On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.     

On the variation of the spectra of matrices

Several “distances” between the spectra of two matrices are discussed and compared. Optimal bounds are given, which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two. The results of Bhatia and Mukherjea and of Bhatia and Friedland on the eigenvalue variation are derived in an elementary way using results of Henrici on the spectral variation.     

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.     

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.     

Effect of the incompressibility condition on the stability of the solutions of ill-posed problems of the theory of elasticity

One considers the energy functional with density depending only on the first invariant of the strain tensor. Under natural constraints on the density and in the absence of exterior actions, the zero displacements yield a minimum to the energy functional both for the compressible and the incompressible media. In the paper one proves the isolation of this minimum for incompressible media and its nonisolation for compressible media.Translated from Problemy Matemticheskogo Analiza, No. 10, pp. 62–71, 1986.     

Regularization of the solution of the cauchy problem for the generalized Moisil-Theodoresco system

We consider the problem of analytic continuation of a solution of the generalized Moisil-Theodoresco system in a spatial domain on the basis of its values on part of the boundary of this domain, that is, the Cauchy problem. We construct an approximate solution of this problem on the basis of the Carleman matrix method.     

A local estimate of the maximum of the module of the deviator of the strain tensor in an elastoplastic body with linear hardening

An upper estimate for the module of the deviator of the strain tensor at any interior point of an elastoplastic body is proved. It depends only on the L₂-norm of the deviator of the strain tensor over the domain occupied by this elastoplastic body, on the L_p-norm of body forces, on the parameters of the elastoplastic medium, and on the distance to the boundary of the body. Bibliography: 5 titles. To dear Olga Alexandrovna Ladyzhenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 167–176. Translated by G. A. Seregin.     

Analysis of the steady motions of the tippe top

The motion of the tippe top on a horizontal plane is considered taking into account sliding friction within the Contensou model. The tippe top is modelled by two spherical segments rigidly joined by a rod directed along the common axis of symmetry of the segments. The dimensions of the spherical segments and the rod are chosen so that, as the axis of symmetry deviates from the upward vertical, the tippe top is supported on the plane at a point on one segment up to a certain critical value and at a point on the other segment at larger deviations (at points on both segments at the critical value). The motion of the tippe top is described by different equations in different regions of configuration space, and the motion is accompanied by impacts on the boundary of these regions. An effective potential of the system is constructed, and the type of its critical points is investigated. Poincaré–Chetayev bifurcation diagrams and generalized Smale diagrams are constructed for steady motions. Plots of the steady-state precessional motions have a discontinuity on the boundary between the regions indicated.     

关于Camassa-Holm方程动量密度的紧支集在半轴上的估计

主要考虑在半轴上Camassa-Holm方程解的动量密度紧支集大小的估计,方法是根据区间长度与区间特征值的关系,通过估计第一Dirichlet特征值来估计动量密度紧支集的长度.因为知道动量密度紧支集外解的性态,所以通过估计动量密度支集的大小可以得到方程解的更多信息.     

直线上分形的Hausdorff测度的一个算法

本文给出了直线上Marion集的Hausdorff测度的一个有效计算方法,并通过几个实例得出如何利用此方法计算出直线上分形的Hausdorff测度的精确值.     

Bi-Hamiltonian Aspects of the Separability of the Neumann System

The Neumann system on the two-dimensional sphere is used as a tool to convey some ideas on the bi-Hamiltonian standpoint on separation of variables. We show that from this standpoint, its separation coordinates and its integrals of motion can be found systematically.     

On the dependence of the critical exponent of the nonlinear heat equation on the initial function

We establish the dependence of the critical exponent of the nonlinear heat equation on the initial function and derive an estimate for the blow-up time of the solution of this problem depending on the nonlinearity exponent and the initial function.     

Dependence of the Nature of Equilibrium States of a Two-Phase Medium on the Parameters of the Problem

Under general assumptions on the densities of the energy of deformation, which depend on parameters, zones of the values of parameters are determined for which one-phase or two-phase equilibrium states are realized. It is proved that the boundary between these zones is Lipschitz. The nature of an equilibrium of a two-phase medium is studied for parameters lying on the line of demarcation of the zones. Bibliography: 1 title.     

On the Modelling of the Adoption of Process Innovations

An alternative view on the modelling of diffusion processesof process innovations has been introduced by Nooteboom (1989). Thebasic idea in that paper is to model the process of adoption ofprocess innovations as a gambling model with odds in favourdepending on expected returns and odds against depending on risk.In this paper it is investigated under which theoretical conditions posed on the expected return function the usual adopted assumptions Nooteboom (1989) with respect to the parameters of the gambling model are in fact reasonable. It is shown that only under certainconditions the assumptions made on the gambling model are true ontheoretical grounds. In all other cases a nonexample shows that theresults as mentioned in literature are in general not true.     

Action of the symmetric groups on the homology of the hypertree posets

The set of hypertrees on n vertices can be endowed with a poset structure. J. McCammond and J. Meier computed the dimension of the unique non-zero homology group of the hypertree poset. We give another proof of their result and use the theory of species to determine the action of the symmetric group on this homology group, which is linked with the anti-cyclic structure of the $\operatorname{PreLie}$ operad. We also compute the action on the Whitney homology of the poset.     

An Economical Algorithm for Numerical Solution of the Problem of Identifying the Right-Hand Side of the Poisson Equation

We propose two economical algorithms for numerical solution of the problem of identifying the right-hand side of the Poisson equation from information on the solution on the boundary of the domain. Both algorithms are based on the method of separation of variables. The method is presented on a discrete level. We use the nonuniform grids along one of the coordinates. There are possible applications for operators with variable coefficients of a special kind.     

随机数学建模方法在钢筋与混凝土粘结强度分析中的应用

本文主要利用随机建模的方法推导了均速加载的速率对粘结承载力影响量化表达的基本形式 .运用随机过程中的 Markov过程的方法得到了加载速率对粘结承载力 (平均粘结强度 )影响的量化表述 ,并综合考虑了钢筋锈蚀率的影响 ,且结合实际试验确定了一些未知参数 ,在此基础上分析了混凝土强度变化可能的影响 .     

Effect of hyperviscosity on the geometry of the vorticity

We consider the two-dimensional modified Navier–Stokes equations by adding a p-Laplacian. We provide upper bounds on the Hausdorff length of the level sets of the vorticity. We express them in terms of the Landau–Lifschitz estimates of the number of degrees of freedom in turbulent flow. The proof, based on very standard methods, relies on a straightforward relation between the size of the nodal set of the vorticity and the growth of hyperdissipation.     

On the spectrum of the Laplacian

In this article we prove a generalization of Weyl’s criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.     

On the zeros of the spectrogram of white noise

In a recent paper, Flandrin [16] proposed filtering based on the zeros of a spectrogram with Gaussian window. His results are based on empirical observations on the distribution of the zeros of the spectrogram of white Gaussian noise. These zeros tend to be uniformly spread over the time–frequency plane, and not to clutter. Our contributions are threefold: we rigorously define the zeros of the spectrogram of continuous white Gaussian noise, we explicitly characterize their statistical distribution, and we investigate the computational and statistical underpinnings of the practical implementation of signal detection based on the statistics of the zeros of the spectrogram. The crux of our analysis is that the zeros of the spectrogram of white Gaussian noise correspond to the zeros of a Gaussian analytic function, a topic of recent independent mathematical interest [24].