279 research outputs found
Geometrical Frustration: A Study of 4d Hard Spheres
The smallest maximum kissing-number Voronoi polyhedron of 3d spheres is the
icosahedron and the tetrahedron is the smallest volume that can show up in
Delaunay tessalation. No periodic lattice is consistent with either and hence
these dense packings are geometrically frustrated. Because icosahedra can be
assembled from almost perfect tetrahedra, the terms "icosahedral" and
"polytetrahedral" packing are often used interchangeably, which leaves the true
origin of geometric frustration unclear. Here we report a computational study
of freezing of 4d hard spheres, where the densest Voronoi cluster is compatible
with the symmetry of the densest crystal, while polytetrahedral order is not.
We observe that, under otherwise comparable conditions, crystal nucleation in
4d is less facile than in 3d. This suggest that it is the geometrical
frustration of polytetrahedral structures that inhibits crystallization.Comment: 4 pages, 3 figures; revised interpretatio
On colouring point visibility graphs
In this paper we show that it can be decided in polynomial time whether or
not the visibility graph of a given point set is 4-colourable, and such a
4-colouring, if it exists, can also be constructed in polynomial time. We show
that the problem of deciding whether the visibility graph of a point set is
5-colourable, is NP-complete. We give an example of a point visibility graph
that has chromatic number 6 while its clique number is only 4
Inconsistency of set theory via evaluation
We introduce in an axiomatic way the categorical theory PR of primitive recursion as the initial cartesian category with Natural Numbers Object. This theory has an extension into constructive set theory S of primitive recursion with abstraction of predicates into subsets and two-valued (boolean) truth algebra. Within the framework of (typical) classical, quantified set theory T we construct an evaluation of arithmetised theory PR via Complexity Controlled Iteration with witnessed termination of the iteration, witnessed termination by availability of Hilbert s iota operator in set theory. Objectivity of that evaluation yields inconsistency of set theory T by a liar (anti)diagonal argument
On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States
We address the problem of constructing positive operator-valued measures
(POVMs) in finite dimension consisting of operators of rank one which
have an inner product close to uniform. This is motivated by the related
question of constructing symmetric informationally complete POVMs (SIC-POVMs)
for which the inner products are perfectly uniform. However, SIC-POVMs are
notoriously hard to construct and despite some success of constructing them
numerically, there is no analytic construction known. We present two
constructions of approximate versions of SIC-POVMs, where a small deviation
from uniformity of the inner products is allowed. The first construction is
based on selecting vectors from a maximal collection of mutually unbiased bases
and works whenever the dimension of the system is a prime power. The second
construction is based on perturbing the matrix elements of a subset of mutually
unbiased bases.
Moreover, we construct vector systems in \C^n which are almost orthogonal
and which might turn out to be useful for quantum computation. Our
constructions are based on results of analytic number theory.Comment: 29 pages, LaTe
Recommended from our members
Experimental measurements of plasma properties for Miller SG-100 torch with Mach I setting
In this work measurements of plasma properties, including the fields of temperature, velocity and plasma composition have been completed for the Miller SG-100 plasma torch using argon-helium mixtures with the Mach 1 nozzle at 1 atm pressure. A computer-controlled system combining both spectroscopic and enthalpy probe diagnostics has been developed to allow temperature measurements covering a range from 2000--16000K which includes the plasma flame region which is of interest. The experimental results expose the dominant effects in different spatial areas of argon-helium plasma jets. In the center near the nozzle exit the temperatures exceed 10,000K, and strong diffusion exists due to the steep radial gradients of temperature and particle number densities. In the jet tail region where the temperatures are well below 10,000K and decay in axial and radial direction, the dominant effects in this area are strong cold gas entrainment associated with turbulence. Substantial discrepancies between temperatures evaluated from spectroscopic and enthalpy probe data are particularly severe in the jet fringes indicating that strong deviations from LTE may exist in the jet fringes. In addition, entrainment of the cold surrounding gas into the plasma jet causes severe discrepancies between spectrometric and enthalpy probe data. The validity of the two diagnostic methods will be discussed. The temperature profiles in argon-helium plasma jets are flatter and wider, and the velocities are higher than in a pure argon plasma jet. These features of argon-helium plasma jets may be beneficial for obtaining better performance in the plasma spraying process. 26 refs., 28 figs., 2 tabs
The strong thirteen spheres problem
The thirteen spheres problem is asking if 13 equal size nonoverlapping
spheres in three dimensions can touch another sphere of the same size. This
problem was the subject of the famous discussion between Isaac Newton and David
Gregory in 1694. The problem was solved by Schutte and van der Waerden only in
1953.
A natural extension of this problem is the strong thirteen spheres problem
(or the Tammes problem for 13 points) which asks to find an arrangement and the
maximum radius of 13 equal size nonoverlapping spheres touching the unit
sphere. In the paper we give a solution of this long-standing open problem in
geometry. Our computer-assisted proof is based on a enumeration of the
so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag
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