Related papers: Real and complexified configuration spaces for sph…
The modelling of discrete regulatory networks combines a graph specifying the pairwise influences between the variables of the system, and a parametrisation from which can be derived a discrete transition system. Given the influence graph…
We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these…
Exact diagonalization (ED) is one of the most reliable and established numerical methods of quantum many-body theory. The main limiting factor of the method is the exponential scaling of Hilbert space dimension with system size.…
In the context of higher gauge theory, we construct a flat and fake flat 2-connection, in the configuration space of $n$ particles in the complex plane, categorifying the Knizhnik-Zamolodchikov connection. To this end, we define the…
In this paper, we explore the interaction between two monoidal structures: a multiplicative one, for the encoding of pairing, and an additive one, for the encoding of choice. We propose a colored PROP to model computation in this framework,…
We propose a geometric explanation for the observation that generic quadratic polynomials over split quaternions may have up to six different factorizations while generic polynomials over Hamiltonian quaternions only have two. Split…
A planar kinematic chain consists of $n$ links connected by joints. In this work we investigate the space of configurations, described in terms of joint angles, that guarantee that the kinematic chain is closed. We give explicit formulas…
We present a new method for solving parametrized linear systems. Under certain assumptions on the parametrization, solutions to the linear systems for all parameters can be accurately approximated by linear combinations of solutions to…
Orthogonal matrices which are linear combinations of permutation matrices have attracted enormous attention in quantum information and computation. In this paper, we provide a complete parametric characterization of all complex, real and…
In this paper we present a complete classification (isomorphism classes with some isomorphism invariants) of complex associative algebras up to dimension five (including both cases: unitary and non-unitary). In some symbolic computations we…
Multipolar expansions are a foundational tool for describing basis functions in quantum mechanics, many-body polarization, and other distributions on the unit sphere. Progress on these topics is often held back by complicated and competing…
The purpose of this paper is twofold. First, we introduce the notions of left-symmetric and left alternative structures on superspaces in characteristic 2. We describe their main properties and classify them in dimension 2. We show that…
This is a brief review of my work on the correspondence between four-dimensional $\mathcal{N} = 1$ supersymmetric field theories realized by brane tilings and two-dimensional integrable lattice models. I explain how to construct integrable…
A kinematic chain in three-dimensional Euclidean space consists of $n$ links that are connected by spherical joints. Such a chain is said to be within a closed configuration when its link lengths form a closed polygonal chain in three…
The observational characteristics of a linear structural equation model can be effectively described by polynomial constraints on the observed covariance matrix. However, these polynomials can be exponentially large, making them impractical…
We compute all isomorphism classes of simplicial arrangements in the real projective plane with up to 27 lines. It turns out that Gr\"unbaums catalogue is complete up to 27 lines except for four new arrangements with 22, 23, 24, 25 lines,…
The total homology of the loop space of the configuration space of ordered distinct n points in R^m has a structure of a Hopf algebra defined by the 4-term relations if m>2. We describe a relation of between the cohomology of this loop…
This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex…
Linear operations on coefficients in the spherical harmonics (SH) transform domain that again yield SH-domain coefficients are an important toolset in many disciplines of research and engineering. They comprise rotations, spatially…
The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere $\SS^q$ ($q\ge 2$). The approximation and localization properties of our…