On the determination of the worst sampling error in a communication system for pulse-amplitude-modulated signals. I: The general model

We present a mathematical model of a communication system perturbed by statistical sampling errors (timing jitter). The aim is to find an ‘optimal’ impulse response for the system, the optimization problem actually being a minimax problem. that is we put the model into a game-theoretical framework. The basic game turns out to be a statistical game similar to those arising from estimation problems in statistics. Earlier results concerning least-favourable sampling error distributions published by Krabs and Vogel are supplemented by estimations of the number of support points of the least-favourable distribution. Furthermore, we state the existence of the saddle points of our game, which formerly has only been proved for some special cases. In the first part we treat the general situation, where one strategy set for the game—the set of all feasible impulse responses—forms a vector space with infinite dimension. In the second part we discuss the problem in the case that the impulse responses are restricted to a finite-dimensional subspace of the whole infinite-dimensional subspace.     

On the determination of the worst sampling error in a communication system for pulse-amplitude-modulated signals by solving an approximation problem

A communication system for pulse-amplitude-modulated signals is considered whose impulse response is bandlimited. For a certain class of sampling errors the worst case is investigated where the sampling error of the system is such that the least mean square transmission error achieved by a suitable choice of the impulse response becomes maximal. This problem is dualized in the sense of two person game theory and thereby transferred to an equivalent approximation problem for whose solution a necessary and sufficient condition is given and applied to special cases.     

Analysis of a bilinear finite element for shallow shells. II: Consistency error

We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane-dominated case, under severe assumptions on the domain, the finite element mesh and the regularity of the solution, an error bound can be obtained if the contribution of transverse shear is neglected. Here is the thickness of the shell, the mesh spacing, and a smoothness parameter. In the bending-dominated case, the uniformly optimal bound is achievable but requires that membrane and transverse shear strains are of order as . In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound .

     

On a variational problem for an infinite particle system in a random medium Part II: The local growth rate

Summary This paper solves the second of two variational problems arising in the study of an infinite system of particles that branch and migrate in a random medium. This variational problem involves a non-linear functional on a subset of the stationary probability measures on [×⁺], describing the interplay between particles and medium. It is shown that the variational problem can be solved in terms of the Lyapunov exponent of a product of random × matrices. This Lyapunov exponent is calculated via a random continued fraction. By analyzing the latter we are able to compute the maximum and the maximizer in the variational problem. It is found that these quantities exhibit interesting non-analyticities and changes of sign as a function of model parameters, which correspond to phase transitions in the infinite particle system. By combining with results from Part I we obtain a complete picture of the phase diagram.     

Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in . In a sense to be discussed below these sharpen known quasi-optimal and estimates for the error on irregular quasi-uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution . We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non-smooth problems.

     

The Markov modulated regulated Brownian motion: A second-order fluid flow model of a finite buffer

A second-order fluid flow model of a queue with finite capacity buffer and variable net input process is presented, based on the previous work of Karandikar and Kulkarni (1995). Queue length is modelled as a Brownian motion whose parameters are controlled by a finite state Markov chain. The process, termed a Markov modulated regulated Brownian motion (MMRBM), provides analytical solutions for steady state queue length distributions, overflow losses and idleness probabilities using boundary regulators. Applications of the model include queues with failure-prone servers and ATM statistical multiplexers with variable traffic loads. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part II: The piecewise linear case

We extend results from Part I about estimating gradient errors elementwise a posteriori, given there for quadratic and higher elements, to the piecewise linear case. The key to our new result is to consider certain technical estimates for differences in the error, , rather than for itself. We also give a posteriori estimators for second derivatives on each element.

     

An error bound for the finite element approximation of a model for phase separation of a multi-component alloy

An error bound is proved for a fully practical piecewise linearfinite element approximation, using a backward-Euler time discretization,of a model for phase separation of a multi-component alloy.Numerical experiments with three components in one and two spacedimensions are also presented.     

On the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides. Part 2: Time-dependent theory

In part 1

1 Math. meth. in the Appl. Sci, 10, 125–144 (1988).

we studied the principle of limiting absorption for local perturbations Ω of the n-dimensional domain Ω₀ = ?^n?1 × (0, π). In this second part we extend our investigations to the time-dependent theory and show that absence of admissible standing waves implies the validity of the principle of limiting amplitude for every frequency ω≥0 if n ≠ 3 and for ω ≠ 2, 3,… if n = 3, respectively. In particular, the principle of limiting amplitude holds for every ω≥0 in the case n ≠ 3 and for every ω ≠ 2, 3,… in the case n = 3 if Ω≠Ω₀ and ν · x ′ ?0 on ?Ω, where x ′ = (x₁,…, x_n?1, 0) and ν is the normal unit vector on ?Ω pointing into the complement of Ω This result stands in remarkable contrast to the fact that both principles are violated in the case of the unperturbed domain Ω₀ at the frequencies ω = 1, 2,… if n?3. The question of the asymptotic behaviour of the solution as t→∞ for n = 3 and ω = 2, 3,… will be discussed in two subsequent papers.     

