scholarly journals Minimization of φ-divergences on sets of signed measures

2006 ◽  
Vol 43 (4) ◽  
pp. 403-442 ◽  
Author(s):  
Michel Broniatowski ◽  
Amor Keziou
Keyword(s):  
Probability Measure ◽  
Vector Space ◽  
Minimization Problem ◽  
Weak Topology ◽  
Linear Constraints ◽  
Measurable Functions ◽  
Class F

We consider the minimization problem of φ-divergences between a given probability measure P and subsets Ω of the vector space M F of all signed measures which integrate a given class F of bounded or unbounded measurable functions. The vector space M F is endowed with the weak topology induced by the class F ∪ B b where B b is the class of all bounded measurable functions. We treat the problems of existence and characterization of the φ-projections of P on Ω. We also consider the dual equality and the dual attainment problems when Ω is defined by linear constraints.

Entropy of a Convolution Operator

2004 ◽  
Vol 11 (01) ◽  
pp. 79-85 ◽  
Author(s):  
Aleksander Urbański
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Absolutely Continuous ◽  
Doubly Stochastic ◽  
Doubly Stochastic Operator ◽  
Zero Entropy

The concept of the entropy of a doubly stochastic operator was introduced in 1999 by Ghys, Langevin, and Walczak. The idea was developed further by Kamiński and de Sam Lazaro, who also conjectured that the entropy of a convolution operator determined by a probability measure on a compact abelian group is equal to zero. We prove that this is true when the group is connected and the convolution operator is determined by a measure absolutely continuous with respect to the normalized Haar measure. Our result provides also a characterization of the set of doubly stochastic operators with non-zero entropy.

scholarly journals Stability of products of equivalence relations

2018 ◽  
Vol 154 (9) ◽  
pp. 2005-2019 ◽  
Author(s):  
Amine Marrakchi
Keyword(s):  
Probability Measure ◽  
Direct Product ◽  
Equivalence Relation ◽  
Equivalence Relations ◽  
Similar Question ◽  
Stable Equivalence ◽  
Local Characterization

An ergodic probability measure preserving (p.m.p.) equivalence relation ${\mathcal{R}}$ is said to be stable if ${\mathcal{R}}\cong {\mathcal{R}}\times {\mathcal{R}}_{0}$ where ${\mathcal{R}}_{0}$ is the unique hyperfinite ergodic type $\text{II}_{1}$ equivalence relation. We prove that a direct product ${\mathcal{R}}\times {\mathcal{S}}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components ${\mathcal{R}}$ or ${\mathcal{S}}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\text{II}_{1}$ factors is also discussed and some partial results are given.

scholarly journals On relationships between two linear subspaces and two orthogonal projectors

2019 ◽  
Vol 7 (1) ◽  
pp. 142-212 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Vector Space ◽  
Matrix Decomposition ◽  
Complex Vector ◽  
Linear Subspaces ◽  
Survey Paper ◽  
Finite Dimensional ◽  
Generic Position ◽  
Ep Matrix ◽  
Orthogonal Projectors

Abstract Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of the m-dimensional complex column vector space 𝔺m with respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

scholarly journals The Bloch-Vector Space for N-Level Systems: the Spherical-Coordinate Point of View

2005 ◽  
Vol 12 (03) ◽  
pp. 207-229 ◽  
Author(s):  
Gen Kimura ◽  
Andrzej Kossakowski
Keyword(s):  
Vector Space ◽  
Quantum State ◽  
Point Of View ◽  
Vector Spaces ◽  
Bloch Vector ◽  
State Spaces ◽  
Spherical Coordinate ◽  
N Level ◽  
Dual Property

Bloch-vector spaces for N-level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We present a characterization of the space by using the spectra of (orthogonal) generators of SU (N). As an application, we find a dual property of the space which provides an overall picture of the space. We also provide three classes of quantum-state representations based on actual measurements and discuss their state-spaces.

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ ALGEBRAS

2005 ◽  
Vol 07 (02) ◽  
pp. 145-165 ◽  
Author(s):  
ALICE FIALOWSKI ◽  
MICHAEL PENKAVA
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Moduli Space ◽  
Deformation Theory ◽  
Three Dimensional ◽  
3 Dimensional ◽  
Exterior Degree ◽  
Versal Deformations

We consider versal deformations of 0|3-dimensional L∞ algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞ algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

A complex biological system: the fly's visual module

2007 ◽  
Vol 366 (1864) ◽  
pp. 345-357 ◽  
Author(s):  
Murilo S Baptista ◽  
Lirio O.B de Almeida ◽  
Jan F.W Slaets ◽  
Roland Köberle ◽  
Celso Grebogi
Keyword(s):  
Probability Measure ◽  
Complex Systems ◽  
Visual System ◽  
Lyapunov Exponents ◽  
Biological System ◽  
Dimensional Space ◽  
Point Of View ◽  
Two Dimensional ◽  
Complex Biological System

Is the characterization of biological systems as complex systems in the mathematical sense a fruitful assertion? In this paper we argue in the affirmative, although obviously we do not attempt to confront all the issues raised by this question. We use the fly's visual system as an example and analyse our experimental results of one particular neuron in the fly's visual system from this point of view. We find that the motion-sensitive ‘H1’ neuron, which converts incoming signals into a sequence of identical pulses or ‘spikes’, encodes the information contained in the stimulus into an alphabet composed of a few letters. This encoding occurs on multilayered sets, one of the features attributed to complex systems . The conversion of intervals between consecutive occurrences of spikes into an alphabet requires us to construct a generating partition . This entails a one-to-one correspondence between sequences of spike intervals and words written in the alphabet. The alphabet dynamics is multifractal both with and without stimulus, though the multifractality increases with the stimulus entropy. This is in sharp contrast to models generating independent spike intervals, such as models using Poisson statistics, whose dynamics is monofractal. We embed the support of the probability measure, which describes the distribution of words written in this alphabet, in a two-dimensional space, whose topology can be reproduced by an M-shaped map. This map has positive Lyapunov exponents, indicating a chaotic-like encoding.

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