Constructions of Almost Secure Frameproof Codes Based on Small-Bias Probability Spaces

2013 ◽  
pp. 53-67 ◽  
Author(s):  
José Moreira ◽  
Marcel Fernández ◽  
Grigory Kabatiansky
Keyword(s):  
Probability Spaces ◽  
Frameproof Codes

Factorization of an adjontable Markov operator

2019 ◽  
Vol 22 (02) ◽  
pp. 1950013
Author(s):  
Carlo Pandiscia
Keyword(s):  
Hilbert Space ◽  
Matrix Algebra ◽  
Unitary Operator ◽  
Markov Operator ◽  
Factorization Property ◽  
Markov Operators ◽  
Dilation Theory ◽  
Probability Spaces

In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization. The method is tested on the two typologies of maps which we know admits a factorization, the Markov operators between commutative probability spaces and adjontable homomorphism. Subsequently, we apply these methods to particular adjontable Markov operator between matrix algebra which fixes the diagonal.

scholarly journals Constructions of almost secure frameproof codes with applications to fingerprinting schemes

2017 ◽  
Vol 86 (4) ◽  
pp. 785-802
Author(s):  
José Moreira ◽  
Marcel Fernández ◽  
Grigory Kabatiansky
Keyword(s):  
Frameproof Codes

On the cohomology equivalence classes for non-singular mappings

1998 ◽  
Vol 18 (6) ◽  
pp. 1385-1397
Author(s):  
ISAAC KORNFELD ◽  
ANDREI KRYGIN
Keyword(s):  
Ergodic Theorem ◽  
Equivalence Classes ◽  
Probability Spaces

The structure of the cohomology equivalence classes for non-singular, not necessarily invertible mappings of probability spaces is studied. In particular, some results of Kochergin and Ornstein–Smorodinsky on the structure of these classes for measure-preserving automorphisms are generalized to the case of non-singular endomorphisms. Our approach is based on Hopf's maximal ergodic theorem and its proof by Garsia.

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

2003 ◽  
Vol 06 (03) ◽  
pp. 337-371 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Probability Theory ◽  
Natural Products ◽  
Tensor Product ◽  
Natural Product ◽  
Free Product ◽  
Noncommutative Probability ◽  
Stochastic Independence ◽  
Boolean Product ◽  
Probability Spaces ◽  
Noncommutative Probability Theory

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

2016 ◽  
Vol 59 (3) ◽  
pp. 533-547 ◽  
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
General Setting ◽  
Maximal Operators ◽  
List Type ◽  
Type Number ◽  
Type Estimate ◽  
Formula Group ◽  
Probability Spaces ◽  
Image Position

AbstractLet $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$. (i)We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$ For each p, the constant e1/p is the best possible.(ii)We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$ and prove that the constant e is optimal. Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

Probability Spaces

2012 ◽  
pp. 3-34
Author(s):  
Santosh S. Venkatesh
Keyword(s):  
Probability Spaces
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