67 research outputs found
Simplicial Ricci Flow
We construct a discrete form of Hamilton's Ricci flow (RF) equations for a
d-dimensional piecewise flat simplicial geometry, S. These new algebraic
equations are derived using the discrete formulation of Einstein's theory of
general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation
is naturally associated to each edge, L, of a simplicial lattice. In defining
this equation, we find it convenient to utilize both the simplicial lattice, S,
and its circumcentric dual lattice, S*. In particular, the RRF equation
associated to L is naturally defined on a d-dimensional hybrid block connecting
with its (d-1)-dimensional circumcentric dual cell, L*. We show that
this equation is expressed as the proportionality between (1) the simplicial
Ricci tensor, Rc_L, associated with the edge L in S, and (2) a certain volume
weighted average of the fractional rate of change of the edges, lambda in L*,
of the circumcentric dual lattice, S*, that are in the dual of L. The inherent
orthogonality between elements of S and their duals in S* provide a simple
geometric representation of Hamilton's RF equations. In this paper we utilize
the well established theories of Regge calculus, or equivalently discrete
exterior calculus, to construct these equations. We solve these equations for a
few illustrative examples.Comment: 34 pages, 10 figures, minor revisions, DOI included: Commun. Math.
Phy
Graphical Designs and Gale Duality
A graphical design is a subset of graph vertices such that the weighted
averages of certain graph eigenvectors over the design agree with their global
averages. We use Gale duality to show that positively weighted graphical
designs in regular graphs are in bijection with the faces of a generalized
eigenpolytope of the graph. This connection can be used to organize, compute
and optimize designs. We illustrate the power of this tool on three families of
Cayley graphs -- cocktail party graphs, cycles, and graphs of hypercubes -- by
computing or bounding the smallest designs that average all but the last
eigenspace in frequency order. We also prove that unless NP = coNP, there
cannot be an efficient description of all minimal designs that average a fixed
number of eigenspaces in a graph.Comment: 30 pages, 14 figures, 1 tabl
Geometrical Frustration and Static Correlations in Hard-Sphere Glass Formers
We analytically and numerically characterize the structure of hard-sphere
fluids in order to review various geometrical frustration scenarios of the
glass transition. We find generalized polytetrahedral order to be correlated
with increasing fluid packing fraction, but to become increasingly irrelevant
with increasing dimension. We also find the growth in structural correlations
to be modest in the dynamical regime accessible to computer simulations.Comment: 21 pages; part of the "Special Topic Issue on the Glass Transition
The double-funnel energy landscape of the 38-atom Lennard-Jones cluster
The 38-atom Lennard-Jones cluster has a paradigmatic double-funnel energy
landscape. One funnel ends in the global minimum, a face-centred-cubic (fcc)
truncated octahedron. At the bottom of the other funnel is the second lowest
energy minimum which is an incomplete Mackay icosahedron. We characterize the
energy landscape in two ways. Firstly, from a large sample of minima and
transition states we construct a disconnectivity tree showing which minima are
connected below certain energy thresholds. Secondly we compute the free energy
as a function of a bond-order parameter. The free energy profile has two
minima, one which corresponds to the fcc funnel and the other which at low
temperature corresponds to the icosahedral funnel and at higher temperatures to
the liquid-like state. These two approaches show that the greater width of the
icosahedral funnel, and the greater structural similarity between the
icosahedral structures and those associated with the liquid-like state, are the
cause of the smaller free energy barrier for entering the icosahedral funnel
from the liquid-like state and therefore of the cluster's preferential entry
into this funnel on relaxation down the energy landscape. Furthermore, the
large free energy barrier between the fcc and icosahedral funnels, which is
energetic in origin, causes the cluster to be trapped in one of the funnels at
low temperature. These results explain in detail the link between the
double-funnel energy landscape and the difficulty of global optimization for
this cluster.Comment: 12 pages, 11 figures, revte
Morphophoric POVMs, generalised qplexes, and 2-designs
We study the class of quantum measurements with the property that the image
of the set of quantum states under the measurement map transforming states into
probability distributions is similar to this set and call such measurements
morphophoric. This leads to the generalisation of the notion of a qplex, where
SIC-POVMs are replaced by the elements of the much larger class of morphophoric
POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The
intrinsic geometry of a generalised qplex is the same as that of the set of
quantum states, so we explore its external geometry, investigating, inter alia,
the algebraic and geometric form of the inner (basis) and the outer (primal)
polytopes between which the generalised qplex is sandwiched. In particular, we
examine generalised qplexes generated by MUB-like 2-design POVMs utilising
their graph-theoretical properties. Moreover, we show how to extend the primal
equation of QBism designed for SIC-POVMs to the morphophoric case.Comment: 27 pages, 5 figure
Applications of finite reflection groups in Fourier analysis and symmetry breaking of polytopes
