Related papers: Periodic Molecular Arrangements from Space Groups
We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach…
This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank $23$ decompositions for the $3\times 3$ matrix multiplication tensor $M_{\langle 3\rangle}$. All our decompositions…
Systems of identical particles with equal charge are studied under a special type of confinement. These classical particles are free to move inside some convex region S and on the boundary of it $\Omega$ (the $S^{d-1}-$ sphere, in our…
Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the $\nabla u=f(u)$ Poisson's equation, which has a subalgebra isomorphic to the $3-$dimensional special Euclidean group ${\rm SE}(3)$ or group of…
The ring configurations for classical two-dimensional atoms are calculated within the Thomson model and compared with the results from `exact' numerical simulations. The influence of the functional form of the confinement potential and the…
We present a novel numerical method to calculate periodic orbits for dynamical systems by an iterative process which is based directly on the action integral in classical mechanics. New solutions are obtained for the planar motion of three…
Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…
In the design and analysis of composite materials based on periodic arrangements of sub-units it is of paramount importance to control the emergent material symmetry in relation to the elastic response. The target material symmetry plays…
Symmetry is at the heart of much of mathematics, physics, and art. Traditional geometric symmetry groups are defined in terms of isometries of the ambient space of a shape or pattern. If we slightly generalize this notion to allow the…
Physical systems exhibiting stochastic or chaotic behavior are often amenable to treatment by random matrix models. In deciding on a good choice of model, random matrix physics is constrained and guided by symmetry considerations. The…
This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's…
The numerical optimized shooting method for finding periodic orbits in nonlinear dynamical systems was employed to determine the existence of periodic orbits in the well-known R\"ossler system. By optimizing the period $T$ and the three…
Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the…
The detailed investigation of electronic and magnetic properties of polymeric coordination materials with accurate ab initio quantum mechanical methods is often computationally extremely demanding because of the large number of atoms in the…
We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties ${\mathcal L}$ of…
The existence, uniqueness, and asymptotic stability of modulo periodic Poisson stable solutions of dynamic equations on a periodic time scale are investigated. The model under investigation involves a term which is constructed via a Poisson…
By folding an autonomous system of rational equations in the plane to a scalar difference equation, we show that the rational system has coexisting periodic orbits of all possible periods as well as stable aperiodic orbits for certain…
A subspace arrangement is a finite collection of affine subspaces in $\mathbb{R}^n$. One of the main problems associated to arrangements asks up to what extent the topological invariants of the union of these spaces, and of their complement…
In this paper we study the rotation and spatial inversion symmetry of regular tetrahedron. We obtain the representation matrix, multiplication table,the order of all group elements, all possible combinations of generator elements, the…
Using a global optimization approach that directly searches for the composition of greatest stability, we have been able to find the particularly stable structures for binary Lennard-Jones clusters with up to 100 atoms for a range of…