Weak Continuity Properties Research Articles

Weak and strong convergence of a modified adaptive generalized Popov's algorithm for solving variational inequality problems

In this paper, we introduce a modified adaptive step-size algorithm by basing a generalized Popov's method to solve variational inequality problems in a real Hilbert space. The algorithm only needs to act on the operator once time in each iteration, which greatly reduces the calculation time and calculation difficulty. Of particular interesting, by using our new same generalized Popov's method, we also weaken the conditions of the operator about weak continuity and quasi-monotone property. Of particular interesting, by using our new same generalized Popov's method, we can obtain a weak convergence result under the condition that the operator A is quasi-monotone and a R-linear convergence result under the condition that the operator A is strongly pseudo-monotone in the variational inequality, respectively. Finally, the feasibility and effectiveness of the algorithm are verified by some numerical experiments.

Low-order divergence-free approximations for the Stokes problem on Worsey–Farin and Powell–Sabin splits

We derive low-order, inf–sup stable and divergence-free finite element approximations for the Stokes problem using Worsey–Farin splits in three dimensions and Powell–Sabin splits in two dimensions. The velocity space simply consists of continuous, piecewise linear polynomials, whereas the pressure space is a subspace of piecewise constants with weak continuity properties at singular edges (3D) and singular vertices (2D). We discuss implementation aspects that arise when coding the pressure space, and in particular, show that the pressure constraints can be enforced at an algebraic level.

Finite-Memory Strategies in POMDPs with Long-Run Average Objectives

Partially observable Markov decision processes (POMDPs) are standard models for dynamic systems with probabilistic and nondeterministic behaviour in uncertain environments. We prove that in POMDPs with long-run average objective, the decision maker has approximately optimal strategies with finite memory. This implies notably that approximating the long-run value is recursively enumerable, as well as a weak continuity property of the value with respect to the transition function.

The problem of multiple commons: A market design approach

There are several locations, each of which is endowed with a resource that is specific to that location. Each agent will go to a single location and harvest some of the resource there. Several agents may go to each location, and thereby cause a congestion effect that is modeled in a general way. We assign harvesting rights based on preferences alone, though we later extend the model to accommodate private endowments of money. We find the best allocation rule in the class of rules that are strategy-proof, anonymous, and that satisfy a weak continuity property. For a special class of preferences, we also find an ascending mechanism, similar to an auction, that implements the rule. The rule coincides with a special pseudo-market equilibrium, wherein agents buy their desired resource with tokens distributed by the social planner.

Some Properties of the Derivatives on Sierpinski Gasket Type Fractals

In this paper, we focus on Strichartz’s derivatives, a family of derivatives including the normal derivative, on post critically finite fractals, which are defined at vertices in the graphs that approximate the fractal. We obtain a weak continuity property of the derivatives for functions in the domain of the Laplacian. For a function with zero normal derivative at any fixed vertex, the derivatives, including the normal derivatives, of the neighboring vertices will decay to zero. The rates of approximations are described, and several nontrivial examples are provided to illustrate that our estimates are optimal. We also study the boundedness property of derivatives for functions in the domain of the Laplacian. A necessary condition for a function having a weak tangent of order one at a vertex is provided. Furthermore, we give a counterexample of a conjecture of Strichartz on the existence of higher-order weak tangents.

Currents carried by the graphs of semi-monotone maps

In this paper we study the structure, weak continuity and approximability properties for the integer multiplicity rectifiable currents carried by the graphs of maximal semi-monotone set-valued maps on an n-dimensional convex domain. Especially, we give an enhanced version of approximation theorem for the subgradients of semi-convex functions.

The Problem of Multiple Commons: A Market Design Approach

There are several locations, each of which is endowed with a resource that is specific to that location. Examples include coastal fisheries, oil fields, etc. Each agent will go to a single location and harvest some of the resource there. Several agents may go to each location. We assign harvesting rights based on preferences alone, though we later extend the model to accommodate private endowments of money. We find the best allocation rule in the class of rules that are strategy-proof, anonymous, and that satisfy a weak continuity property. We also find an ascending mechanism, similar to an auction, that implements the rule. The rule coincides with a special simulated price equilibrium, wherein agents buy their desired resource with tokens distributed by the social planner. Equilibrium price vectors form a lower semi-lattice and thus there is a unique minimal price vector. The equilibria associated with the minimal price vector are called min-price Walrasian equilibria. These equilibria form an essentially single-valued correspondence, and this correspondence is the rule we characterize.

A tent pitching scheme motivated by Friedrichs theory

Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet. Motivated by this continuity property, an explicit space–time finite element scheme of the tent pitching type, with spaces that conform to the continuity property, is designed. Numerical results for a model one-dimensional wave propagation problem are presented.

