Related papers: A numerical characterization of Dunkl systems
Modular and quasimodular solutions of specific second order differential equation in the upper-half plane which originates from a study of supersingular j-invariants are given explicitly. A characterization of the differential equation is…
In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators…
An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the…
Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in…
The isotropic Dunkl oscillator model in three-dimensional Euclidean space is considered. The system is shown to be maximally superintegrable and its symmetries are obtained by the Schwinger construction using the raising/lowering operators…
Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra.…
Recently we proposed a generic construction of the additional integrals of motion for the St\"ackel systems applying addition theorems to the angle variables. In this note we show some trivial examples associated with angle variables for…
Complex network theory has been used to study complex systems. However, many real-life systems involve multiple kinds of objects . They can't be described by simple graphs. In order to provide complete information of these systems, we…
In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic curves on the plane. In particular, we consider St\"ackel…
We characterize in simple terms the Pak-Stanley labels $\lambda(R)$ of the regions $R$ of the $m$-Catalan arrangement. We also propose a simple algorithm that returns $R$ from $\lambda(R)$. Finally, we characterize in close terms the labels…
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study…
A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator model in the plane, whose Hamiltonian…
We study the Z/2-equivariant K-theory of the complement of the complexification of a real hyperplane arrangement. We compute the rational K and KO rings, and give two different combinatorial descriptions of the subring of the integral KO…
We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which…
Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parametrized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a…
We develop a statistical theory to characterize correlations in weighted networks. We define the appropriate metrics quantifying correlations and show that strictly uncorrelated weighted networks do not exist due to the presence of…
An algebraic formalism for the study of a system of charged particles interacting with an external quantum field is developed. The notion of monoidal categories with duality is used for the description of composite systems and corresponding…
These are the lecture notes of a series of lectures on Dunkl operators. We discuss the underlying algebraic structure of the degenerate double affine Hecke algebra, intertwiners and shift operators. We apply this to Macdonald theory. We…
We give a common generalization of (1) Hassett's weighted stable curves, and (2) Hacking-Keel-Tevelev's stable hyperplane arrangements.
In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the…