591 research outputs found
On the existence of most-preferred alternatives in complete lattices
If a preference ordering on a complete lattice is quasisupermodular, or just satisfies a rather weak analog of the condition, then it admits a maximizer on every subcomplete sublattice if and only if it admits a maximizer on every subcomplete subchainlattice optimization; quasisupermodularity
Experimental study of energy-minimizing point configurations on spheres
In this paper we report on massive computer experiments aimed at finding
spherical point configurations that minimize potential energy. We present
experimental evidence for two new universal optima (consisting of 40 points in
10 dimensions and 64 points in 14 dimensions), as well as evidence that there
are no others with at most 64 points. We also describe several other new
polytopes, and we present new geometrical descriptions of some of the known
universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic
Optimal Packings of Superballs
Dense hard-particle packings are intimately related to the structure of
low-temperature phases of matter and are useful models of heterogeneous
materials and granular media. Most studies of the densest packings in three
dimensions have considered spherical shapes, and it is only more recently that
nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs
(whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a
versatile family of convex particles (p >= 0.5) with both cubic- and
octahedral-like shapes as well as concave particles (0 < p < 0.5) with
octahedral-like shapes. In this paper, we provide analytical constructions for
the densest known superball packings for all convex and concave cases. The
candidate maximally dense packings are certain families of Bravais lattice
packings. The maximal packing density as a function of p is nonanalytic at the
sphere-point (p = 1) and increases dramatically as p moves away from unity. The
packing characteristics determined by the broken rotational symmetry of
superballs are similar to but richer than their two-dimensional "superdisk"
counterparts, and are distinctly different from that of ellipsoid packings. Our
candidate optimal superball packings provide a starting point to quantify the
equilibrium phase behavior of superball systems, which should deepen our
understanding of the statistical thermodynamics of nonspherical-particle
systems.Comment: 28 pages, 16 figure
Maximizing a preference relation on complete chains and lattices
Maximization of a preference relation on a given family of subsets of its domain defines a choice function. Assuming the domain to be a poset or a lattice, and considering subcomplete chains or sublattices as potential feasible sets, we study conditions ensuring the existence of optima, as well as properties of the choice function conducive to monotone comparative statics. Concerning optimization on chains, quite a number of characterization results are obtained; when it comes to lattices, we mostly obtain sufficient conditions
Area laws for the entanglement entropy - a review
Physical interactions in quantum many-body systems are typically local:
Individual constituents interact mainly with their few nearest neighbors. This
locality of interactions is inherited by a decay of correlation functions, but
also reflected by scaling laws of a quite profound quantity: The entanglement
entropy of ground states. This entropy of the reduced state of a subregion
often merely grows like the boundary area of the subregion, and not like its
volume, in sharp contrast with an expected extensive behavior. Such "area laws"
for the entanglement entropy and related quantities have received considerable
attention in recent years. They emerge in several seemingly unrelated fields,
in the context of black hole physics, quantum information science, and quantum
many-body physics where they have important implications on the numerical
simulation of lattice models. In this Colloquium we review the current status
of area laws in these fields. Center stage is taken by rigorous results on
lattice models in one and higher spatial dimensions. The differences and
similarities between bosonic and fermionic models are stressed, area laws are
related to the velocity of information propagation, and disordered systems,
non-equilibrium situations, classical correlation concepts, and topological
entanglement entropies are discussed. A significant proportion of the article
is devoted to the quantitative connection between the entanglement content of
states and the possibility of their efficient numerical simulation. We discuss
matrix-product states, higher-dimensional analogues, and states from
entanglement renormalization and conclude by highlighting the implications of
area laws on quantifying the effective degrees of freedom that need to be
considered in simulations.Comment: 28 pages, 2 figures, final versio
Universal optimality of the and Leech lattices and interpolation formulas
We prove that the root lattice and the Leech lattice are universally
optimal among point configurations in Euclidean spaces of dimensions and
, respectively. In other words, they minimize energy for every potential
function that is a completely monotonic function of squared distance (for
example, inverse power laws or Gaussians), which is a strong form of robustness
not previously known for any configuration in more than one dimension. This
theorem implies their recently shown optimality as sphere packings, and broadly
generalizes it to allow for long-range interactions.
The proof uses sharp linear programming bounds for energy. To construct the
optimal auxiliary functions used to attain these bounds, we prove a new
interpolation theorem, which is of independent interest. It reconstructs a
radial Schwartz function from the values and radial derivatives of and
its Fourier transform at the radii for integers
in and in . To prove this
theorem, we construct an interpolation basis using integral transforms of
quasimodular forms, generalizing Viazovska's work on sphere packing and placing
it in the context of a more conceptual theory.Comment: 95 pages, 6 figure
On the existence of most-preferred alternatives in complete lattices
If a preference ordering on a complete lattice is
quasisupermodular, or just satisfies a rather weak analog of the condition, then it admits a maximizer on every subcomplete sublattice if and only if it admits a maximizer on every subcomplete subchai
Non-Abelian Analogs of Lattice Rounding
Lattice rounding in Euclidean space can be viewed as finding the nearest
point in the orbit of an action by a discrete group, relative to the norm
inherited from the ambient space. Using this point of view, we initiate the
study of non-abelian analogs of lattice rounding involving matrix groups. In
one direction, we give an algorithm for solving a normed word problem when the
inputs are random products over a basis set, and give theoretical justification
for its success. In another direction, we prove a general inapproximability
result which essentially rules out strong approximation algorithms (i.e., whose
approximation factors depend only on dimension) analogous to LLL in the general
case.Comment: 30 page
Metric and latticial medians
This paper presents the -linked- notions of metric and latticial medians and it explains what is the median procedure for the consensus problems, in particular in the case of the aggregation of linear orders. First we consider the medians of a v-tuple of arbitrary or particular binary relations.. Then we study in depth the difficult (in fact NP-difficult) problem of finding the median orders of a profile of linear orders. More generally, we consider the medians of v-tuples of elements of a semilattice and we describe the median semilattices, i.e. the semilattices were medians are easily computable.Ce texte présente les notions -reliées- de médianes métriques et latticielles et explique le rôle de la procédure médiane dans les problèmes de consensus, notamment dans le cas de l'agrégation d'ordres totaux.. Après avoir étudié les médianes d'un v-uple de relations binaires arbitraires ou particulières, on étudie en détail le problème -difficile (NP-difficile)- d'obtention des ordres médians d'un profil d'ordres totaux. Plus généralement on considère les médianes de v-uples d'éléments d'un demi-treillis (ou d'un treillis) et l'on décrit les demi-treillis à médianes,i.e. ceux où l'obtention des médianes est aisée
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