Topology Of Uniform Convergence Research Articles

Cardinal Functions, Bornologies and Strong Whitney convergence

Let $C(X)$ be the set of all real valued continuous functions on a metric space $(X,d)$. Caserta introduced the topology of strong Whitney convergence on bornology for $C(X)$, which is a generalization of the topology of strong uniform convergence on bornology introduced by Beer-Levi. The purpose of this paper is to study various cardinal invariants of the function space $C(X)$ endowed with the topologies of strong Whitney and Whitney convergence on bornology. In the process, we present simpler proofs of a number of results from the literature. In the end, relationships between cardinal invariants of strong Whitney convergence and strong uniform convergence on $C(X)$ have also been studied.

Scrambled sets and chain recurrence points of generic graph maps

In this paper two generic properties are presented for graph maps. Let be the space of continuous maps from the topological graph X into itself, equipped with the topology of uniform convergence. Then there is a residual subset such that for every , all the scrambled sets of f, as well as the set of chain recurrence points of f, are nowhere dense and of zero Hausdorff dimension.

Pseudo-Fubini Real-Entire Functions on the Plane

In this note, it is proved the existence of a mathfrak {c}-dimensional vector space of real-entire functions all of whose nonzero members are non-integrable in the sense of Lebesgue but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space can be chosen to be dense in the space of all real C^infty -functions on the plane endowed with the topology of uniform convergence on compacta for all derivatives of all orders. If the condition of being entire is dropped, then a closed infinite dimensional subspace satisfying the same properties can be obtained.

Most continuous and increasing functions have two different fixed points

"We study the space of all continuous and increasing self-mappings of the real interval $[0,1]$ equipped with the topology of uniform convergence. In particular, we show that most such functions have at least two different fixed points."

Continuous maps that preserve Hausdorff measure

We show that the space of homeomorphisms on the unit interval [0,1], equipped with the topology of uniform convergence, contains a dense subspace of functions that preserve the Hausdorff measure of any subset of certain one-dimensional self-similar sets. We extend the results to a class of Cantor dust type self-similar sets in R2.

Accelerated exponential Euler scheme for stochastic heat equation: convergence rate of the density

AbstractThis paper studies the numerical approximation of the density of the stochastic heat equation driven by space-time white noise via the accelerated exponential Euler scheme. The existence and smoothness of the density of the numerical solution are proved by means of Malliavin calculus. Based on a priori estimates of the numerical solution we present a test-function-independent weak convergence analysis, which is crucial to show the convergence of the density. The convergence order of the density in uniform convergence topology is shown to be exactly $1/2$ in the nonlinear drift case and nearly $1$ in the affine drift case. As far as we know, this is the first result on the existence and convergence of density of the numerical solution to the stochastic partial differential equation.

THE CLP-COMPACT-OPEN TOPOLOGY ON KC(X,Y)

In this paper, we introduce clp-compact-open topology on KC(X,Y) and compare this topology with compact-open topology and the topology of uniform convergence. Then, we examine metrizability, completeness and countability properties of the clp-compact-open topology on KC(X,Y).

On the complete metrisability of spaces of contractive semigroups

The space of unitary C_{0}-semigroups on a separable infinite-dimensional Hilbert space, when viewed under the topology of uniform weak operator convergence on compact subsets of {mathbb {R}}_{+}, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [4]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive C_{0}-semigroups constitute Polish spaces and thus positively addresses the open problem.

Convergence of exclusion processes and the KPZ equation to the KPZ fixed point

We show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the Kardar-Parisi-Zhang (KPZ) equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of Wu [Tightness and local fluctuation estimates for the KPZ line ensemble, 2021], Dimitrov and Matetski [Ann. Probab. 49 (2021), pp. 2477–2529], the KPZ line ensemble converges to the Airy line ensemble. For the KPZ equation we are able to start from a continuous function plus a finite collection of narrow wedges. For nearest neighbour exclusions, we can take (discretizations) of continuous functions with | h ( x ) | ≤ C ( 1 + | x | ) |h(x)|\le C(1+\sqrt {|x|}) for some C > 0 C>0 , or one narrow wedge. For non-nearest neighbour exclusions, we are restricted at the present time to a class of (random) initial data, dense in continuous functions in the topology of uniform convergence on compacts. The method is by comparison of the transition probabilities of finite range exclusion processes and the totally asymmetric simple exclusion processes using energy estimates.

Cosine Manifestations of the Gelfand Transform

The goal of the paper is to provide a detailed explanation on how the (continuous) cosine transform and the discrete(-time) cosine transform arise naturally as certain manifestations of the celebrated Gelfand transform. We begin with the introduction of the cosine convolution $$\star _c$$ , which can be viewed as an “arithmetic mean” of the classical convolution and its “twin brother”, the anticonvolution. D’Alembert’s property of $$\star _c$$ plays a pivotal role in establishing the bijection between $$\varDelta (L^1(G),\star _c)$$ and the cosine class $$\mathcal {COS}(G),$$ which turns out to be an open map if $$\mathcal {COS}(G)$$ is equipped with the topology of uniform convergence on compacta $$\tau _{ucc}$$ . Subsequently, if $$G = \mathbb {R},\mathbb {Z}, S^1$$ or $$\mathbb {Z}_n$$ we find a relatively simple topological space which is homeomorphic to $$\varDelta (L^1(G),\star _c).$$ Finally, we witness the “reduction” of the Gelfand transform to the aforementioned cosine transforms.

