Inverse power law potentials in rectangular configurations

Summary Calculations based on a (distance)^– intermolecular potential (>3) enable study of the effects on adsorption of the geometry of the solid. This paper gives the closed form solution for the adsorptive potential about a homogeneous solid rectangular corner; and, through systematic superposition, closed form solutions for the following configurations also: the rectangular corner of a cavity; laminae and rectangular cracks occupying a quarter plane; semi-infinite rectangular prisms and prismatic cavities; rectangular parallelepipeds and brick-shaped cavities. These various results are developed in detail for the cases =6 and =4. The paradox that potentials for >3 seem to be obtainable more readily than Newtonian potentials (=1) is explained by the existence only for >3 of simple fundamental solutions for infinite homogeneous solid configurations.

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Zusammenfassung Berechnungen, denen ein intermolekulares Potential der Form (Abstand)^– (>3) zugrunde gelegt ist, ermöglichen eine Untersuchung von Effekten der Adsorption auf die Geometrie des Festkörpers. Die vorliegende Arbeit gibt die Lösung in geschlossener Form für das Adsorptionspotential um eine feste, homogene, rechtwinklige Ecke an. Ausserdem werden durch systematische Superposition Lösungen in geschlossener Form für die folgenden Konfigurationen angegeben: die rechtwinklige Innenecke einer Mulde; viertelunendliche, ebene Platten und rechteckige Spalten; halbunendliche, reckteckige Prismen und prismatische Mulden; Quader und quaderförmige Höhlen. Diese Ergebnisse sind ausführlich dargestellt für die Fälle =4. Das Paradoxon. dass Potentiale mit >3 scheinbar leichter zugänglich sind als das Gravitationspotential (=1), wird dadurch erklärt, dass nur für >3 einfache Grundlösungen für unendliche, homogene Festköperkonfigurationen existieren.

     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

On the complexity of approximating a KKT point of quadratic programming

We present a potential reduction algorithm to approximate a Karush—Kuhn—Tucker (KKT) point of general quadratic programming (QP). We show that the algorithm is a fully polynomial-time approximation scheme, and its running-time dependency on accuracy (0, 1) is O((l/) log(l/) log(log(l/))), compared to the previously best-known result O((l/)²). Furthermore, the limit of the KKT point satisfies the second-order necessary optimality condition of being a local minimizer. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research support in part by NSF grants DDM-9207347 and DMI-9522507, and the Iowa Business School Summer Grant.     

Cone Adaptation Strategies for a Finite and Exact Cutting Plane Algorithm for Concave Minimization

In this paper we are concerned with pure cutting plane algorithms for concave minimization. One of the most common types of cutting planes for performing the cutting operation in such algorithm is the concavity cut. However, it is still unknown whether the finite convergence of a cutting plane algorithm can be enforced by the concavity cut itself or not. Furthermore, computational experiments have shown that concavity cuts tend to become shallower with increasing iteration. To overcome these problems we recently proposed a procedure, called cone adaptation, which deepens concavity cuts in such a way that the resulting cuts have at least a certain depth with 0, where is independent of the respective iteration, which enforces the finite convergence of the cutting plane algorithm. However, a crucial element of our proof that these cuts have a depth of at least was that we had to confine ourselves to -global optimal solutions, where is a prescribed strictly positive constant. In this paper we examine possible ways to ensure the finite convergence of a pure cutting plane algorithm for the case where = 0.     

Decomposition based and branch and bound global optimization approaches for the phase equilibrium problem

An increasingly popular approach when solving the phase and chemical equilibrium problem is to pose it as an optimization problem. However, difficulties are encountered due to the highly nonlinear nature of the models used to represent the behavior of the fluids, and because of the existence of multiple local solutions. This work shows how it is possible to guarantee -global solutions for a certain important class of the phase and chemical equilibrium problem, namely when the liquid phase can be modeled using neither the Non-Random Two-Liquid (NRTL) equation, or the UNIversal QUAsi Chemical (UNIQUAC) equation. Ideal vapor phases are easily incorporated into the global optimization framework. A numberof interesting properties are described which drastically alter the structure of the respective problems. For the NRTL equation, it is shown that the formulation can be converted into a biconvex optimization problem. The GOP algorithm of Floudas and Visweswaran [8, 9] can then be used to obtain -global solutions in this case. For the UNIQUAC equation, the new properties show how the objective function can be transformed into the difference of two convex functions (i.e. a D.C. programming problem is obtained), where the concave portion is separable. A branch and bound algorithm based on that of Falk and Soland [6] is used to guarantee convergence to an -global solution. Examples are presented which demonstrate the performance of both algorithms.     

Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison

We consider the following global optimization problems for a Lipschitz functionf implicitly defined on an interval [a, b]. Problem P: find a globally-optimal value off and a corresponding point; Problem Q: find a set of disjoint subintervals of [a, b] containing only points with a globally-optimal value and the union of which contains all globally optimal points. A two-phase algorithm is proposed for Problem P. In phase I, this algorithm obtains rapidly a solution which is often globally-optimal. Moreover, a sufficient condition onf for this to be the case is given. In phase II, the algorithm proves the-optimality of the solution obtained in phase I or finds a sequence of points of increasing value containing one with a globally-optimal value. The new algorithm is empirically compared (on twenty problems from the literature) with a best possible algorithm (for which the optimal value is assumed to be known), with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen. For small, the new algorithm requires only a few percent more function evaluations than the best possible one. An extended version of Piyavskii's algorithm is proposed for problem Q. A sufficient condition onf is given for the globally optimal points to be in one-to-one correspondance with the obtained intervals. This result is achieved for all twenty test problems.The research of the authors has been supported by AFOSR grants 0271 and 0066 to Rutgers University. Research of the second author has been also supported by NSERC grant GP0036426, FCAR grant 89EQ4144 and partially by AFOSR grant 0066. We thank Nicole Paradis for her help in drawing the figures.     

An improved univariate global optimization algorithm with improved linear lower bounding functions

Recently linear lower bounding functions (LLBF's) were proposed and used to find -global minima. Basically an LLBF over an interval is a linear function which lies below a given function over the interval and matches the function value at one end point. By comparing it with the best function value found, it can be used to eliminate subregions which do not contain -global minima. To develop a more efficient LLBF algorithm, two important issues need to be addressed: how to construct a better LLBF and how to use it efficiently. In this paper, an improved LLBF for factorable functions overn-dimensional boxes is derived, in the sense that the new LLBF is always better than those in [3] for continuously differentiable functions. Exploration of the properties of the LLBF enables us to develop a new LLBF-based univariate global optimization algorithm, which is again better than those in [3]. Numerical results on some standard test functions indicate the high potential of our algorithm.This work was supported in part by VLSI Technology Inc. and Tyecin Systems Inc. through the University of California MICRO proram with grant number 92-024.     

Almost flat locally finite coverings of the sphere

For any subset A of the unit sphere of a Banach space X and for [0,2) the notion of -flatness is introduced as a measure of non-flatness of A. For any positive , construction of locally finite tilings of the unit sphere by -flat sets is carried out under suitable -renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed.     

A one-phase algorithm for semi-infinite linear programming

We present an algorithm for solving a large class of semi-infinite linear programming problems. This algorithm has several advantages: it handles feasibility and optimality together; it has very weak restrictions on the constraints; it allows cuts that are not near the most violated cut; and it solves the primal and the dual problems simultaneously. We prove the convergence of this algorithm in two steps. First, we show that the algorithm can find an-optimal solution after finitely many iterations. Then, we use this result to show that it can find an optimal solution in the limit. We also estimate how good an-optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding an-optimal solution under some assumptions. This algorithm is generalized to solve a class of nonlinear semi-infinite programming problems. Applications to convex programming are discussed.     

The generalized inverse of a perturbed singular matrix

It is shown that, whenA is singular, (A+A ₁)^–1 can be expanded into a Laurent's series in. The coefficients of the expansion are given in an explicit form. The case where A+A ₁ vanishes identically in is also studied and a generalized inverse ofA+A ₁ is given.

