On some continuity and differentiability properties of paths of Gaussian processes

The following path properties of real separable Gaussian processes ξ with parameter set an arbitrary interval are established. At every fixed point the paths of ξ are continuous, or differentiable, with probability zero or one. If ξ is measurable, then with probability one its paths have essentially the same points of continuity and differentiability. If ξ is measurable and not mean square continuous or differentiable at every point, then with probability one its paths are almost nowhere continuous or differentiable, respectively. If ξ harmonizable or if it is mean square continuous with stationary increments, then its paths are absolutely continuous with probability one if and only if ξ is mean square differentiable; also mean square differentiability of ξ implies path differentiability with probability one at every fixed point. If ξ is mean square differentiable and stationary, then on every interval with probability one its paths are either differentiable everywhere or nondifferentiable on countable dense subsets. Also a class of harmonizable processes is determined for which of the following are true: (i) with probability one paths are either continuous or unbounded on every interval, and (ii) mean square differentiability implies that with probability one on every interval paths are either differentiable everywhere or nondifferentiable on countable dense subsets.     

Local properties of some Gaussian processes

The purpose of this paper is twofold. First we obtain exact formulae for the Hausdorff dimensions of the level sets and graph, and of the image of a fixed time set, for a Gaussian process with stationary increments and monotone incremental variance. Inequalities for these dimensions have been obtained by Kahane in the case where the process is the sum of a trigonometric series with random coefficients. Secondly we obtain some precise results for the brownian motion process. We consider when an image set has positive Lebesgue measure and when the zero set has positive capacity with respect to a given kernel. We show that certain conditions, which Kahane has shown to be sufficient to ensure these properties, are also necessary.     

Approximate solution for some stochastic differential equations involving both Gaussian and Poissonian white noises

By combining the Kramers-Moyal expansion with fractional Brownian motion of order n, in a formal symbolic calculus, one can obtain an approximation for the solution of some stochastic differential equations involving both Gaussian and Poissonian white noises, in terms of rotating Gaussian white noises on the grid defined by the complex roots of the unity. Illustrative examples are outlined.     

On some strongly functions defined by α-open

In this paper is to introduce and investigate new classes of generalizations of non-continuous functions, obtain some of their properties and to hold decompositions of strong α-irresolute in topological spaces.     

Spectral properties of some problems in mechanics of strongly nonhomogeneous media

Spectral properties of the following three homogenized problems in the mechanics of strongly nonhomogeneous media are considered: the “double porosity” problem, the oscillation problem for a mixture of two viscous compressible liquids, and the oscillation of a medium consisting of an elastic frame and a viscous liquid. For all three cases results on the structure of the spectrum and on the existence of so-called “spectral gaps” are obtained.     

Spectral properties of some problems in mechanics of strongly inhomogeneous media

Spectral properties of the following three homogenized problems in mechanics of strongly inhomogeneous media are considered: the problem of “double porosity,” the problem of vibration of a mixture of two viscous compressible fluids, and the problem of vibration of a medium consisting of an elastic frame and a viscous fluid. Interesting results about the structure of spectra and the presence of so-called “spectral gaps” are obtained for each of these cases. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

On equivalent stability properties in semi-infinite optimization

This paper studies noncompact feasible sets of a semi-infinite optimization problem which are defined by finitely many equality constraints and infinitely many inequality constraints. The main result is the equivalence of the overall validity of the Extended Mangasarian Fromovitz Constraint Qualification with certain (topological) stability conditions. Furthermore, two perturbation theorems being of independent interest are presented.This work was supported by the Deutsche Forschungsgemeinschaft under grant Gu 304/1-2.     

On bent functions with some symmetric properties

This paper discusses a kind of bent functions that have some symmetric properties about some variables. Section 2 mainly discusses the bent functions symmetric about some two variables and gives the necessary and sufficient condition for these functions. Section 3 gives algebraic expressions of some bent functions.     

On non-autonomous strongly damped wave equations with a uniform attractor and some averaging

In this paper we study the existence of a uniform attractor for strongly damped wave equations with a time-dependent driving force. If the time-dependent function is translation compact, then in a certain parameter region, the uniform attractor of the system has a simple structure: it is the closure of all the values of the unique, bounded complete trajectory of the wave equation. And it attracts any bounded set exponentially. At the same time, we consider the strongly damped wave equations with rapidly oscillating external force gε(x,t)=g(x,t,t/ε) having the average g⁰(x,t) as ε→⁰⁺. We prove that the Hausdorff distance between the uniform attractor Aε of the original equation and the uniform attractor A₀ of the averaged equation is less than O(ε1/2). We mention, in particular, that the obtained results can be used to study the usual damped wave equations.     

