An optimal design problem with perimeter penalization

We study the optimal design problem of finding the minimal energy configuration for a mixture of two conducting materials when a perimeter penalization of the unknown domain is added. We show that in this situation an optimal domain exists and that, under suitable assumptions on the data, it is an open set.This work is part of the project EURHomogenization, contract SC1-CT91-0732 of the program SCIENCE of the Commission of the European Communities.     

Binary hyperidentities of lattices

Summary An equational identity of a given type involves two kinds of symbols: individual variables and the operation symbols. For example, the distributive identity: x (y + z) = x y + x z has three variable symbols {x, y, z} and two operation symbols {+, }. Here the variables range over all the elements of the base set while the two operation symbols are fixed. However, we shall say that an identity ishypersatisfied by a varietyV if, whenever we also allow the operation symbols to range over all polynomials of appropriate arity, the resulting identities are all satisfied byV in the usual sense. For example, the ring of integers Z; +, satisfies the above distributive law, but it does not hypersatisfy the same formal law because, e.g., the identityx + (y z) = (x + y) (x + z) is not valid. By contrast, is hypersatisfied by the variety of all distributive lattices and is thus referred to as a distributive latticehyperidentity. Thus a hyperidentity may be viewed as an equational scheme for writing a class of identities of a given type and the original identities themselves are obtained as special cases by substituting specific polynomials of appropriate arity for the operation symbols in the scheme. In this paper, we provide afinite equational scheme which is a basis for the set of all binary lattice hyperidentities of type 2, 2, .This research was supported by the NSERC operating grant # 8215     

Optimal control for the dam problem

Consider the variational inequality for the rectangular dam problem and assume that fluid can be withdrawn from the bottom at a rate proportional tok(x). Denote byp(x, y) the pressure of the fluid in the dam corresponding to a particular choice ofk. Consideringk(x) as a control variable varying in a class {0k(x)N, k(x)dxM}, we introduce the functionalJ(k)=g(y)p(x, y) whereg(y) is a given positive and monotone nondecreasing function. We characterize the controlsk ₀ which minimizeJ(k). This work is partially supported by National Science Foundation MCS-8300293     

Brownian confinement and pinning in a Poissonian potential. I

Summary We study the behavior of ad dimensional Brownian motion in a soft repulsive Poissonian potential over long time intervals [0,t]. We introduce certaint and configuration dependent scales, which grow almost linearly witht. For typical configurations with probability tending to 1 ast goes to , the size of displacements of the process is bounded above by these scales, (confinement effect). The proof involves calculations beyond leading order. To this end we use a coarse grained picture of the environment (method of enlargement of obstacles) and of the path (a backbone of excursions between clearings and forest parts in the environment). These coarse grained pictures are also used in the sequel [11] to the present article, when proving the pinning effect.This article was processed by the author using the LATEX style filepljour1m from Springer-Verlag.     

Stieltjes Polynomials and Gauss-Kronrod Quadrature for Jacobi Weight Functions

Summary. Let w _,beta (t)=(1–t)(1+t), ,>–1, denote the Jacobi weight function. For 0,<5/2 it is shown that on compact subintervals of (–1,1), the corresponding Stieltjes polynomials and their derivatives are asymptotically equal to certain Jacobi polynomials. This also leads to asymptotic representations of those weights of Gauss-Kronrod quadrature formulae, which correspond to nodes in a compact subinterval of (–1,1). On the other hand, it is demonstrated that for the parameters satisfying min(,)0 and max(,)>5/2 the Stieltjes polynomial has only few real zeros and that Gauss-Kronrod quadrature is not possible. So far, results of the above type have been known only for ultraspherical weight functions for 02. For >3, the impossibility of Gauss-Kronrod quadrature has been proved by the authors recently. Mathematics Subject Classification (1991):33C10, 33C45, 42C05, 65D32The second author is sponsored by a Heisenberg scholarship of the Deutsche Forschungsgemeinschaft.     

