scholarly journals Some properties of the containing spaces and saturated classes of spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 487 ◽  
Author(s):  
S.D. Iliadis
Keyword(s):  
Open Subset ◽  
Closed Subset ◽  
Dense Subset ◽  
Simple Condition ◽  
Specific Subset ◽  
Universal Element ◽  
The Given ◽  
Ordered Pairs

<p>Subjects of this paper are: (a) containing spaces constructed in [2] for an indexed collection S of subsets, (b) classes consisting of ordered pairs (Q,X), where Q is a subset of a space X, which are called classes of subsets, and (c) the notion of universality in such classes.</p> <p>We show that if T is a containing space constructed for an indexed collection S of spaces and for every X ϵ S, Q<sup>X</sup> is a subset of X, then the corresponding containing space TI<sub>Q</sub> constructed for the indexed collection Q ={Q<sup>X</sup> : X ϵ S} of spaces, under a simple condition, can be considered as a specific subset of T. We prove some “commutative” properties of these specific subsets.</p> <p>For classes of subsets we introduce the notion of a (properly) universal element and define the notion of a (complete) saturated class of subsets. Such a class is “saturated” by (properly) universal elements. We prove that the intersection of (complete) saturated classes of subsets is also a (complete) saturated class.</p> <p>We consider the following classes of subsets: (a) IP(Cl), (b) IP(Op), and (c) IP(n.dense) consisting of all pairs (Q;X) such that: (a) Q is a closed subset of X, (b) Q is an open subset of X, and (c) Q is a never dense subset of X, respectively. We prove that the classes IP(Cl) and IP(Op) are complete saturated and the class IP(n.dense) is saturated. Saturated classes of subsets are convenient to use for the construction of new saturated classes by the given ones.</p>

scholarly journals An answer to Furstenberg’s problem on topological disjointness

2019 ◽  
Vol 40 (9) ◽  
pp. 2467-2481 ◽  
Author(s):  
WEN HUANG ◽  
SONG SHAO ◽  
XIANGDONG YE
Keyword(s):  
Open Subset ◽  
Dense Subset ◽  
Minimal System ◽  
Open Neighbourhood ◽  
Countable Dense Subset ◽  
Weakly Mixing ◽  
Transitive System

In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any non-empty open subset $U\subset X$, there is $x\in D\cap U$ such that $\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^{n},T^{(n)})$ and $(X,T^{n})$ for any $n\in \mathbb{N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_{K})$ is disjoint from all minimal systems.

The universal multiplicity theory for analytic operator-valued functions

1995 ◽  
Vol 118 (2) ◽  
pp. 315-320 ◽  
Author(s):  
Jón Arason ◽  
Robert Magnus
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Open Subset ◽  
Closed Subset ◽  
Complex Banach Space ◽  
Singular Set ◽  
Analytic Operator ◽  
Set Of Points

An analytic operator-valued function A is an analytic map A: D → L(E, E), where D = D(A) is an open subset of the complex plane C and E = E(A) is a complex Banach space. For such a function A the singular set σ(A) of A is defined as the set of points z ∈ D such that A(z) is not invertible. It is a relatively closed subset of D.

scholarly journals RELATIONS Vs FUNCTIONS

2020 ◽  
pp. 35-39
Keyword(s):  
Cartesian Product ◽  
Clear Picture ◽  
Product Domain ◽  
The Given ◽  
Ordered Pairs

This paper aims to understand the concepts of relations and functions as well with the inclusive of ordered pairs ,Cartesian product ,domain ,codomain ,range of a function. This paper also focusing on some of the special types of functions and also facilitates better idea on when a relation can be a function. This article also provides clear picture on what magic involved in the given relation to identify whether a function using graphing functions.

scholarly journals FRÉCHET INTERMEDIATE DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS ON ASPLUND SPACES

2009 ◽  
Vol 79 (2) ◽  
pp. 309-317 ◽  
Author(s):  
J. R. GILES
Keyword(s):  
Large Class ◽  
Open Subset ◽  
Lipschitz Function ◽  
Dense Subset ◽  
Lipschitz Functions ◽  
Nonempty Open Subset ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Fréchet Differentiable

AbstractThe deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

scholarly journals Reconstruction of the Sturm-Liouville operators on a graph with $\delta'_s$ couplings

2011 ◽  
Vol 42 (3) ◽  
pp. 329-342 ◽  
Author(s):  
ChuanFu Yang
Keyword(s):  
Uniqueness Theorem ◽  
Liouville Operator ◽  
Dense Subset ◽  
Star Graph ◽  
Central Vertex ◽  
Inverse Nodal Problems ◽  
Sturm Liouville ◽  
The Given ◽  
The Uniqueness Theorem ◽  
Liouville Operators

Inverse nodal problems consist in constructing operators from the given zeros of their eigenfunctions. In this work, we deal with the inverse nodal problems of reconstructing the Sturm- Liouville operator on a star graph with $\delta'_s $ couplings at the central vertex. The uniqueness theorem is proved and a constructive procedure for the solution is provided from a dense subset of zeros of the eigenfunctions for the problem as a data.

scholarly journals INVERSE NODAL PROBLEM FOR A CLASS OF NONLOCAL STURM‐LIOUVILLE OPERATOR

2010 ◽  
Vol 15 (3) ◽  
pp. 383-392 ◽  
Author(s):  
Chuan-Fu Yang
Keyword(s):  
Boundary Condition ◽  
Dense Subset ◽  
Liouville Problem ◽  
Integral Boundary ◽  
Inverse Nodal Problem ◽  
Classical Boundary Condition ◽  
Classical Boundary ◽  
Sturm Liouville ◽  
Nodal Problem ◽  
The Given