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Zusammenfassung Es wird gezeigt, daß wennA singulär ist (A+A ₁)^–1 auf eine Laurent-Serie in erweitert werden kann. Die Koeffizienten der Erweiterung werden in expliziter Form gegeben. Der Fall wo A+A ₁ sich identisch in auflöst wird ebenfalls untersucht und eine verallgemeinerte Umkehrung vonA+A ₁ wird angegeben.

     

Lower semicontinuity of attractors of gradient systems and applications

Summary For 0₀, let T(t), t0, be a family of semigroups on a Banach space X with local attractors A. Under the assumptions that T₀(t) is a gradient system with hyperbolic equilibria and T(t) converges to T₀(t) in an appropriate sense, it is shown that the attractors {A, 0₀} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between A and A₀, in some examples.Research supported by U.S. Army Research Office DAAL-03-86-K-0074 and the National Science Foundation DMS-8507056.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

Boundary-layer averaging for standard systems with lag

Thekth-order asymptotic solution of a standard system with lag is constructed along trajectories calculated according to the averaging scheme of A. N. Filatov. If the perturbation parameter 1, then the use of the step method for finding the solution is connected with cumbersome calculations because the number of required steps is inversely proportional to . We suggest another approach in which the step method is used onlyk times fort [0,k] and justify the asymptotic method.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10 pp. 1362–1368, October, 1994.     

An algebraic approach to the vector ε-algorithm

The vector -algorithm is obtained from the scalar -algorithm by taking the pseudo-inverse of a vector instead of the inverse of a scalar. Thus the vector -algorithm is known only through its rules contrarily to the scalar -algorithm and some other extrapolation algorithms.The aim of this paper is to provide an algebraic approach to the vector -algorithm.     

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x().     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Pointwise displacement errors in linear shell theory resulting from errors in the stress-strain relations

It is rigorously proved that relative errors of order in the stress-strain relations of linear shell theory result in relative pointwise errors in the solution displacement field of order .

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Zusammenfassung Für die Theorie dünner Schalen wird bewiesen, daß ein relativer Fehler der Größe in den Spannungs-Dehnungs-Beziehungen einen relativen lokalen Fehler der Größe in der Lösung für das Verschiebungsfeld erzeugt.

     

How to approximate the solutions of certain free boundary problems for the Laplace equation by using the contraction principle

We show how the free boundary of an ideal fluid, subject to a generalized Bernoulli condition, can (under appropriate circumstances) be approximated. Our method is based on a class of free-boundary perturbation operatorsT , 0<<1, which are all contracting relative to a suitable norm and class of boundaries, and whose fixed points converge to the desired free boundary solution as 0+.

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Zusammenfassung Wir zeigen, wie der freie Rand einer idealen Flüssigkeit, welcher einer verallgemeinerten Bernoulli-Bedingung genügt, unter geeigneten Umständen approximiert werden kann. Unsere Methode stützt sich auf eine Klasse freier RandperturbationsoperatorenT , 0<<1, welche relativ zu einer geeigneten Norm und Ränderklasse kontrahierend sind und deren Fixpunkte gegen die gewünschte Lösung der freien Randaufgabe mit 0+ konvergieren.

     

Characterization of Dirichlet Forms by their Excessive and Co–Excessive Elements

It is proved that if S, T are two elliptic Dirichlet operators on an ordered Hilbert space such that the excessive (resp. coexcessive) elements with respect to S and T are the same then there exists > 0 with T = S. Particularly if , are two elliptic Dirichlet forms on L² ( ) having the same domain of definition and the same -excessive (resp. -coexcessive) elements for any > 0 then = .     

On the distribution of comparisons in sorting algorithms

We considered the following natural conjecture: For every sorting algorithm every key will be involved in(logn) comparisons for some input. We show that this is true for most of the keys and prove matching upper and lower bounds. Every sorting algorithm for some input will involven–n /2+1 keys in at leastlog₂ n comparisons,>0. Further, there exists a sorting algorithm that will for every input involve at mostn–n ^/c keys in greater thanlog₂ n comparisons, wherec is a constant and>0. The conjecture is shown to hold for natural algorithms from the literature.