Family Of Continuous Functions Research Articles

Continuous nowhere Hölder functions on ℤ_{𝕡}

K. Weierstrass (1872) was probably the first to present the existence of continuous nowhere differentiable functions (although B. Bolzano, in 1822, was the first to come up with such a construction). Almost a century later, V. Gurariĭ [Dokl. Akad. SSSR 167 (1966), pp. 971–973] observed that the family of continuous functions on [ 0 , 1 ] [0,1] that are differentiable at no point contains, except for the null function, an infinite dimensional vector space. Moreover, and among other recent contributions in this direction, S. Hencl [Proc. Amer. Math. Soc. 128 (2000), p. 3505–3511] generalized the previously mentioned result by proving the existence of isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Here, we continue this ongoing research with the study of continuous nowhere Hölder functions, no longer defined in subsets of R \mathbb {R} , but in subsets of the p p -adic field Q p \mathbb {Q}_p .

New fractal functions on the sphere

In this article, a family of continuous functions on the unit sphere \(S\subseteq {\mathbb {R}}^{3}\) is considered as a generalization of spherical harmonics. The family is fractalized using a linear and bounded operator of functions on the sphere. Particular values of the scale vector in the iterated function system (IFS) may yield classical functions system on the sphere. We have shown that for different values of the scale vector in the IFS, Bessel sequences, frames, and Riesz bases can be established for the space \({\mathcal {L}}^{2}(S)\) of square integrable functions on the sphere.

An algebra of polyanalytic functions

The most important uniform algebra is the family of continuous functions on a compact subset $K$ of the complex plane $\mathbb C $ which are analytic on the interior ${\rm int} (K)$. For polyanalytic functions and compact sets $K$ which are <i>regular</i>

Edge stabilization in the homology of graph braid groups

We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains, which contains strictly more information than the homology level action. We show that the resulting differential graded module is almost never formal over the ring of edges.

Family of Dynamical Systems Whose Topological Entropy Is Nowhere Upper Semicontinuous

We construct a continuous family of continuous mappings of a compact metric space such that the topological entropy of these mappings as a function of the parameter is nowhere upper semicontinuous.

A Galois connection related to restrictions of continuous real functions

Given a family of continuous real functions G, let RG be a binary relation defined as follows: a continuous function f:R→R is in the relation with a closed set E⊆R if and only if there exists g∈G such that f↾E=g↾E. We consider a Galois connection between families of continuous functions and hereditary families of closed sets of reals naturally associated to RG. We study complete lattices determined by this connection and prove several results showing the dependence of the properties of these lattices on the properties of G. In some special cases we obtain exact description of these lattices.

Application of Homotopy to the Ageing Process of Human Body Within the Framework of Algebraic Topology

This paper presents the ageing process of the human body from the homotopy concept in the algebraic topology. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. The complex connected cylindrical shape of the body x=S^1*I, described by the functions f,h: x&amp;rarr;x is a Cartesian product of the vertical interval I=[0,&amp;beta;] and a circle S^1. The initial state of the body x=S^1*I is the topological shape of the infant. The ageing process called the homotopy is the family of continuous functions f_t(x) on the interval I=[0,&amp;alpha;] where &amp;alpha;&amp;gt;&amp;beta; and t&amp;ge;&amp;beta;. It is an increasing sequence of the function f_t(x) of the body x=S^1*I. The homotopy relates the topological shape of the infant to the topological shape of the adult. The homology theory assigns to the human body (x) a sequence of abelian groups H_n(x) for n=0,1,2&amp;hellip;and to the continuous function a sequence of homomorphism. The homology group characterizes the number and continuity of the compact surface of human body. The study excludes genetic, hormonal, environmental and other factors in the ageing process and recognizes the fact that ageing continues throughout life, through childhood, and adolescence into adulthood. Topologically the infant is equal to the adult since the infant continuously grows into the adult. The study found an algebraic way of relating homotopy to the process of ageing of human body. It was also established that this could offer other topologists useful topological tips in the application of homotopy to other physical continuous processes.&#x0D; &#x0D; &amp;nbsp;&#x0D; &#x0D; &amp;nbsp;

On properties of topological pressure

We consider a parametric family of continuous maps of a compact metric space which continuously depend on a parameter ranging over a metric space. The topological pressure of maps in any such family is studied as a function of the parameter from the viewpoint of the Baire classification of functions.

Algebrability of [formula omitted]-continuous functions

Considering the natural topology or S-density topology on the domain and on the range we obtain different families of continuous functions f:R→R. We prove that certain differences of those families are strongly c-algebrable, i.e., they contain c-generated free subalgebras. In particular, we obtain strengthenings of recent results of J. Hejduk, A. Loranty, R. Wiertelak, S. Lindner and M. Terepeta.

Nonlinear Chebyshev approximation to set-valued functions

In this paper, we give a characterization of best Chebyshev approximation to set-valued functions from a family of continuous functions with the weak betweeness property. As a consequence, we obtain a characterization of Kolmogorov type for best simultaneous approximation to an infinity set of functions. We introduce the concept of a set-sun and give a characterization of it. In addition, we prove a property of Amir–Ziegler type for a family of real functions and we get a characterization of best simultaneous approximation to two functions

The shark teeth is a topological IFS-attractor

We show that the space called the shark teeth is a topological IFS-attractor, that is, for every open cover of \(X = \bigcup\nolimits_{i = 1}^n {f_i (X)}\), its image under every suitable large composition from the family of continuous functions {f 1, ..., f n } lies in some set from the cover. In particular, there exists a space that is not homeomorphic to any IFS-attractor but is a topological IFS-attractor.