An efficient heuristic algorithm for minimum matching

We give a new heuristic algorithm for minimum matching problems and apply it to the Euclidean problem with random vertices in 2 dimensions. The algorithm is based on simulated annealing and performs in practice faster than previous heuristic algorithms yielding suboptimal solutions of the same good quality. From configurations with up toN=20.000 vertices in the unit square we estimate that the length of a minimum matching scales asymptotically asLN with (=0.3123±0.0016.

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Zusammenfassung Wir stellen einen neuen heuristischen Algorithmus für minimale Matching-Probleme vor und wenden diesen auf das euklidische Problem mit zufÄlliger Punkteverteilung in 2 Dimensionen an. Auf Simulated Annealing basierend lÄuft der Algorithmus schneller als frühere heuristische Algorithmen und erreicht dabei suboptimale Lösungen gleich guter QualitÄt. Aus Konfigurationen mit bis zuN=20.000 Punkten im Einheitsquadrat schÄtzen wir, da\ für die LÄnge des minimalen Matchings asymptotischLN mit=0.3123±0.0016 gilt.

     

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas

The Broadwell model of the Boltzmann equation for a simple discrete velocity gas is investigated on two asymptotic problems. (a) The decay of solutions inxR ast+. (b) The hydrodynamical limit in the compressible Euler level as the mean free path0.     

On the Convergence of Inhomogeneous Markov Chains Approximating Equilibrium Placements of Flexible Objects

This paper deals with the minimization of local forces in two-dimensional placements of flexible objects within rigid boundaries. The objects are disks of the same size but, in general, of different materials. Potential applications include the design of new amorphous polymeric and related granular materials as well as the design of package cushioning systems. The problem is considered on a grid structure with a fixed step size w and for a fixed diameter of the discs, i.e., the number of placed disks may increase as the size of the placement region increases. The near-equilibrium configurations have to be calculated from uniformly distributed random initial placements. The final arrangements of disks must ensure that any particular object is deformed only within the limits of elasticity of the material. The main result concerns -approximations of the probability distribution on the set of equilibrium placements. Under a natural assumption about the configuration space, we prove that a run-time of n+log^O(1)(1/} is sufficient to approach with probability 1 – the minimum value of the objective function, where depends on the maximum of the escape depth of local minima within the underlying energy landscape. The result is derived from a careful analysis of the interaction among probabilities assigned to configurations from adjacent distance levels to minimum placements. The overall approach for estimating the convergence rate is relatively independent of the particular placement problem and can be applied to various optimization problems with similar properties of the associated landscape of the objective function.     

On the thermomechanics of materials that have multiple natural configurations

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasticity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.Received: February 17, 2004     

On the thermomechanics of materials that have multiple natural configurations Part I: Viscoelasticity and classical plasticity

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasicity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.     

On the thermomechanics of materials that have multiple natural configurations Part I: Viscoelasticity and classical plasticity

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasicity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.     

Metrics with finite sets of primitive extensions

In this paper, we give a complete characterization for the class of rational finite metrics with the property that the set () of primitive extensions of is finite. Here, for a metric on a setT, a positive extensionm of to a setV T is calledprimitive if none of the convex combinations of other extensions of toV is less than or equal tom. Our main theorem asserts that the following the properties are equivalent: (i) () is finite; (ii) Up to an integer factor, is a submetric of the path metric d ^H of a graphH with |(d ^H )=1; (iii) A certain bipartite graph associated with contains neither isometrick-cycles withk6 nor induced subgraphsK _3,3 ^– . We then show that () is finite if and only if the dimension of the tight span of is at most two. We also present other results, discuss applications to multicommodity flows, and raise open problems.This research was supported by grant 97-01-00115 from the Russian Foundation of Basic Research and a grant from the Sonderforschungsbereich 343, Bielefeld Universität, Bielefeld, Germany.     

