Quasistatic evolution of a brittle thin film

This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a Γ-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in L∞. Mathematics Subject Classification (2000) 74K30, 49J45, 74K30, 35R35, 49Q20     

Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures

We define a notion of quasistatic evolution for the elliptic approximation of the Mumford-Shah functional proposed by Ambrosio and Tortorelli. Then we prove that this regular evolution converges to a quasi static growth of brittle fractures in linearly elastic bodies.Received: 1 January 2003, Accepted: 29 January 2004, Published online: 12 May 2004Mathematics Subject Classification (2000): 35R35, 74R10, 35J25     

On a further condition in the macroscopic extended model for ultrarelativistic gases

Abstract An exact macroscopic extended model, with many moments, for ultrarelativistic gas has been recently proposed in literature. However, a further condition has not been imposed, even if it is evident in the case of a charged gas and when the electromagnetic field acts as an external force; in the present paper we exploit it and prove that it results in many identities and in residual conditions which allow to determine the arbitrary single variable functions present in the general theory. The result is that they are polynomials determined except for a corresponding number of constants. These are arbitrary constants, so that the macroscopic model remains still more general than the kinetic model. Keywords: Extended Thermodynamics, Fluid Models, Ultra-Relativistic Gas, Entropy Principle Mathematics Subject Classification (2000): 74A15, 74A20     

Locally minimax tests in symmetrical distributions

In this paper we give an extension of the theory of local minimax property of Giri and Kiefer (1964, Ann. Math. Statist., 35, 21–35) to the family of elliptically symmetric distributions which contains the multivariate normal distribution as a member.This work was partially supported by the Canadian N.S.E.R.C. grant     

Boundary Integral Operators over Lipschitz Surfaces for a Stokes Equation in R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we show the existence of the solutions of the screen problems of the Stokes equation in R ⁿ , n ≥ 3. For the purpose of it, we use the fractional layer potentials (for the Stokes equation) in a bounded Lipschitz domain The author is supported by the Korea Research Grant Foundation Grant (MOEHRD)KRF-2005-214-C00179.     

Bayesian stratified random sampling using auxiliary information

Summary Smith (1976,J. R. Statist. Soc., A,139, 183–204) has argued that survey statisticians should attempt to model finite population structures in the same way that statisticians in other disciplines have to provide models of finite or infinite populations. Following this argument, we suggest in this paper that an obvious model for a stratified population when auxiliary information regarding variate values is available, is the one way analysis of covariance model with unequal variances and we consider the problem of estimating the finite population mean. Finally a possible extension of this result is discussed.     

Liouville theorems in halfspaces for parabolic hypoelliptic equations

Abstract We prove some one-side Liouville-type theorems in halfspaces for a class of evolution hypoelliptic equations. The operators we deal with are left translation invariant, and homogeneous of degree two, on homogeneous Lie groups on . Keywords: Parabolic operators, Liouville Theorems, Liouville Theorems in halfspaces Mathematics Subject Classification (2000): 35K65, 35H10, 35B99     

On the symmetry function of a convex set

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}We attempt a broad exploration of properties and connections between the symmetry function of a convex set S and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point , let sym(x,S) denote the symmetry value of x in S: , which essentially measures how symmetric S is about the point x, and define x ^* is called a symmetry point of S if x ^* achieves the above maximum. The set S is a symmetric set if sym (S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym (S) and/or sym (x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym (x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the L?wner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x ^* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x ^* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory. Dedicated to Clovis Gonzaga on the occasion of his 60th birthday.     

Stability properties of an inverse parabolic problem with unknown boundaries

We treat the stability issue for an inverse problem arising from non-destructive evaluation by thermal imaging. We consider the determination of an unknown portion of the boundary of a thermic conducting body by overdetermined boundary data for a parabolic initial-boundary value problem. We obtain that when the unknown part of the boundary is a priori known to be smooth, the data are as regular as possible and all possible measurements are taken into account, the problem is exponentially ill-posed. Then, we prove that a single measurement with some a priori information on the unknown part of the boundary and minimal assumptions on the data, in particular on the thermal conductivity, is enough to have stable determination of the unknown boundary. Given the exponential ill-posedness, the stability estimate obtained is optimal. AMS 2000 Mathematics Subject Classification. Primary 35R30, Secondary 35B60, 33C90     

