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

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.     

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.     

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.     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

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 → ∞.     

On Martos' and Chaehes-Coopeb's approach vis-a-vis

In this paper a linear fractional programming problem is studied in presence of “singular-points”. It is proved that “singular points”, if present, exist at an extreme point of S: = {x ? R ⁿ | Ax = b, x ≧0}

It is also shown that a “singular point” is adjacent to an optimal point of S and a characterization of a non-basic vector is obtained, whose entry into the optimal basis in Martos' approach yields the “singular point”.     

Resonance spectrum for one-dimensional layered media

We consider the “weighted” operator P_k=????_x a(x)? _x on the real line with a step-like coefficient which appears when propagation of waves through a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of P_k. If the coefficient is periodic on a finite interval (locally periodic) with k identical cells, then the resonance spectrum of P_k has band structure. In the article, we study a transition to semi-infinite medium by taking the limit k→?∞?. The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem (k=∞) with k???1 or k resonances in each band. We prove that as k→?∞?, the resonance spectrum converges to the real axis.     

Weighted isoperimetric inequalities on ℝn and applications to rearrangements

We study isoperimetric inequalities for a certain class of probability measures on ?ⁿ together with applications to integral inequalities for weighted rearrangements. Furthermore, we compare the solution to a linear elliptic problem with the solution to some “rearranged” problem defined in the domain {x: x₁ < α (x₂, …, x_n)} with a suitable function α (x₂, …, x_n). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation

A detailed analysis is given to the solution of the cubic Schrödinger equation iq_t + q_xx + 2|q|²q = 0 under the boundary conditions as |x|→∞. The inverse-scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and “pulsates” in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [q(x + T,t) = q(x,t) as |x|→∞] is discussed.     

Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X₀, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X₀, X) of the subdiagram type whenever X₀ is nonlinear. It remains unsolved whether rigidity holds when (X₀, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X₀, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure \(\overline{\omega} : \mathscr{C}(S) \rightarrow S\) on a uniruled projective manifold \((X,\,{\cal K})\) equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve ? emanating from a general point x ∈ S, there exists an immersed neighborhood N_? of ? which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ? X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ? 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X₀ into X which induces an isomorphism on second homology groups. By studying ?*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X₀ ? X. Under the additional hypothesis that all holomorphic sections in Γ(X₀, T_x∣x₀) lift to global holomorphic vector fields on X, we prove that the admissible pair (X₀, X) is rigid. As examples we check that (X₀, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ? ?²ⁿ, n ? 3, and when X₀ ? X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

     

Poles and zeros of best rational approximants of |x|

The asymptotic distribution (forn→∞) of poles and zeros of best rational approximantsr _n ^* ∈R _nn of the function |x| on [?1, 1] as well as the asymptotic distribution of extreme points of the error function |x|?r _n ^* (x) on [?1, 1] is investigated. The precision of the asymptotic formulae corresponds to that of the strong error formula $\lim _{n \to \infty } e^{\pi \sqrt n } E_{nn} (|x|,[ - 1,1]) = 8$ , which has been proved in [St1]. Here,E _nn (|x|, [?1, 1]) denotes the minimal approximation error in the uniform norm on [?1, 1]. The accuracy of the asymptotic distribution functions is so high that the location of individual poles, zeros, and extreme points can be distinguished forn sufficiently large.     

Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation Δu = |u|^σ?1 u is considered in the exterior of a ball in ? ⁿ , _n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n?2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type.     

Monotonically Computable Real Numbers

Area number x is called k‐monotonically computable (k‐mc), for constant k > 0, if there is a computable sequence (x_n)_{n ∈ ℕ} of rational numbers which converges to x such that the convergence is k‐monotonic in the sense that k · |x — x_n| ≥ |x — x_m| for any m > n and x is monotonically computable (mc) if it is k‐mc for some k > 0. x is weakly computable if there is a computable sequence (x_s)_{s ∈ ℕ} of rational numbers converging to x such that the sum $\sum _{s \in \mathbb{N}}$|x_s — x_{s + 1}| is finite. In this paper we show that a mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers.     