A Property of the Ansatz of Hirota's Method for Quasilinear Parabolic Equations

By using the recently discovered new invariant properties of the ansatz of R. Hirota's method, we prove that the classes of linear fractional solutions to some nonlinear equations are closed. This allows us to construct new solutions for a chosen class of dissipative equations. This algorithm is similar to the method of dressing the solutions of integrable equations. The equations thus obtained imply a compatibility condition and are known as a nonlinear Lax pair with variable coefficients. So we propose a method for constructing such pairs. To construct solutions of a more complicated form, we propose to use the property of zero denominators and factorized brackets, which has been discovered experimentally. The expressions thus constructed are said to be quasi-invariant. They allow us to find true relations between the functions contained in the ansatz, to correct the ansatz, and to construct a solution. We present some examples of new solutions constructed following this approach. Such solutions can be used for majorizing in comparison theorems and for modeling phase processes and process in neurocomputers. A program for computing solutions by methods of computer algebra is written. These techniques supplement the classical methods for constructing solutions by using their group properties.     

More on directed colimits of models

M. Richter has proved that whenever a classK of -structures has a finitary first-order axiomatization then the inclusionK Str preserves all existing directed colimits (see [7]). We will generalize this result to classes of -structures having an infinitary first-order axiomatization in a larger signature . We will also show that, as categories, these classes have a natural characterization.In honour of Dieter Pumplün on the occassion of his 60th birthday     

On the thermomechanics of materials that have multiple natural configurations Part II: Twinning and solid to solid phase transformation

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasticity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.     

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,) over a small category . We show a natural bijection between the set of equivalence classes of such torsors and [B({}),B(G,)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with _i (Y)=0 for all i1,2.     

Two characterization theorems for integral operators

Let (X,) be a separable -finite measure space. A bounded operator A on L²(X) is called an integral operator if it is induced by an equation: Af(x) = k(x,y)f(y)d(y), where k is a measurable function on X × X such that |k(x,y)f(y)|d(y) < a.e. for every f in L²(X).In this paper, some results on Carleman operators, due to von Neumann, Tarjonski and Weidmann, are extended to the case of the general integral operator.     

Asymptotic Analysis and Modeling of the Jointing of a Massive Body with Thin Rods

Asymptotic formulas are obtained for solutions of the anisotropic elasticity problem for a body with cavities into which thin rods are inserted, the outer ends of the rods being rigidly fixed. The surface of the body and the lateral surface of the rods are assumed load-free, but the entire elastic junction is subject to mass forces. The elastic materials are inhomogeneous and the stiffness of the rods may differ greatly from that of the body, their ratio being of the order h with an arbitrary exponent ; for = 0, the junction is homogeneous. Together with the asymptotic formulas, we construct and justify an asymptotic model of the junction. This model is applicable for a wide range of the exponent and preserves the parameter h in the conjugation conditions but is represented by a regularly perturbed problem. Since the leading asymptotic term involves fields with strong singularities, we have to give correct statements of the limit problem for a body with one-dimensional rods. For this purpose, we use the theory of self-adjoint extensions of operators or the technique of weighted spaces with separated asymptotics. The justification of our asymptotic expansions utilizes weighted anisotropic Korn inequalities, which take into account the mutual position of the rods and provide the best possible a priori estimates of the solutions. In contrast to other investigations, we describe, in explicit terms, the dependence of the bounds in the error estimates on the right-hand sides of the original problem. We also discuss the relationship between the asymptotic ansatz formulas and the weighted norms in the asymptotically precise Korn inequality.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 95–214, 2004.     

Analytic representation of the equation of state in classical statistical mechanics

For a system of classical particles with short-range pairwise interactions, analytic representations of the spherically-symmetrical (uniform) compression modulus are constructed in terms of convergent series in the usual and complementary densities and in terms of contour integrals. Similar representations are given for the equation of state and the specific logarithm of the configuration integral. The uniform compression modulus, which appears to be single-valued in a vicinity of the phase transition, suffices to determine all of the interesting thermodynamic quantities. The techniques which we develop are tested by application to an exactly solvable model, the Van der Waals substance, which is a model substance whose exact equation of state is given by the Van der Waals equation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 135–158, July, 1996.