scholarly journals Logarithmic (translationally) rapidly varying sequences and selection principles

2020 ◽  
Vol 107 (121) ◽  
pp. 45-51
Author(s):  
Dragan Djurcic ◽  
Nebojsa Elez ◽  
Valentina Timotic
Keyword(s):  
Information Theory ◽  
Basic Properties ◽  
Proper Subclass ◽  
Selection Principles

We introduce a proper subclass of the class of rapidly varying sequences (logarithmic (translationally) rapidly varying sequences), motivated by a notion in information theory (self-information of the system). We prove some of its basic properties. In the main result, we prove that Rothberger?s and Kocinac?s selection principles hold, when this class is on the second coordinate, and on the first coordinate we have the class of positive and unbounded sequences

scholarly journals NATURAL METRIC FOR QUANTUM INFORMATION THEORY

2009 ◽  
Vol 07 (05) ◽  
pp. 1009-1019 ◽  
Author(s):  
P. W. LAMBERTI ◽  
M. PORTESI ◽  
J. SPARACINO
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum Information Theory ◽  
Quantum States ◽  
Square Root ◽  
Mixed States ◽  
Basic Properties ◽  
Illustrative Application

We study in detail a very natural metric for quantum states. This new proposal has two basic ingredients: entropy and purification. The metric for two mixed states is defined as the square root of the entropy of the average of representative purifications of those states. Some basic properties are analyzed and its relation to other distances is investigated. As an illustrative application, the proposed metric is evaluated for one-qubit mixed states.

Formal development of ordinal number theory

1955 ◽  
Vol 20 (2) ◽  
pp. 95-104
Author(s):  
Steven Orey
Keyword(s):  
Number Theory ◽  
Ordinal Number ◽  
Ordered Sets ◽  
Stringent Condition ◽  
Easy Consequence ◽  
Basic Properties ◽  
Ordinal Numbers ◽  
Order Types ◽  
Proper Subclass ◽  
Definition Of

In this paper we shall develop a theory of ordinal numbers for the system ML, [6]. Since NF, [2], is a sub-system of ML one could let the ordinal arithmetic developed in [9] serve also as the ordinal arithmetic of ML. However, it was shown in [9] that the ordinal numbers of [9], NO, do not have all the usual properties of ordinal numbers and that theorems contradicting basic results of “intuitive ordinal arithmetic” can be proved.In particular it will be a theorem in our development of ordinal numbers that, for any ordinal number α, the set of all smaller ordinal numbers ordered by ≤ has ordinal number α; this result does not hold for the ordinals of [9] (see [9], XII.3.15). It will also be an easy consequence of our definition of ordinal number that proofs by induction over the ordinal numbers are permitted for arbitrary statements of ML; proofs by induction over NO can be carried through only for stratified statements with no unrestricted class variables.The class we shall take as the class of ordinal numbers, to be designated by ‘ORN’, will turn out to be a proper subclass of NO. This is because in ML there are two natural ways of defining the concept of well ordering. Sets which are well ordered in the sense of [9] we shall call weakly well ordered; sets which satisfy a certain more stringent condition will be called strongly well ordered. NO is the set of order types of weakly well ordered sets, while ORN is the class of order types of strongly well ordered sets. Basic properties of weakly and strongly well ordered sets are developed in Section 2.

Constructing and Deconstructing Concepts

2016 ◽  
Vol 63 (5) ◽  
pp. 249-262 ◽  
Author(s):  
Charles A. Doan ◽  
Ronaldo Vigo
Keyword(s):  
Information Theory ◽  
Model Performance ◽  
Unsupervised Classification ◽  
Family Resemblance ◽  
Category Membership ◽  
Formal Models ◽  
Human Beings ◽  
One Dimensional ◽  
Relational Patterns ◽  
Optimal Classification

Abstract. Several empirical investigations have explored whether observers prefer to sort sets of multidimensional stimuli into groups by employing one-dimensional or family-resemblance strategies. Although one-dimensional sorting strategies have been the prevalent finding for these unsupervised classification paradigms, several researchers have provided evidence that the choice of strategy may depend on the particular demands of the task. To account for this disparity, we propose that observers extract relational patterns from stimulus sets that facilitate the development of optimal classification strategies for relegating category membership. We conducted a novel constrained categorization experiment to empirically test this hypothesis by instructing participants to either add or remove objects from presented categorical stimuli. We employed generalized representational information theory (GRIT; Vigo, 2011b , 2013a , 2014 ) and its associated formal models to predict and explain how human beings chose to modify these categorical stimuli. Additionally, we compared model performance to predictions made by a leading prototypicality measure in the literature.

Sign in / Sign up

Export Citation Format

Share Document