Orientation-preserving condition and polyconvexity on a surface: Application to nonlinear shell theory

We propose in this paper a definition of a “polyconvex function on a surface”, inspired by the definitions set forth in other contexts by J. Ball (1977) [3] and by J. Ball, J.C. Currie, and P.J. Olver (1981) [5]. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, a property which eventually allows to establish the existence of minimizers. We also indicate how this new approach compares with the classical nonlinear shell theories, such as those of W.T. Koiter and P.M. Naghdi.     

Construction of polyconvex,anisotropic free‐energy functions

The existence of minimizers of some variational principles in finite elasticity is based on the concept of quasiconvexity, introduced by Morrey [6]. This integral inequality is rather complicated to handle. Thus, the sufficient condition for quasiconvexity, the polyconvexity condition in the sense of Ball [1], is a more important concept for practical applications, see also Ciarlet [4] and Dacorogna [5]. In the case of isotropy there exist some models which satisfy this condition. Furthermore, there does not exist a systematic treatment of anisotropic, polyconvex free‐energies in the literature. In the present work we discuss some aspects of the formulation of polyconvex, anisotropic free‐energy functions in the framework of the invariant formulation of anisotropic constitutive equations and focus on transverse isotropy.     

Explicit formulas for Hankel norm approximations of infinite-dimensional systems

Recently Hankel norm approximation for finite dimensional systems has been studied extensively, both for continuous time systems and discrete time systems. One of the approaches is to combine the work of J.A. Ball and J.W. Helton with results on Wiener-Hopf factorization to produce explicit formulas for the solutions. This has been done by J.A. Ball and A.C.M. Ran for the finite-dimensional case and for a class of infinite-dimensional systems by A.C.M. Ran. Here we show that this approach can be extended to a much larger class of infinite-dimensional systems; those with an exponentially stable semigroup and unbounded input and output operators which satisfy a smoothness condition. This class is larger than the nuclear class for which this problem was solved by K. Glover, R.F. Curtain and J.R. Partington by an approximation approach.     

Un théorème d'existence pour une coque non linéairement élastique ≪ en flexion ≫

We establish an existence theorem for the two-dimensional equations of a nonlinearly elastic “flexural” shell, recently justified by V. Lods and B. Miara by the method of formal asymptotic expansions applied to the corresponding three-dimensional equations of nonlinear elasticity. To this end, we show that the associated energy has at least one minimizer over the corresponding set of admissible deformations. The strain energy is a quadratic expression in terms of the “exact” change of curvature tensor, between the deformed and undeformed middle surfaces; the set of admissible deformations is formed by the deformations of the undeformed middle surface that preserve its metric and satisfy boundary conditions of clamping or simple support.     

Polyconvex potentials,invertible deformations,and thermodynamically consistent formulation of the nonlinear elasticity equations

It is shown that the nonstationary finite-deformation thermoelasticity equations in Lagrangian and Eulerian coordinates can be written in a thermodynamically consistent Godunov canonical form satisfying the Friedrichs hyperbolicity conditions, provided that the elastic potential is a convex function of entropy and of the minors of the elastic deformation Jacobian matrix. In other words, the elastic potential is assumed to be polyconvex in the sense of Ball. It is well known that Ball’s approach to proving the existence and invertibility of stationary elastic deformations assumes that the elastic potential essentially depends on the second-order minors of the Jacobian matrix (i.e., on the cofactor matrix). However, elastic potentials constructed as approximations of rheological laws for actual materials generally do not satisfy this requirement. Instead, they may depend, for example, only on the first-order minors (i.e., the matrix elements) and on the Jacobian determinant. A method for constructing and regularizing polyconvex elastic potentials is proposed that does not require an explicit dependence on the cofactor matrix. It guarantees that the elastic deformations are quasiisometries and preserves the Lame constants of the elastic material.     

A necessary and sufficient condition of convexity for SOC reformulation of trust-region subproblem with two intersecting cuts

We consider the extended trust-region subproblem with two linear inequalities. In the “nonintersecting” case of this problem, Burer et al. have proved that its semi-definite programming relaxation with second-order-cone reformulation (SDPR-SOCR) is a tight relaxation. In the more complicated “intersecting” case, which is discussed in this paper, so far there is no result except for a counterexample for the SDPR-SOCR. We present a necessary and sufficient condition for the SDPR-SOCR to be a tight relaxation in both the “nonintersecting” and “intersecting” cases. As an application of this condition, it is verified easily that the “nonintersecting” SDPR-SOCR is a tight relaxation indeed. Furthermore, as another application of the condition, we prove that there exist at least three regions among the four regions in the trust-region ball divided by the two intersecting linear cuts, on which the SDPR-SOCR must be a tight relaxation. Finally, the results of numerical experiments show that the SDPR-SOCR can work efficiently in decreasing or even eliminating the duality gap of the nonconvex extended trust-region subproblem with two intersecting linear inequalities indeed.     