Some combinatorial principles for trees and applications to tree families in Banach spaces

Suppose that is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree‐families. More precisely, assuming that for any infinite chain β of S the sequence is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence is nearly (resp., convexly) unconditional. In the case where is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence is unconditional. Finally, in the more general setting where for any chain β, is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences .

On the Generality of Assuming that a Family of Continuous Functions Separates Points

For an algebra $${\mathcal{A}}$$ of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that $${\mathcal{A}}$$ separates points in the sense that for each distinct pair $${x, y \in X}$$ , there exists an $${f \in \mathcal{A}}$$ such that $${f(x) \neq f(y)}$$ . If $${\mathcal{A}}$$ does not separate points, it is known that there exists an algebra $${\widehat{\mathcal{A}}}$$ on a compact Hausdorff space $${(\widehat{X}, \widehat{\tau})}$$ that does separate points such that the map $${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$$ is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume $${\mathcal{A}}$$ separates points. The construction of $${{\widehat{\mathcal{A}}}}$$ and $${(\widehat{X}, \widehat{\tau})}$$ does not require that $${\mathcal{A}}$$ has any algebraic structure nor that $${(X, \tau)}$$ has any properties, other than being a topological space. In this work we develop a framework for determining the degree to which separation of points may be assumed without loss of generality for any family $${\mathcal{A}}$$ of bounded, complex-valued, continuous functions on any topological space $${(X, \tau)}$$ . We also demonstrate that further structures may be preserved by the mapping $${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$$ , such as boundaries of weak peak points, the Lipschitz constant when the functions are Lipschitz on a compact metric space, and the involutive structure of real function algebras on compact Hausdorff spaces.

On the almost monotone convergence of sequences of continuous functions

Abstract A sequence (f n)n of functions f n: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f n+1(x) ≤ f n (x) (f n+1(x) ≥ f n(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.

Generalized Strong Boundary Points and Boundaries of Families of Continuous Functions

If \({\mathcal{A}}\) is a family of continuous functions on a locally compact Hausdorff space X, a boundary for \({\mathcal{A}}\) is a subset \({B \subset X}\) such that every \({f \in \mathcal{A}}\) attains its maximum modulus on B. In this work we generalize the concept of strong boundary points for families of functions and show that the collection of these generalized strong boundary points is always a boundary for \({\mathcal{A}}\) . We give conditions under which all boundaries for \({\mathcal{A}}\) consist of generalized strong boundary points and under which these points coincide with classical strong boundary points. When \({\mathcal{A}}\) has sufficient algebraic structure it is proven that this construction provides a unique boundary for \({\mathcal{A}}\) consisting of boundary points, and we conclude by demonstrating how this approach provides an alternate technique for proving the existence of the Choquet and Shilov boundaries in certain function algebras.

Universal families and quantum control in infinite dimensions

In a topological space, a family of continuous mappings is called universal if its action, in at least one element of the space, is dense. If the mappings are unitary or trace-preserving completely positive, the notion of universality is closely related to the notion of controllability in either closed or open quantum systems. Quantum controllability in infinite dimensions is discussed in this setting and minimal generators are found for full control universal families. Some of the requirements of the operators needed for control in infinite dimensions follow from the properties of the infinite unitary group. Hence, a brief discussion of this group and the appropriate mathematical spaces is also included.

The lower quantization coefficient of the [formula omitted]-conformal measure is positive

Let F = { f ( i ) : 1 ≤ i ≤ N } be a strongly Hölder family of continuous functions and let { ϕ i : 1 ≤ i ≤ N } be a conformal iterated function system. We prove that the lower quantization coefficient of the F -conformal measure associated with { ϕ i } is positive. This settles an open question of Lindsay and Mauldin.

Characterization of the classical extensions of the family of continuous functions as Dedekind hulls

In various fields of mathematics, extensions of the family C of all bounded continuous functions on a topological space T arise; many of them have become classical. Such extensions include the families BM and BM 0 of functions measurable in the sense of Borel and Baire, the families B and B 0 of classes of functions with the Baire property and with the Baire property relative to cozero sets, the family R µ of classes of Riemann µ -integrable functions, the family L µ of classes of Lebesgue µ -measurable functions, the family UM of universally measurable functions, etc.

On a class of free boundary problems of type $div(a(X) \nabla u) = -div(\chi (u)H(X))$

We consider a class of two-dimensional free boundary problems of type $div(a(X) \nabla u) = -div(\chi (u)H(X))$, where $H$ is a Lipschitz vector function satisfying $div(H(X))\geq 0$. We prove that the free boundary $\partial [u>0] \cap\Omega$ is represented locally by a family of continuous functions.

Local and Global Monotonicity

We give characterizations of sets $E\subset[0,1]$ for which the local monotonicity of each function $f:[0,1]\to\mathbb{R}$ from a given class $\mathcal{F}$, at all points $x\in E$, implies the global monotonicity of $f$ on $[0,1]$. We consider as $\mathcal{F}$ -- the families of continuous functions, differentiable functions, absolutely continuous functions, functions of class $C^n$ ($n=1,2,...,\infty$), real analytic functions and polynomials.