k=2t·3时不存在以(Fn-k,Fn,Fn)为边长的Fibonacci三角形

Fibonacci三角形是边长为Fibonacci数、面积为整数的三角形.利用平方剩余的方法得到:当k=2'·3时,不存在边长为(Fn-k,Fn,Fn)的Fibonacci三角形(k＜2).     

尾数恰含k个零的正Fibonacci数

根据正Fibonacci数Fn的标准分解式中,因子2和因子5的指数的性质,利用初等数论的知识,讨论了尾数恰含k个零的正Fibonacci数Fn的下标n的特征,并证明了:对于任意大的正整数k,都存在着尾数恰含k个零的正Fibonacci数.     

正Fibonacci数的标准分解式中因子2的指数

Fibonacci数列 {Fn}定义如下 :F0 =0 ,F1 =1,Fn + 1 =Fn +Fn -1 (n =1,2 ,… ) ,我们把 {Fn}中每一项Fn 叫做一个Fibonacci数 ,当n≥ 1时 ,称Fn 为正Fibonacci数 .关于正Fibonacci数的奇偶性及其中偶Fibonacci数中因子 2的指数 ,笔者在文 [1]中已有部分结果 (见下文中引理 1) ,即正Fibonacci数Fn 的奇偶性 ,由其下标n是否含因子 3来确定 ,且当n是一个奇数的 3倍时 ,Fn 的标准分解式中 ,因子 2的指数确定为1.本文所做的工作 ,是利用同余的知识 ,对于n是一个正偶数的 3倍时 ,Fn 的标准分解式中因子 2的指数给出一个准确的结果 .定理 1…     

利用Fibonacci数列解题

Fibonacci数列本身就有很大的魅力 ,吸引着许多数学爱好者去学习和研究 .这里我们将视角定位在如何利用该数列去解决一些数学竞赛中的问题 .Fibonacci数列是指由下面的递推式定义的数列 {Fn}:F0 =F1 =1,Fn + 2 =Fn+ 1 +Fn ,n =0 ,1,2 ,…可以利用特征方程的方法求出其通项公式 ,也可以用数学归纳法证出其许许多多的性质 .但在这里我们更多的是用到其本身 ,而不是它的性质 .例 1(第 5 2届波兰数学竞赛试题 )　考虑数列 {xn}:x1 =a ,x2 =b ,xn + 2 =xn + 1 +xn,n =1,2 ,… ,这里a ,b∈R .对任意c∈R ,如果存在k ,l∈N ,k≠l ,使得xk =xl=…     

Fibonacci数列的模数列的周期性

对于Fibonacci数列{Fn}以及给定的正整数m,由Fn关于模m的最小非负剩余an,构成一个新的数列{an},称为Fibonacci数列的模数列.本文利用初等数论的知识和数学归纳法,证明了Fibonacci数列的模数列是周期数列,并且是纯周期数列.     

正Fibonacci数的标准分解式中因子5的指数

根据Fibonacci数列的定义,利用初等数论的知识和数学归纳法,讨论了正Fibonacci数Fn的标准分解式中因子5的指数与下标n的关系,得到下列结论:正Fibonacci数Fn的标准分解式中因子5的指数,与下标n的标准分解式中因子5的指数一致.     

Fibonacci数的一组整除特征

Fibonacci数列 {Fn}定义如下 :F0 =0 ,F1=1 ,Fn +1=Fn+Fn - 1(n =1 ,2 ,… ,) ,我们把{Fn}中每一项Fn 叫做一个Fibonacci数 .本文将讨论Fibonacci数Fn 被某些整数整除的特征 .在其证明过程中所用到的关于整除、最大公约数、最小公倍数以及同余的一些简单性质 ,恕不一一列作引理 .此外 ,证明过程中还用到下列数据 :F0 =0 ,F1=1 ,F3=2 ,F4 =3,F5=5,F9=34,F10 =55,F15=6 1 0 ,F16 =987,F2 7=1 96 41 8,F2 8=31 781 1 ,等等 ,这些数据 ,都不难利用Fibonacci数列的定义直接计算得到 .以下的引理是后面定理的证明过程所必须的 .引理 1 […     