Contraction mappings underlying undiscounted Markov decision problems

This paper is concerned with the properties of the value-iteration operator0 which arises in undiscounted Markov decision problems. We give both necessary and sufficient conditions for this operator to reduce to a contraction operator, in which case it is easy to show that the value-iteration method exhibits a uniform geometric convergence rate. As necessary conditions we obtain a number of important characterizations of the chain and periodicity structures of the problem, and as sufficient conditions, we give a general “scrambling-type” recurrency condition, which encompasses a number of important special cases. Next, we show that a data transformation turns every unichained undiscounted Markov Renewal Program into an equivalent undiscounted Markov decision problem, in which the value-iteration operator is contracting, because it satisfies this “scrambling-type” condition. We exploit this contraction property in order to obtain lower and upper bounds as well as variational characterizations for the fixed point of the optimality equation and a test for eliminating suboptimal actions.     

Two maps on one surface

Two embeddings of a graph in a surface S are said to be “equivalent” if they are identical under an homeomorphism of S that is orientation‐preserving for orientable S. Two graphs cellularly embedded simultaneously in S are said to be “jointly embedded” if the only points of intersection involve an edge of one graph transversally crossing an edge of the other. The problem is to find equivalent embeddings of the two graphs that minimize the number of these edge‐crossings; this minimum we call the “joint crossing number” of the two graphs. In this paper, we calculate the exact value for the joint crossing number for two graphs simultaneously embedded in the projective plane. Furthermore, we give upper and lower bounds when the surface is the torus, which in many cases give an exact answer. In particular, we give a construction for re‐embedding (equivalently) the graphs in the torus so that the number of crossings is best possible up to a constant factor. Finally, we show that if one of the embeddings is replaced by its “mirror image,” then the joint crossing number can decrease, but not by more than 6.066%. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 198–216, 2001     

A free boundary problem related to thermal insulation

We study a free boundary problem arising from the theory of thermal insulation. The outstanding feature of this set optimization problem is that the boundary of the set being optimized is not a level surface of a harmonic function, but rather a hypersurface along which a harmonic function satisfies a Robin condition. We show that minimal sets exist, satisfy uniform density estimates, and, under some geometric conditions, have “locally flat” boundaries.     

Exact controllability of Naghdi shells

We consider the second order evolution equations of the displacement and rotation fields in any points of the middle surface of a shell in Naghdi's formulation. We establish that, with two controls, the exact controllability on part of the lateral boundary is achieved under the sufficient geometrical condition that the middle surface of the shell is “not too far” from a plane in a sense that is made more precise in the proof.     

A model of concept formation

At first we model the way an intelligence “I” constructs statements from phrases, and then how “I” interlocks these statements to form a string of statements to attain a concept. These strings of statements are called progressions. That is, starting with an initial stimulating relation between two phrases, we study how “I” forms the first statement of the progression and continues from this first statement to form the remaining statements in these progressions to construct a concept. We assume that “I” retains the progressions that it has constructed. Then we show how these retained progressions provide “I” with a platform to incrementally constructs more and more sophisticated conceptual structures. The reason for the construction of these conceptual structures is to achieve additional concepts. Choice plays a very important role in the progression and concept formation. We show that as “I” forms new concepts, it enriches its conceptual structure and makes further concepts attainable. This incremental attainment of concepts is a way in which we humans learn, and this paper studies the attainability of concepts from previously attained concepts. We also study the ability of “I” to apply its progressions and also the ability of “I” to electively manipulate its conceptual structure to achieve new concepts. Application and elective manipulation requires of “I” ingenuity and insight. We also show that as “I” attains new concepts, the conceptual structures change and circumstances arise where unanticipated conceptual discoveries are attainable. As the conceptual structure of “I” is developed, the logical and structural relationships between concepts embedded in this structure also develop. These relationships help “I” understand concepts in the context of other concepts and help “I₁” communicate to another “I₂” information and concept structures. The conceptual structures formed by “I” give rise to a directed web of statement paths which is called a convolution web. The convolution web provides “I” with the paths along which it can reason and obtain new concepts and alternative ways to attain a given concept.This paper is an extension of the ideas introduced in [1]. It is written to be self-contained and the required background is supplied as needed.     

