Quenched large deviation principle for words in a letter sequence

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established.     

网络外部性条件下差异化双边平台买者参与问题

双边平台存在的基础是参与的用户,因此用户的分配是双边市场上各利益主体最关心的问题。为研究用户分配问题,本文在具有差异性的竞争双边市场上,建立了两阶段的双边平台竞争模型。第一阶段使用Rubinstein讨价还价模型对卖者的参与人数进行分配,利用讨价还价顺序描述卖者进入的先后顺序;第二阶段分别在平台利润最大化和社会福利最大化两种情况下,刻画买者对平台的规模偏好,并求均衡的市场份额。通过此模型分析竞争性平台差异化的情况下,网络外部性系数对平台均衡市场份额以及市场利益相关者利得的影响。研究表明:在考虑平台利润最大化的情况下,买者参与人数受卖者规模与网络外部性系数的影响,平台规模并非越大越好;在考虑社会福利最大化的情况下,买者的网络外部性系数与买者参与人数不相关,但与总社会福利正相关。     

Numerical Simulation of Crack Growth

In recent years the X-FEM based on the partition of unity method and the strong discontinuity approach (SDA) have shown to be powerful tools to model crack growth. Both methods model the crack surface by introducing additional d.o.f.. In the X-FEM the nodes in the mesh around a crack are globally enhanced with new d.o.f. while in in the SDA the new d.o.f. are commonly introduced as internal ones. Thus the jump displacement fields are constant across elements. Therefore the d.o.f. can be condensed on element level which results in jumps in the displacement field at element edges. In this contribution the strong discontinuity approach is used approximating the displacement jump linearly across the crack length similar as e.g. in [3]. New additional nodes of the cracked elements that lie on the element edges are introduced but are not considered as internal nodes but remain global. Thus crack path continuity is automatically given. These global d.o.f. approximate the discontinuous part of the displacement field. The sum of the aforementioned part and the continuous displacement field represent the total displacement field including a possible jump. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

天然心瓣关闭机理研究

本文在二维模型假定的基础上,从一个新的角度,通过理论分析和实验研究,解决了心脏瓣膜在射血加速相的部分关闭机理以及旋涡在瓣关闭中的作用这两个心瓣关闭机理研究中长期以来一直悬而未决的问题,并指出了窦(或瓣叶后空腔)对有效瓣膜关闭的决定性作用.此外,本文还研究了瓣长、频率等对瓣运动的影响.估计本文的研究对人工心瓣的研制有一定参考作用.     

The Convergence of the Spectral Scheme for Solving Two-Dimensional Vorticity Equations

Much work has been done for the spectral scheme of the P.D.E. The author proposed a technique to prove the strict error estimation of the spectral scheme for the K.D.V.-Burgers equation. In this paper, the technique is generalized to two-dimensional vorticity equations. Under some conditions, the error estimation implies the convergence. The more smooth the solution of the vorticity equations, the more accurate the approximate solution.     

Problem of an elastic half-plane with a crack emerging orthogonally at the boundary

The problem of the half-plane, in which a finite crack emerges orthogonally at the boundary, is studied. On the edges of the crack a self-balancing load is applied. A detailed investigation is carried out for an integral equation with respect to the unknown derivative of the displacement jump, to which the problem can be reduced. The exact solution of the integral equation is constructed with the aid of the Mellin transform and the Riemann boundary value problem for the halfplane. The asymptotic behavior of the solution at both ends of the crack is elucidated. First the asymptotic behavior of the solution at the point of emergence of the crack is obtained and the dependence of this asymptotic behavior on the type of the load is established. For a special form of the load one obtains a simple expression of the stress intensity coefficient. In the case of a general load, the asymptotic behavior is used for the construction of an effective approximate solution on the basis of the method of orthogonal polynomials. As a result, the problem reduces to an infinite algebraic system, solvable by the reduction method.Translated from Dinamicheskie Sistemy, No. 4, pp. 45–51, 1985.     

On a lower and upper bound for the curvature of ellipses with more than two foci

Let a set of points in the Euclidean plane be given. We are going to investigate the levels of the function measuring the sum of distances from the elements of the pointset which are called foci. Levels with only one focus are circles. In case of two different points as foci they are ellipses in the usual sense. If the set of the foci consists of more than two points then we have the so-called polyellipses. In this paper we investigate them from the viewpoint of differential geometry. We give a lower and upper bound for the curvature involving explicit constants. They depend on the number of the foci, the rate of the level and the global minimum of the function measuring the sum of the distances. The minimizer will be characterized by a theorem due to E. Weiszfeld together with a new proof. Explicit examples will also be given. As an application we present a new proof for a theorem due to P. Erd?s and I. Vincze. The result states that the approximation of a regular triangle by circumscribed polyellipses has an absolute error in the sense that there is no way to exceed it even if the number of the foci are arbitrary large.     

