Resonances of an elastic plate in a compressible confined fluid

We present a theoretical study of the resonances of a fluid–structureproblem, an elastic plate placed in a duct in the presence ofa compressible fluid. The case of a rigid plate has been largelystudied. Acoustic resonances are then associated to resonantmodes trapped by the plate. Due to the elasticity of the plate,we need to solve a quadratic eigenvalue problem in which theresonance frequencies k solve the equations (k) = k², where are the eigenvalues of a self-adjoint operator of the formA + kB. First, we show how to study the eigenvalues locatedbelow the essential spectrum by using the min–max principle.Then, we study the fixed-point equations. We establish sufficientconditions on the characteristics of the plate and of the fluidto ensure the existence of resonances. Such conditions are validatednumerically.     

An unfitted finite-element method for elliptic and parabolic interface problems

Bhupen Deka Department of Mathematics, Assam University, Silchar-788011, India ^{A finite-element discretization, independent of the locationof the interface, is proposed and analysed for linear ellipticand parabolic interface problems. We establish error estimatesof optimal order in the H1-norm and almost optimal order inthe L2-norm for elliptic interface problems. An extension toparabolic interface problems is also discussed and an optimalerror estimate in the L2(0, T;H1())-norm and an almost optimalorder estimate in the L2(0, T;L2())-norm are derived for thespatially discrete scheme. A fully discrete scheme based onthe backward Euler method is analysed and an optimal order errorestimate in the L2(0, T;H1())-norm is derived. The interfacesare assumed to be of arbitrary shape and smooth for our purpose.}     

Adaptive frame methods for elliptic operator equations: the steepest descent approach

Massimo Fornasier^¶ Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università "La Sapienza" in Roma, Via Antonio Scarpa, 16/B, I-00161 Roma, Italy Rob Stevenson^|| Department of Mathematics, Utrecht University, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands ^{This paper is concerned with the development of adaptive numericalmethods for elliptic operator equations. We are particularlyinterested in discretization schemes based on wavelet frames.We show that by using three basic subroutines an implementable,convergent scheme can be derived, which, moreover, has optimalcomputational complexity. The scheme is based on adaptive steepestdescent iterations. We illustrate our findings by numericalresults for the computation of solutions of the Poisson equationwith limited Sobolev smoothness on intervals in 1D and L-shapeddomains in 2D.}     

Lame's Equation and Its Use in Elliptic Cone Problems

A new solution is obtained for Lamé's equation in the form of a perturbation seriesabout k = 0; numerical accuracy appears to be high when v ismoderate and k not too close to 1, so that the solution is ofpractical value in problems involving elliptic cones or infinitesectors.     

Stability in the Sense of Bounded Average Power

An R^m-valued sequence (x_k): = (x_k : k = 1, 2, ...), e.g. generatedrecursively by x_k = f_k (x_k–_k, U_k), is called ‘averagepth power bounded’ if (1/K) is bounded uniformly in K= 1, 2,.... (The case p = 2 may correspond to ‘power’in the physical sense.) This is a notion of stability. Givenestimates of the form: f_k (x, u) < a x + ¶ _k conditionsare obtained on the coefficient sequence (a_k) and the inputestimates e_k:=¶_k (u_k) which ensure this form of stabilityfor the output (x_k). In particular, a condition (utilized inan application to adaptive control) is obtained which imposes(i) a bound b on (a_k) and a ‘sparsity measure’ m(K) on #{kK: a_k>} as K ( >1) (ii) average pth power boundednesson (e_k), and (iii) a growth condition on (e_k) related to b andm (•). This condition is sharp.     

A domain embedding preconditioner for the Lagrange multiplier system

Finite element approximations for the Dirichlet problem associated to a second-order elliptic differential equation are studied. The purpose of this paper is to discuss domain embedding preconditioners for discrete systems. The essential boundary condition on the interior interface is removed by introducing Lagrange multipliers. The associated discrete system, with a saddle point structure, is preconditioned by a block diagonal preconditioner. The main contribution of this paper is to propose a new operator, constructed from the -inner product, for the block of the preconditioner corresponding to the multipliers.

     

On the Bargmann Transform and the Wigner Transform

We observe that the Bargmann transform arises from the polardecomposition of a simple restriction operator. By virtue ofthe Bargmann transform, the Wigner transform is imbedded inthe geometric symmetry group of phase space. From this imbedding,we easily obtain that the spectrum of the Wigner transform consistsof {e^ik/12:k=0, 1, ..., 23}. 1991 Mathematics Subject Classification81R30, 47B38, 81S30.     

Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

Sorin Micu ^{This paper studies the numerical approximation of the boundarycontrol for the wave equation in a square domain. It is knownthat the discrete and semi-discrete models obtained by discretizingthe wave equation with the usual finite-difference or finite-elementmethods do not provide convergent sequences of approximationsto the boundary control of the continuous wave equation as themesh size goes to zero. Here, we introduce and analyse a newsemi-discrete model based on the space discretization of thewave equation using a mixed finite-element method with two differentbasis functions for the position and velocity. The main theoreticalresult is a uniform observability inequality which allows usto construct a sequence of approximations converging to theminimal L2-norm control of the continuous wave equation. Wealso introduce a fully discrete system, obtained from our semi-discretescheme, for which we conjecture that it provides a convergentsequence of discrete approximations as both h and t, the timediscretization parameter, go to zero. We illustrate this factwith several numerical experiments.}     

The Index of Toeplitz Operators on Free Transformation Group C*-Algebras

Let be a discrete group acting on a compact manifold X, letV be a -equivalent Hermitian vector bundle over X, and let Dbe a first-order elliptic self-adjoint -equivalent differentialoperator acting on sections of V. This data is used to defineToeplitz operators with symbols in the transformation groupC*-algebra C(X), and it is shown that if the symbol of sucha Toeplitz operator is invertible, then the operator is Fredholm.In the case where is finite and acts freely on X, a geometric-topologicalformula for the index is stated that involves an explicitlyconstructed differential form associated to the symbol. 2000Mathematics Subject Classification 47A53 (primary), 19K56, 47B35,46L87 (secondary).     

On the Stable Rank of Algebras of Operator Fields Over an N-Cube

Let A be a unital maximal full algebra of operator fields withbase space [0, 1]^k and fibre algebras . It is shown in this paper that the stable rankof A is bounded above by the quantity , where ‘sr’ means stable rank. Usingthe above estimate, the stable ranks of the C*-algebras of the(possibly higher rank) discrete Heisenberg groups are computed.2000 Mathematics Subject Classification 47L99.     

Contributions to the Theory of Waves in Non-linear Dispersive Systems

This paper makes contributions to the general theory of wavepropagation in conservative systems under conditions when theproportional change in amplitude or wavenumber over a distanceof one wavelength is very small. For linear systems, such propagationis governed by the well-known theory of group velocity; thereis "frequency dispersion", in the sense that energy in componentsof different frequency is propagated at different group velocities.For non-linear systems without frequency dispersion, e.g. acousticsystems, a different, but also well-known, modification of thewaveform occurs. It may be called "amplitude dispersion", inthat different values of an amplitude variable like the pressureare propagated at different speeds. A much more general theory of non-linear systems, where frequencydispersion and amplitude dispersion would be expected to bein competition, has been given by Whitham (1965b). Energy doesnot play a key role in the theory, because it is easily transferredbetween components of different frequencies. The fundamentalequation follows from Hamilton's principle in an averaged form. In examples given by Whitham, changes in, for example, wavenumber(or amplitude) are propagated at two different velocities, becausethe fundamental equation is hyperbolic. However, in the limitingcase of infinitesimal amplitude, the equation is parabolic andonly one velocity of propagation (the group velocity) occurs.Thus, Whitham showed that non-linearity can "split" the groupvelocity. This paper is concerned with the inference of detailed conclusionsfrom Whitham's theory, to enable comparisons with experimentthat will show the range of applicability of the theory. Itattempts to obtain these in the simplest case, namely, thatof one-dimensional propagation when Whitham's "pseudo-frequencies"are absent. If the relationship between frequency and wavenumber k forinfinitesimal amplitude is = f(k), then for finite amplitudethe equation is shown to be hyperbolic or elliptic respectively,according as [–f(k)]f^*(k) takes positive or negative values.For gravity waves on deep water this product is negative andthese, it is inferred, may be good for comparison of theorywith experiment in the elliptic case. A new non-linear non-perturbationaltheory of waves under the combined action of gravity and surfacetension is used to indicate that waves at 9.6 c/s on mercurymay be suitable for comparison with experiment in the hyperboliccase. When non-linear effects are only moderate, approximate transformationsof Whitham's equation to the axisymmetric Laplace and wave equationsrespectively, in the elliptic and hyperbolic cases, are usedto obtain particular solutions for comparison with experiment.A feature of these solutions is the appearance of discontinuitiesin wavelength. For example, when a wavemaker creates gravity waves of fixedfrequency whose amplitude first increases and then decreases,the theory predicts that the length of waves in the group decreasesahead of the point of maximum amplitude and increases behindit. This produces in turn a concentration of energy towardsthe centre of the group, which continues during the whole periodbefore a discontinuity in wavelength actually forms. This solutionin the elliptic case is obtained with the aid of the theoryof imaginary characteristics.     

