376,099 research outputs found
The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve
This article sketches the proofs of two theorems about sphere packings in
Euclidean 3-space. The first is K. Bezdek's strong dodecahedral conjecture: the
surface area of every bounded Voronoi cell in a packing of balls of radius 1 is
at least that of a regular dodecahedron of inradius 1. The second theorem is L.
Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of
congruent balls such that each ball is touched by twelve others consists of
hexagonal layers. Both proofs are computer assisted. Complete proofs of these
theorems appear in the author's book "Dense Sphere Packings" and a related
preprintComment: The citations and title have been update
On the weak-hash metric for boundedly finite integer-valued measures
It is known that the space of boundedly finite integer-valued measures on a
complete separable metric space becomes itself a complete separable metric
space when endowed with the weak-hash metric. It is also known that convergence
under this topology can be characterised in a way that is similar to the weak
convergence of totally finite measures. However, the original proofs of these
two fundamental results assume that a certain term is monotonic, which is not
the case as we give a counterexample. We manage to clarify these original
proofs by addressing specifically the parts that rely on this assumption and
finding alternative arguments.Comment: Minor typos corrected, Bulletin of the Australian Mathematical
Society, 201
Mackey-complete spaces and power series -- A topological model of Differential Linear Logic
In this paper, we have described a denotational model of Intuitionist Linear
Logic which is also a differential category. Formulas are interpreted as
Mackey-complete topological vector space and linear proofs are interpreted by
bounded linear functions. So as to interpret non-linear proofs of Linear Logic,
we have used a notion of power series between Mackey-complete spaces,
generalizing the notion of entire functions in C. Finally, we have obtained a
quantitative model of Intuitionist Differential Linear Logic, where the
syntactic differentiation correspond to the usual one and where the
interpretations of proofs satisfy a Taylor expansion decomposition
Renyi-Wehrl entropies as measures of localization in phase space
We generalize the concept of the Wehrl entropy of quantum states which gives
a basis-independent measure of their localization in phase space. We discuss
the minimal values and the typical values of these R{enyi-Wehrl entropies for
pure states for spin systems. According to Lieb's conjecture the minimal values
are provided by the spin coherent states. Though Lieb's conjecture remains
unproven, we give new proofs of partial results that may be generalized for
other systems. We also investigate random pure states and calculate the mean
Renyi-Wehrl entropies averaged over the natural measure in the space of pure
quantum states.Comment: 18 pages, no figures, some improved versions of main proofs, added
J.referenc
Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry
Employing five commuting sets of five-qubit observables, we propose specific
160-661 and 160-21 state proofs of the Bell-Kochen-Specker theorem that are
also proofs of Bell's theorem. A histogram of the 'Hilbert-Schmidt' distances
between the corresponding maximal bases shows in both cases a noise-like
behaviour. The five commuting sets are also ascribed a finite-geometrical
meaning in terms of the structure of symplectic polar space W(9,2).Comment: 10 pages, 2 figure
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