scholarly journals Preservation of the Borel class under open-LC functions

2011 ◽  
Vol 213 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Alexey Ostrovsky
Keyword(s):  
Borel Class

Some Remarks on Extreme Derivates

1969 ◽  
Vol 12 (4) ◽  
pp. 385-388 ◽  
Author(s):  
A. M. Bruckner
Keyword(s):  
Lipschitz Condition ◽  
Real Variable ◽  
Kernel Functions ◽  
Class 1 ◽  
Borel Class ◽  
Borel Measurable ◽  
Class Of Functions

In 1957 Hájek [1] proved that the extreme bilateral derivates of an arbitrary finite real valued function of a real variable, are Borel measurable of class ≦ 2. It was later shown by Staniszewska [3] that Hájek's result is the best possible (even among the class of functions satisfying a Lipschitz condition). Staniszewska exhibited a Eipschitz function whose extreme bilateral derivates are not in Borel class 1. Staniszewska's proof makes use of a result of Zahorski's [4] concerning kernel functions.

scholarly journals Computing Borel's regulator

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Zacky Choo ◽  
Wajid Mannan ◽  
Rubén J. Sánchez-García ◽  
Victor P. Snaith
Keyword(s):  
Infinite Series ◽  
Borel Class

AbstractWe present an infinite series formula based on the Karoubi–Hamida integral, for the universal Borel class evaluated on

scholarly journals On the Borel class of the derived set operator. II

1983 ◽  
Vol 79 ◽  
pp. 367-372 ◽  
Author(s):  
Douglas Cenzer ◽  
R. Daniel Mauldin
Keyword(s):  
Derived Set ◽  
Borel Class

A Borel reductibility theory for classes of countable structures

1989 ◽  
Vol 54 (3) ◽  
pp. 894-914 ◽  
Author(s):  
Harvey Friedman ◽  
Lee Stanley
Keyword(s):  
Point Of View ◽  
Algebraic Structures ◽  
Main Thrust ◽  
Similarity Type ◽  
Borel Class ◽  
Natural Classes

AbstractWe introduce a reducibility preordering between classes of countable structures, each class containing only structures of a given similarity type (which is allowed to vary from class to class). Though we sometimes work in a slightly larger context, we are principally concerned with the case where each class is an invariant Borel class (i.e. the class of all models, with underlying set = ω, of an Lω1ω sentence; from this point of view, the reducibility can be thought of as a (rather weak) sort of Lω1ω-interpretability notion). We prove a number of general results about this notion, but our main thrust is to situate various mathematically natural classes with respect to the preordering, most notably classes of algebraic structures such as groups and fields.

Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces

2011 ◽  
Vol 54 (2) ◽  
pp. 302-310 ◽  
Author(s):  
Ondřej Kurka
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Strictly Convex ◽  
Borel Class ◽  
Norm Attaining ◽  
Convex Spaces

AbstractLet X be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of X.

scholarly journals A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Dong Hyun Cho
Keyword(s):  
Simple Formula ◽  
Fourier Transforms ◽  
Convolution Product ◽  
Multivariate Normal ◽  
Normal Distributions ◽  
Conditional Expectations ◽  
Change Of Scale ◽  
Convolution Products ◽  
Borel Class ◽  
Generalized Cylinder

Using a simple formula for conditional expectations over continuous paths, we will evaluate conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the measures on the Borel class of L2[0,T]. We will then investigate their relationships. Particularly, we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we will establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and change of scale formulas, we use multivariate normal distributions so that the conditioning function does not contain present positions of the paths.

scholarly journals Borel and Volume Classes for Dense Representations of Discrete Groups

2021 ◽  
Author(s):  
James Farre
Keyword(s):  
Three Dimensional ◽  
Hyperbolic Manifolds ◽  
Discrete Groups ◽  
Bounded Cohomology ◽  
Hyperbolic Volume ◽  
Uniformly Separated ◽  
Point Line ◽  
Borel Class ◽  
Dense Representation ◽  
The Continuum

Abstract We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.

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