Time inhomogeneous generalized Mehler semigroups and skew convolution equations

Abstract A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space ℍ is defined through p s,t f(x) = ∫ℍ f(U(t,s)x + y)μ t,s (d y), s,t ∈ ℝ, t ≥ s, x ∈ ℍ, for every bounded measurable function f on ℍ, where (U(t,s)) t≥s is an evolution family of bounded operators on ℍ and (μ t,s ) t≥s is a family of probability measures on (ℍ,ℬ(ℍ)) satisfying the following time inhomogeneous skew convolution equations: μ t,s = μ t,r * (μ r,s ∘ U(t,r)-1), t ≥ r ≥ s. This kind of semigroups typically arise as the “transition semigroups” of non-autonomous (possibly non-continuous) Ornstein–Uhlenbeck processes driven by some proper additive process. Suppose that μ t,s converges weakly to δ0 as t ↓ s or s ↑ t. We show that μ t,s has further weak continuity properties in t and s. As a consequence, we prove that for every t ≥ s, μ t,s is infinitely divisible. Natural stochastic processes associated with (μ t,s ) t≥s are constructed and are applied to get probabilistic proofs for the weak continuity and infinite divisibility. Then we analyze the structure, existence and uniqueness of the corresponding evolution systems of measures (= space-time invariant measures) of (p s,t ) t≥s . We also establish a dimension free Harnack inequality for (p s,t ) t≥s and present some of its applications.

Fixed point theorems for correspondences with properties weaker than lower semicontinuity

In this paper, we study existence of fixed points for correspondences having weak continuity properties. The obtained results extend or improve the corresponding results present in literature. We use continuous selection technique and also well-known KKM principle in order to establish our fixed point theorems.

Global attractors for multivalued semiflows with weak continuity properties

A method is proposed to prove the global attractor existence for multivalued semiflows with weak continuity properties. An application to the reaction–diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented.

Weak continuity properties of topologized groups

We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if (G, ·, τ) is a regular right (left) semitopological group with dev(G) < Nov(G) such that all left (right) translations are feebly continuous, then (G, ·, τ) is a topological group. This extends several results in literature.

Weak Continuity and Lower Semicontinuity Results for Determinants

Weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper. In particular, it is shown that if {un} is bounded in Open image in new window and if u ∈ BV(Ω;ℝN) are such that un→u in L1(Ω;ℝN) and

Perturbation of semi-linear evolution equations under weak assumptions at initial time

We prove perturbation results for abstract semi-linear evolution equations in a Banach space. The main feature is that only very weak assumptions are needed at initial time. This allows to prove weak continuity properties and to deal with rather general domain perturbation problems for semi-linear parabolic and hyperbolic boundary value problems with various boundary conditions. The theory also implies the well known theory on parameter dependent equations.

Continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator

In this paper, we obtain some strong and weak type continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator on Hardy spaces, Hardy type spaces and weak Hardy type spaces.

Well-posedness of the full Ericksen–Leslie model of nematic liquid crystals

The Ericksen–Leslie model of nematic liquid crystals is a coupled system between the Navier–Stokes and the Ginzburg–Landau equations. We show here the local well-posedness for this problem for any initial data regular enough, by a fixed point approach relying on some weak continuity properties in a suitable functional setting. By showing the existence of an appropriate local Lyapunov functional, we also give sufficient conditions for the global existence of the solution, and some stability conditions.

A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy

We construct a variational approximation scheme for the equations of three-dimensional elastodynamics with polyconvex stored energy. The scheme is motivated by some recently discovered geometric identities (Qin [18]) for the null Lagrangians (the determinant and cofactor matrix), and by an associated embedding of the equations of elastodynamics into an enlarged system which is endowed with a convex entropy. The scheme decreases the energy, and its solvability is reduced to the solution of a constrained convex minimization problem. We prove that the approximating process admits regular weak solutions, which in the limit produce a measure-valued solution for polyconvex elastodynamics that satisfies the classical weak form of the geometric identities. This latter property is related to the weak continuity properties of minors of Jacobian matrices, here exploited in a time-dependent setting.

On the Necessity and Sufficiency of the Patch Test for Convergence of Nonconforming Finite Elements

This paper proposes a weak patch test that improves both the patch test and the generalized patch test. Under a weak superapproximation assumption, which can be obtained from certain approximation and weak continuity property, it is shown that the weak patch test is equivalent to the generalized patch test. Thus if a nonconforming finite element passes the patch test and satisfies some approximation and weak continuity property, its convergence is then guaranteed. Furthermore, the consistency term of such a nonconforming element can be proved to be of order O(h). It is shown that if a nonconforming element does not pass the patch test and a condition relative to subdividing a domain by the element is true, then one can find a family of triangulations for which the element is divergent.

Weak continuity properties in constructive analysis

Weak continuity properties in constructive analysis

Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces

Let X X and Y Y be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of X X onto the Teichmüller space of Y Y is induced by a quasiconformal homeomorphism of X X onto Y Y . These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.