A generalization of topology of uniform convergence on C(X)

We introduce the UI-topology, a generalized version of the topology of uniform convergence or the U-topology on C(X) through a z-ideal in C(X). It is realized that the UI-topology coincides with the mI-topology initiated and investigated by Azarpanah et al. in 2012, if and only if amongst others (C(X),UI) is a topological ring. Connected ideals in the latter topological space are completely determined. On writing I as the set of all z-ideals in C(X) and for I∈I, [I]={J∈I:UI-topology =UJ-topology}, it is observed that each such equivalence class has a largest member and some equivalence classes have smallest members too. Analogous facts with the mI-topologies are also found to be true. The paper ends with a characterization of P-spaces via these equivalence classes.

Recognizing the topologies of spaces of metrics with the topology of uniform convergence

Recognizing the topologies of spaces of metrics with the topology of uniform convergence

Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to be upper semicontinuous concerning the second argument everywhere, as in other related works. A condition involving the contingent derivative relative to the non-decreasing function (recently introduced and applied to initial value problems by R.L. Pouso, I.M. Marquez Albes, and J. Rodriguez-Lopez) is imposed on the set where the upper semicontinuity and the assumption to have compact convex values fail. Based on previously obtained results for periodic problems in the single-valued cases, the existence of solutions is proven. It is also pointed out that the solution set is compact in the uniform convergence topology. In particular, the existence results are obtained for periodic impulsive differential inclusions (with multivalued impulsive maps and finite or possibly countable impulsive moments) without upper semicontinuity assumptions on the right-hand side, and also the existence of solutions is derived for dynamic inclusions on time scales with periodic boundary conditions.

The topology of uniform convergence on Lindelöf subsets

This paper is devoted to analyzing the topology of uniform convergence on Lindelöf subspaces of X on the set of real-valued continuous functions from X, C(X). The set C(X) endowed with this topology is denoted by Cl,u(X). We compare this topology with the pointwise convergence topology, the compact-open topology and the uniform convergence topology. Also, we analyze the relations among Cl,u(X), the open-Lindelöf topology on C(X), Cl(X), and the space C(X) with the topology of uniform convergence on countable subsets, Cs(X). Moreover, we determine when Cl,u(X) satisfies some topological properties such as metrizability, separability, countable cellularity, Čech-completeness and the Baire property.

Dynamical behavior of recurrent neural networks with different external inputs

The main aim of this paper is to study the dynamics of a recurrent neural networks with different input currents in terms of asymptotic point. Under certain circumstances, we studied the existence, the uniqueness of bounded solutions and their homoclinic and heteroclinic motions of the considered system with rectangular currents input. Moreover, we studied the unpredictable behavior of the continuous high-order recurrent neural networks and the discrete high-order recurrent neural networks. Our method was primarily based on Banach’s fixed-point theorem, topology of uniform convergence on compact sets and Gronwall inequality. For the demonstration of theoretical results, we give examples and their numerical simulations.

The space of contractive C0-semigroups is a Baire space

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive C0-semigroups under the topology of uniform weak operator convergence on compact subsets of R+. Eisner and Serény raised in [6] and [3] the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The transfer result in turn relies upon classification of topological properties via infinite games. Our approach is sufficiently general and can be applied to other contexts, e.g. the space of contractions under the ▪-topology.

Minimal usco maps and cardinal invariants of the topology of uniform convergence on compacta

Minimal usco maps and cardinal invariants of the topology of uniform convergence on compacta

Two families of hypercyclic nonconvolution operators

Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $\lambda,b\in\mathbb{C}$, let $C_{\lambda,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_{\lambda,b} f(z)=f(\lambda z+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{\lambda,b}=C_{\lambda,b} \circ D$ by showing that whenever $|\lambda|\geq 1$, the collection of operators \begin{align*} \{\psi(T_{\lambda,b}): \psi(z)\in H(\mathbb{C}), \psi(0)=0 \text{ and } \psi(T_{\lambda,b}) \text{ is continuous}\} \end{align*} forms an algebra under the usual addition and multiplication of operators which consists entirely of hypercyclic operators (i.e., each operator has a dense orbit). We also show that the collection of operators \begin{align*} \{C_{\lambda,b}\circ\varphi(D): \varphi(z) \text{ is an entire function of exponential type with } \varphi(0)=0\} \end{align*} consists entirely of hypercyclic operators.

Approximation by simple partial fractions in unbounded domains

For unbounded simply connected domains in the complex plane, bounded by several simple curves with regular asymptotic behaviour at infinity, we obtain necessary conditions and sufficient conditions for simple partial fractions (logarithmic derivatives of polynomials) with poles on the boundary of to be dense in the space of holomorphic functions in (with the topology of uniform convergence on compact subsets of ). In the case of a strip bounded by two parallel lines, we give estimates for the convergence rate to zero in the interior of of simple partial fractions with poles on the boundary of and with one fixed pole. Bibliography: 16 titles.

Quasicontinuous functions and the topology of uniform convergence on compacta

Let X be a Hausdorff topological space, Q(X,R) be the space of all quasicontinuous functions on X with values in R and ?UC be the topology of uniform convergence on compacta. If X is hemicompact, then (Q(X,R), ?UC) is metrizable and thus many cardinal invariants, including weight, density and cellularity coincide on (Q(X,R), ?UC). We find further conditions on X under which these cardinal invariants coincide on (Q(X,R), ?UC) as well as characterizations of some cardinal invariants of (Q(X,R), ?UC). It is known that the weight of continuous functions (C(R,R), ?UC) is ?0. We will show that the weight of (Q(R,R), ?UC) is 2c.