Continuous Real-valued Functions Research Articles

Formula omitted]-factorizable topological groups

In this paper, we define a new class of PM-factorizable topological groups. A topological group G is called PM-factorizable if, for every continuous real-valued function f on G, one can find a perfect homomorphism p:G→L onto a metrizable topological group L and a continuous real-valued function h on L such that f=h∘p. The relations between PR-factorizable, PM-factorizable and M-factorizable topological groups are studied. Also, some new characterizations of PR-factorizable, PM-factorizable and M-factorizable topological groups are obtained.

Neural Networks of the Rational r-th Powers of the Multivariate Bernstein Operators

In this study, a novel neural network for the multivariance Bernstein operators' rational powers was developed. A positive integer is required by these networks. In the space of all real-valued continuous functions, the pointwise and uniform approximation theorems are introduced and examined first. After that, the Lipschitz space is used to study two key theorems. Additionally, some numerical examples are provided to demonstrate how well these neural networks approximate two test functions. The numerical outcomes demonstrate that as input grows, the neural network provides a better approximation. Finally, the graphs used to represent these neural network approximations show the average error between the approximation and the test function.

Bochner Representable Operators on Spaces of Continuous Functions

Let C_b(X) be the Banach lattice of all bounded continuous real-valued functions on a completely regular Hausdorff space X and beta denote the natural strict topology on C_b(X). For a Banach space (E,Vert cdot Vert _E), a linear operator T:C_b(X)rightarrow E is said to be tight if Vert T(u_alpha )Vert _Erightarrow 0 whenever (u_alpha ) is a uniformly bounded net in C_b(X) such that u_alpha rightarrow 0 uniformly on all compact sets in X. It is shown that a linear operator T:C_b(X)rightarrow E is nuclear tight if and only if T is a nuclear operator between the locally convex space (C_b(X),beta ) and a Banach space E and if and only if T is Bochner representable, that is, there exist a positive Radon measure mu on X and a E-valued mu -Bochner integrable function g on X so that T(u)=int _X u(x)g(x)dmu for all uin C_b(X).

Quasi SV-frames and their properties

Let P be a prime ideal of the ring of continuous real-valued functions on a completely regular frame L, i.e., L. We study many new results about the residue class domains L/P with an emphasis on determining when the ordered L/P is a valuation domain (i.e., when given any two non-zero elements of L/P , one divides the other). A prime ideal P of L is called a valuation prime ideal if L/P is a valuation domain. A frame L is called an SV-frame if every prime ideal of L is a valuation prime ideal. We introduce and study two new generalizations of the SV-frames. The first is that of a quasi SV-frame in which every real maximal ideal of L that is not a minimal prime ideal contains a non-maximal prime ideal P such that L/P is a valuation domain. In the second, we define a frame L to be an almost SV-frame if every maximal ideal of L contains a minimal valuation prime ideal. A point I ∈ Pt(βL) is called a special βF -point if O I = {δ ∈ L : coz δ ∈ I} is a valuation prime ideal of L. It is shown that I is a special βF -point if and only if the pseudo-prime ideals of L containing O I that are not primary form a chain under set inclusion.

Approximation by pointwise bounded sets of continuous functions

Assuming that X is a Tychonoff space, we obtain necessary and sufficient conditions for a function f∈RX to belong to the closure in RX of a pointwise bounded family F in C(X), the ring of real-valued continuous functions on X. In other words, if M(X) stands for the bidual of the space Cp(X), equipped with the pointwise topology, we provide necessary and sufficient conditions for a function f∈RX to belong to M(X). As a noteworthy fact, if M, S and R are, respectively, the Michael, the Sorgenfrey, and the real line, it is shown that C(M)⊈M(R) but C(S)⊆M(R).

On bivariate fractal approximation

In this paper, the notion of dimension preserving approximation for real-valued bivariate continuous functions, defined on a rectangular domain , has been introduced and several results, similar to well-known results of bivariate constrained approximation in terms of dimension preserving approximants, have been established. Further, some clue for the construction of bivariate dimension preserving approximants, using the concept of fractal interpolation functions, has been added. In the last part, some multi-valued fractal operators associated with bivariate \(\alpha \)-fractal functions are defined and studied.

A Functional Characterization of the Hurewicz Property

For a Tychonoff space $X$, we denote by $C_p(X)$ the space of all real-valued continuous functions on $X$ with the topology of pointwise convergence.  We study a functional characterization of the covering property of Hurewicz.

