Measure-Preservation and the Existence of a Root of <Emphasis Type="Italic">p</Emphasis>-Adic 1-Lipschitz Functions in Mahler’s Expansion

In this paper, we provide necessary and sufficient conditions for 1-Lipschitz functions that are uniformly differentiable mod p on ?_p to be measure-preserving, in Mahler’s expansion, and show that these conditions can be modified to guarantee the existence of a root of p-adic Lipschitz functions.     

Approximation of almost Sahoo–Riedel’s points by Sahoo–Riedel’s points

In this paper, we present an extention of Hyers–Ulam stability of Sahoo–Riedel’s points for real-valued differentiable functions on [a, b] and then we obtain stability results of Flett’s points for functions in the class of continuously differentiable functions on [a, b] with f′(a) = f′(b).     

Local reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(? ₁, ..., ? _N ) generated by finitely many compactly supported functions ? ₁, ..., ? _N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(? ₁, ..., ? _N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(?) generated by a compactly supported refinable function ?, we prove that for almost all \((x_0, x_1)\in [0,1]^2\), any signal in V(?) can be locally reconstructed from its samples from \(\{x_0, x_1\}+{\mathbb Z}\) with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(?) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.     

Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

For the self-adjoint Schrödinger operator ? defined on ? by the differential operation ?(d/dx)² + q(x) with a distribution potential q(x) uniformly locally belonging to the space W ₂ ^?1, we describe classes of functions whose spectral expansions corresponding to the operator ? absolutely and uniformly converge on the entire line ?. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.     

Note on the locally product and almost locally product structures

This paper is concerned with a study of some of the properties of locally product and almost locally product structures on a differentiable manifold X _n of class C ^k . Every locally product space has certain almost locally product structures which transform the local tangent space to X _n at an arbitrary point P in a set fashion: this is studied in Theorem (2.2). Theorem (2.3) considers the nature of transformations that exist between two co-ordinate systems at a point whenever an almost locally product structure has the same local representation in each of these co-ordinate systems. A necessary and sufficient condition for X _n to be a locally product manifold is obtained in terms of the pseudo-group of co-ordinate transformations on X _n and the subpseudo-groups [cf., Theoren (2.1)]. Section 3 is entirely devoted to the study of integrable almost locally product structures.     

Classification of smooth structures on a homotopy complex projective space

We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective n-space \(\mathbb {C}\textbf {P}^{n}\), where n=3 and 4. Let M²ⁿ be a closed smooth 2n-manifold homotopy equivalent to \(\mathbb {C}\textbf {P}^{n}\). We show that, up to diffeomorphism, M⁶ has a unique differentiable structure and M⁸ has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover N²ⁿ of \(\mathbb {C}\textbf {P}^{n}\) for n=4,7 or 8 and six distinct differentiable structures on N¹⁰.     

A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field

We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function f is analytic on a compact interval [a, b] ? ?, f and its analytic continuation f?_∞ satisfy the same properties that can be expressed in the language of real closed ordered fields. If f is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.     

Complex convexity and monotonicity in quasi-Banach lattices

In this paper we study the monotonicity and convexity properties in quasi-Banach lattices. We establish relationship between uniform monotonicity, uniform ?-convexity, H-and PL-convexity. We show that if the quasi-Banach lattice E has α-convexity constant one for some 0 < α < ∞, then the following are equivalent: (i) E is uniformly PL-convex; (ii) E is uniformly monotone; and (iii) E is uniformly ?-convex. In particular, it is shown that if E has α-convexity constant one for some 0 < α < ∞ and if E is uniformly ?-convex of power type then it is uniformly H-convex of power type. The relations between concavity, convexity and monotonicity are also shown so that the Maurey-Pisier type theorem in a quasi-Banach lattice is proved.Finally we study the lifting property of uniform PL-convexity: if E is a quasi-Köthe function space with α-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Köthe-Bochner function space E(X) is uniformly PL-convex if and only if both E and X are uniformly PL-convex.     

Local approximations of differentiable functions

Spaces of differentiable functions constructed from the spaces L_p, 0 < p < ∞, and the interdependence of the differential properties of functions with their local approximations are considered.     

Nonlinear commutators for the fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian and applications

We prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p-Laplacian. This implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weakly fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded \(\frac{n}{s}\)-harmonic maps converge strongly outside at most finitely many points.     

Inequalities between best polynomial approximations and some smoothness characteristics in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> and widths of classes of functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ_k(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L₂. On the basis of this, for differentiable functions in the classes L₂^r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω_k. For classes of functions defined by their smoothness characteristics Λ_k(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.     

