The Theory of Constructive Signal Analysis

Let h(x) = e^?αxk(x), where and λ0=0. The closure theorem, V_h = L¹(?), is proved for various α and k (V_h is the L¹-closed variety generated by h). The Tauberian condition, |?| > 0, is not used, since generally this condition is difficult to compute directly. The functions h arise naturally in time series and analytic number theory. The technique of proof is constructive and depends on the semigroup {γ_j} generated by {λ_j}. The semigroup theory which consolidates and completes the results herein will be developed separately as “A closure problem for signals in semigroup invariant systems.”     

Construction of Subsets of Finite Positive Hausdorff Measure on the Hyperspace (𝔉(X), ρH)

Subsets of finite positive Hausdorff measure will be constructed for all Hausdorff functions h_α(t)=t^α, α>0 a real number.     

Die Konstruktion der mittleren Linie bei Oberflächenprofilen und einige Zusammenhänge zwischen den zur Kennzeichnung von Oberflächen benutzten Grössen

Summary Attention is directed first to the analogy between the problem of evaluating collectives and that of finding a numerical expression for an existing surface.For the valueh _RMS /h _AVE , approximations are made using Fourier series of the surface profil based on the center line.h _AVE =center line average (=h _CLA ). Forh _AVE , an upper bound calculated from the Fourier coefficients is stated. With this, and ash _RMS may be determined in a definite form, a lower bound forh _RMS /h _AVE is obtained. Provided that the surface profil is the superimposition of two harmonics, a lower, bound forh _AVE and then an upper bound forh _RMS /h _AVE could be found. In order to get a base line for which the areas embraced by the profile above and below the line are equal and a minimum, a graphical method is constructed.

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Mitteilung aus der Physikalisch-Technischen Bundesanstalt.     

On covered prime elements and complete homomorphisms of frames

Abstract

It is a standard fact that the right adjoints h_* of general frame homomor- phisms h : L → M send prime elements to prime elements. Unlike this, the preservation of covered prime elements by complete frame homomorphism is a special fact. Indeed, if L is totally ordered this happens (for arbitrary M) if and only if L is well-ordered. On the other hand, if M is both a frame and a co-frame there is no restriction on L.

This work was motivated by the fact that, in an earlier paper, we erroneously claimed h_* preserves covered primeness for any complete frame homomorphism h.     

On the Slowly Well Orderedness of εo

For α < ε₀, N_α denotes the number of occurrences of ω in the Cantor normal form of α with the base ω. For a binary number-theoretic function f let B(K; f) denote the length n of the longest descending chain (α₀, …, α_n–1) of ordinals <ε₀ such that for all i < n, Nα_i ≤ f (K, i). Simpson [2] called ε₀ as slowly well ordered when B (K; f) is totally defined for f (K; i) = K · (i+ 1). Let |n| denote the binary length of the natural number n, and |n|_k the k-times iterate of the logarithmic function |n|. For a unary function h let L(K; h) denote the function B (K; h₀(K; i)) with h₀(K, i) = K + |i| · |i|_h(i). In this note we show, inspired from Weiermann [4], that, under a reasonable condition on h, the functionL (K; h) is primitive recursive in the inverse h^–1 and vice versa.     

Thermodynamic Limits for Hamiltonians Defined as Pseudodifferential Operators

Abstract

If P(h) is a h-pseudodifferential operator in R ⁿ associated to an holomorphic semi-bounded symbol in some neighborhood of the real phase space, with bounded derivatives, we describe the symbol of e ^?tP(h), by inequalities where the constants depend on the bounds for the derivatives of the symbol of P(h), but not on the dimension n. Some applications to thermodynamic limits (free energy) are given.     

Projective supercharacter theory

Abstract

The idea of supercharacters for ordinary characters of a finite group G was introduced by Diaconis and Isaacs and further extended to Brauer characters by Chen and Lewis. The twin concepts of supercharacters and superclasses are further extended here to α-characters of G for α a complex-valued 2-cocycle of G. An α-quasi-supercharacter theory of G arises when the set of α-quasi-supercharacters of G are compatible with the set of α-regular quasi-superclasses of G. The structure of solvable groups that have exactly two α-quasi-supercharacter theories is determined.

Communicated by Mark L. Lewis     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

New Extensions of Popoviciu’s Inequality

Popoviciu’s inequality is extended to the framework of h-convexity and also to convexity with respect to a pair of quasi-arithmetic means. Several applications are included.

     

A generalized h-fractional Gronwall's inequality and its applications for nonlinear h-fractional difference systems with ‘maxima’

ABSTRACT

This paper gives a generalized h-fractional Gronwall's inequality. Applying this result, we prove the uniqueness and give bounds on solutions for a nonlinear h-fractional difference system with ‘maxima’. Finally, we give an example to illustrate one of our main results.     

