Completion problems forj pq-inner functions. I.

This paper studies various completion problems for a subclass ofj _pq-inner functions. Special attention is drawn to so-calledA-normalizedj _pq-elementary factors of full-rank, which are closely related to the matricial Schur problem. Finally, as an application an inverse problem for Carathéodory sequences is answered.     

Complete Flat Affine and Lorentzian Manifolds

Following in the tradition of Hilbert's 18th problem of classifying crystallographic groups, we provide a survey of a series of results which have culminated in the study of flat Lorentz manifolds. In particular, Milnor asked whether all complete flat affine manifolds have virtually polycyclic fundamental groups. Margulis answered this question negatively by constructing complete flat Lorentz manifolds with free fundamental groups. In this paper, we follow the effort to classify and understand these interesting counterexamples to Milnor's question, and their generalizations.     

The Cohen--Lusk Conjecture

The question of Cohen and Lusk about the partial gluing of an orbit under a map of a free Z _p-space to ⁿ is answered in part.     

Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids

Boundary-value problems describing the stationary flow of a generalized Newtonian liquid are considered. The regularity of solutions to such problems is studied near the boundary. The W ₂ ² -estimate for a solution and the partial regularity of the strain velocity tensor are established. In the two-dimensional case, the complete regularity of the strain velocity tensor is also proved. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 239–265.     

Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.     

Discrete time analytic semigroups and the geometry of Banach spaces

We study different notions of discrete maximal regularity for discrete-time abstract Cauchy problems in Banach spaces. First we look at l ₂-discrete maximal regularity and show that Hilbert spaces are the only Banach spaces, among spaces with an unconditional basis, in which the analyticity of the associated discrete-time semigroup is a sufficient condition to obtain this kind of regularity. We then turn to different notions of regularity, in a l ₁ and in a l _∞ sense. We link the existence of particular semigroups such that the associated Cauchy problem has one of these maximal regularities to the geometry of the underlying Banach space (more precisely, to the existence of a complemented subspace isomorphic to c ₀ or l ₁). Finally, we give some elements to compare these regularities.     

Linear Equations and Regularity Conditions on Semigroups

R. Croisot in 1953, stated a very interesting problem of classification of all types of the regularity of semigroups defined by equations of the form a = a^mxaⁿ, with m,n \ge 0, m + n \ge 2. He proved that any of these equations determines either the ordinary regularity, left, right or complete regularity (see also the book by A. H. Clifford and G. B. Preston, Section 4.1). A similar problem, concerning all types of the regularity of semigroups and their elements defined by equations of the form a = a^pxa^qya^r, with p,q,r \ge 0, was treated by S. Lajos and G. Szász in 1975. The purpose of this paper is to generalize the results by Croisot, Lajos and Szász considering all types of the regularity of semigroups and their elements determined by more general equations called linear. We determine all types of the regularity of elements defined by linear equations, and prove that there are exactly 14 types of the regularity of semigroups defined by such equations. We also give implication diagrams for linear equations and regularity conditions.     

Classification of multiple unilateral weighted shifts by $$\mathbb{A}_{\aleph _0 }$$

In this paper, it is characterized when a multiple unilateral weighted shift belongs to the classes \mathbbA_n ( 1 \leqslant n \leqslant à₀ )\mathbb{A}_n \left( {1 \leqslant n \leqslant \aleph _0 } \right). As a result, we perfect and generalize the previous conclusions given by H. Bercovici, C. Foias, and C. Pearcy. Moreover, we remark that Question 21 posed by Shields has been negatively answered.     

On Regular Riesz Subspaces

The paper is devoted to investigations of properties of regular Riesz subspaces and connections between regularity and some topological properties. The problem if a topological closure preserves regularity is solved in the class of discrete Riesz spaces. We also characterize Dedekind complete Riesz spaces possessing the same classes of -regular and regular Riesz subspaces Moreover, various examples of regular and non regular Riesz spaces are presented.     

A New Inequality for Weakly （K1, K2）-Quasiregular Mappings

We obtain a new inequality for weakly （K1,K2）-quasiregular mappings by using the McShane extension method. This inequality can be used to derive the self-improving regularity of （K1, K2）-Quasiregular Mappings.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

A note on the regularity criteria for the Navier-Stokes equations

We consider the regularity problem for 3D Navier-Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient ∇p/(1+|u|^δ₁+|∇u|^δ₂) for the Navier-Stokes equations is established.     

Discrete Time Analytic Semigroups and the Geometry of Banach Spaces

Abstract. We study different notions of discrete maximal regularity for discrete-time abstract Cauchy problems in Banach spaces. First we look at l ₂ -discrete maximal regularity and show that Hilbert spaces are the only Banach spaces, among spaces with an unconditional basis, in which the analyticity of the associated discrete-time semigroup is a sufficient condition to obtain this kind of regularity. We then turn to different notions of regularity, in a l ₁ and in a l _∞ sense. We link the existence of particular semigroups such that the associated Cauchy problem has one of these maximal regularities to the geometry of the underlying Banach space (more precisely, to the existence of a complemented subspace isomorphic to c ₀ or l ₁ ). Finally, we give some elements to compare these regularities.     

Optimality of mappings and some separation axioms

Optimal mappings and optimally irresolute mappings are introduced. Fundamental properties of them are obtained. Preservation of well-known separation axioms (semi-ℐ₂, ℐ₂, semi-normality, normality, s-regularity, regularity) under such types of mappings (with some additional conditions) is studied.      

Elliptic boundary-value problem in a two-sided improved scale of spaces

We study a regular elliptic boundary-value problem in a bounded domain with smooth boundary. We prove that the operator of this problem is a Fredholm one in a two-sided improved scale of functional Hilbert spaces and that it generates there a complete collection of isomorphisms. Elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces and some their modi.cations. An a priori estimate for a solution is obtained and its regularity is investigated.     

A Volume Constrained Variational Problem with Lower-Order Terms

We study a one-dimensional variational problem with two or more level set constraints. The existence of global and local minimizers turns out to be dependent on the regularity of the energy density. A complete characterization of local minimizers and the underlying energy landscape is provided. The Γ -limit when the phases exhaust the whole domain is computed.     

A Volume Constrained Variational Problem with Lower-Order Terms

We study a one-dimensional variational problem with two or more level set constraints. The existence of global and local minimizers turns out to be dependent on the regularity of the energy density. A complete characterization of local minimizers and the underlying energy landscape is provided. The Γ -limit when the phases exhaust the whole domain is computed.     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

Global attractors in stronger norms for a class of parabolic systems with nonlinear boundary conditions

For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space. The approach relies on maximal L_p-regularity with temporal weights for the linearized problem. An inherent smoothing effect due to the weights is employed for obtaining gradient estimates. In several applications we can improve the convergence to an attractor by one regularity level.     

The Discrete Planar L0-Minkowski Problem

In the discrete setting, the L₀-Minkowski problem extends the question posed and answered by the classical Minkowski's existence theorem for polytopes. In particular, the planar extension, which we address in this paper, concerns the existence of a convex polygonal body which contains the origin, whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area.