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We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra $\mathfrak {gl}(n,m)$, depending on a parameter. We show that for generic values of the parameter all the orbits are…
We present a generalisation of the theory of iterated function systems and associated fractals to the setting of noncommutative geometry. Along the way, we discuss some ideas surrounding locally compact noncommutative metric spaces.
In this paper we discuss a new method to blend fractal attractors using the code map for the IFS formed by the Hutchinson--Barnsley operators of a finite family of hyperbolic IFSs. We introduce a parameter called blending coefficient to…
We examine various consequences of the existence of exceptional representations of an irreducible Weyl group. (These are notes from a talk in the MIT Lie groups seminar.)
It is shown that it is now possible to include van der Waals interactions via a nonempirical implementation of density functional theory to describe the correlation energy in electronic structure calculations on infinite systems of no…
In our previous work, On the localization of Hutchinson-Barnsley fractals, Chaos Solitons Fractals, 173 (2023), 113-674, we presented a method for finding a finite family of closed balls whose union contains the attractor of a given…
We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups $\widehat{PGL}(N)$, which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups $(\widehat{W}\times…
Let W be a Weyl group, presented as a crystallographic reflection group on a Euclidean vector space V, and C an open Weyl chamber. In a recent paper, Waldspurger proved that the images (id-w)(C), for Weyl group elements w, are all disjoint,…
We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized…
Random Wavelet Series form a class of random processes with multifractal properties. We give three applications of this construction. First, we synthesize a random function having any given spectrum of singularities satisfying some…
This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These…
In this short note we discuss the interplay between finite Coxeter groups and construction of wavelet sets, generalized multiresolution analysis and sampling.
The construction of fractal generalized zone plates (FraGZPs) from a set of periodic diffractive optical elements with circular symmetry is proposed. This allows us to increase the number of foci of a conventional fractal zone plate…
We study some aspects of noncommutative differential geometry on a finite Weyl group in the sense of S. Woronowicz, K. Bresser {\it et al.}, and S. Majid. For any finite Weyl group $W$ we consider the subalgebra generated by flat…
The natural kinship between classical theories of interpolation and approximation is well explored. In contrast to this, the interrelation between interpolation and approximation is subtle and this duality is relatively obscure in the…
We present some work relating to fractal transformations on masked iterated function systems and demonstrate how well known algorithms for generating fractal transformations can be modifed for these systems. We also demonstrate that these…
We demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities. As an interesting example we study the statistical properties of complex frequencies of the fully chaotic…
The theory of graphons has proven to be a powerful tool in many areas of graph theory. In this paper, we introduce several foundational aspects of the theory of digraphons -- asymmetric two-variable functions that arise as limits of…
General invariants of a geometric mapping of a symmetric affine connection space are obtained in this paper. These invariants are generalizations of the previous obtained basic invariants (see [16]). Moreover, these invariants are related…
In this contribution we investigate several extensions of the powerset that comprise arbitrarily nested subsets, and call them superpower set. This allows the definition of graphs with possibly infinitely nested nodes. additionally we…