Nuclear tight operators on spaces of continuous functions

Let C b ( X ) C_b(X) denote the Banach space of all bounded continuous real-valued functions on a completely regular Hausdorff space X X . We characterize nuclear tight operators T : C b ( X ) → E T:C_b(X)\rightarrow E in terms of their representing Radon vector measures. As an application, it is shown that some natural kernel operators on C b ( X ) C_b(X) are tight and nuclear.

A characterization of Cesàro average utility

Let X be a connected metric space, and let ⪰ be a weak order defined on a suitable subset of XN. We characterize when ⪰ has a Cesàro average utility representation. This means that there is a continuous real-valued function u on X such that, for all sequences x=(xn)n=1∞ and y=(yn)n=1∞ in the domain of ⪰, we have x⪰y if and only if the limit as N→∞ of the average value of u(x1),…,u(xN) is higher than limit as N→∞ of the average value of u(y1),…,u(yN). This has applications to decision theory, game theory, and intergenerational social choice.

First and second countable generators

A subset G of Cp(X) - the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence - is a generator (resp., (0,≠0)-generator, or (0,1)-generator) if whenever a point x is not in a closed set C of X, there is a g in G such that g(x)∉g(C)‾ (resp., g(x)≠0 and g(C)⊆{0}, or g(x)=1 and g(C)⊆{0}).The motivation behind the questions: Which spaces have a first countable(0,≠0)-generator containing the zero function,0? Are they precisely the separable spaces? Which spaces have a second countable(0,≠0)-generator containing0? Are they precisely the cosmic spaces? is elucidated, and partial solutions presented.

Multivariate fractal interpolation functions: some approximation aspects and an associated fractal interpolation operator

In the classical (non-fractal) setting, the natural kinship between theories of interpolation and approximation is well explored. In contrast to this, in the context of fractal interpolation, the interrelation between interpolation and approximation is subtle, and this duality is relatively obscure. The notion of α-fractal functions provides a proper foundation for the approximation-theoretic facet of univariate fractal interpolation functions (FIFs). However, no comparable approximation-theoretic aspects of FIFs have been developed for functions of several variables. The current article intends to open the door for intriguing interactions between approximation theory and multivariate FIFs. To this end, in the first part of this article, we develop a general framework for constructing multivariate FIFs, which is amenable to provide a multivariate analogue of an α-fractal function. Multivariate α-fractal functions provide a parameterized family of fractal approximants associated with a given multivariate continuous function. Some elementary aspects of the multivariate fractal (not necessarily linear) interpolation operator that sends a continuous function defined on a hyperrectangle to its fractal analogue are studied. As in the univariate setting, the notion of α-fractal functions serves as a basis for fractalizing various results in multivariate approximation theory, including that of multivariate splines. For our part, we provide some approximation classes of multivariate fractal functions and prove a few results on the constrained fractal approximation of real-valued continuous functions of several variables.

Computable Classifications of Continuous, Transducer, and Regular Functions

We develop a systematic algorithmic framework that unites global and local classification problems for functional separable spaces and apply it to attack classification problems concerning the Banach space C[0,1] of real-valued continuous functions on the unit interval. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipshitz functions is $\Sigma^0_2$-complete. We show that a function $f\colon [0,1] \rightarrow \mathbb{R}$ is (binary) transducer if and only if it is continuous regular; interestingly, this peculiar and nontrivial fact was overlooked by experts in automata theory. As one of many consequences, our $\Sigma^0_2$-completeness result covers the class of transducer functions as well. Finally, we show that the Banach space $C[0,1]$ of real-valued continuous functions admits an arithmetical classification among separable Banach spaces. Our proofs combine methods of abstract computability theory, automata theory, and functional analysis.

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.

Topological pressure and large deviations for multivariate potentials

Given a topological dynamical system (X,T), this paper investigates the topological pressure and large deviation estimate for multivariate potentials Φ:={Snφ}n≥1, where φ is a real-valued continuous function on Xk andSnφ(x)=1nk−1∑i1=0n−1∑i2=0n−1⋅⋅⋅∑ik=0n−1φ(Ti1x,Ti2x,⋅⋅⋅,Tikx),x∈X. As an application, this paper finally discusses the properties of the topological pressure for the following multivariate functiong˜s(x,y):=−s12(g(x)−g(y))2,s≥0, where g:X→X is a continuous function.

