Related papers: Canonical random variables for multivariate, algeb…
Free probabilistic considerations of type B first appeared in a paper by Biane, Goodman and Nica in 2003. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko…
One can often encounter claims that classical (Kolmogorovian) probability theory cannot handle, or even is contradicted by, certain empirical findings or substantive theories. This note joins several previous attempts to explain that these…
A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space $\mathcal{H}$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group $G$ on…
We discuss the canonical structure of a spacetime version of the radial gauge, i.e. Gau{\ss}ian normal spacetime coordinates. While it was found for the spatial version of the radial gauge that a "local" algebra of observables can be…
We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
This paper presents a categorical account of conditional probability, covering both the classical and the quantum case. Classical conditional probabilities are expressed as a certain "triangle-fill-in" condition, connecting marginal and…
We compare two approaches to embedding joint distributions of random variables recorded under different conditions (such as spins of entangled particles for different settings) into the framework of classical, Kolmogorovian probability…
We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the…
Changes of variables giving the dual model are constructed explicitly for sigma-models without isotropy. In particular, the jacobian is calculated to give the known results. The global aspects of the abelian case as well as some of those of…
We review the construction of the cross product algebra $\A\rtimes\U$ from two dually paired Hopf algebras $\U$ and $\A$. The canonical element in $\U \otimes \A$ is then introduced, and its properties examined. We find that it is useful…
We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and the shuffle algebra based on the same space. This isomorphism uses the commutative…
Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated…
In this paper, we introduce a non-commutative space of stochastic distributions, which contains the non-commutative white noise space, and forms, together with a natural multiplication, a topological algebra. A special inequality which…
We give an expository review of applications of computational algebraic statistics to design and analysis of fractional factorial experiments based on our recent works. For the purpose of design, the techniques of Gr\"obner bases and…
This paper provides robust estimators for the first canonical correlation and directions of random elements on Hilbert separable spaces by using robust association and scale measures combined with basis expansion and/or penalizations as a…
We propose a novel algebraic framework for treating probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on…
The linear canonical transforms of position and momentum are used to construct the tomographic probability representation of quantum states where the fair probability distribution determines the quantum state instead of the wave function or…
Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear…
We study the formulation of quantum statistical mechanics in noncommutative spaces. We construct microcanonical and canonical ensemble theory in noncommutative spaces. We consider for illustration some basic and important examples in the…