Derivation of the particle dynamics from kinetic equations

The microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov are considered. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversible. Also, the so called approximate microscopic solutions are constructed. These solutions are continuous and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the timereversible particle dynamics. Usually one tries to derive the kinetic equations from the particle dynamics. On the contrary, we postulate the Boltzmann-Enskog equation or another kinetic equation and treat their microscopic solutions as the particle dynamics. So, instead of the derivation of the kinetic equations from the microdynamics we suggest a kind of derivation of the microdynamics from the kinetic equations.     

Nonstationary boundary problem for model kinetic equations at critical parameters

Nonstationary solutions of the model kinetic equation at critical values of the motion of the wall (the boundary of the half-space occupied by gas) are studied. The characteristic equation is obtained by separating the variables. The eigenfunctions and the eigenvalue spectrum are found in the distribution space. A solution to the equation is expandable over the eigenfunction basis. The Rayleigh problem is considered as an application. The distribution function is continuous in the plane of the wall-motion parameters, including the closed curve of critical parameter values. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 116. No. 2, pp. 305–320. August. 1998.     

Influence of the solvent on the durability of cellulose acetate films formed from solutions

The authors examine the influence of the nature of the solvent on the time dependence of the strengths of films obtained from solutions of cellulose acetate in various solvents. They found that the durability of the films depends on the nature of the solvent. They have established that as the thermodynamic affinity of the solvent in the film-forming solutions gets worse, the durability of the films passes through a maximum; the nature of the solvent influences the parameter in the equation for the durability, which is responsible for changes in the structure of the polymer.V. I. Lenin Tadzhik State University, Dushanbe. A. M. Gor'kii Ural State University, Sverdlovsk. Translated from Mekhanika Polimerov, No. 2, pp. 360–362, March–April, 1974.     

The canonical transformation method in the periodic Anderson model

We investigate a version of the periodic Anderson model in which both the d- and f-electron subsystems are strongly correlated. The one-site hybridization of the electron quantum states in each subsystem and the possibility of the d-electron hopping between lattice sites are taken into account. To construct the canonical transformation S-matrix, we use the system of one-site orthonormalized functions belonging to the zero Hamiltonian matrix of rank 16. We solve the problem exactly and determine the thermodynamic properties of the system in the approximation where the width of the conductance band vanishes. We use the diagram technique to investigate the delocalization of electrons in each subsystem and the renormalization of the one-particle Green's functions. We find the quasiparticle energy spectrum of delocalized electrons in the chain diagram approximation. We show that there are eight energy subbands in the symmetrical case. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 464–478, December, 1999.     

Singular Perturbation Solution of Time Dependent Mass-Transfer with Non-linear Chemical Reaction

In this work the author presents two analyses of time dependentprocesses in which diffusion and non-linear reaction take placesimultaneously. The first is an analysis of the distributionof a single reactant in a stagnant medium when the reactionrate is proportional to the concentration squared. In the secondanalysis the time dependent distributions of two reactants aretraced, when the reaction rate is porportional to the productof the concentrations. The processes under consideration areassumed to take place in unbounded containers and attentionis focused on the cases in which the initial concentrationshave non-zero space average. The analyses are carried out assumingthat the space variations of the initial distributions and thediffusivities are large while the reaction constant(s) is small.Therefore, initially the changes in the distribution due todiffusion are much faster than the rate of absorption in thereaction. However, eventually the processes become reactiondominated. The two distinct behaviours are represented by correspondingasymptotic expansions which are matched in the usual manner.On the other hand, if the initial average concentrations vanish,then, in as much as the processes are diffusion dominated initially,they are diffusion dominated thoughout. The distributions canthen be solved-for by regular perturbations. The treatment of the single component case shows that when theinitial average concentration does not vanish, the residualamount of reactant is inversely proportional to the time thatelapsed since the beginning of the process multiplied by thereaction constant. This residual amount is altogether independentof the bulk quantity supplied initially. The decay of the binaryprocess is similar provided the bulk quantities supplied initiallyare well proportioned. If there is a disproportion, the decayis exponential rather than algebraic. In all events, late inthe process the residual amount(s) is independent of the diffusivity.     

