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1.
Numerical estimates of the Kolmogorov-Sinai entropy based on a finite amount of data decay towards zero in the relevant limits. Rewriting differences of block entropies as averages over decay rates, and ignoring all parts of the sample where these rates are uncomputable because of the lack of neighbours, yields improved entropy estimates. In the same way, the scaling range for estimates of the information dimension can be extended considerably. The improvement is demonstrated for experimental data. (c) 1996 American Institute of Physics. 相似文献
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3.
Mathieu Dutour Sikirić Achill Schürmann Frank Vallentin 《Discrete and Computational Geometry》2010,44(4):904-911
The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the
contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits. 相似文献
4.
For a centrally symmetric convex
and a covering lattice L for K, a lattice polygon P is called a covering polygon, if
. We prove that P is a covering polygon, if and only if its boundary bd(P) is covered by (L ∩ P) + K. Further we show that this characterization is false for non-symmetric planar convex bodies and
in Euclidean d–space, d ≥ 3, even for the unit ball K = B
d.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
5.
H. Lutz J.D. Gunton H.K. Schurmann J.E. Crow T. Mihalisin 《Solid State Communications》1974,14(11):1075-1078
The depression of Tc for Ni films is found to be proportional to n?λ where n is the number of atomic layers and λ = 1.33±0.13. This suggests that λ may equal where v is the correlation length exponent. 相似文献
6.
Achill Schürmann 《Periodica Mathematica Hungarica》2006,53(1-2):257-264
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be a discrete subset of Euclidean $d$-space. We allow
subsequently continuous movements of single elements, whenever the minimum
distance to other elements does not decrease. We discuss the question, if it is
possible to move all elements of $X$ in this way, for example after removing a
finite subset $Y$ from $X$. Although it is not possible in general, we show the
existence of such finite subsets $Y$ for many discrete sets $X$, including all
lattices. We define the \textit{instability degree} of $X$ as the minimum
cardinality of such a subset $Y$ and show that the maximum instability degree
among lattices is attained by perfect lattices. Moreover, we discuss the
$3$-dimensional case in detail. 相似文献
7.
We show that the shapes of convex bodies containing m translates of a convex body K, so that their Minkowskian surface area is minimum, tends for growing m to a convex body L.Received: 7 January 2002 相似文献
8.
Achill Schürmann 《Advances in Mathematics》2010,225(5):2546-2564
We introduce a parameter space for periodic point sets, given as unions of m translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension d?8 and d=24. 相似文献
9.
We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies. 相似文献
10.
We describe algorithms which address two classical problems in lattice
geometry: the lattice covering and the simultaneous lattice
packing-covering problem. Theoretically our algorithms solve the two
problems in any fixed dimension d in the sense that they approximate
optimal covering lattices and optimal packing-covering lattices within
any desired accuracy. Both algorithms involve semidefinite
programming and are based on Voronoi's reduction theory for positive
definite quadratic forms, which describes all possible Delone
triangulations of ℤd. In practice, our implementations reproduce known results in dimensions d ≤ 5 and in particular solve the two problems in
these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are
closely related to the lattice E*6. For d = 7,8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort
to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices
are locally optimal. 相似文献