Lower bounds to eigenvalues of piecewise continuous elastic systems

Summary The method of truncation introduced by Bazley and Fox has been extended to include situations where it is desirable to model mechanical systems as having discontinuities in either geometrical and/or material properties. The results are then applied to typical problems in the areas of stability and vibrations. Numerical results are presented.

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Zusammenfassung Die von Bazley und Fox vorgeschlagene truncation method wird auf Modelle mechanischer Systeme mit geometrischen und/oder materiellen Unstetigkeiten erweitert. Die Ergebnisse werden auf typische Stabilitäts- und Schwingungsprobleme angewandt. Numerische Schranken werden gegeben.

     

Methods for computing lower bounds to eigenvalues of self-adjoint operators

Summary. New approaches for computing tight lower bounds to the eigenvalues of a class of semibounded self-adjoint operators are presented that require comparatively little a priori spectral information and permit the effective use of (among others) finite-element trial functions. A variant of the method of intermediate problems making use of operator decompositions having the form is reviewed and then developed into a new framework based on recent inertia results in the Weinstein-Aronszajn theory. This framework provides greater flexibility in analysis and permits the formulation of a final computational task involving sparse, well-structured matrices. Although our derivation is based on an intermediate problem formulation, our results may be specialized to obtain either the Temple-Lehmann method or Weinberger's matrix method. Received December 12, 1992 / Revised version received October 5, 1994     

Upper and lower bounds for sloshing frequencies by intermediate problems

This article presents very good, rigorous, numerical estimates for the sloshing eigenvalues of two families of two-dimensional regions. For each region the eigenvalues are proportional to the square of the frequencies of small lateral oscillations of an ideal fluid under the influence of gravity in a canal or in a horizontal cylindrical tank that has the region as cross-section. Our estimates are obtained by using conformal transformations that carry rectangles onto the regions, and then by employing intermediate problems with a generalized special choice to find upper and lower bounds to eigenvalues. The transformed problems are analogous to certain nonuniform string eigenvalue problems we estimate in a similar way.

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Résumé On calcule des bornes inférieures et supérieures aux fréquences d'oscillation latérale d'un liquide idéal sous l'influence de la gravité dans un réservoir cylindrique ou dans un canal. Les carrés des fréquences sont proportionnnels aux valeurs propres d'un opérateur aux dérivées partielles dans un ensemble de deux dimensions qui prend la forme de la section du liquide. Pour déterminer les bornes aux valeurs propres, on se sert des fonctions analytiques qui transforment les sections en rectangles ainsi que de problèmes intermédiaires choisis spécialement sous un aspect général. Les problèmes aux valeurs propres résultant des transformations sont analogues à certains problèmes de cordes vibrantes pour lesquels on peut calculer des bornes aux valeurs propres de façon semblable.

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This work was supported by the Department of the Navy, Naval Sea Command under Contract No. N 00024-81-C-5301.     

Upper and lower bounds for the frequencies of rectangular free plates

Zusammenfassung Eine neuentwickelte Methode für untere Schranken und das Rayleigh-Ritzverfahren für obere Schranken werden von den Verfassern dazu angewandt, die Eigenfrequenzen der Schwingungen von dünnen gleichförmigen, rechteckigen Platten mit freien Rändern abzuschätzen. Die Anwendbarkeit des Verfahrens für untere Schranken wird hervorgehoben, und es werden Berechnungen für eine Symmetrieklasse von rechteckigen sowie für eine Unterklasse von quadratischen Platten angegeben. Die dadurch entstehenden Schranken führen zu einer Verbesserung bisher veröffentlichter Resultate und weisen auf die Brauchbarkeit der Methode für untere Schranken hin.

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Nomenclature The following symbols are not defined in the text ² Laplacian differential operator (= ²/x ² + ²/y ²) - ⁴ Biharmonic differential operator (= ⁴/x ⁴ + 2 ⁴/x ² y ² + ⁴/y ⁴) - u Generic name for displacement functions - Generic name for eigenvalues - a,b Plate side lengths - Circular frequency of free vibration - Mass density of plate material - h Plate thickness - D Plate flexural rigidity - First variation - d Element of surface area - _ij Kronecker's delta - {} Matrix or vector of the included elements The research reported in this article has been sponsored by the Department of the Navy under Contract NOw-62-0604-c with the Bureau of Naval Weapons.     

Error bounds for systems of lower semicontinuous functions in Asplund spaces

In this paper, using the Fréchet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems.     

Lower bounds for lower Ramsey numbers

For any graph G, let i(G) and μ;(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers m and n, the lower Ramsey number s(m, n) is the largest integer p so that every graph of order p has i(G) ≤ m or μ;(G) ≤ n. In this paper we give several new lower bounds for s (m, n) as well as determine precisely the values s(1, n).     