Cette thèse présente une étude des applications des groupes de réflexion finis aux problems liés aux réseaux bidimensionnels et aux polytopes tridimensionnels. Plusieurs familles de fonctions orbitales, appelées fonctions orbitales de Weyl, sont associées aux groupes de réflexion cristallographique. Les propriétés exceptionnelles de ces fonctions, telles que l’orthogonalité continue et discrète, permettent une analyse de type Fourier sur le domaine fondamental d’un groupe de Weyl affine correspondant. Dans cette considération, les fonctions d’orbite de Weyl constituent des outils efficaces pour les transformées discrètes de type Fourier correspondantes connues sous le nom de transformées de Fourier–Weyl. Cette recherche limite notre attention aux fonctions d’orbite de Weyl symétriques et antisymétriques à deux variables du groupe de réflexion cristallographique A2. L’objectif principal est de décomposer deux types de transformations de Fourier–Weyl du réseau de poids correspondant en transformées plus petites en utilisant la technique de division centrale. Pour les cas non cristallographiques, nous définissons les indices de degré pair et impair pour les orbites des groupes de réflexion non cristallographique avec une symétrie quintuple en utilisant un remplacement de représentation-orbite. De plus, nous formulons l’algorithme qui permet de déterminer les structures de polytopes imbriquées. Par ailleurs, compte tenu de la pertinence de la symétrie icosaédrique pour la description de diverses molécules sphériques et virus, nous étudions la brisure de symétrie des polytopes doubles de type non cristallographique et des structures tubulaires associées. De plus, nous appliquons une procédure de stellation à la famille des polytopes considérés. Puisque cette recherche se concentre en partie sur les fullerènes icosaédriques, nous présentons la construction des nanotubes de carbone correspondants. De plus, l’approche considérée pour les cas non cristallographiques est appliquée aux structures cristallographiques. Nous considérons un mécanisme de brisure de symétrie appliqué aux polytopes obtenus en utilisant les groupes Weyl tridimensionnels pour déterminer leurs extensions structurelles possibles en nanotubes.This thesis presents a study of applications of finite reflection groups to the problems related to two-dimensional lattices and three-dimensional polytopes. Several families of orbit functions, known as Weyl orbit functions, are associated with the crystallographic reflection groups. The exceptional properties of these functions, such as continuous and discrete orthogonality, permit Fourier-like analysis on the fundamental domain of a corresponding affine Weyl group. In this consideration, Weyl orbit functions constitute efficient tools for corresponding Fourier-like discrete transforms known as Fourier–Weyl transforms. This research restricts our attention to the two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2. The main goal is to decompose two types of the corresponding weight lattice Fourier–Weyl transforms into smaller transforms using the central splitting technique. For the non-crystallographic cases, we define the even- and odd-degree indices for orbits of the non-crystallographic reflection groups with 5-fold symmetry by using a representation-orbit replacement. Besides, we formulate the algorithm that allows determining the structures of nested polytopes. Moreover, in light of the relevance of the icosahedral symmetry to the description of various spherical molecules and viruses, we study symmetry breaking of the dual polytopes of non-crystallographic type and related tube-like structures. As well, we apply a stellation procedure to the family of considered polytopes. Since this research partly focuses on the icosahedral fullerenes, we present the construction of the corresponding carbon nanotubes. Furthermore, the approach considered for the non-crystallographic cases is applied to crystallographic structures. We consider a symmetry-breaking mechanism applied to the polytopes obtained using the three-dimensional Weyl groups to determine their possible structural extensions into nanotubes
Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods
Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym-
metry consisting of only pentagons and hexagons faces. It has been the object
of interest for chemists and mathematicians due to its widespread application
in various fields, namely including electronic and optic engineering, medical sci-
ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity
of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751
isomers. The Fries and Clar numbers are stability predictors of a Fullerene
molecule. These number can be computed by solving a (possibly N P -hard)
combinatorial optimization problem. We propose several ILP formulation of
such a problem each yielding a solution algorithm that provides the exact value
of the Fries and Clar numbers. We compare the performances of the algorithm
derived from the proposed ILP formulations. One of this algorithm is used to
find the Clar isomers, i.e., those for which the Clar number is maximum among
all isomers having a given size. We repeated this computational experiment for
all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774
isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path
following (PDIPF) interior point (IP) algorithm for separable convex quadratic
minimum cost flow network problem. In each iteration of PDIPF algorithm, the
main computational effort is solving the underlying Newton search direction
system. We concentrated on finding the solution of the corresponding linear
system iteratively and inexactly. We assumed that all the involved inequalities
can be solved inexactly and to this purpose, we focused on different approaches
for distributing the error generated by iterative linear solvers such that the
convergences of the PDIPF algorithm are guaranteed. As a result, we achieved
theoretical bases that open the path to further interesting practical investiga-
tion
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