一个问题的解决

本文将解决文[1]末提出的如下问题： 问题1 求函数y-^n∑i=1Fi｜x-Fi｜的最小值，其中x∈R,{Fn}n≥0为Fibonacci数列，它由F0=0,F1=1,Fn＋2=Fn＋1＋Fn（n∈N）确定。     

Fibonacci数列与三道不等式的指数推广

著名的斐波那契 (Fibonacci)数列具有以下一个重要性质 :设 F1 =F2 =1 ,Fn 2 =Fn 1 Fn,n≥ 1 ,则Fn 3 =2 Fn 1 Fn.文 [1 ] [2 ] [3] [4]曾先后涉及到三道不等式 ,笔者发现其字母指数恰按斐波那契数列呈现 .请看 :问题 1　 (第 2 6届 USAMO赛题 )证明对所有正实数 a、b、c     

广义Fibonacci立方体的网络容错性质分析

主要研究广义Fibonacci立方体的容错直径和宽直径,证明了n维Fibonacci立方体网络的k-1容错直径和k宽直径都是n-1,其中k=[n/3].     

Variation of the structure and permeability of tensioned fiber layers during impregnation with a thermoplastic polymer melt

The influence of displacements of tensioned fibers on the impregnation of fibrous layers with a polymer melt and on the final composite structure is studied. Using computer simulation, it is shown that, during impregnation, the structure of tensioned fibrous layers changes considerably depending on the initial arrangement and tensioning of fibers. The consolidated regions formed under the melt front move inside the impregnated layer with the advancing melt front. Displacement of the tensioned fibers as well as the formation of “washouts” favors the impregnation of internal layers, but cause significant inhomogeneity of the polymer structure. The surface (on the side of the melt flow) regions are more saturated with the polymer than the internal ones. A difference in the melt percolation mechanisms at various impregnation regimes is revealed. The effective permeability coefficients of a tensioned fiber layer are not constant but depend on the conditions and regimes of impregnation. Submitted to the 11th Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 259–270, March–April, 2000.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

带有免疫和传染年龄的传染病模型

建立了带有免疫和传染年龄的传染病模型,这种传染病带有病原体Ⅰ或Ⅱ,病原体Ⅰ可发展为病原体Ⅱ,得到了无病平衡态全局稳定和局部稳定的条件.当病原体Ⅰ不发展为病原体Ⅱ时,得到了病原体Ⅰ类平衡态的稳定性依赖于病原体Ⅱ类的基本再生指数.     

Pole placement controllers for linear time-delay systems with commensurate point delays

This paper investigates the exact and approximate spectrum assignment properties associated with realizable output-feedback pole-placement type controllers for single-input single-output linear time-invariant time-delay systems with commensurate point delays. The controller synthesis problem is discussed through the solvability of a set of coupled diophantine equations of polynomials. An extra complexity is incorporated to the above design to cancel extra unsuitable dynamics being generated when solving the above diophantine equations. Thus, the complete controller tracks any arbitrary prefixed (either finite or delaydependent) closed-loop spectrum. However, if the controller is simplified by deleting the above mentioned extra complexity, then the robust stability and approximated spectrum assignment are still achievable for a certain sufficiently small amount of delayed dynamics. Finally, the approximate spectrum assignment and robust stability problems are revisited under plant disturbances if the nominal controller is maintained. In the current approach, the finite spectrum assignment is only considered as a particular case to the designer‘s choice of a (delay-dependent) arbitrary spectrum assignment objective.     

Utilization and maintenance in a model with scrapping

We consider a productive asset, called equipment or capital good, and we examine the properties of, as well as the interactions between, the operating policies, which are determined by its optimal utilization and maintenance, and the capital policy of scrapping, which defines the optimal time when the productive asset is retired from its current use. Starting with an abstract model and using the approach of optimal control, initially we characterize the various types of equipment by assigning to them a single total profit index, which indicates how the above policies affect the flow of operating revenues plus capital gains or losses. This index is a function of market-determined prices. So using it we then investigate how the operating and capital policies are influenced by the rate of discount, the price of new equipment, and the rise or fall of the price of new equipment relative to the value of its output. Among other interesting results, we find that the effects of these prices on the nature and interactions of optimal policies depend crucially on whether the equipment is of the usual profit making type, where output is the main source of revenue, or of the antique type, where the main source of revenue is capital gains.     

消除毛细管电泳槽道中弯道导致的扩散效应的新方法

首先分析了毛细管通道中流动的弯道效应及其产生的原因,接着在建立电渗流场数学模型的基础上,用有限差分法对弯道处内外壁面上不同电荷分布时的扩散进行了数值模拟．根据计算结果,提出了一种基于改变弯道处内外壁面上电荷分布的新方法,以此使流场的弯道效应最弱．同时还建立了分析和确定弯道处最佳电荷分布的优化方法．结果表明,该方法能极大地消除毛细管通道中的流动弯道效应．     

Modelling and simulation of micro-well formation

Physico-chemical processes on the micro-scale require new modelling concepts because some effects become dominating that are negligible for macroscopic systems. This is illustrated by a new method for the production of micro-wells based on the placement of a small drop of toluene on a plate of polystyrene. After droplet evaporation, a micro-well is left. A mathematical model has been developed to understand the elementary processes of the micro-well formation. The model accounts for: (1) growth of the drop on the substrate, (2) evaporation process of the solvent, (3) dissolution of the substrate, (4) flow rate in the evaporating drop caused by the pinning effect, including the vertical velocity profile, and (5) increase in the concentration of dissolved material followed by precipitation. In the modelling and simulation process, it could be shown that the method of drop production also has a significant influence on the shape of the micro-wells.     

A perturbed risk model with dependence between premium rates and claim sizes

This paper considers a dependent risk model with diffusion for the surplus of an insurer, in which a current premium rate will be adjusted after a claim occurs and the adjusted rate is determined by the amount of the claim. At the same time, the diffusion is changed correspondingly. Using Rouché’s theorem, we first derive the closed-form solution for the Laplace transform of the survival probability in the dependent risk model. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. For the exponential claim sizes, we present the explicit recursion expression for the survival probability, by which we can exactly solve the survival probability step-by-step. We also illustrate the influence of the model parameters in the dependent risk model on the survival probability by numerical examples.     

Multi‐dimensional Lotka–Volterra systems for carcinogenesis mutations

In the paper we consider three classes of models describing carcinogenesis mutations. Every considered model is described by the system of (n+1) equations, and in each class three models are studied: the first is expressed as a system of ordinary differential equations (ODEs), the second—as a system of reaction–diffusion equations (RDEs) with the same kinetics as the first one and with the Neumann boundary conditions, while the third is also described by the system of RDEs but with the Dirichlet boundary conditions. The models are formulated on the basis of the Lotka–Volterra systems (food chains and competition systems) and in the case of RDEs the linear diffusion is considered. The differences between studied classes of models are expressed by the kinetic functions, namely by the form of kinetic function for the last variable, which reflects the dynamics of malignant cells (that is the last stage of mutations). In the first class the models are described by the typical food chain with favourable unbounded environment for the last stage, in the second one—the last equation expresses competition between the pre‐malignant and malignant cells and the environment is also unbounded, while for the third one—it is expressed by predation term but the environment is unfavourable. The properties of the systems in each class are studied and compared. It occurs that the behaviour of solutions to the systems of ODEs and RDEs with the Neumann boundary conditions is similar in each class; i.e. it does not depend on diffusion coefficients, but strongly depends on the class of models. On the other hand, in the case of the Dirichlet boundary conditions this behaviour is related to the magnitude of diffusion coefficients. For sufficiently large diffusion coefficients it is similar independently of the class of models, i.e. the trivial solution that is unstable for zero diffusion gains stability. Copyright © 2009 John Wiley & Sons, Ltd.     

A necessary and sufficient condition for the existence of an exponential attractor

We give a necessary and sufficient condition for the existence of an exponential attractor. The condition is formulated in the context of metric spaces. It also captures the quantitative properties of the attractor, i.e., the dimension and the rate of attraction. As an application, we show that the evolution operator for the wave equation with nonlinear damping has an exponential attractor.