A remark on polyconvex envelopes of radially symmetric functions in dimension 2 × 2

We study polyconvex envelopes of a class of functions related to the function of Kohn and Strang introduced in [4]. We present an example of a function of this class for which the polyconvex envelope may be computed explicitly and we also point out some general features of the problem.     

A CONSTITUTIVE EQUATION SATISFYING THE NULL CONDITION FOR NONLINEAR COMPRESSIBLE ELASTICITY

This paper constructs a polyconvex stored energy function, satisfying the null condition, for isotropic compressible elastic materials with given Lame constants. The difference between this stored energy function and St Venant-Kirchhoff's is a three order term.     

The dependence of the height of a Lorenz curve of a Zipf function on the size of the system

The Lorenz curve of a Zipf function describes, graphically, the relation between the fraction of the items and the fraction of the sources producing these items. Hence it generalizes the so-called 80/20-rule to general fractions.In this paper we examine the relation of such Lorenz curves with the size of the system (expressed by the total number of sources). We prove that the height of such a Lorenz curve is an increasing function of the total number of sources.In other words, we show that the share of items as a function of the corresponding share of sources increases with increasing size of the system. This conclusion is opposite (but not in contradiction) to a conclusion of Aksnes and Sivertsen (studied in an earlier paper of Egghe) but where “share of sources” is replaced by “number of sources”.     

Some Observations on the Theory of Cryptographic Hash Functions

In this paper, we study issues related to the notion of “secure” hash functions. Several necessary conditions are considered, as well as a popular sufficient condition (the so-called random oracle model). We study the security of various problems that are motivated by the notion of a secure hash function. These problems are analyzed in the random oracle model, and we prove that the obvious trivial algorithms are optimal. As well, we look closely at reductions between various problems. In particular, we consider the important question “does collision resistance imply preimage resistance?”. We provide partial answers to this question – both positive and negative! – based on uniformity properties of the hash function under consideration.     

具有有穷个模态辞的模态系统

<正> 在真值逻辑系统中如果加入“可能”“必然”等模熊概念,所得的逻辑系统叫做模态系统(modal system).如果该真值系就为伟统的二值系统,特名曰传统模态系统(下文的讨论不限于传统模态系统).纯由命题变元以及“～”(非)“◇”(可能)“口”(必然)三运算而组成的命题叫做模态辞(modality).若只经奇数次～运算的名曰负模态辞,经偶数次(包括0次)～运算的名曰正模态辞.     

Bauer's maximum principle and hulls of sets

We use the Bauer maximum principle for quasiconvex, polyconvex and rank-one convex functions to derive Krein-Milman-type theorems for compact sets in . Further we show that in general the set of quasiconvex extreme points is not invariant under transposition and it is different from the set of rank-one convex extreme points. Finally, a set in with different polyconvex, quasiconvex and rank-one convex hulls is constructed. Received September 14, 1999 / Accepted January 14, 2000 /Published online July 20, 2000     

The non-emptiness of the core of a partition function form game

The purpose of this paper is to provide a necessary and sufficient condition for the non-emptiness of the core for partition function form games. We generalize the Bondareva–Shapley condition to partition function form games and present the condition for the non-emptiness of “the pessimistic core”, and “the optimistic core”. The pessimistic (optimistic) core describes the stability in assuming that players in a deviating coalition anticipate the worst (best) reaction from the other players. In addition, we define two other notions of the core based on exogenous partitions. The balanced collections in partition function form games and some economic applications are also provided.     

An MRA approach to surface completion and image inpainting

The objective of this paper is to introduce a multi-resolution approximation (MRA) approach to the study of continuous function extensions with emphasis on surface completion and image inpainting. Along the line of the notion of diffusion maps introduced by Coifman and Lafon with some “heat kernels” as integral kernels of these operators in formulating the diffusion maps, we apply the directional derivatives of the heat kernels with respect to the inner normal vectors (on the boundary of the hole to be filled in) as integral kernels of the “propagation” operators. The extension operators defined by propagations followed by the corresponding sequent diffusion processes provide the MRA continuous function extensions to be discussed in this paper. As a case study, Green's functions of some “anisotropic” differential operators are used as heat kernels, and the corresponding extension operators provide a vehicle to transport the surface or image data, along with some mixed derivatives, from the exterior of the hole to recover the missing data in the hole in an MRA fashion, with the propagated mixed derivative data to provide the surface or image “details” in the hole. An error formula in terms of the heat kernels is formulated, and this formula is applied to give the exact order of approximation for the isotropic setting.     

Evolution of nonparametric surfaces with speed depending on curvature II. The mean curvature case

We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t ≧ 0. In addition, we establish estimates of the rate at which solutions tend to zero as t → ∞.