A Graph-Theoretic Approach to the Unique Midset Property of Metric Spaces

A metric space X has the unique midset property if there isa topology-preserving metric d on X such that for every pairof distinct points x, y there is one and only one point p suchthat d(x, p) = d(y, p). The following are proved. (1) The discretespace with cardinality n has the unique midset property if andonly if n 2, 4 and n c, where c is the cardinality of thecontinuum. (2) If D is a discrete space with cardinality notgreater than c, then the countable power D^N of D has the uniquemidset property. In particular, the Cantor set and the spaceof irrational numbers have the unique midset property. A finite discrete space with n points has the unique midsetproperty if and only if there is an edge colouring of the completegraph K_n such that for every pair of distinct vertices x, ythere is one and only one vertex p such that (xp) = (yp). Letump(K_n) be the smallest number of colours necessary for sucha colouring of K_n. The following are proved. (3) For each k 0, ump(K_2k+1) = k. (4) For each k 3, k ump(K_2k) 2k–1.     

Properties of the weighted logarithmic matrix norms

Mingzhu Liu Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Email: ghu{at}hit.edu.cn, ghuca{at}yahoo.ca Email: mzliu{at}hit.edu.cn Received on January 31, 2006; Accepted on December 9, 2006 In this paper, we are concerned with the properties of the weightedlogarithmic matrix norms. A relation between the elliptic logarithmicmatrix norm and the weighted logarithmic matrix norm is given.Based on Lyapunov equations, two weighted logarithmic matrixnorms are constructed which are less than 1-logarithmic matrixnorm and -logarithmic matrix norm, respectively. Then, an iterativescheme is presented to obtain the logarithmically -efficientmatrix norm. Numerical examples are given to illustrate theresults.     

On a question of Sarkozy and Sos for bilinear forms

We prove that, if 2 k₁ k₂, then there is no infinite sequence of positive integers such that the representation functionr(n) = #{(a, a'): n = k₁a + k₂a', a, a' } is constant for nlarge enough. This result completes the previous work of Diracand Moser for the special case k₁ = 1 and answers a questionposed by Sárkozy and Sós.     

Spectral properties of banded Toeplitz matrices * By Albrecht Bottcher and Sergei M. Grudsky: * 411 pp., US$ 95.00, ISBN 0-89871-599-7 * (Society for Industrial and Applied Mathematics, Philadelphia, 2005)

Recall that an infinite Toeplitz matrix, , say, is one for which the values t_jk dependonly on j – k, so we may write t_jk = a_{j – k}; thusthe entries are constant on diagonals sloping down from topleft to bottom right:

Sucha matrix is said to be banded if there are only finite manynon-zero a_j, say . The propertiesof such a matrix, when defining a linear operator T on a sequencespace, are often closely related to the properties of the Laurentpolynomial

for zlying on the unit circle in the complex plane. For example,suppose that the matrix acts on the most familiar sequence space,the Hilbert space ²; then it has been known for many years thatthe operator norm of T is the maximum     

Elliptic Genera for Z/k-Manifolds I

Using Z/k-manifolds we give a geometric interpretation of thering homomorphism from cobordismwith Z/k coefficients to elliptic cohomology with Z/k coefficients,induced by the map of spectra MSO Ell associated to the naturaltransformation of cohomology theories MSO* Ell*.     

Approximation of exact controllability problem involving parabolic differential equations

In this paper we examine computation of optimal control u^* ofthe exact controllability problem (referred to as the constraintproblem) governed by the following type of linear parabolicdifferential equations: (y/t) + Ay = u in Q y = 0 on y(0) = y₀ on where A is the second-order elliptic differential operator, is a bounded domain in ^k with smooth boundary , Q = (0, T)x , = (0, T) x and T > 0. This is achieved by approximatingu^* through a sequence {u_n} of controls corresponding to unconstrainedproblems involving a penalty function arising from the controllabilityconstraint.