On the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides. Part 1: Time-independent theory

Let Ω be a local perturbation of the n-dimensional domain Ω₀ = Ropf;^{n ? 1} × (0, π). In a previous paper⁸ we have introduced the notion of an admissible standing wave. We shall prove that the principle of limiting absorption holds for the Dirichlet problem of the reduced wave equation in Ω at ω ≥ 0 if Ω does not allow admissible standing waves with frequency ω. From Reference 8, this condition is satisfied for every ω ≥ 0 if Ω ≠ Ω₀, and v · x ′ ≤ 0 on δΩ, where x′ = ( x ₁,…, x_{n ? 1}, 0) and v is the normal unit vector on δΩ pointing into the complement of Ω. In contrast to this, the principle of limiting absorption is violated in the case of the unperturbed domain Ω₀ at the frequencies ω = 1,2,… if n ≤ 3. The second part of our investigation, which will appear in a subsequent paper, is devoted to the principle of limit amplitude.     

On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case

We give a constructive method for giving examples of doping functions and geometry of the device for which the nonelectroneutral voltage driven equations have multiple solutions. We show in particular, by performing a singular perturbation analysis of the current driven equations that if the electroneutral voltage driven equations have multiple solutions then the nonelectroneutral voltage driven equations have multiple solutions for sufficiently small normed Debye length. We then give a constructive method for giving examples of data for which the electroneutral voltage driven equations have multiple solutions.

     

On the number of conjugacy classes in a finite group II

In this work we obtain new properties connected with the number of conjugacy classes of elements of a finite group, through the analysis of the numberr _G(gN) of conjugacy classes of elements ofG that intersect the cosetgN, whereN is a normal subgroup ofG andg any element ofG. The results obtained about this number are not only used in the general problem of classifying finite groups according to the number of conjugacy classes, but they also allow us to improve and generalize known results relating to conjugacy classes due to P. Hall, M. Cartwright, A. Mann, G. Sherman, A. Vera-López and L. Ortíz de Elguea. Examples are given which illustrate our improvements. This work has been supported by the University of the Basque Country.     

On the Cauchy problem for a model equation for shallow water waves of moderate amplitude

We prove the local well-posedness for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime.     

On the hydrodynamic interaction of two spheres oscillating in a viscous fluid. II. Three dimensional case

Asymptotic solution of the Navier-Stokes equations is derived for the problem of the viscous flow induced by two spheres oscillating in a direction perpendicular to their central line.The main purpose of the paper is to study the effect of the hydrodynamic interaction of the spheres on the forces acting upon them in this complicated three dimensional flow.It is shown that the drag on each of the two spheres decreases as the distance between them increases and also that each sphere experiences a repulsive force in a direction perpendicular to the oscillating stream.

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Zusammenfassung Das asymptotische Verfahren für die Lösung der Navier-Stokesschen Gleichungen wird angewendet für die Strömung, die bei der Schwingung von zwei Kugeln senkrecht zur Verbindungslinie ihrer Mittelpunkte entsteht.Das wichtigste Ziel der Arbeit ist die Untersuchung der hydrodynamischen Beeinflussung der Kugeln durch die Kräfte, die bei dieser komplizierten dreidimensionalen Strömung entstehen.Es wurde gefunden, daß der Widerstand der Kugeln sich mit wachsendem Abstand zwischen ihnen verringert. In Richtung senkrecht zur Schwingung wirkt auf jede Kugel eine abstoßende Kraft.

     

On the whole spectrum of Timoshenko beams. Part II: further applications

The problem of free vibrations of the Timoshenko beam model has been addressed in the first part of this paper. A careful analysis of the governing equations has shown that the vibration spectrum consists of two parts, separated by a transition frequency, which, depending on the applied boundary conditions, might be itself part of the spectrum. Here, as an extension, the case of a doubly clamped beam is considered. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided: this allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. This case is a meaningful example of more general ones, where the wave-numbers equation cannot be written in a factorized form and hence must be solved by general root-finding methods for nonlinear transcendental equations. These theoretical results can be used as further benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. finite element models.     

On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings

The purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of [H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150-159; B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957-961; P.L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Paris, Ser. A 284 (1977), 1357-1359; S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287-292; Z.H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003) 351-358; R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486-491; H.K. Xu, M.G. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optimiz. 22 (2001) 767-773; Y.Y. Zhou, S.S. Chang, Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optimiz. 23 (2002) 911-921].     

A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain, II

In this paper we analyze a new dual mixed formulation of the elastodynamic system in polygonal domains by using an implicit scheme for the time discretization. After the analysis of stability of the fully discrete scheme, L^∞ in time, L² in space a priori error estimates for the approximation of the displacement, the strain, the pressure and the rotational are derived. Numerical tests are presented which confirm our theoretical results.     

On the bifurcation of large amplitude solutions for a system of wave and beam equations

We consider the existence of solutions for linearly coupled system of wave and beam equation with a sublinear perturbation. The main concept is the matrix spectrum which is a natural extension of the usual spectrum. Using the standard ‘change of degree’ argument we shall find necessary and sufficient conditions for the existence of asymptotic bifurcation with respect to the matrix spectrum both in one parameter and four parameter cases. Obviously the results can be modified for more general systems of partial differential equations.