Acceleration of the convergence of the Laguerre series in the problem of inverting the Laplace transform

When the Laplace transform is inverted numerically, the original function is sought in the form of a series in the Laguerre polynomials. To accelerate the convergence of this series, the Euler-Knopp method is used. The techniques for selecting the optimal value of the parameter of the transform on the real axis and in the complex plane are proposed.     

多晶体光滑表面疲劳微裂纹形核机理研究

建立晶体学模型来研究多晶体承受均匀外载时光滑表面疲劳微裂纹形核的机理．由于晶粒取向的不相同,即使承受均匀外载,多晶体内将产生不均匀的应力,变形也不均匀A·D2在疲劳载荷的作用下,表面粗糙度随疲劳周次逐渐增加,变形逐渐集中到若干部位,即局部化,形成微裂纹．     

Examples of the influence of the geometry on the propagation of progressive waves

In this paper, we give examples of the influence of the domain of propagation on progressive waves. More precisely, we numerically investigate the propagation of reaction diffusion waves in cylinders with variable radius. We show that, when the radius rapidly expands from a very small radius to a larger one, depending on the viscosity and the nonlinearity, the travelling wave may be blocked. The aim of this paper is to give numerical illustrations and quantifications of this effect, and to propose some conjectures which could be interesting subjects for further mathematical investigations.This work is linked to the study of spreading depression (SD), a propagative mechanism in brain and various tissues which has been observed in vivo and in vitro in many species since their discovery in 1944 by Leao. As a matter of fact, their direct observation in Man is still controversial. The complex structure of gray and white matter in humans may block the propagation of SD over large distances in brain and thus explain the difficulty of observing it. Medical consequences of the current numerical studies are detailed in [M.A. Dronne, et al., Influence of brain geometry on spreading depressions: A computationnal study, Progress in Biophysics and Molecular Biology 97 (1) (2008) 54–59] and a first mathematical approach given in [M.A. Dronne, E. Grenier, H. Gilquin, Modelization of spreading depressions following Nedergaard, preprint, 2003].     

On the algorithm of the solution of the signorini problem

An algorithm is proposed for solving the Signorini problem /1/ in the formulation of a unilateral variational problem for the boundary functional in the zone of possible contact /2/. The algorithm is based on a dual formulation of Lagrange maximin problems for whose solution a decomposition approach is used in the following sense: a Ritz process in the basis functions that satisfy the linear constraint of the problem, the differential equation in the domain, is used in solving the minimum problem (with fixed Lagrange multipliers); the maximum problem is solved by the method of descent (a generalization of the Frank-Wolf method) under convexity constraints on the Lagrange multipliers. The algorithm constructed can be conisidered as a modification of the well-known algorithm to find the Udzawa-Arrow-Hurwitz saddle points /3, 4/. The convergence of the algorithm is investigated. A numerical analysis of the algorithm is performed in the example of a classical contact problem about the insertion of a stamp in an elastic half-plane under approximation of the contact boundary by isoparametric boundary elements. The comparative efficiency of the algorithm is associated with the reduction in the dimensionality of the boundary value problem being solved and the possibility of utilizing the calculation apparatus of the method of boundary elements to realize the solution.     

On the choice of the best members of the Kim family and the improvement of its convergence

The best members of the Kim family, in terms of stability, are obtained by using complex dynamics. From this elements, parametric iterative methods with memory are designed. A dynamical analysis of the methods with memory is presented in order to obtain information about the stability of them. Numerical experiments are shown for confirming the theoretical results.     

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p,n→∞ with p/n→c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p,n.     

Bound on the number of eigenvalues near the boundary of the pseudospectrum

We show an estimate of the number of eigenvalues in a neighbourhood of a finite part of the boundary of the semiclassical pseudospectrum of pseudodifferential non-selfadjoint operators in terms of a corresponding volume in phase space.

     

生态系统的参数确定问题

通过不同观测数据研究捕食者—被捕食者生态系统参数确定问题.研究了四种情形1.观察数据无误差,并已知一个参数值.这种情况下,参数可由其相轨线和最小二乘法精确确定.2.观察数据无误差,但所有参数未知.此时仅靠相轨线的研究,无论给出多少组精确数据,都无法精确确定这些参数.通过原非线性模型的数值计算和网格搜索法,至少需要4组数据,同样得到了精度较高的参数值.3.当观测数据有误差时,根据解的周期性,引入标准周期的概念,在一个标准周期里讨论参数的确定问题,并利用标准周期内的捕食者与被捕食者的数量均值与系统的平衡点的关系对参数进行修正,然后使用网格法进行搜索,进一步提高了参数的精度.4.当观测时间也有误差时,先选取相对最优的随机正态数对观测时刻进行修正,然后再利用3.的方法估计参数.     

On the degree of the inverse of an automorphism of the affine space

We study the degree of the inverse of an automorphism f of the affine n-space over a -algebra k, in terms of the degree d of f and of other data. For n = 1, we obtain a sharp upper bound for deg (f^{− 1}) in terms of d and of the nilpotency index of the ideal generated by the coefficients of f^′'. For n = 2 and arbitrary d≥ 3, we construct a (triangular) automorphism f of Jacobian one such that deg(f^{− 1}) > d. This answers a question of A. van den Essen (see [3]) and enables us to deduce that some schemes introduced by authors to study the Jacobian conjecture are not reduced. Still for n = 2, we give an upper bound for deg (f^{− 1}) when f is triangular. Finally, in the case d = 2 and any n, we complete a result of G. Meisters and C. Olech and use it to give the sharp bound for the degree of the inverse of a quadratic automorphism, with Jacobian one, of the affine 3-space.     