Diophantine approximation generalized

In this paper we study the set of x ∈ [0, 1] for which the inequality |x − x _n | < z _n holds for infinitely many n = 1, 2, .... Here x _n ∈ [0, 1) and z _n s> 0, z _n → 0, are sequences. In the first part of the paper we summarize known results. In the second part, using the theory of distribution functions of sequences, we find the asymptotic density of n for which |x − x _n | < z _n , where x is a discontinuity point of some distribution function of x _n . Generally, we also prove, for an arbitrary sequence x _n , that there exists z _n such that the density of n = 1, 2, ..., x _n → x, is the same as the density of n = 1, 2, ..., |x − x _n | < z _n , for x ∈ [0, 1]. Finally we prove, using the longest gap d _n in the finite sequence x ₁, x ₂, ..., x _n , that if d _n ≤ z _n for all n, z _n → 0, and z _n is non-increasing, then |x − x _n | < z _n holds for infinitely many n and for almost all x ∈ [0, 1].     

Compositions of sums of absolute powers

We obtain an explicit formula for then-dimensional volumes of certain bodies, calledoddballs hereinafter. An oddball is a bodyG = {x εR ⁿ :f(x) ≤ 1}, wheref:R ⁿ →R is anoddball function. Oddball functions are defined by way of the following construction: We begin with the class of functionsf of the formf(x ₁, ...,x _k ) = |x ₁|^α + |x ₂|^β + ... + |x _k|^γ. Herek may be any positive integer, and is not fixed. The Greek exponents are arbitrary positive real numbers. We extend this class by permitting any finite number of substitutions among functions in the class. Finally, we extend the substitution-enlarged class by permitting linear formsy _i = Σ _j b _ij x _j to replacex _i 's, the transformations being nonsingular. Thus, if det(b _ij ) ≠ 0, the oddball function $$f(x_1 ,x_2 ,x_3 ,x_4 ,x_5 ,x_6 ) = ((|y_1 |^\alpha + |y_2 |^\beta )^\tau + (|y_3 |^\gamma + |y_4 |^\phi + |y_5 |^\psi )^\delta )^\mu + |y_6 |^\eta $$ is a fairly typical example. We also consider the number of lattice points in certain types of oddballs, as well as their latticepacking densities. Neither do oddballs include thesuperballs discussed elsewhere by this and other authors, nor is every oddball a superball.     

Asymptotic Solutions of Hamilton–Jacobi Equations with Semi-Periodic Hamiltonians

We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations in ? ⁿ . We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem “converges” to an asymptotic solution for any lower semi-almost periodic initial function.     

Modulated waves in a semiclassical continuum limit of an integrable NLS chain

A one‐dimensional integrable lattice system of ODEs for complex functions Q_n(τ) that exhibits dispersive phenomena in the phase is studied. We consider wave solutions of the local form Q_n(τ) ∼ qexp(i(kn + ωτ + c)), in which q, k, and ω modulate on long time and long space scales t = ετ and x = εn. Such solutions arise from initial data of the form Q_n(0) = q(nε) exp(iϕ(nε)/ε), the phase derivative ϕ′ 0 giving the local value of the phase difference k. Formal asymptotic analysis as ε → 0 yields a first‐order system of PDEs for q and ϕ′ as functions of x and t. A certain finite subchain of the discrete system is solvable by an inverse spectral transform. We propose formulae for the asymptotic spectral data and use them to study the limiting behavior of the solution in the case of initial data |Q_n| < 1, which yield hyperbolic PDEs in the formal limit. We show that the hyperbolic case is amenable to Lax‐Levermore theory. The associated maximization problem in the spectral domain is solved by means of a scalar Riemann‐Hilbert problem for a special class of data for all times before breaking of the formal PDEs. Under certain assumptions on asymptotic behaviors, the phase and amplitude modulation of the discrete systems is shown to be governed by the formal PDEs. Modulation equations after breaking time are not studied. Full details of the WKB theory and numerical results are left to a future exposition. © 2000 John Wiley & Sons, Inc.     

Resonance phenomena in a semi-infinite string with a periodic end

We study the propagation of linear waves, generated by a compactly supported time-harmonic force distribution, in a semi-infinite string under the assumption that the material properties depend p-period-ically on the space variable outside a sufficiently large interval [0, a]. The spectrum of the self-adjoint extension A of the spatial part of the differential operator consists of a finite or countable number of bands and a (possibly empty) discrete set of eigenvalues located in the gaps of the continuous spectrum. We show that resonances of order t or t^½, respectively, occur if either ω² is an eigenvalue of A or (i) ω² is a boundary point of the continuous spectrum of A and (ii) the corresponding time-independent homogeneous problem has a non-trivial solution which is p-periodic or p-semiperiodic for x > a (‘standing wave’).