried out on a computer. However, naive implementations of interval algorithms often lead to overestimation, i.e., too conservative enclosures which can make the results meaningless. In this contribution, we summarize the basic routines of ValEncIA-IVP which computes interval enclosures of all reachable states of dynamical systems described by ordinary differential equations ODEs. ValEncIA-IVP , VAL idation of state ENC losures using I nterval A rithmetic for I nitial V alue P roblems, can be applied to the simulation of systems with both uncertain parameters and uncertain initial conditions. Advanced techniques for reduction of overestimation are demonstrated for a simplified catalytic reactor. Afirst approach to using VanEncIA-IVP for the simulation of sets of differential algebraic equations is outlined. Finally, an outlook on the integration of ValEncIA-IVP in an interval arithmetic framework for computation of optimal and robust control strategies for continuous-time processes is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

积分中值定理中间点比较及有关平均不等式

中值定理中间点是区间端点的平均.设f (x)、g(x)在同一区间[a,b]内严格单调并可积,p(x)、q(x)恒正可积,按积分中值定理各有唯一的中间点ξf ,p(a,b)和ξg,q(a,b) .当f递增(减)且f (g- 1)凸(凹)时,有ξg,p(a,b) <ξf,p(a,b) ;当p(x)q(x) 递增(减)且q(x) ∫bap(x) dx >( <) 0时,有ξf,q(a,b) <ξf ,p(a,b) .由此可证明和发现一系列有关平均的不等式.     

Error estimates for a Galerkin method for a class of model equations for long waves

The Galerkin method, together with a second order time discretization, is applied to the periodic initial value problem for $$\frac{\partial }{{\partial t}}(u - (a(x)u_x )_x ) + (f(x,u))_x = 0$$ . Heref(x, ·) may be highly nonlinear, but a certain cancellation effect is assumed for∫f(x, u) _x u. Optimal order error estimates inL ₂,H ₁, andL _∞ are derived for a general class of piecewise polynomial spaces.     

带有Gilbert阻尼项的Landau-Lifshitz铁磁链方程解的最佳衰减率

主要研究带有Gilbert阻尼项的Landau-Lifshitz铁磁链方程的柯西问题.当初值的一阶导数适当小时,基于加权能量估计,证明了强解的整体存在性并且给出了解的最优的L2和L∞衰减估计.