Related papers: Periodic Molecular Arrangements from Space Groups
We construct a family of finitely generated infinite periodic groups. The basic example is a 2-group, called the tetrahedron group. We generalize the construction by suggesting a family of infinite finitely generated dice groups. We provide…
We present local distributed, stochastic algorithms for \emph{alignment} in self-organizing particle systems (SOPS) on two-dimensional lattices, where particles occupy unique sites on the lattice, and particles can make spatial moves to…
Understanding the geometry and topology of configuration or conformational spaces of molecules has relevant applications in chemistry and biology such as the proteins folding problem, drug design and the structure activity relationship…
A combinatorial model of molecular conformational space that was previously developped (J. Gabarro-Arpa, Comp. Biol. and Chem. 27, (2003) 153-159), had the drawback that structures could not be properly embedded beacause it lacked explicit…
A spin space group provides a suitable way to fully exploit the symmetry of a spin arrangement with a negligible spin-orbit coupling. There has been a growing interest in applying spin symmetry analysis with the spin space group in the…
Freezing of charged colloids on square or triangular two-dimensional periodic substrates has been recently shown to realize a rich variety of orientational orders. We propose a theoretical framework to analyze the corresponding structures.…
We construct a class of periodic tilings of the plane, which corresponds to toroidal arrangements of trivalent atoms, with pentagonal, hexagonal and heptagonal rings. Each tiling is characterized by a set of four integers and determines a…
We consider an infinitesimal volume where there are many rigid molecules of the same kind, and discuss the description and classification of the local anisotropy in this volume by tensors. First, we examine the symmetry of a rigid molecule,…
In this paper, we investigate periodic vibrations of a group of particles with a dihedral configuration in the plane governed by the Lennard-Jones and Coulomb forces. Using the gradient equivariant degree, we provide a full topological…
Probabilities to find a chosen number of electrons in flexible domains of space are calculated for highly correlated wave functions. Quantum mechanics can produce higher probabilities for chemically relevant arrangements of electrons in…
This paper proposes a methodology to stabilize relative equilibria in a model of identical, steered particles moving in three-dimensional Euclidean space. Exploiting the Lie group structure of the resulting dynamical system, the…
The past decade has witnessed a surge of interest in exploring emergent particles in condensed matter systems. Novel particles, emerged as excitations around exotic band degeneracy points, continue to be reported in real materials and…
The traditional display of elements in the periodic table is convenient for the study of chemistry and physics. However, the atomic number alone is insufficient for training statistical machine learning models to describe and extract…
All leptons, quarks, and gauge bosons can be placed in the periodic table of elementary particles. The periodic table is derived from dualities of string theory and a Kaluza-Klein substructure for the six extra spatial dimensions. As a…
The geometry of atomic arrangement underpins the structural understanding of molecules in many fields. However, no general framework of mathematical/computational theory for the geometry of atomic arrangement exists. Here we present…
An algorithm for detecting unstable periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp.…
We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples…
Modeling potential alloys requires the exploration of all possible configurations of atoms. Additionally, modeling the thermal properties of materials requires knowledge of the possible ways of displacing the atoms. One solution to finding…
A theory of structure is formulated for systems of many structureless classical particles with stable local interactions in Euclidean space. Such systems are shown to have their structure in thermodynamic equilibrium determined exactly by a…
We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational…