Convex-Cyclic Weighted Composition Operators and Their Adjoints

We characterize the convex-cyclic weighted composition operators $$W_{(u,\psi )}$$ W ( u , ψ ) and their adjoints on the Fock space in terms of the derivative powers of $$ \psi $$ ψ and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and, hence, with the weak and strong topologies, and answers a question raised by T. Carrol and C. Gilmore in [5].

On a metric of the space of idempotent probability measures

<p>In this paper we introduce a metric on the space I(X) of idempotent probability measures on a given compact metric space (X; ρ), which extends the metric ρ. It is proven the introduced metric generates the pointwise convergence topology on I(X).</p>

On the continuity of functors of the type C(X, Y)

We consider the category P, the objects of which are pairs of topological spaces (X, Y). Each such pair (X, Y) is assigned the space of continuous maps Cτ(X, Y) with some topology τ. By imposing some restrictions on objects and morphisms of category P, we define a subcategory K ⊂ P, for which the above map is a functor from K to the category Top of topological spaces and continuous maps. The following question is investigated. What are the additional conditions on K, under which the above functor is continuous? Along the way the problem of finding the limit of the inverse spectrum in the category P is solved. We show, that it reduces to finding the limits of the corresponding direct spectrum and inverse spectrum in the category Top. Point convergence topology, compact-open topology and graph topology are considered as the topology τ.

Large Function Algebras with Certain Topological Properties

LetFbe a family of continuous functions defined on a compact interval. We give a sufficient condition so thatF∪{0}contains a densec-generated free algebra; in other words,Fis denselyc-strongly algebrable. As an application we obtain densec-strong algebrability of families of nowhere Hölder functions, Bruckner-Garg functions, functions with a dense set of local maxima and local minima, and nowhere monotonous functions differentiable at all but finitely many points. We also study the problem of the existence of large closed algebras withinF∪{0}whereF⊂RXorF⊂CX. We prove that the set of perfectly everywhere surjective functions together with the zero function contains a2c-generated algebra closed in the topology of uniform convergence while it does not contain a nontrivial algebra closed in the pointwise convergence topology. We prove that an infinitely generated algebra which is closed in the pointwise convergence topology needs to contain two valued functions and infinitely valued functions. We give an example of such an algebra; namely, it was shown that there is a subalgebra ofRRwith2cgenerators which is closed in the pointwise topology and, for any functionfin this algebra, there is an open setUsuch thatf-1(U)is a Bernstein set.

A New Characterization of Compact Sets in Fuzzy Number Spaces

We give a new characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space endowed with the level convergence topology. Our results imply that some previous compactness criteria are wrong. A counterexample also is given to validate this judgment.

Collineation group as a subgroup of the symmetric group

Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_\psi $ of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc., 1974, 8(3), 426] that if ψ is finite then H contains the alternating subgroup $\mathfrak{A}_\psi $ of $\mathfrak{S}_\psi $. We show in Theorem 3.1 that H = $\mathfrak{S}_\psi $, if ψ is infinite.

On the Fuzzy Number Space with the Level Convergence Topology

We characterize compact sets of𝔼1endowed with the level convergence topologyτℓ. We also describe the completion(𝔼1̂,𝒰̂)of𝔼1with respect to its natural uniformity, that is, the pointwise uniformity𝒰, and show other topological properties of𝔼1̂, as separability. We apply these results to give an Arzela-Ascoli theorem for the space of(𝔼1,τℓ)-valued continuous functions on a locally compact topological space equipped with the compact-open topology.

EXTREMAL EQUILIBRIA FOR DISSIPATIVE PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES

We consider a reaction diffusion equation ut = Δu + f(x, u) in ℝN with initial data in the locally uniform space [Formula: see text], q ∈ [1, ∞), and with dissipative nonlinearities satisfying s f(x, s) ≤ C(x)s2 + D(x) |s|, where [Formula: see text] and [Formula: see text] for certain [Formula: see text]. We construct a global attractor [Formula: see text] and show that [Formula: see text] is actually contained in an ordered interval [φm, φM], where [Formula: see text] is a pair of stationary solutions, minimal and maximal respectively, that satisfy φm ≤ lim inft→∞ u(t; u0) ≤ lim supt→∞ u(t; u0) ≤ φM uniformly for u0 in bounded subsets of [Formula: see text]. A sufficient condition concerning the existence of minimal positive steady state, asymptotically stable from below, is given. Certain sufficient conditions are also discussed ensuring the solutions to be asymptotically small as |x| → ∞. In this case the solutions are shown to enter, asymptotically, Lebesgue spaces of integrable functions in ℝN, the attractor attracts in the uniform convergence topology in ℝN and is a bounded subset of W2,r(ℝN) for some r > N/2. Uniqueness and asymptotic stability of positive solutions are also discussed. Applications to some model problems, including some from mathematical biology are given.