Related papers: On some generalizations of real numbers representa…
Many fundamental questions in theoretical computer science are naturally expressed as special cases of the following problem: Let $G$ be a complex reductive group, let $V$ be a $G$-module, and let $v,w$ be elements of $V$. Determine if $w$…
Interacting systems of particles with generalized statistics are considered on both classical and quantum level. It is shown that all possible quantum states and corresponding processes can be represented in terms of certain specific…
To represent real $m$-dimensional vectors, a positional vector system given by a non-singular matrix $M \in \mathbb{Z}^{m \times m}$ and a digit set $\mathcal{D} \subset \mathbb{Z}^m$ is used. If $m = 1$, the system coincides with the well…
Graphs arise naturally in many real-world applications including social networks, recommender systems, ontologies, biology, and computational finance. Traditionally, machine learning models for graphs have been mostly designed for static…
We discuss a formal system of mathematics. We use it to construct the natural numbers.
This note summarizes in an informal way some geometric properties of Anosov representations into the symplectic group, which were presented in a talk at the conference What is Next. The mathematical legacy of Bill Thurston, held in June…
A new method is developed to represent probabilistic relations on multiple random events. Where previously knowledge bases containing probabilistic rules were used for this purpose, here a probability distribution over the relations is…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
The recently introduced model of representations has been defined and motivated somewhat ex-nihilo. In this document, I will show that representations are related to a more ''classical'' model through a 2-adjunction. The target model is…
We discuss representations of monogenic functions over very regular groups.
This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The…
Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of…
We introduce notions of linear reduction and linear equivalence of bijections for the purposes of study bijections between Young tableaux. Originating in Theoretical Computer Science, these notions allow us to give a unified view of a…
We review various combinatorial problems with underlying classical or quantum integrable structures. (Plenary talk given at the International Congress of Mathematical Physics, Aalborg, Denmark, August 10, 2012.)
An abstract mathematical framework is presented in this paper as a unification of several deformed or generalized algebra proposed recently in the context of generalized statistical theories intended to treat certain complex thermodynamic…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
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…
Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect…
Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems.…
Summarization is one of the key features of human intelligence. It plays an important role in understanding and representation. With rapid and continual expansion of texts, pictures and videos in cyberspace, automatic summarization becomes…