New Directions in Duality Theory for Modal Logic

AbstractIn this work we present some new contributions towards two different directions in the study of modal logic. First we employ tense logics to provide a temporal interpretation of intuitionistic quantifiers as “always in the future” and “sometime in the past.” This is achieved by modifying the Gödel translation and resolves an asymmetry between the standard interpretation of intuitionistic quantifiers.Then we generalize the classic Gelfand–Naimark–Stone duality between compact Hausdorff spaces and uniformly complete bounded archimedean $\ell $ -algebras to a duality encompassing compact Hausdorff spaces with continuous relations. This leads to the notion of modal operators on bounded archimedean $\ell $ -algebras and in particular on rings of continuous real-valued functions on compact Hausdorff spaces. This new duality is also a generalization of the classic Jónsson-Tarski duality in modal logic.Abstract taken directly from the thesis.E-mail: lcarai@unisa.itURL: https://www.proquest.com/openview/5d284dbfb954383da9364149fa312b6f/1?pq-origsite=gscholar&cbl=18750&diss=y

MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

AbstractIt is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions

Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game);The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X).In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset;the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X).If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).

Spectral residual method for nonlinear equations on Riemannian manifolds

In this paper, the spectral algorithm for nonlinear equations (SANE) is adapted to the problem of finding a zero of a given tangent vector field on a Riemannian manifold. The generalized version of SANE uses, in a systematic way, the tangent vector field as a search direction and a continuous real-valued function that adapts this direction and ensures that it verifies a descent condition for an associated merit function. To speed up the convergence of the proposed method, we incorporate a Riemannian adaptive spectral parameter in combination with a non-monotone globalization technique. The global convergence of the proposed procedure is established under some standard assumptions. Numerical results indicate that our algorithm is very effective and efficient solving tangent vector field on different Riemannian manifolds and competes favorably with a Polak–Ribiére–Polyak method recently published and other methods existing in the literature.

On the pseudouniform topology on C(X)

We denote by Cs(X) the set C(X) of real-valued continuous functions defined on X endowed with the topology of the uniform convergence on the closed separable subspaces of X. In this paper we continue the study of Cs(X) initiated in Pseudouniform topologies onC(X)given by ideals (Pichardo-Mendoza et al. (2013) [6]). We prove that Cs(X) is a k-space if and only if Cs(X) is metrizable, and that compactness, sequential compactness and countable compactness coincide in subspaces of Cs(X). In addition, we study the cellularity, density, weight and character of Cs(X). We prove that (1) d((R2λ)s)=λ if λ=λω, and (2) the Continuum Hypothesis is equivalent to the statement: Every non-separable space X satisfies χ(Cs(X))=w(Cs(X)).

Pre-image of functions in $C(L)$

Let $C(L)$ be the ring of all continuous real functions on a frame $L$ and $S\subseteq{\mathbb R}$. An $\alpha\in C(L)$ is said to be an overlap of $S$, denoted by $\alpha\blacktriangleleft S$, whenever $u\cap S\subseteq v\cap S$ implies $\alpha(u)\leq\alpha(v)$ for every open sets $u$ and $v$ in $\mathbb{R}$. This concept was first introduced by A. Karimi-Feizabadi, A.A. Estaji, M. Robat-Sarpoushi in {\it Pointfree version of image of real-valued continuous functions} (2018). Although this concept is a suitable model for their purpose, it ultimately does not provide a clear definition of the range of continuous functions in the context of pointfree topology. In this paper, we will introduce a concept which is called pre-image, denoted by ${\rm pim}$, as a pointfree version of the image of real-valued continuous functions on a topological space $X$. We investigate this concept and in addition to showing ${\rm pim}(\alpha)=\bigcap\{S\subseteq{\mathbb R}:~\alpha\blacktriangleleft S\}$, we will see that this concept is a good surrogate for the image of continuous real functions. For instance, we prove, under some achievable conditions, we have ${\rm pim}(\alpha\vee\beta)\subseteq {\rm pim}(\alpha)\cup {\rm pim}(\beta)$, ${\rm pim}(\alpha\wedge\beta)\subseteq {\rm pim}(\alpha)\cap {\rm pim}(\beta)$, ${\rm pim}(\alpha\beta)\subseteq {\rm pim}(\alpha){\rm pim}(\beta)$ and ${\rm pim}(\alpha+\beta)\subseteq {\rm pim}(\alpha)+{\rm pim}(\beta)$.