Boundary Solutions of Differential Inclusions and Recovering the Initial Data

The solutions of differential inclusions staying at the boundary of attained sets are investigated using the contingent cones to attained sets. The possibility of recovering the starting point and the time which elapsed since the beginning is shown when some fragments of an attained set are known.     

A polyhedral study on 0–1 knapsack problems with disjoint cardinality constraints: Strong valid inequalities by sequence-independent lifting

We study the set of 0–1 integer solutions to a single knapsack constraint and a set of non-overlapping cardinality constraints (MCKP), which generalizes the classical 0–1 knapsack polytope and the 0–1 knapsack polytope with generalized upper bounds. We derive strong valid inequalities for the convex hull of its feasible solutions using sequence-independent lifting. For problems with a single cardinality constraint, we derive two-dimensional superadditive lifting functions and prove that they are maximal and non-dominated under some mild conditions. We then show that these functions can be used to build strong valid inequalities for problems with multiple disjoint cardinality constraints. Finally, we present preliminary computational results aimed at evaluating the strength of the cuts obtained from sequence-independent lifting with respect to those obtained from sequential lifting.     

On upper bounds for high order Neumann eigenvalues of convex domains in Euclidean space

We derive sharp upper bounds for eigenvalues of the Laplacian under Neumann boundary conditions on convex domains with given diameter in Euclidean space. We use the Brunn-Minkowski theorem in order to reduce the problem to a question about eigenvalues of certain classes of Sturm-Liouville problems.

     

Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors

This paper extends the concept of generalized polarization tensors (GPTs), which was previously defined for inclusions with homogeneous conductivities, to inhomogeneous conductivity inclusions. We begin by giving two slightly different but equivalent definitions of the GPTs for inhomogeneous inclusions. We then show that, as in the homogeneous case, the GPTs are the basic building blocks for the far-field expansion of the voltage in the presence of the conductivity inclusion. Relating the GPTs to the Neumann-to-Dirichlet (NtD) map, it follows that the full knowledge of the GPTs allows unique determination of the conductivity distribution. Furthermore, we show important properties of the the GPTs, such as symmetry and positivity, and derive bounds satisfied by their harmonic sums. We also compute the sensitivity of the GPTs with respect to changes in the conductivity distribution and propose an algorithm for reconstructing conductivity distributions from their GPTs. This provides a new strategy for solving the highly nonlinear and ill-posed inverse conductivity problem. We demonstrate the viability of the proposed algorithm by preforming a sensitivity analysis and giving some numerical examples.     

On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Dirichlet-to-Neumann (or Steklov) problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.     

Blow up for nonlinear parabolic equations with time degeneracy under dynamical boundary conditions

We study the blow up behaviour of nonlinear parabolic equations including a time degeneracy, under dynamical boundary conditions. For some exponential and polynomial degeneracies, we develop some energy methods and some spectral comparison techniques and derive upper bounds for the blow up times.     

On the interface boundary conditions between two interacting incompressible viscous fluid flows

We consider two incompressible viscous fluid flows interacting through thin non-Newtonian boundary layers of higher Reynolds? number. We study the asymptotic behaviour of the problem, with respect to the vanishing thickness of the layers, using Γ-convergence methods. We derive general interfacial boundary conditions between the two fluid flows. These boundary conditions are specified for some particular cases including periodic or fractal structures of layers.     

Equivalent Conditions for Local Error Bounds

In this paper we present two classes of equivalent conditions for local error bounds in finite dimensional spaces. We formulate conditions of the first class by using subderivatives, subdifferentials and strong slopes for nearby points outside the referenced set, and show that these conditions actually characterize a uniform version of the local error bound property. We demonstrate this uniformity for the max function of a finite collection of smooth functions, and as a consequence we show that quasinormality constraint qualifications guarantee the existence of local error bounds. We further present the second class of equivalent conditions for local error bounds by using the various limits defined on the boundary of the referenced set. In presenting these conditions, we exploit the variational geometry of the referenced set in a systematic way and unify some existing results in the literature.     

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.     

一类抛物型方程有限元算法的计算准则

用有限元法分析瞬态温度场,很有可能得到“振荡”和“超界”的计算结果.这两种现象不符合热传导规律.为解决此问题,我们提出时间单调性和空间单调性的概念,推导出三维无源热传导方程的数值解的时间单调性的几组充分条件.对某些特殊边值问题,使用规则单元网格,可以得到合理结果时Δt/Δx2的上下界公式.文中还研究了空间单调性.最后我们还讨论了集中质量阵的算法.针对以热传导方程为代表的这一类抛物型方程的有限元算法,我们创造性地给出几组计算准则.     