Sharp-interface limit of the Allen-Cahn action functional in one space dimension

We analyze the sharp-interface limit of the action minimization problem for the stochastically perturbed Allen-Cahn equation in one space dimension. The action is a deterministic functional which is linked to the behavior of the stochastic process in the small noise limit. Previously, heuristic arguments and numerical results have suggested that the limiting action should “count” two competing costs: the cost to nucleate interfaces and the cost to propagate them. In addition, constructions have been used to derive an upper bound for the minimal action which was proved optimal on the level of scaling. In this paper, we prove that for d = 1, the upper bound achieved by the constructions is in fact sharp. Furthermore, we derive a lower bound for the functional itself, which is in agreement with the heuristic picture. To do so, we characterize the sharp-interface limit of the space-time energy measures. The proof relies on an extension of earlier results for the related elliptic problem. Mathematics Subject Classification (2000) 49J45, 35R60, 60F10     

Rank-one convex functions on 2×2 symmetric matrices and laminates on rank-three lines

We construct a function on the space of symmetric 2× 2 matrices in such a way that it is convex on rank-one directions and its distributional Hessian is not a locally bounded measure. This paper is also an illustration of a recently proposed technique to disprove L¹ estimates by the construction of suitable probability measures (laminates) in matrix space. From this point of view the novelty is that the support of the laminate, besides satisfying a convex constraint, needs to be contained on a rank-three line, up to arbitrarily small errors. Mathematics Subject Classification (2000): 35J50 – 26D15 – 49N60 –74G65     

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R B comes from R (respectively, B). This rule implicitly partitions space into a red set and a blue set that are separated by a red-blue decision boundary. In this paper we develop output-sensitive algorithms for computing this decision boundary for point sets on the line and in ². Both algorithms run in time O(n log k), where k is the number of points that contribute to the decision boundary. This running time is the best possible when parameterizing with respect to n and k.     

Liouville-type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space:

Blow-up of solutions to a nonlinear dispersive rod equation

In this paper, firstly we find the best constant for a convolution problem on the unit circle via a variational method. Then we apply the best constant on a nonlinear rod equation to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Mathematics Subject Classification (2000) 30C70, 37L05, 35Q58, 58E35     

On the existence of periodic solutions in the thermistor problem with degenerate thermal conductivity

Abstract This paper deals with the existence of weak periodic solutions for a parabolic-elliptic system proposed as a model for a time dependent thermistor with degenerate thermal conductivity. Applying the maximal monotone mappings theory, we prove an existence result for weak periodic solutions. Keywords: Nonlinear parabolic-elliptic system of degenerate type, Periodic solutions, Thermistor problem Mathematics Subject Classification (2000): 35B10, 35J60, 35K65     

Growth Curve Model with Hierarchical Within-Individuals Design Matrices

This paper deals with some inferential problems under an extended growth curve model with several hierarchical within-individuals design matrices. The model includes the one whose mean structure consists of polynomial growth curves with different degrees. First we consider the case when the covariance matrix is unknown positive definite. We derive a LR test for examining the hierarchical structure for within individuals design matrices and a model selection criterion. Next we consider the case when a random coefficients covariance structure is assumed, under certain assumption of between-individual design matrices. Similar inferential problems are also considered. The dental measurement data (see, e.g., Potthoff and Roy (1964, Biometrika, 51, 313-326)) is reexamined, based on extended growth curve models.     

Approximation by finite elements, existence and uniqueness for a model of stratified thermoviscoplastic materials

Abstract  abstract In the present paper we consider for a < x < b, 0 < t < T, the system of partial differential equations

Modeling Influence of Raltegravir Intensification on Viral Dynamics: Stability and Hopf Bifurcation

In this paper, we propose an ordinary differential equation model with logistic target cell growth to describe influence of raltegravir intensifi- cation on viral dynamics. The basic reproduction number R 0 is established. The infection-free equilibrium E 0 is globally attractive if R 0 < 1, while virus is uniformly persistent if R 0 > 0. In addition, we find that Hopf bifurcation can occur at around the positive equilibrium within certain parameter ranges. Numerical simulations are performed to illustrate theoretical results.     

Bisexual Galton‐Watson Branching Process in Varying Environments

Abstract

In this paper we introduce a bisexual Galton‐Watson branching process (BGWP) in which the offspring probability distribution is different in each generation. We obtain some relations among the probability generating functions (pgf) involved in the model and, making use of mean growth rates and fractional linear functions (flf), we provide sufficient and necessary conditions for its almost sure extinction.