基于上限理论的孔隙水压力作用下隧道洞口段含裂缝仰坡稳定性分析

研究了孔隙水压力作用下隧道洞口段含裂缝仰坡的稳定性。采用极限分析上限法,构建了坡顶含竖向裂缝的对数螺旋转动破坏机制,推导了可反映边坡临界坡高的稳定系数计算公式,将计算结果与未考虑孔隙水压力作用下的含裂缝边坡稳定性极限分析结果进行对比,验证了所提研究方法的合理性。通过算例分析,研究了坡顶裂缝最不利位置分布及仰坡整体安全系数。结果表明:坡顶裂缝开裂深度、土体内摩擦角、坡角越大及水位分布越浅,裂缝位置越靠近坡顶边缘处;孔隙水压力系数、坡顶裂缝开裂深度越大,仰坡稳定性系数越小;坡顶裂缝越深、孔隙水压力系数越大、边坡越陡,越不利于仰坡稳定;而坡内水位分布越低,越有利于仰坡稳定。     

Bending of a semi-infinite cantilevered plate weakened by a cut with contacting edges

We solve the problem of the bending of a semi-infinite cantilevered plate containing a cut perpendicular to a clamped edge. Contact of the edges of the cut is taken into account in the two-dimensional formulation on the basis of the model of contacting edges on the face of the plate. We study the effect of the boundary on the distribution of the contact reaction and compute the coefficients of force and moment intensity and determine the breakinge load. We compare the results obtained with the solution of the problem not taking account of the contact of the edges of the cut. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 83–86.     

Generalized singly-implicit Runge-Kutta methods with arbitrary knots

The aim of this paper is to derive Butcher's generalization of singly-implicit methods without restrictions on the knots. Our analysis yields explicit computable expressions for the similarity transformations involved which allow the efficient implementation of the first phase of the method, i.e. the solution of the nonlinear equations. Furthermore, simple formulas for the second phase of the method, i.e. computation of the approximations at the next nodal point, are established. Finally, the matrix which governs the stability of the method is studied.     

An asymptotic form of the energy functional for an elastic body with a crack and a rigid inclusion. The plane problem

The plane problem in the linear theory of elasticity for a body with a rigid inclusion located within it is investigated. It is assumed that there is a crack on part of the boundary joining the inclusion and the matrix and complete bonding on the remaining part of the boundary. Zero displacements are specified on the outer boundary of the body. The crack surface is free from forces and the stress state in the body is determined by the bulk forces acting on it. The variation in the energy functional in the case of a variation in the rigid inclusion and the crack is investigated. The deviation of the solution of the perturbed problem from the solution of the initial problem is estimated. An expression is obtained for the derivative of the energy functional with respect to a zone perturbation parameter that depends on the solution of the initial problem and the form of the vector function defining the perturbation. Examples of the application of the results obtained are studied.     

Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice

The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.     

Micromechanics of polymer fracture

Small-angle x-ray scattering has been used to investigate the formation of embryonic submicroscopic cracks in polymers under a load. The main characteristics of crack formation in various loading regimes are analyzed. It is shown that there is a relation between the submicrocrack concentration and the deformation of the loaded polymer. The principal parameters of crack formation determining the strength properties of the polymer are found to be the transverse dimension of the initial submicrocracks relative to the loading axis, which is determined by the structural heterogeneity of the material, and the submicrocrack concentration in the prefracture state. The principles of the micromechanics of polymer fracture are formulated on the basis of the results of an analysis of the quantitative relationship between these parameters. The dominant role of the surface in the fracture process is demonstrated by comparing the parameters of crack formation in the interior and at the surface of the loaded polymer.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 5, pp. 792–801, September–October, 1974.     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

Structure equations and constraint manifolds on Lorentz plane

Calculating the structure equation of a chain is important to represent the position of the end link on the chain. Furthermore, the structure equation helps to determine the constraint manifold of the chain. The constraint manifold satisfies to make geometric interpretations about the form that is obtained. What is more, the constraint forced on the positions of the end link by the rest of the chain is represented by the manifold. In Lorentz space, the structure equations change according to the causal characters of the first link. In this paper, we attain the structure equations of a planar open chain in terms of the causal character of the first link in this space. Later, the constraint manifolds of the chain by using these equations are given. Some geometric comments about these manifolds are explained.