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Variational methods for the solution of problems of equilibrium and vibrations
 Bull. Amer. Math. Soc
, 1943
"... problem " is a phrase of indefinite meaning. Pure mathematicians sometimes are satisfied with showing that the nonexistence of a solution implies a logical contradiction, while engineers might consider a numerical result as the only reasonable goal. Such one sided views seem to reflect human l ..."
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problem " is a phrase of indefinite meaning. Pure mathematicians sometimes are satisfied with showing that the nonexistence of a solution implies a logical contradiction, while engineers might consider a numerical result as the only reasonable goal. Such one sided views seem to reflect human limitations rather than objective values. In itself mathematics is an indivisible organism uniting theoretical contemplation and active application. This address will deal with a topic in which such a synthesis of theoretical and applied mathematics has become particularly convincing. Since Gauss and W. Thompson, the equivalence between boundary value problems of partial differential equations on the one hand and problems of the calculus of variations on the other hand has been a central point in analysis. At first, the theoretical interest in existence proofs dominated and only much later were practical applications envisaged by two physicists, Lord Rayleigh and
Asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle, preprint
, 1996
"... We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper b ..."
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We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small. 1.
SOME INEQUALITIES AND ASYMPTOTIC FORMULAS FOR EIGENVALUES ON RIEMANNIAN MANIFOLDS
, 2009
"... In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture f ..."
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In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the onedimensional case.
A Review of Methods for Constrained Eigenvalue Problems
"... Numerous studies are concerned with vibrations or buckling of constrained systems realising that complex systems can be analysed when starting from unconstrained or known problems. Over the years various constrained eigenvalue formulations have been published and put into practical use. The most imp ..."
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Numerous studies are concerned with vibrations or buckling of constrained systems realising that complex systems can be analysed when starting from unconstrained or known problems. Over the years various constrained eigenvalue formulations have been published and put into practical use. The most important ones are the Lagrangian multiplier method (modal synthesis method or component modes synthesis method), the receptance method and the modal constraint method. In this paper the similarities and merits of the various methods are discussed. It is striking that so far the similarities of the various constrained eigenvalue expressions have not been reported nor has the similarity of the eigenvalue expressions of these methods with Weinstein’s determinant for intermediate problems of the first type been noticed. The eigenvalue formulations of the Lagrangian multiplier method and that of the receptance appear to be similar to Weinstein’s determinant for intermediate problems of the first type. The modal constraint method is based on an extension of Weinstein’s method for intermediate problems of the first type and offers some significant advantages, i.e. the resulting eigenvalue formulation of the modal constraint method has a standard form in contrast with that of the other mentioned methods in that they have the known and unknown eigenvalues in the denominator of the eigenvalue formulations. Further, zero modal displacements, persistent and multiple eigenvalues do require special attention using these methods whereas this is not the case for the modal constraint method. Based on the similarities of the various constrained eigenvalue expressions a number of interesting conclusions are drawn.