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 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.     

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.

     

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     

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].     

Time-independence of the solution of the problem of determining the hydraulic conductivity field

We consider the determination of the hydraulic conductivity field of a nonhomogeneous layer with linear seepage of an incompressible fluid in a nonelastic layer. In mathematical terms, the problem is formulated as a Cauchy problem for a partial differential equation of a special form. A theorem is proved which establishes that the solutions obtained for different times are identical.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 67–70, 1985.     

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 growth of the Betti sequence of the canonical module

We study the growth of the Betti sequence of the canonical module of a Cohen–Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. We answer the question of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen–Macaulay ring possessing a canonical module to be Gorenstein. GJL was partly supported by a grant from the National Security Agency. An erratum to this article can be found at     

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.     

关于我国体质指数BMI的分布研究

生活水平的提高,使肥胖症患的人数逐年增加。大量研究证明,肥胖是诱发高血压、冠心病、糖尿病、高脂血症等慢性病的一种主要危险因素,对肥胖症的预防是主要的。确定肥胖的一个简单而又有效的办法是利用世界卫生组织推荐的体质指数ＢＭＩ（Ｂｏｄｙ Ｍａｓｓ Ｉｎｄｅｘ）。本基于上海市延吉地区近万名３０岁以上成年人的调查结果,对体质指数的分布进行了研究,并利用被污染的正态分布族模型对其进行了拟合。     

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.     

On the regularity of the solution of the biharmonic variational inequality

It is shown that the solution of the biharmonic variational inequality has bounded second derivatives provided that the obstacle and the data are smooth.During the preparation of a portion of the paper, the author was a guest of the Scuola Normale Superiore in Pisa, supported by the German Research Association (DFG).     

A note on the interpretation of the Bahadur bound and the rate of convergence of the maximum likelihood estimator

Some interpretation of the Bahadur bound and the rate of convergence of the maximum likelihood estimator is provided using a theorem of Fu (1982) and the geometrical methods discussed in Kass (1984). We focus on replicated nonlinear regression and show that, in the sense of rate of convergence of the least-squares estimator in a small neighborhood of the true model, the most important characteristic that distinguishes one family of models from another is its statistical curvature (which is a multiple of the ‘intrinsic curvature’ of Bates and Watts, 1980).     

Time-convexity of the entropy in the multiphasic formulation of the incompressible Euler equation

We study the multiphasic formulation of the incompressible Euler equation introduced by Brenier: infinitely many phases evolve according to the compressible Euler equation and are coupled through a global incompressibility constraint. In a convex domain, we are able to prove that the entropy, when averaged over all phases, is a convex function of time, a result that was conjectured by Brenier. The novelty in our approach consists in introducing a time-discretization that allows us to import a flow interchange inequality previously used by Matthes, McCann and Savaré to study first order in time PDE, namely the JKO scheme associated with non-linear parabolic equations.     

Variational characterization of the uniqueness of the optimal state for the minimal-time problem

In this paper, we give a variational characterization of the uniqueness of the optimal state in a proper linear control process for the time-optimal problem; we extend to control processes with time-variable coefficients a characterization of normality given by Hajek in Ref. 1.This work was supported by CNR-GNAFA, Rome, Italy.     

On the Volume of the Union of Balls

We prove that if some balls in the Euclidean space move continuously in such a way that the distances between their centers decrease, then the volume of their union cannot increase. The proof is based on a formula expressing the derivative of the volume of the union as a linear combination of the derivatives of the distances between the centers with nonnegative coefficients. Received September 6, 1996, and in revised form March 26, 1997.     

On the Structure of the Complements of Chebyshev Sets

A set is called a Chebyshev set if it contains a unique best approximation element. We study the structure of the complements of Chebyshev sets, in particular considering the following question: How many connected components can the complement of a Chebyshev set in a finite-dimensional normed or nonsymmetrically normed linear space have? We extend some results from [A. R. Alimov, East J. Approx, 2, No. 2, 215--232 (1996)]. A. L. Brown's characterization of four-dimensional normed linear spaces in which every Chebyshev set is convex is extended to the nonsymmetric setting. A characterization of finite-dimensional spaces that contain a strict sun whose complement has a given number of connected components is established.     

On the Chebyshev Center and the Nonemptiness of the Intersection of Nested Sets

Mathematical Notes - We show that if every bounded set in a Banach space has a Chebyshev center, then the intersection of nested closed bounded sets in this space is nonempty in the case of a...     

Optimal design of the damping set for the stabilization of the wave equation

We consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in RN, N=1,2. In a first theoretical part, we reformulate the problem into an equivalent non-convex vector variational one using a characterization of divergence-free vector fields. Then, by means of gradient Young measures, we obtain a relaxed formulation of the problem in which the original cost density is replaced by its constrained quasi-convexification. This implies that the new relaxed problem is well-posed in the sense that there exists a minimizer and, in addition, the infimum of the original problem coincides with the minimum of the relaxed one. In a second numerical part, we address the resolution of the relaxed problem using a first-order gradient descent method. We present some numerical experiments which highlight the influence of the over-damping phenomena and show that for large values of the damping potential the original problem has no minimizer. We then propose a penalization technique to recover the minimizing sequences of the original problem from the optimal solution of the relaxed one.