Continuity of harmonically conjugate functions in Jordan domains

We study sufficient conditions for the continuity of harmonically conjugate functions in dependence of the geometrical structure of various classes of Jordan domains. The results obtained are generalizations of the well-known Dini continuity condition for functions on the disk.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1402–1405, October, 1992.     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Continuity of the SLE trace in simply connected domains

We prove that the SLE _κ trace in any simply connected domain G is continuous (except possibly near its endpoints) if κ < 8. We also prove an SLE analog of Makarov’s Theorem about the support of harmonic measure.     

Dynamics in dumbbell domains III. Continuity of attractors

In this paper we conclude the analysis started in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597] and continued in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)] concerning the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain as the channel shrinks to a line segment. In [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)], we have analyzed the behavior of the limiting problem. In this paper we show that the attractors are upper semicontinuous and, moreover, if all equilibria of the limiting problem are hyperbolic, then they are lower semicontinuous and therefore, continuous. The continuity is obtained in Lp and H¹ norms.     

Dynamics in dumbbell domains I. Continuity of the set of equilibria

We analyze the dynamics of a reaction-diffusion equation with homogeneous Neumann boundary conditions in a dumbbell domain. We provide an appropriate functional setting to treat this problem and, as a first step, we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds.     

Complexity spaces as quantitative domains of computation

We study domain theoretic properties of complexity spaces. Although the so-called complexity space is not a domain for the usual pointwise order, we show that, however, each pointed complexity space is an ω-continuous domain for which the complexity quasi-metric induces the Scott topology, and the supremum metric induces the Lawson topology. Hence, each pointed complexity space is both a quantifiable domain in the sense of M. Schellekens and a quantitative domain in the sense of P. Waszkiewicz, via the partial metric induced by the complexity quasi-metric.     

Continuity of embeddings of weighted Sobolev spaces in Lebesgue spaces on anisotropically irregular domains

In our earlier publications, the domains satisfying the flexible σ-cone condition were classified with respect to an anisotropy parameter λ. In the present paper we establish the continuity of embeddings of weighted Sobolev spaces in Lebesgue spaces in these classes of domains. For each class of domains with parameter λ ≠ (1, ..., 1), the theorems obtained are stronger than those in the general case of domains satisfying the flexible σ-cone condition.     

Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains

We study regularity of Bergman and Szeg? projections on Sobolev type weighted-sup spaces. The paper covers the case of strongly pseudoconvex domains with C⁴ boundary and, partially, domains of finite type in the sense of D’Angelo. Received: 6 October 2005     

A quantitative discrete H2-regularity estimate for the Shortley-Weller scheme in convex domains

Summary We investigate the discreteH ²-regularity properties of the Shortley-Weller discretization of Poisson's equation (with homogeneous Dirichlet boundary condition) in bounded convex domains ². It is shown that the regularity constant is 1 independent of the mesh sizeh if theH ²-seminorm is defined in a way assigning less weight to the (unsymmetric) differences near the boundary.     

Continuity in topological groups

We prove that, when G is a group equipped with a Baire and metrizable topology, if there is a second category dense subset S of G such that the right translations ρs and ρs⁻¹ are continuous for all s∈S and each left translation λs, s∈G, is almost-continuous (defined below) on a residual subset of G, then G is a topological group. Among other consequences, this yields that when G is a second countable locally compact right topological group, its topological centre is a topological group.     

集值映射的伪(*)连续与弱(*)连续性

本文引入了集值映射的伪（＊）连续性与弱（＊）连续性概念的定义,研究了伪（＊）连续、（＊）连续及弱（＊）连续的等价关系,最后研究了乘积空间中的集值映射成为伪（＊）连续和弱（＊）连续的充要条件。     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Absolute Continuity in Periodic Waveguides

We study second order elliptic operators with periodic coefficientsin two-dimensional simply connected periodic waveguides withthe Dirichlet or Neumann boundary conditions. It is proved thatunder some mild smoothness restrictions on the coefficients,such operators have purely absolutely continuous spectra. Theproof follows a method suggested previously by A. Morame totackle periodic operators with variable coefficients in dimension2. 2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.     

On the Continuity and Absolute Continuity of Random Closed Sets

Abstract

In the case of real-valued random variables, the concept of absolute continuity is well-defined in terms of the absolute continuity of the probability law of a random variable with respect to the usual Lebesgue measure, since both are acting on the same Borel sigma algebra on the real line. Naturally, the same extends to random vectors with real components. A satisfactory and commonly accepted definition of absolute continuity of random closed sets is not available, while in various applications this would help in clarifying the kind of randomness of a random set. We introduce here a definition that is shown to be an extension of the concept related to real-valued random variables, such that also for random sets it is true that absolute continuity implies continuity. Significant examples and counter examples are presented to illustrate the role of our definition in concrete cases. The relationship between our definition and others in well-accepted literature is shown.     

Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so‐called Ma‐Trudinger‐Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover, our new condition, again combined with the strict convexity of the nonfocal domains, allows us to prove that all injectivity domains are strictly convex too. These results apply, for instance, on any small C⁴‐deformation of the 2‐sphere. © 2009 Wiley Periodicals, Inc.     

Extension by Continuity in Pointfree Topology

Bourbaki's extension theorem (General Topology, Part 1, I, 8.5) is known to be one of the most important results concerning the extension of continuous maps. This paper presents a pointfree generalization of this together with various applications to illustrate its remarkable usefulness.     

Continuity in the density topology II

The paper deals with real functions of real variable continuous in the density topology on the domain and on the range. The set on which such functions preserve upper outer density is described. Also the class of homeomorphisms continuous in the density topology is considered. At last some problems of differentiability are analysed.