Improved Hashin–Shtrikman Bounds for Elastic Moment Tensors and an Application

This paper is devoted to the derivation of trace bounds for elastic moment tensors. Starting from the integral equation formulation of the elastic moment tensor, we establish that its trace can be obtained as a sum of minimal energies. We then recover the so-called Hashin–Shtrikman bounds, and show that these bounds can be tightened for inclusions which have some local thickness. As an application, we show that the volume of the inclusion can be estimated by the elastic moment tensor. Y.C. is partially supported by the grants RTN MULTIMAT and ANR EchoScan. H.K. is partially supported by the grant KOSEF R01-2006-000-10002-0.     

Bounds and estimates on the effective properties for nonlinear composites

In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational principle together with the bounds presented by Lukkassen, Persson and Wall in 1995. Moreover, we also present some examples where the bounds are so tight that they may be used as a good estimate of the effective behavior.     

Integrating heuristic information into exact methods: The case of the vertex p-centre problem

We solve the vertex p-centre problem optimally using an exact method that considers both upper and lower bounds as part of its search engine. Tight upper bounds are generated quickly via an efficient three-level heuristic, which are then used to derive potential ‘lower bounds’ accordingly. These two pieces of information when used together make our chosen exact method more efficient at obtaining optimal solutions relatively quickly. The proposed implementation produced excellent results when tested on the OR Library data set. This integrated approach can be adopted for those exact methods that consider both upper and lower bounds within their search engine and hence provide a wider spectrum of applicability in other hard combinatorial problems.     

The boundary and the shape of binary images

In this paper we will consider an unknown binary image, of which the length of the boundary and the area of the image are given. These two values together contain some information about the general shape of the image. We will study two properties of the shape in particular. First, we will prove sharp lower bounds for the size of the largest connected component. Second, we will derive some results about the size of the largest ball containing only ones, both in the case that the connected components are all simply connected and in the general case.     

Mean–variance optimal portfolios in the presence of a benchmark with applications to fraud detection

We first study mean–variance efficient portfolios when there are no trading constraints and show that optimal strategies perform poorly in bear markets. We then assume that investors use a stochastic benchmark (linked to the market) as a reference portfolio. We derive mean–variance efficient portfolios when investors aim to achieve a given correlation (or a given dependence structure) with this benchmark. We also provide upper bounds on Sharpe ratios and show how these bounds can be useful for fraud detection. For example, it is shown that under some conditions it is not possible for investment funds to display a negative correlation with the financial market and to have a positive Sharpe ratio. All the results are illustrated in a Black–Scholes market.     

Connection probabilities and RSW‐type bounds for the two‐dimensional FK Ising model

We prove Russo‐Seymour‐Welsh‐type uniform bounds on crossing probabilities for the FK Ising (FK percolation with cluster weight q = 2) model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model [24], which allows us to get precise estimates on boundary connection probabilities. We stay in a discrete setting; in particular, we do not make use of any continuum limit, and our result can be used to derive directly several noteworthy properties—including some new ones—among which are the fact that there is no infinite cluster at criticality, tightness properties for the interfaces, and the existence of several critical exponents, in particular the half‐plane, one‐arm exponent. Such crossing bounds are also instrumental for important applications such as constructing the scaling limit of the Ising spin field [6] and deriving polynomial bounds for the mixing time of the Glauber dynamics at criticality [17]. © 2011 Wiley Periodicals, Inc.     

On the domain derivative for scattering by impenetrable obstacles in chiral media

For scattering of electromagnetic waves in a chiral medium bysome perfectly conducting inclusions, we study the dependenceof the scattered field on the boundary of the inclusions andshow its Fréchet differentiability in appropriate spaces.Further, we derive a characterization of the derivative as asolution to some corresponding chiral boundary value problem.Our proof contains a new approach to rigorously derive thischaracterization.     

Extremal graph characterization from the upper bound of the Laplacian spectral radius of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. In this paper, we obtain two upper bounds on the spectral radius of the Laplacian matrix of weighted graphs and characterize graphs for which the bounds are attained. Moreover, we show that some known upper bounds on the Laplacian spectral radius of weighted and unweighted graphs can be deduced from our upper bounds.     

A posteriori error analysis for linearization of nonlinear elliptic problems and their discretizations

The paper is devoted to a posteriori quantitative analysis for errors caused by linearization of non-linear elliptic boundary value problems and their finite element realizations. We employ duality theory in convex analysis to derive computable bounds on the difference between the solution of a non-linear problem and the solution of the linearized problem, by using the solution of the linearized problem only. We also derive computable bounds on differences between finite element solutions of the nonlinear problem and finite element solutions of the linearized problem, by using finite element solutions of the linearized problem only. Numerical experiments show that our a posteriori error bounds are efficient.     

Finding lower bounds on the complexity of secret sharing schemes by linear programming

Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants.By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.