Tree structures associated to a family of functions

2005 ◽  
Vol 70 (3) ◽  
pp. 681-695 ◽  
Author(s):  
Spiros A. Argyros ◽  
Pandelis Dodos ◽  
Vassilis Kanellopoulos
Keyword(s):  
Polish Space ◽  
Uniform Boundedness ◽  
Index Theorem ◽  
Continuous Functions ◽  
Baire Class ◽  
Compact Set ◽  
Limit Function ◽  
Tree Structures ◽  
Pointwise Limit ◽  
First Baire Class

The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by ∝(f) the separation rank of a real-valued function f in B1(X), with X a Polish space. Notice that in [3], Bourgain has provided a positive answer to this problem in the case of K satisfying with X a compact metric space. The key ingredient in Bourgain's approach is that whenever a sequence of continuous functions pointwise converges to a function f, then the possible discontinuities of the limit function reflect a local ℓ1-structure to the sequence (fn)n. More precisely the complexity of this ℓ1-structure increases as the complexity of the discontinuities of f does. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A.S. Kechris and A. Louveau have invented the rank rND(f) which permits the link between the c0-structure of a sequence (fn)n of uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal's c0-theorem [11] and the c0-index theorem [2] are consequences of this interaction.Passing to the case where either (fn)n are not continuous or X is a non-compact Polish space, this nice interaction is completely lost.

scholarly journals Note on limits of simply continuous and cliquish functions

1994 ◽  
Vol 17 (3) ◽  
pp. 447-450 ◽  
Author(s):  
Janina Ewert
Keyword(s):  
Metric Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Baire Property ◽  
Baire Class ◽  
Class 1 ◽  
Pointwise Limit ◽  
Separable Metric Space

The main result of this paper is that any functionfdefined on a perfect Baire space(X,T)with values in a separable metric spaceYis cliquish (has the Baire property) iff it is a uniform (pointwise) limit of sequence{fn:n≥1}of simply continuous functions. This result is obtained by a change of a topology onXand showing that a functionf:(X,T)→Yis cliquish (has the Baire property) iff it is of the Baire class 1 (class 2) with respect to the new topology.

Separate approximate continuity implies measurability

1973 ◽  
Vol 73 (3) ◽  
pp. 461-465 ◽  
Author(s):  
Roy O. Davies
Keyword(s):  
Continuum Hypothesis ◽  
Measurable Cardinal ◽  
Continuous Functions ◽  
Affirmative Answer ◽  
Baire Class ◽  
Continuous Real Function ◽  
Approximate Continuity ◽  
Class 1 ◽  
Measure Spaces ◽  
First Baire Class

It is known that a real-valued function f of two real variables which is continuous in each variable separately need not be continuous in (x, y), but must be in the first Baire class (1). Moreover if f is continuous in x for each y and merely measurable in y for each x then it must be Lebesgue-measurable (7), and this result can be extended to more general product spaces (2). However, the continuum hypothesis implies that this result fails if continuity is replaced by approximate continuity, as can be seen from the proof of Theorem 2 of (2). This makes Mišik's question (5) very natural: is a function which is separately approximately continuous in both variables necessarily Lebesgue-measurable? Our main aim is to establish an affirmative answer. It will be shown that such a function must in fact be in the second Baire class, although not necessarily in the first Baire class (unlike approximately continuous functions of one variable (3)). Finally, we show that the existence of a measurable cardinal would imply that a separately continuous real function on a product of two topological finite complete measure spaces need not be product-measurable.

scholarly journals Baire and automata

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Pierre Simonnet ◽  
Benoit Cagnard
Keyword(s):  
Computer Science ◽  
Rational Function ◽  
Continuous Functions ◽  
Transfinite Induction ◽  
Baire Class ◽  
Class 1 ◽  
Pointwise Limit ◽  
International Audience ◽  
Büchi Automaton

International audience In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science.

Some properties of the space of fuzzy-valued continuous functions on a compact set

2009 ◽  
Vol 160 (11) ◽  
pp. 1620-1631 ◽  
Author(s):  
Jin-Xuan Fang ◽  
Qiong-Yu Xue
Keyword(s):  
Continuous Functions ◽  
Compact Set

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

scholarly journals Product formalisms for measures on spaces with binary tree structures: representation, visualization, and multiscale noise

2020 ◽  
Vol 8 ◽  
Author(s):  
Devasis Bassu ◽  
Peter W. Jones ◽  
Linda Ness ◽  
David Shallcross
Keyword(s):  
Data Science ◽  
Theoretical Foundation ◽  
Continuous Functions ◽  
Noise Model ◽  
Data Types ◽  
Data Set ◽  
Tree Structures ◽  
Simple Concept ◽  
Wide Range ◽  
Computer Scientists

Abstract In this paper, we present a theoretical foundation for a representation of a data set as a measure in a very large hierarchically parametrized family of positive measures, whose parameters can be computed explicitly (rather than estimated by optimization), and illustrate its applicability to a wide range of data types. The preprocessing step then consists of representing data sets as simple measures. The theoretical foundation consists of a dyadic product formula representation lemma, and a visualization theorem. We also define an additive multiscale noise model that can be used to sample from dyadic measures and a more general multiplicative multiscale noise model that can be used to perturb continuous functions, Borel measures, and dyadic measures. The first two results are based on theorems in [15, 3, 1]. The representation uses the very simple concept of a dyadic tree and hence is widely applicable, easily understood, and easily computed. Since the data sample is represented as a measure, subsequent analysis can exploit statistical and measure theoretic concepts and theories. Because the representation uses the very simple concept of a dyadic tree defined on the universe of a data set, and the parameters are simply and explicitly computable and easily interpretable and visualizable, we hope that this approach will be broadly useful to mathematicians, statisticians, and computer scientists who are intrigued by or involved in data science, including its mathematical foundations.

Callias-type operators in C∗-algebras and positive scalar curvature on noncompact manifolds

2018 ◽  
Vol 12 (04) ◽  
pp. 897-939 ◽  
Author(s):  
Simone Cecchini
Keyword(s):  
Scalar Curvature ◽  
Riemannian Metrics ◽  
Index Theorem ◽  
Complete Riemannian Manifold ◽  
Positive Scalar Curvature ◽  
Type Operator ◽  
Spin Manifold ◽  
Compact Set ◽  
Positive Scalar ◽  
C Algebras

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert [Formula: see text]-module bundles and prove an index theorem for such operators. As an application, we derive an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when [Formula: see text] is a closed spin manifold, we show that if the cylinder [Formula: see text] carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on [Formula: see text] must vanish.

Lexicographic products as compact spaces of the first Baire class

2019 ◽  
Vol 267 ◽  
pp. 106871
Author(s):  
Antonio Avilés ◽  
Stevo Todorcevic
Keyword(s):  
Baire Class ◽  
Compact Spaces ◽  
First Baire Class
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