Strict upper and lower bounds of quantities in linear second-order systems

This paper aims to present a model verification technique for general numerical computations of linear second-order systems. Strict upper and lower bounds on quantities of interest are eventually obtained. The model verification technique consists of two major steps. The first is the representation of the error in quantities of interest through an adjoint correction; the second is the application of the strict upper bound derived for the time-discretization error. Moreover, the bounds on quantities of interest are further improved through an optimization procedure. Academic examples are studied to verify the proposed bounds and to explore the potential application of these bounds to adaptive time-stepping.     

Mixed lower bounds for quantum transport

Let H be a self-adjoint operator on a separable Hilbert space , . Given an orthonormal basis of , we consider the time-averaged moments 〈|X|_ψ^p〉(T) of the position operator associated to . We derive lower bounds for the moments in terms of both spectral measure μ_ψ and generalized eigenfunctions u_ψ(n,x) of the state ψ. As a particular corollary, we generalize the recently obtained lower bound in terms of multifractal dimensions of μ_ψ and give some equivalent forms of it which can be useful in applications. We establish, in particular, the relations between the L^q-norms (q>1/2) of the imaginary part of Borel transform of probability measures and the corresponding multifractal dimensions.     

New lower bounds for multiple coverings

A code , where Z ₂ = {0,1}, is said to be a binary μ-fold R-covering code, if for any word there are at least μ distinct codewords which differ from v in at most R coordinates. The size of the smallest binary μ-fold R-covering code of length n is denoted by K(n, R, μ). In this paper we use integer programming and exhaustive search to improve 57 lower bounds on K(n, R, μ) for 6 ≤ n ≤ 16, 1 ≤ R ≤ 4 and 2 ≤ μ ≤ 4.      

New lower bounds for Hopcroft's problem

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set ofn points andm hyperplanes in $\mathbb{R}^d $ , is any point contained in any hyperplane? We define a general class ofpartitioning algorithms, and show that in the worst case, for allm andn, any such algorithm requires time Ω(n logm + n ^2/3m^2/3 + m logn) in two dimensions, or Ω(n logm + n ^5/6m^1/2 + n^1/2m^5/6 + m logn) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log^*(n+m)) of the best known upper bound, due to Matou?ek. Previously, the best known lower bound, in any dimension, was Ω(n logm + m logn). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called amonochromatic cover, and derive lower bounds on its size in the worst case. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover. As a related result, using a straightforward adversary argument, we derive aquadratic lower bound on the complexity of Hopcroft's problem in a surprisingly powerful decision tree model of computation.     

Range of lower bounds

Each of n jobs is to be processed without interruption on a single machine. Each job becomes available for processing at time zero. The objective is to find a processing order of the jobs which minimizes the sum of weighted completion times added with maximum weighted tardiness. In this paper we give a general case of the theorem that given in [6]. This theorem shows a relation between the number of efficient solutions, lower bound LB and optimal solution. It restricts the range of the lower bound, which is the main factor to find the optimal solution. Also, the theorem opens algebraic operations and concepts to find new lower bounds.     

Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity

Lower bounds for the eigenvalues of some elliptic equations and elliptic systems over bounded regions are obtained. The bounds are universal in that they depend only upon the volume of the region. Specific applications include the clamped plate, the buckling problem for the clamped plate and the equations of linear elasticity. Our results are consequences of extensions of the methods of Li and Yau (Comm. Mat. Phys. 88 (1983) 309–318) who obtained such results for the eigenvalues of the fixed membrane problem.     

Ranking lower bounds for the bin-packing problem

In this paper, we evaluate different known lower bounds for the bin-packing problem including linear programming relaxation (LP), Lagrangean relaxation (LR), Lagrangean decomposition (LD) and the Martello–Toth (MT) [Martello, S., Toth, P., Knapsack Problems: Algorithms and Computer Implementations, Wiley, New York, 1990] lower bounds. We give conditions under which Lagrangean bounds are superior to the LP bound, show that Lagrangean decomposition (LD) yields the same bound as Lagrangean relaxation (LR) and prove that the MT lower bound is a Lagrangean bound evaluated at a finite set of Lagrange multipliers; hence, it is no better than the LR and LD lower bounds.     

On error bounds for lower semicontinuous functions

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f. Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002     

Constructive lower bounds for off-diagonal Ramsey numbers

We describe an explicit construction whicy, for some fixed absolute positive constant ε, produces, for every integers>1 and all sufficiently largem, a graph on at least vertices containing neither a clique of sizes nor an independent set of sizem. Part of this work was done at the Institute for Advanced Study, Princeton, NJ 08540, USA. Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research supported in part by a grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

Complexity lower bounds for machine computing models

The article is the text of a survey report on methods of obtaining lower bounds on the computational complexity for abstract computers. Besides the methods for obtaining the lower bounds, related methods for the simulation of some machines by others, with the preservation of some complexity measures at the expense of increase in others (trade-off results), are presented. Methods of crossing sequences, tails, overlaps, and related methods are examined. A new proof of an old result is sometimes given to illustrate the working of a method, or a new result is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 118, pp. 4–24, 1982.