Pseudo-amenability of certain semigroup algebras

In this paper, we investigate the pseudo-amenability of semigroup algebra ? ¹(S), where S is an inverse semigroup with uniformly locally finite idempotent set. In particular, we show that for a Brandt semigroup \(S={\mathcal{M}}^{0}(G,I)\), the pseudo-amenability of ? ¹(S) is equivalent to the amenability of G.     

Regular variation of a random length sequence of random variables and application to risk assessment

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence C(N) = (C ₁,…,C _N ) of random length N, where C(N) comes from the product of a matrix A(N) of random size N × N and a random sequence X(N) of random length N. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the extremal behavior of ∥C(N)∥, for some norm ∥?∥. We propose a generalization of Breiman’s Lemma that gives way to a tail estimate of ∥C(N)∥ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our main result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.     

A characteristic property of the algebra <Emphasis Type="Italic">C</Emphasis>(Ω)<Subscript><Emphasis Type="Italic">ß</Emphasis></Subscript>

We study some properties of the algebras of continuous functions on a locally compact space whose topology is defined by the family of all multiplication operators (β-uniform algebras). We introduce the notion of a β-amenable algebra and show that a β-uniform algebra is β-amenable if and only if it coincides with the algebra of bounded functions on a locally compact space (an analog of M. V. She?nberg’s theorem for uniform algebras).     

Locally Unmixed Modules and Linearly Equivalent Ideal Topologies

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated R-module. The purpose of this paper is to show that N is locally unmixed if and only if, for any N-proper ideal I of R generated by ht _N I elements, the topology defined by (I N)⁽ⁿ⁾, n ≥ 0, is linearly equivalent to the I-adic topology.     

Homogeneous Ricci almost solitons

It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or locally isometric to a product R×N(c), where N(c) is a space of constant curvature.     

The absolute convergence of the Fourier-Haar series for two-dimensional functions

It is well-known that if an one-dimensional function is continuously differentiable on [0, 1], then its Fourier-Haar series converges absolutely. On the other hand, if a function of two variables has continuous partial derivatives f _x ^′ and f _y ^′ on T ², then its Fourier series does not necessarily absolutely converge with respect to a multiple Haar system (see [1]). In this paper we state sufficient conditions for the absolute convergence of the Fourier-Haar series for two-dimensional continuously differentiable functions.     

On the equiconvergence rate with the Fourier integral of the spectral expansion associated with the self-adjoint extension of the Sturm-Liouville operator with uniformly locally integrable potential

In the space L ₂(?), we consider the self-adjoint extension \(\mathcal{L}\) of the Sturm-Liouville operator ly = ?y″ + q(x)y whose potential q is uniformly locally integrable on ?, i.e., satisfies the condition

$\omega _q (h) = \mathop {\sup }\limits_{x \in \mathbb{R}} \int\limits_x^{x + h} {\left| {q(t)} \right|dt < + \infty ,h > 0.} $

. We study the problem on the equiconvergence rate of the spectral expansion associated with \(\mathcal{L}\) of a function f ∈ L ₁(?) with the Fourier integral on the entire real line. We obtain uniform estimates of the equiconvergence rate under some additional conditions on f or q.

     

Polynomial approximations of a class of stochastic multiscale elasticity problems

We consider a class of elasticity equations in \({\mathbb{R}^d}\) whose elastic moduli depend on n separated microscopic scales. The moduli are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger–Reissner mixed problem that allows for computing the stress directly and the multiscale mixed problem with a penalty term for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters which represent the probabilistic law of the stochastic equations. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information. The solutions of these multiscale homogenized problems are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of N gpc modes. Assuming summability properties for the coefficients of the elastic moduli’s expansion, we deduce bounds and summability properties for the solutions’ gpc expansion coefficients. These bounds imply explicit rates of convergence in terms of N when the gpc modes used for the Galerkin approximation are chosen to correspond to the best N terms in the gpc expansion. For the mixed problem with a penalty term for nearly incompressible materials, we show that the rate of convergence for the best N term approximation is independent of the Lamé constants’ ratio when it goes to \({\infty}\). Correctors for the homogenization problem are deduced. From these we establish correctors for the solutions of the parametric multiscale problems in terms of the semidiscrete Galerkin approximations. For two-scale problems, an explicit homogenization error which is uniform with respect to the parameters is deduced. Together with the best N term approximation error, it provides an explicit convergence rate for the correctors of the parametric multiscale problems. For nearly incompressible materials, we obtain a homogenization error that is independent of the ratio of the Lamé constants, so that the error for the corrector is also independent of this ratio.