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This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented…
The primary purpose of this paper is to present the concept of dichotomy in image illumination modeling based on the power function. In particular, we review several mathematical properties of the power function to identify the limitations…
Representation theorems for formal systems often take the form of an inductive translation that satisfies certain invariants, which are proved inductively. Theory morphisms and logical relations are common patterns of such inductive…
In these lecture notes I give an introduction to deformation quantization. The quantization problem is discussed in some detail thereby motivating the notion of star products. Starting from a deformed observable algebra, i.e. the star…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact…
It is well-known that a random variable, i.e., a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an…
A central question in cognitive science is whether conceptual representations converge onto a shared manifold to support generalization, or diverge into orthogonal subspaces to minimize task interference. While prior work has discovered…
The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker…
The connection between q-analogs of special functions and representations of quantum algebras has been developed recently. It has led to advances in the theory of q-special functions that we here review.
We study a certain family of finite-dimensional simple representations over quantum affine superalgebras associated to general linear Lie superalgebras, the so-called fundamental representations: the denominators of rational $R$-matrices…
In our earlier work math.QA/9808015 some results on integral representations of functions in quantum disc were announced. It was then shown in math.QA/9808037 that the validity of those results is related to the invariance of kernels of…
For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…
Quasiprobability representations are well-established tools in quantum information science, with applications ranging from the classical simulability of quantum computation to quantum process tomography, quantum error correction, and…
A study on a method for the establishment of a phase space representation of quantum theory is presented. The approach utilizes the properties of Gaussian distribution, the properties of Hermite polynomials, Fourier analysis and the current…
A generalization of the quotient integral formula is presented and some of its properties are investigated. Also the relations between two function spaces related to the spacial homogeneous spaces are derived by using general quotient…
The integral representation theorem for martingales has been widely used in probability theory. In this work, we propose and prove a general representation theorem for a class of set-valued submartingales. We also extend the stochastic…
An assessment is given as to the extent to which pure unitary evolution, as distinct from environmental decohering interaction, can provide the transition necessary for an observer to interpret perceived quantum dynamics as classical. This…
Given a covering of the projective line with ramifications defined over a number field, we define a plain model of the algebraic curve realizing the Riemann existence theorem for this covering, and bound explicitly the defining equation of…
We define a very general notion of regularity for functions taking values in an alternative real $*$-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over…