Divisibility of multilinear mappings with applications to affine differential geometry

For real finite-dimensional vector spaces V, W we call a bilinear symmetric mapping h?:?V?×?V?→?W non-degenerate if the components of h with respect to a certain basis are linearly independent and non-degenerate. We say that a symmetric trilinear mapping C?:?V?×?V?×?V?→?W is divisible by h if there exists a linear form α such that C(v,?v,?v)?=?α(v)h(v,?v) for every v?∈?V. In affine differential geometry of affine immersions h is the second fundamental form and C – the cubic form of the immersion. The immersion has pointwise planar normal sections if h(v,?v)?∧?C(v,?v,?v)?=?0 for every tangent vector v. We prove that it implies that C is divisible by h if h is non-degenerate and the codimension is greater than two. For immersions with Wiehe's or Sasaki's choice of transversal bundles divisibility of C by h implies vanishing of C.     

The General Solution of the Eisenhart Equation and Projective Motions of Pseudo-Riemannian Manifolds

The solution of the Eisenhart equation for pseudo-Riemannian manifolds (Mⁿ,g) of arbitrary signature and any dimension is obtained. Thereby, pseudo-Riemannian h-spaces (i.e., spaces admitting nontrivial solutions h ≠ cg of the Eisenhart equation) of all possible types determined by the Segrè characteristic χ of the bilinear form h are found. Necessary and sufficient conditions for the existence of an infinitesimal projective transformation in (Mⁿ,g) are given. The curvature 2-form of a (rigid) h-space of type χ = {r₁, …, r_k} is calculated and necessary and sufficient conditions for this space to have constant curvature are obtained.

     

On the first eigenfunction of the fixed membrane: Some extensions of results of Payne and Stakgold

Zusammenfassung Mit Hilfe der zwei Maximumprinzipien von E. Hopf [3], [4] werden Schranken für grad² w als Funktion des Quotientenw=u/h konstruiert. Dabei istu die erste Eigenfunktion einer am Rande eingespannten schwingenden MembranD, undh ist eine positive Lösung der Helmholz'schen Gleichung inD. Der Spezialfallh 1 führt auf die Resultate von L. E. Payne und I. Stakgold [7] zurück.

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Résumé A l'aide des deux principes de maximum de E. Hopf [3], [4], on obtient des bornes pour grad² w en fonction du quotientw=u/h, oùu est la première fonction propre d'une membrane vibranteD fixée sur son contour, et oùh est une solution positive de l'équation de Helmholz dansD. Dans le cas particulierh 1, on retrouve les résultats de L. E. Payne et I. Stakgold [7].

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This research was supported by the Swiss Nationalfonds.     

Product (α1, α2)-modulation spaces

We study fundamental properties of product (α₁, α₂)-modulation spaces built by (α₁, α₂)-coverings of ℝⁿ¹ × ℝⁿ². Precisely we prove embedding theorems between these spaces with different parameters and other classical spaces. Furthermore, we specify their duals. The characterization of product modulation spaces via the short time Fourier transform is also obtained. Families of tight frames are constructed and discrete representations in terms of corresponding sequence spaces are derived. Fourier multipliers are studied and as applications we extract lifting properties and the identification of our spaces with (fractional) Sobolev spaces with mixed smoothness.

     

Internal gravity–capillary solitary waves in finite depth

We consider a two‐dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a time‐like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)‐plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian‐Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0²‐resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian‐Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0²‐resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.     

h-Divisible Modules

Abstract

We investigate classes of h-divisible modules over domains and a class of domains over which every module has a divisible envelope.     

On h-convexity

We introduce a class of h-convex functions which generalize convex, s-convex, Godunova-Levin functions and P-functions. Namely, the h-convex function is defined as a non-negative function which satisfies f(αx+(1−α)y)?h(α)f(x)+h(1−α)f(y), where h is a non-negative function, α∈(0,1) and x,y∈J. Some properties of h-convex functions are discussed. Also, the Schur-type inequality is given.     

On Generators and Representations of the Sporadic Simple Groups

In this paper we determine the irreducible projective representations of sporadic simple groups over an arbitrary algebraically closed field F, whose image contains an almost cyclic matrix of prime-power order. A matrix M is called cyclic if its characteristic and minimum polynomials coincide, and we call M almost cyclic if, for a suitable α ∈F, M is similar to diag(α·Id _h , M ₁), where M ₁ is cyclic and 0 ≤ h ≤ n. The paper also contains results on the generation of sporadic simple groups by minimal sets of conjugate elements.     

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

In this paper, we study two compact finite difference schemes for the Schrödinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H¹ error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ² + h⁴) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

     

A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions

We study the problem of parameter estimation for Ornstein–Uhlenbeck processes driven by symmetric α-stable motions, based on discrete observations. A least squares estimator is obtained by minimizing a contrast function based on the integral form of the process. Let h be the length of time interval between two consecutive observations. For both the case of fixed h and that of h → 0, consistencies and asymptotic distributions of the estimator are derived. Moreover, for both of the cases of h, the estimator has a higher order of convergence for the Ornstein–Uhlenbeck process driven by non-Gaussian α-stable motions (0 < α < 2) than for the process driven by the classical Gaussian case (α = 2).