On the Young Conjugate Functions and the Behavior of the Maximal Terms of the Derivatives of Dirichlet Series

We obtain a criterion for the growth of the maximal term of the Dirichlet series with positive exponents in terms of the behavior of the maximal terms of the successive derivatives of this series. This criterion is obtained from a more general result concerning Young conjugate functions proved in this paper.     

Effect of the properties of the components and the geometric characteristics of the structure on the thermal-conductivity coefficients of glass-reinforced plastics

A method is proposed for estimating the thermal-conductivity coefficients of orthotropic composite glass-reinforced plastics with a structure composed of continually repeating units, in which the contours of the reinforcing elements can be described by a sinusoidal law. The arguments in the formulas are the relative content and thermophysical characteristics of the components and the parameters of the microstructure of the materials considered.Mekhanika Polimerov, Vol. 4, No. 1, pp. 18–23, 1968     

Simulation of the effect of learning on the topology of the functional connectivity of neural networks

We introduce a procedure for simulating adaptive learning in neural networks and the effect this learning has on the way in which the functional connections between the nodes of the network are established. The procedure combines two mechanisms: firstly, the gradual dilution of the network through the elimination of synaptic weights in increasing order of magnitude, thus reducing the costs of the network structure. Secondly, to train the network as it is diluted so as not to compromise its performance pursuant to the proposed task. Considering different levels of learning difficulty, we compare the topology of the functional connectivities that result from the application of this procedure with those obtained using fMRI in healthy volunteers. According to our results, the topology of functional connectivities in healthy subjects can be interpreted as the product of a learning process with a specific degree of difficulty.     

Combined use of the method of thermoplasticity and the method of the shrinkage function for the study of the process of thermal straightening of shipbuilding panels

We present an approach to the combined use of the general method of thermoplasticity and the approximate method of the shrinkage function for predicting total distortions of large-sized spatial structures in welding or thermal straightening. The proposed approach is particularly efficient in the case of a large number of welded joints in the structure or local heatings in thermal straightening. An example of using the approach in the study of the efficiency of the process of thermal straightening of shipbuilding panels is presented.     

On the optimality of the greedy solutions of the general knapsack problems

Summary: In this paper we submit a unified discussion of some closely related results which were achieved independently in number theory and integer programming, and we partially generalize them. In the unified discussion we treat together two problems where the greedy method has different characters, in the first one it is an internal-point algorithm, in the second one it is an outer-point method. We call a knapsack problem "pleasant" if the greedy solution is optimal for every right-hand side. A sufficient and two finite necessary and sufficient conditions for the pleasantness of a problem are discussed. The sufficient condition can be checked very easily. The paper is finished with an error analysis of some nonpleasant problems     

On the connectedness of the complement of the zero-divisor graph of a poset

Abstract

In this paper, connectedness is completely characterized for the complements of the zero-divisor graphs of partially ordered sets. These results are applied to annihilating ideal graphs and intersection graphs of submodules, generalizing some of the work that has recently appeared in the literature.     

On the local semiconcavity of the solutions of the eikonal equation

We show that if the Hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave. Furthermore, in the 1D case, we show that every viscosity solution of the eikonal equation is semiconcave if and only if the Hamiltonian is Lipschitz continuous with respect to the state variable.     

On the lack of null-controllability of the heat equation on the half-line

We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.

     

On the minimization of the product of the powers of several integrals

This paper considers the minimization of the product of the powers ofn integrals, each of which depends on a functiony(x) and its derivative . The necessary conditions for the extremum are derived within the frame of the Mayer-Bolza formulation of the calculus of variations, and it is shown that the extremal arc is governed by a second-order differential equation involvingn undetermined multipliers related to the unknown values of the integrals. After the general solution is combined with the definitions of the multipliers and the end conditions, a system ofn+2 algebraic equations is obtained; it involvesn+2 unknowns, that is, then undetermined multipliers and two integration constants.The procedure discussed here can be employed in the study of shapes which are aerodynamically optimum at supersonic, hypersonic, and free-molecular flow velocities, that is, wings and fuselages having the maximum lift-to-drag ratio or the minimum drag. The problem of a slender body of revolution having the minimum pressure drag in Newtonian hypersonic flow is developed as an example. First, a general solution is derived for any pair of conditions imposed on the length, the thickness, the wetted area, and the volume. Then, a particular case is treated, that in which the thickness and the wetted area are given, while the length and the volume are free; the shape minimizing the pressure drag is a cone.This research, supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensed version of the investigation described in Ref. 1. The author is indebted to Messrs. H. Y. Huang, J. C. Heideman, and J. N. Damoulakis for analytical and numerical assistance.     

Reversal of the reinforcing effect of fillers and the relation between the flexibility of the macromolecules and the mechanical properties of filled polymers

The author describes various effects which he has detected involving the reversal of the reinforcing action of fillers in polymers in different subregions of the glassy state. It is suggested that these effects can be explained in terms of, among other things, the flexibility of the macromolecules. The concept of an optimal flexibility at which the maximum value of a given mechanical property can be realized is introduced. It is shown that it is possible, in principle, to reinforce a polymer in any subregion of the glassy state and even below T_b, as well as to obtain equal-strength, equal-modulus, equally reinforced, etc., composites containing different (large and small) amounts of fillers and plasticizers.     

On the distribution of the length of the longest increasing subsequence of random permutations

The authors consider the length, , of the longest increasing subsequence of a random permutation of numbers. The main result in this paper is a proof that the distribution function for , suitably centered and scaled, converges to the Tracy-Widom distribution of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest descent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel for the Poissonization of the distribution function of .