Fourth-order parabolic equations with weak BMO coefficients in Reifenberg domains

We study optimal W2,p-regularity for fourth-order parabolic equations with discontinuous coefficients in general domains. We obtain the global W2,p-regularity for each 1<p<∞ under the assumption that the coefficients have suitably small BMO semi-norm of weak type and the boundary of the domain is δ-Reifenberg flat. The situation of our main theorem arises when the conductivity on fractals is controlled by a random variable in the time direction.     

Fuzzy local ω-systems

The natural language computing today demands for the study of ω-languages. Therefore in this respect it is convenient to consider fuzzy ω-languages. In this paper, the concept of fuzzy local ω-language, Büchi fuzzy local ω-language, and some closure properties of fuzzy local ω-languages are presented. We introduce deterministic fuzzy finite automaton with different acceptance mode on fuzzy ω-languages and establish the relationship between these various classes of fuzzy ω-languages. We have defined deterministic fuzzy local automaton and also establish relationships between deterministic fuzzy local automaton, fuzzy local ω-language and Büchi fuzzy local ω-language. Further we show that every fuzzy regular ω-language is a projection of a Büchi fuzzy local ω-language.     

A partition theorem for the complete graph on ω

This paper is a proof of the following theorem:

ω^ω→(ω^ω, 3)²

p-cyclic matrices and the symmetric successive overrelaxation method

In this paper, the new functional equation, [λ?(1?ω)²]^p=λ[λ+1?ω]^p-2(2?ω)²ω^pμ^p, which connects the eigenvalues μ of a particular weakly cyclic (of index p) Jacobi matrix B to the eigenvalues λ of its associated symmetric successive overrelaxation (SSOR) matrix S_ω, is derived. This functional equation is then applied to the problem of determining bounds for the intervals of convergence and divergence of the SSOR iterative method for classes of H-matrices.     

Estimation of the algorithmic complexity of classes of computable models

We estimate the algorithmic complexity of the index set of some natural classes of computable models: finite computable models (Σ ₂ ⁰ -complete), computable models with ω-categorical theories (Δ _ω ⁰ -complex Π _ω+2 ⁰ -set), prime models (Δ _ω ⁰ -complex Π _ω+2 ⁰ -set), models with ω ₁-categorical theories (Δ _ω ⁰ -complex Σ _ω+1 ⁰ -set. We obtain a universal lower bound for the model-theoretic properties preserved by Marker’s extensions (Δ _ω ⁰ .     

Elliptic Obstacle Problems with Measurable Coefficients in Non-Smooth Domains

We establish a global weighted W ^{1, p} -regularity for solutions to variational inequalities and obstacle problems for divergence form elliptic systems with measurable coefficients in bounded non-smooth domains.     

Remarks on regularity up to the boundary for solutions to variational problems in plasticity theory

We discuss the problem of the global W ₂ ¹ -regularity for the stress tensor of a perfect elastic-plastic body at equilibrium. In particular, we construct an example showing that the method proposed by the author to establish the local W ₂ ¹ -regularity, in general, does not work in investigations of regularity up to the boundary if the given body is nonconvex. Bibliography: 3 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 227–232.     

On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L ₁-approximations of classes W ^r F, r ∈ ?, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes W ^r H ^ω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ?, k ∈ ?, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.     

On isometrically universal spaces, mappings, and actions of groups

In this paper we first consider some well-known classes of separable metric spaces which are isometrically ω-saturated (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559]) and, therefore, contain isometrically universal spaces. We put some problems concerning such spaces most of which are related with the properties of the isometrically universal Urysohn space. Furthermore, using the defined notions of isometrically universal mappings and G-spaces (which are analogies of the notion of isometrically universal spaces) we introduce the notions of an isometrically ω-saturated class of mappings and an isometrically ω-saturated class of G-spaces (in which there are “many” isometrically universal elements). We prove that all results of Sections 6.1 and 7.1 of [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559] can be reformulated for isometrically ω-saturated classes of spaces and G-spaces, respectively. In particular, we prove that if D and R are isometrically ω-saturated classes of spaces, then the class of all mappings with the domain in D and range in R is an isometrically ω-saturated class of mappings and, therefore, in this class there are isometrically universal elements. As a corollary of this result we have that since the class of all mappings is isometrically ω-saturated, in this class there are isometrically universal mappings. Similarly, if G is an arbitrary separable metric group and P is an isometrically ω-saturated class of spaces, then the class of all G-spaces (X,F), where X is an element of P, is an isometrically ω-saturated class of G-spaces and, therefore, in this class there are isometrically universal elements. In particular, for any separable metric group G, in the class of all G-spaces there are isometrically universal G-spaces. We also pose some problems concerning isometrically universal mappings and G-spaces some of which concern the Urysohn space.     

Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables

We present a hierarchy of semidefinite programming (SDP) relaxations for solving the concave cost transportation problem (CCTP), which is known to be NP-hard, with p suppliers and q demanders. In particular, we study cases in which the cost function is quadratic or square-root concave. The key idea of our relaxation methods is in the change of variables to CCTPs, and due to this, we can construct SDP relaxations whose matrix variables are of size O((min {p, q}) ^ω ) in the relaxation order ω. The sequence of optimal values of SDP relaxations converges to the global minimum of the CCTP as the relaxation order ω goes to infinity. Furthermore, the size of the matrix variables can be reduced to O((min {p, q}) ^ω-1 ), ω ≥  2 by using Reznick’s theorem. Numerical experiments were conducted to assess the performance of the relaxation methods.     

Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors AV in space V (has H²-regularity, see notation in Section 2) and AH in space H (has L²-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.     

Cofinal types of ultrafilters

We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are not Tukey above [ω₁]^<ω. We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.     

The cardinal characteristic for relative γ-sets

For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γ-set in X, i.e., there exists an ω-cover of X with no γ-subcover of Z. We give a characterization of p(ω²) and p(ωω) in terms of definable free filters on ω which is related to the pseudo-intersection number p. We show that for every uncountable standard analytic space X that either p(X)=p(ω²) or p(X)=p(ωω). We show that the following statements are each relatively consistent with ZFC: (a) p=p(ωω)<p(ω²) and (b) p<p(ωω)=p(ω²)     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

Products of Fitting Classes of Finite Groups

Fitting classes are considered. In particular, it is shown that a product of n-multiply ω-local Fitting classes is an n-multiply ω-local Fitting class.     

Criteria of solvability for multidimensional Riccati equations

We study the solvability problem for the multidimensional Riccati equation ??u=|?u|^q+ω, whereq>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation ?Δu?ωu=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions onR ⁿ in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type?Lu=f(x, u, ?u)+ω where , andL is a uniformly elliptic operator.     

The critical Fujita exponent for a diffusion equation with a potential term

We study the initial-boundary-value problem of the diffusion equation u _t = Δu ^m ? V (x)u ^m + u ^p in a conelike domain D = [1,∞) × Ω, where V (x) ~ ω ₂ |x| ^?2 with ω ₂ > 0. Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω, and let l denote the positive root of l ² + (n ? 2)l = ω ₁ + ω ₂. We prove that if m < p ≤ m + 2/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + 2/(n + l), then the problem has global solutions for some \( {u_0}\gneq 0 \) .     

Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H.