On the instability of resonances in parallel-plane waveguides

It has been observed¹³ that the propagation of acoustic waves in the region Ω₀= ?² × (0, 1), which are generated by a time-harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown⁹ in the case of Dirichlet's boundary condition U = 0 on ?Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω₀. By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω₀ ?B, where B is a smooth bounded domain with B??Ω₀. In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n · x ′ ? 0 on ? B, where x ′ = (x₁, x₂, 0) and n is the normal unit vector pointing into the interior B of ? B. In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω₀ will be discussed in a subsequent paper.     

Resonance phenomena in infinite strings with periodic ends

We study the propagation of linear waves, generated by a compactly supported time-harmonic force distribution, in an infinite string under the assumption that the material properties are p₁-periodic for x > a and p₂-periodic for x < ? a. As has been pointed out in two preceding papers devoted to related configurations ([4], [5]), the combination of a time-periodic force and a periodic spatial structure may lead to resonance phenomena. We show that the present configuration also permits resonances of orders t and t^1/2 for a discrete set of frequencies. The occurrence of resonances is closely related to the presence of non-trivial solutions of the corresponding time-independent homogeneous problem which satisfy certain asymptotic properties (‘standing waves’).     

On the instability of resonances in parallel-plane waveguides under local perturbations of the boundary

We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω₀ ? ?² × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω₀. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω₀ at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ²U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation.     

Dynamical Systems Induced by Continuous Time Difference Equations and Long-time Behavior of Solutions

The paper considers the continuous time difference equation x(t + 1) = f(x(t)), t?R ⁺ (?) with f being a continuous interval map. Although the long-time behavior of continuous solutions of Eq. (?) has been extensively described under certain added conditions, there are a number of relatively simple and yet pivotal results for Eq. (?) with no restrictions on f , that were not published. The paper is to compensate for this gap. Herein, properties of the solutions are derived from that of the ω -limit sets of trajectories of the dynamical system induced by Eq. (?) . In particular, if the ω -limit set corresponding a solution x(t) is a cycle or the closure of an almost periodic trajectory, x(t) tends (in the Hausdorff metric for graphs) to a certain upper semicontinuous function.     

The identification problem for the exponential Radon transform

The exponential Radon transform, which arises in single photon emission computed tomography, is defined by ? ?(μ:ω,s) = ∫_R?(sω + tomega;^?) e^μt dt?. Here ? is a compactly supported distribution in the plane which represents the location and intensity of a radio-pharmaceutical in a body of constant, but unknown, attenuation μ, and ω is a direction. The identification problem is to determine the attenuation μ from the data ?? with ? unknown. We will show that μ can be determined from the data if and only if ? is not a radial distribution and give formulae for computing μ when ? is not radial.     

A GAME-THEORETIC PROOF OF ANALYTIC RAMSEY THEOREM

We give a simple game-theoretic proof of Silver's theorem that every analytic set is Ramsey. A set P of subsets of ω is called Ramsey if there exists an infinite set H such that either all infinite subsets of H are in P or all out of P. Our proof clarifies a strong connection between the Ramsey property of partitions and the determinacy of infinite games.     

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 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.     

Resonance phenomena for water waves in channels of arbitrary cross‐section

This article studies the evolutionary problem for linear gravity waves on the surface of water in a uniform, symmetric channel which is excited by an antisymmetric pressure force of frequency ω at the free surface. It is shown that there is a countably infinite set of frequencies {ω₀, ω₁, …} which give rise to resonance phenomena: the amplitude of the wave motion grows like t^1/2 as t→∞ in a sense which is precisely specified. Under pressure forcing at any other frequency the solution obeys the principle of limiting amplitude. These results are obtained by combining methods developed for problems in acoustic waveguides with regularity theory for elliptic boundary‐value problems in non‐smooth domains. Copyright © 1999 John Wiley & Sons, Ltd.     

STABILITY AND ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

Itiswellknownthattheexistenceofalmostperiodicsolutionsiscloselyrelatedtothestabilityofsolutions.Forfunctionaldifferentialequationswithinfinitedelay,Y.Hin.[5'6]studiedtheproblemsontheexistenceofalmostperiodicsolutionsandthestability.However,therearefewpapersll2]dealingwithneutralfunctionaldifferentialequationswithinfinitedelay.Inthepresentpaper,forneutralfunctionaldifferentialequationswithinfinitedelay,weprovetheinherencetheoremfortheuniformlystableoperatorD(t),definethestabilitywithrespecttot…     

Existence and modulation of traveling waves in particles chains

We consider an infinite particle chain whose dynamics are governed by the following system of differential equations: where q_n(t) is the displacement of the n^th particle at time t along the chain axis and denotes differentiation with respect to time. We assume that all particles have unit mass and that the interaction potential V between adjacent particles is a convex C∞ function. For this system, we prove the existence of C∞, time‐periodic, traveling‐wave solutions of the form q_n(t) = q(wt kn + where q is a periodic function q(z) = q(z+1) (the period is normalized to equal 1), ω and k are, respectively, the frequency and the wave number, is the mean particle spacing, and can be chosen to be an arbitrary parameter. We present two proofs, one based on a variational principle and the other on topological methods, in particular degree theory. For small‐amplitude waves, based on perturbation techniques, we describe the form of the traveling waves, and we derive the weakly nonlinear dispersion relation. For the fully nonlinear case, when the amplitude of the waves is high, we use numerical methods to compute the traveling‐wave solution and the non‐linear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modulation equations for the wave parameters α, β, k, and ω. © 1999 John Wiley & Sons, Inc.     