Inverse nodal problem consists in constructing operators from the given nodes (zeros) of their eigenfunctions. In this work, the Sturm‐Liouville problem with one classical boundary condition and another nonlocal integral boundary condition is considered. We prove that a dense subset of nodal points uniquely determine the boundary condition parameter and the potential function of the Sturm‐Liouville equation. We also provide a constructive procedure for the solution of the inverse nodal problem.

scholarly journals On super-exponential divergence of periodic points for partially hyperbolic systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaolong Li ◽  
Katsutoshi Shinohara
Keyword(s):  
Growth Rate ◽  
Lower Limit ◽  
Open Subset ◽  
Exponential Growth ◽  
Dense Subset ◽  
Hyperbolic Systems ◽  
Periodic Points ◽  
Residual Subset ◽  
Partially Hyperbolic ◽  
Smooth Closed Manifold

<p style='text-indent:20px;'>We say that a diffeomorphism <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> is super-exponentially divergent if for every <inline-formula><tex-math id="M2">\begin{document}$ b&gt;1 $\end{document}</tex-math></inline-formula> the lower limit of <inline-formula><tex-math id="M3">\begin{document}$ \#\mbox{Per}_n(f)/b^n $\end{document}</tex-math></inline-formula> diverges to infinity, where <inline-formula><tex-math id="M4">\begin{document}$ \mbox{Per}_n(f) $\end{document}</tex-math></inline-formula> is the set of all periodic points of <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> with period <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. This property is stronger than the usual super-exponential growth of the number of periodic points. We show that for any <inline-formula><tex-math id="M7">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional smooth closed manifold <inline-formula><tex-math id="M8">\begin{document}$ M $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M9">\begin{document}$ n\ge 3 $\end{document}</tex-math></inline-formula>, there exists a non-empty open subset <inline-formula><tex-math id="M10">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M11">\begin{document}$ \mbox{Diff}^1(M) $\end{document}</tex-math></inline-formula> such that diffeomorphisms with super-exponentially divergent property form a dense subset of <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id="M13">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-topology. A relevant result about the growth rate of the lower limit of the number of periodic points for diffeomorphisms in a <inline-formula><tex-math id="M14">\begin{document}$ C^r $\end{document}</tex-math></inline-formula>-residual subset of <inline-formula><tex-math id="M15">\begin{document}$ \mbox{Diff}^r(M)\ (1\le r\le \infty) $\end{document}</tex-math></inline-formula> is also shown.</p>

Logical Conditioning and Entropy Inference Based on Observational Data

2001 ◽  
Vol 08 (03) ◽  
pp. 201-239 ◽  
Author(s):  
Alberto Solana-Ortega ◽  
Vicente Solana
Keyword(s):  
Observational Data ◽  
Entropy Method ◽  
The Other ◽  
Simple Condition ◽  
Logical Probability ◽  
Inference Problem ◽  
Additional Information ◽  
Inference Problems ◽  
Relative Entropy Method ◽  
The Given

The logical conditioning inference problem is studied when a simple condition setting a threshold for a potential observation of a scalar observable uncertain quantity is introduced as an additional information to the pieces of evidence originally available to make inference, which include observational data. Two alternative ways of incorporating this condition can be considered. They lead to two conditioned evidences from which it is possible to start and, consequently, two different inference problems arise. Due to the need to coherently represent observational data and to express unambiguously the given evidences, these problems are formalized in a plausible logic language with observational data, within the logical probability framework. They are solved by applying the relative entropy method with fractile constraints. A comparison of the solutions obtained indicates that one of them is a particular instance of the other. It is concluded that the broadest one constitutes the general solution to the logical conditioning inference problem.

Random packing of an interval

1976 ◽  
Vol 8 (03) ◽  
pp. 477-501 ◽  
Author(s):  
David Mannion
Keyword(s):  
Finite Number ◽  
Open Subset ◽  
Closed Subset ◽  
Random Number ◽  
Closed Interval ◽  
Random Packing ◽  
Final Configuration ◽  
Packing Process

At each stage of the packing of a closed interval K, a random number of random open intervals (the packing objects) are placed in that part of K which is as yet unoccupied. No overlapping between the packing objects is allowed. The packing prescription is such that the packing process terminates after at most a finite number of stages. Attention is focused on the final configuration, K = K – + G, where G is a random open subset of K, and is that part of K which is eventually occupied by packing objects, while K –, a random closed subset of K, is that part of K which remains unoccupied.

scholarly journals NON-COMMUTATIVE REPRESENTATIONS OF FAMILIES OF k2 COMMUTATIVE POLYNOMIALS IN 2k2 COMMUTING VARIABLES

2013 ◽  
Vol 23 (07) ◽  
pp. 1685-1753
Author(s):  
HARRY DYM ◽  
J. WILLIAM HELTON ◽  
CALEB MEIER
Keyword(s):  
Dense Subset ◽  
Open Dense Subset ◽  
Condensed Representation ◽  
K Matrix ◽  
The Given

Given a collection [Formula: see text] of k2 commutative polynomials in 2k2 variables, the objective is to find a condensed representation for these polynomials in terms of a single non-commutative (nc) polynomial p(X, Y) in two k × k matrix variables X and Y. In this paper, we develop algorithms that will generically determine whether the given family [Formula: see text] has a nc representation and will produce such a representation if it exists. In particular, we determine an open, dense subset of the space of nc polynomials in two variables that satisfies the following property: if a family [Formula: see text] of polynomials admits a nc representation in this subset, then our algorithms will determine this representation.

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