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We report briefly on an approach to quantum theory entirely based on symmetry grounds which improves Geometric Quantization in some respects and provides an alternative to the canonical framework. The present scheme, being typically…
In this paper we study some algebraic properties of the rack structure as well as the representation theory of it, following the ideas given by M. Elhamdadi and E. M. Moutuou in \cite{Elhamdadi}. We establish a correspondence between the…
In the present work, we demonstrate how the pseudoinverse concept from linear algebra can be used to represent and analyze the boundary conditions of linear systems of partial differential equations. This approach has theoretical and…
The main purpose of this paper is providing a simple method to generate the matrices of irreducible representations because it is useful to reduce the computational time of solving the eigenvalue problems. The only information we need to…
Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family…
Quantum networks are often modelled using Schroedinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article…
This paper introduces a generalised 3rd-order Spectral Representation Method for the simulation of multi-dimensional stochastic fields with asymmetric non-linearities. The simulated random fields satisfy a prescribed Power Spectrum and…
We use group schemes to construct optimal packings of lines through the origin. In this setting, optimal line packings are naturally characterized using representation theory, which in turn leads to a necessary integrality condition for the…
The article covers developments in the representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and…
Let G be an arbitrary finite weighted digraph with weights in the set of complex rational functions. A general procedure is proposed which allows for the reduction of G to a smaller graph with a less complicated structure having the same…
The inductive biases of graph representation learning algorithms are often encoded in the background geometry of their embedding space. In this paper, we show that general directed graphs can be effectively represented by an embedding model…
We give new, explicit and asymptotically sharp, lower bounds for dimensions of irreducible modular representations of finite symmetric groups.
Isotopic pairs and their representations are considered in a general framework of the vector superalgebra. Numerous examples of finite-dimensional and infinite-dimensional isotopic pairs are discussed. Several types of their representations…
The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as…
In this paper we make a clear relationship between the automorphic representations and the quantization through the Geometric Langlands Correspondence. We observe that the discrete series representation are realized in the sum of…
In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by…
We discuss a method of parameter reduction in complex models known as the Manifold Boundary Approximation Method (MBAM). This approach, based on a geometric interpretation of statistics, maps the model reduction problem to a geometric…
A relative Picard theory in the context of graded manifolds is introduced. A Berezinian calculus and a theory of connections over SUSY-curves are systematically developed, and used to prove a Gauss-Bonnet theorem for line bundles in that…
We study natural linear representations of self-similar groups over finite fields. In particular, we show that if the group is generated by a finite automaton, then obtained matrices are automatic. This shows a new relation between two…
This paper considers the use of singular perturbation approximations for a class of linear quantum systems arising in the area of linear quantum optics. The paper presents results on the physical realizability properties of the approximate…