Bootstrapping Autoregressive and Moving Average Parameter Estimates of Infinite Order Vector Autoregressive Processes

We consider anr-dimensional multivariate time series {y_t, tZ} which is generated by an infinite order vector autoregressive process. We show that a bootstrap procedure which works by generating time series replicates via an estimated finitek-order vector autoregressive process (k→∞ at an appropriate rate with the sample size) gives asymptotically valid approximations to the joint distribution of the growing set of estimated autoregressive coefficients and to the corresponding set of estimated moving average coefficients (impuls responses).     

Periodic forcing of solutions of a boundary value problem for a second-order differential equation in Hilbert space

We show that if u is a bounded solution on $\text{R}$⁺ of u″(t) ?Au(t) + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈L_loc²($\text{R}$⁺;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t) ? ω(t)   0 and u′(t) ? ω′(t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in $\text{D(A)}$ and that a smoothing effect takes place.     

On instability under small periodic perturbation of two-dimensional hyperbolic systems

Under consideration is a mixed problem for a strictly hyperbolic linear first-order system in the half-strip Π = {(x, t): 0 < x < 1, t > 0} generating a group of unitary operators. In the case of a periodic perturbation a method is proposed for finding the frequencies for which the perturbed system develops parametric resonance. The method is illustrated with a system of two equations.     

Propagation of round-off error in a model of quadratic rational rotations

We investigate the propagation of round-off error for a discrete map modeling a one-dimensional linear oscillator viewed stroboscopically in phase space, with uniform, non-dissipative round-off. The probability P(r,t) of a net displacement r during t time steps can be reduced, essentially, to a weighted sum over contributions from a small number of infinite scaling sequences of periodic orbits. We show that the successive members of each scaling sequence can be built up by application of a set of substitution rules. This implies recursion relations, not only for the geometry of the orbits, but also for P(r,t) and its moments, allowing these quantities to be calculated exactly as algebraic numbers. For asymptotically large t, the moments have power-law increase, modulated by log-periodic or (in one particularly interesting case) log-quasi-periodic oscillations.     

SOME RAMSEY THEORY IN BOOLEAN ALGEBRA FOR COMPLEXITY CLASSES

It is known that for two given countable sets of unary relations A and B on ω there exists an infinite set H ? ω on which A and B are the same. This result can be used to generate counterexamples in expressibility theory. We examine the sharpness of this result.     

Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure

We consider propagation of waves through a spatio-temporal doubly periodic material structure with rectangular microgeometry in one spatial dimension and time. Both spatial and temporal periods in this dynamic material are assumed to be of the same order of magnitude. A “double Floquet” solution is obtained in the special case when the wave equation t(ρut)−z(kuz)=0 allows for the separation of variables. We also consider a checkerboard microgeometry where variables cannot be separated. The squares in a space-time checkerboard are assumed to be filled with materials having equal impedance but different phase speeds. Within certain parameter ranges, we observe numerically the formation of distinct and stable limiting characteristic paths (“limit cycles”) that attract neighbouring characteristics after a few time periods. The average speed of propagation along the limit cycles remains the same throughout certain ranges of parameters of the microgeometry (the “plateau effect”). We formulate, as a hypothesis, the statement saying that a checkerboard structure is on a plateau if and only if it yields stable limit cycles. A dynamic material is a thermodynamically open system, as it is involved in a permanent exchange of energy and momentum with the environment. Material assemblages that produce the limit cycles are special in this aspect. Specifically, to make a wave travel through such an assemblage, we find analytically that an external agent may need to supply infinite energy and this may be so regardless of the wave frequency. For spatio-temporal laminates, however, an accumulation of energy (parametric resonance) may emerge only for frequencies that are not too low relative to some characteristic frequency of the system.     

A recursive construction of t‐wise uniform permutations

We present a recursive construction of a (2t + 1)‐wise uniform set of permutations on 2n objects using a combinatorial design, a t‐wise uniform set of permutations on n objects and a (2t + 1)‐wise uniform set of permutations on n objects. Using the complete design in this procedure gives a t‐wise uniform set of permutations on n objects whose size is at most t²ⁿ, the first non‐trivial construction of an infinite family of t‐wise uniform sets for . If a non‐trivial design with suitable parameters is found, it will imply a corresponding improvement in the construction. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 531–540, 2015     

Eine einfache Verallgemeinerung der Rayleigh-Grenzschicht

Summary A simple generalization of the theory of the compressible boundary layer near an infinite flat plate to the case with suction or blowing out is given if at the timet=0 the plate is set into motion in its own plane with velocityu _w t ⁿ. The normal velocity at the wall shall vary with time according tov _wt ^–1/2. In that case one gets similar boundary layer profiles for allt>0, which can be reduced to the profiles without suction or blowing (v _w=0) by a simple parallel displacement and stretching of the coordinates. As an example the Rayleigh boundary layer (n=0,u _w=const) is discussed.     

Ein resonanzphänomen in der theorie akustischer und elektromagnetischer wellen

We discuss the asymptotic behaviour of acoustic and electromagnetic waves, generated by given time-harmonic exterior forces with frequency ω, in the unbounded region between the parallel planes X₃ = 0 and X₃ = 1, and show that the principle of limiting amplitude is violated if ω = πn(n = 1, 2,…). For these values of the frequency, forces with compact support can be chosen such that the amplitudes of the waves increase with a logarithmic rate as t → ∞.