Multiphase double‐porosity homogenization for perimeter functionals

In this paper, we study a homogenization problem for perimeter energies in highly contrasted media; the analysis of the previous paper is carried out by removing the hypothesis that the perforated medium Rⁿ ? E is composed of disjoint compact components. Assuming E to be the union of a finite number N of connected components E¹, … ,E^N, the Γ‐limit F is a multiphase energy with a ‘decoupled’ surface part, obtained by homogenization from the surface tensions in each E ^j, a trivial bulk term obtained as a weak limit, and a further interacting term between the phases, involving an asymptotic formula for a family minimum problems on invading an asymptotic formula for a family of minimum problems on invading domains with prescribed boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

Direct numerical Euler method for finding extrema of nonlocal functionals

A direct numerical method is considered, which is a generalization of the Euler method for nonlocal functionals depending on functions with differently signed deviations, and also for nonlocal functionals depending on functions of two independent variables.     

Large time behavior and Lyapunov functionals for a nonlocal differential equation

A new approach is used to describe the large time behavior of the nonlocal differential equation initially studied in T.-N. Nguyen (On the \({\omega}\)-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016). Our approach is based upon the existence of infinitely many Lyapunov functionals and allows us to extend the analysis performed in T.-N. Nguyen (On the \({\omega}\)-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016).     

Gamma-convergence and the least squares method

We study the asymptotic behaviour of the sequence of first order equations _tu––a_h(y)_xu=0 by means of the related energies

Patrolling a perimeter

In this paper we study the problem of patrolling a perimeter. The general situation considered here can correspond to different tactical problems and it is studied from the point of view of game theory. To put the ideas in a context we describe it as follows. An intruder seeks to carry out a sabotage on the perimeter of a protected zone. He has to perform the action along n consecutive days and has to position himself each day at one of m strategic points placed on this border. The first day he can take his place at any of the m points, but on successive days he can move only to adjacent points. Furthermore, the perimeter is protected by a patroller, who will select each day one of the m points to inspect. The strategic situation is modeled as a two-person zero-sum game, which is developed on a cyclic set of m points over n time units. We prove some interesting properties of the strategies, solve the game in closed form under certain constraints and obtain bounds for the value of the game in several non-solved cases.     

The perimeter of words

We define $\left[k\right]=\{1,2,\dots ,k\}$ to be a (totally ordered) alphabet on $k$ letters. A word $w$ of length $n$ on the alphabet $\left[k\right]$ is an element of ${\left[k\right]}^n$. A word can be represented by a bargraph (i.e., by a column-convex polyomino whose lower edges lie on the $x$-axis) in which the height of the $i$th column equals the size of the $i$th part of the word. Thus these bargraphs have heights which are less than or equal to $k$. We consider the perimeter, which is the number of edges on the boundary of the bargraph. By way of Cramer’s method and the kernel method, we obtain the generating function that counts the perimeter of words. Using these generating functions we find the average perimeter of words of length $n$ over the alphabet $\left[k\right]$. We also show how the mean and variance can be obtained using a direct counting method.     

Gamma-convergence for Ginzburg-Landau functional with degenerate 3-well potential in one dimension

We consider the Ginzburg-Landau functional in one dimension, endowed with epsilon-dependent 3-well potential which degenerates as small parameter epsilon tends to zero. By using the approach in G. Alberti, S. Muller: A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 , 761-825 (2001), we obtain Gamma-convergence as small parameter epsilon tends to zero. We also recover the underlying geometric properties shared by all minimizing sequences. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence

We show that the nonlinear bending theory of shells arises as a Γ-limit of three-dimensional nonlinear elasticity. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Random sets of finite perimeter

An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of L¹ functions is suggested. A Crofton formula for flat sections of the perimeter is shown. Finally, random processes of particles with finite perimeter are introduced and it is shown that their union sets are random sets with locally finite perimeter.     

Informativeness of functionals

We introduce the concept of informativeness of a continuous functional on a metric spaceX with respect to a setMX and a metric x. We pose the problem of finding the most informative functional. For some sets of continuous functions, this problem is solved by reduction to a subset of functionals given by the value of a function at a certain point.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1156–1163, September, 1994.     

Degrees of functionals

In this paper we will discuss some problems of degree-theoretic nature in connection with recursion in normal objects of higher types.Harrington [2] and Loewenthal [6] have proved some results concerning Post's problem and the Minimal Pair Problem, using recursion modulo subindividuals. Our degrees will be those obtained from Kleene-recursion modulo individuals. To solve our problems we then have to put some extra strength to ZFC. We will first assume V = L, and then we restrict ourselves to the situation of a recursive well-ordering and Martin's axiom.We assume familiarity with recursion theory in higher types as presented in Kleene [3]. Further backround is found in Harrington [2], Moldestad [9] and Normann [11]. We will survey the parts of these papers that we need.In Section 1 we give the general background for the arguments used later. In Section 2 we prove some lemmas assuming V = L. In section 3, assuming V = L we solve Post's problem and another problem using the finite injury method. We will thereby describe some of the methods needed for the more complex priority argument of Section 4 where we give a solution of the minimal pair problem for extended r.e. degress of functionals.In Section 5 we will see that if Martin's Axiom holds and we have a minimal well-ordering of tp (1) recursive in ³E, we may use the same sort of arguments as in parts 3 and 4.     

A new characterization of Laplace functionals and probability generating functionals

Summary The well-known and widely used Laplace resp. probability generating functionals are characterized by properties of positive definiteness and continuity. The methods applied come from Harmonic Analysis on semigroups, and allow also intrinsic characterizations for the transforms of infinitely divisible random measures and point processes.     

Computation of Feynman integrals of functionals that are functions of quadratic functionals

If the arguments of a function G: ^{d 1} are taken as quadratic functional defined on a space C of continuous functions, we obtain a functional G: C ¹.We give a formula for computing analytic Feynman integrals of such functionals. We also propose a method of approximate computation of sequential Feynman integrals based on replacing the kernel of an integral operator by a degenerate kernel.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 58–61.     

Subdifferentiation of integral functionals

Mathematical Programming - We examine how the subdifferentials of nonconvex integral functionals can be deduced from the subdifferentials of the corresponding integrand or at least be estimated...     

Minimization of entropy functionals

Entropy (i.e. convex integral) functionals and extensions of these functionals are minimized on convex sets. This paper is aimed at reducing as much as possible the assumptions on the constraint set. Primal attainment, dual equalities, dual attainment and characterizations of the minimizers are obtained with weak constraint qualifications. These results improve several aspects of the literature on the subject.