On some properties of sets with positive measures

Summary Steinhaus (2) gave eleven theorems on distance sets of sets with positive measure. The author of this paper has shown here that all these theorems hold good if ratio sets are considered for sets with positive measure instead of their distance sets.

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Riassunto Steinhaus (2) dà vari teoremi sugli insiemi delle distanze relative agli insiemi con misura positiva. In questa Nota l'autore dimostra che tutti i sudetti teoremi valgono ancora se per gli insiemi con misura positiva si considerano, invece degli insiemi delle distanze, gli insemi dei rapporti.

     

Wavelet optimal estimations for a density with some additive noises

Using wavelet methods, Fan and Koo study optimal estimations for a density with some additive noises over a Besov ball $B_{r,q}^s(L)(r,q⩾1)$ and over $L^2$ risk (Fan and Koo, 2002 [13]). The $L^{\infty }$ risk estimations are investigated by Lounici and Nickl (2011) [19]. This paper deals with optimal estimations over $L^p(1⩽p⩽\infty )$ risk for moderately ill-posed noises. A lower bound of $L^p$ risk is firstly provided, which generalizes Fan–Koo and Lounici–Nickl's theorems; then we define a linear and non-linear wavelet estimators, motivated by Fan–Koo and Pensky–Vidakovic's work. The linear one is rate optimal for $r⩾p$, and the non-linear estimator attains suboptimal (optimal up to a logarithmic factor). These results can be considered as an extension of some theorems of Donoho et al. (1996) [10]. In addition, our non-linear wavelet estimator is adaptive to the indices s, r, q and L.     

Capacity of mismatched Gaussian channels with and without feedback

Summary Continuous time communication channels with additive noise are considered under an average power constraint. The noises are assumed to be Gaussian processes equivalent (or mutually absolutely continuous) to a Brownian motion. We study the problem whether the capacity of the channel is increased by feedback or not. It is given a sufficient condition under which the capacity is not increased by feedback. It is also given an example of a channel whose capacity is doubled by feedback.Research partially supported by ONR contract N00014-84-0212     

On some properties of base-matroids

Let M=(E,F) be a rank-n matroid on a set E and B one of its bases. A closed set θ⊆E is saturated with respect to B, or B-saturated, when |θ∩B|=r(θ), where r(θ) is the rank of θ.The collection of subsets I of E such that |I∩θ|?r(θ), for every closed B-saturated set θ, turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by M_B. In this paper we prove some properties of M_B, in particular that it satisfies the base-axiom of a matroid.Moreover, we determine a characterization of the matroids M which are isomorphic to M_B for every base B of M.Finally, we prove that the poset of the closed B-saturated sets ordered by inclusion is isomorphic to the Boolean lattice B_n.     

On a stochastic interacting model with stepping-stone noises

In this note, we investigate the existence of a weak mild solution to a stochastic interacting model (described by a system of stochastic partial differential equations) with stepping-stone noises by adopting a weak convergence argument.     

On some properties of generalized quasiisometries with unbounded characteristic

We consider a family of open discrete mappings f:D ?[`(\mathbb R<sup>n</sup>)] f:D \to \overline {{{\mathbb R}^n}} that distort, in a special way, the p-modulus of a family of curves that connect the plates of a spherical condenser in a domain D in \mathbb R<sup>n</sup> {{\mathbb R}^n} ; p > n-1; p < n; and bypass a set of positive p-capacity. We establish that this family is normal if a certain real-valued function that controls the considered distortion of the family of curves has finite mean oscillation at every point or only logarithmic singularities of order not higher than n - 1: We show that, under these conditions, an isolated singularity x <sub>0</sub> ∈ D of a mapping f:D\{ x<sub>0</sub> } ?[`(\mathbb Rⁿ)] f:D\backslash \left\{ {{x_0}} \right\} \to \overline {{{\mathbb R}^n}} is removable, and, moreover, the extended mapping is open and discrete. As applications, we obtain analogs of the known Liouville and Sokhotskii–Weierstrass theorems.