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51.
In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume
of the body. More precisely, he showed that
#(K?\Bbb Zn) £ n! vol(K)+n\#(K\cap{\Bbb Z}^n)\le n! {\rm vol}(K)+n
, whenever
K ì \Bbb RnK\subset{\Bbb R}^n
is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely
#(K?\Bbb Zn) < vol(K) + ((?n+1)/2) (n-1)! F(K)\#(K\cap{\Bbb Z}^n) < {\rm vol}(K) + ((\sqrt{n}+1)/2) (n-1)! {\rm F}(K)
. The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where
t ? \Bbb Rn\\Bbb Znt\in{\Bbb R}^n\setminus{\Bbb Z}^n
and P is an n-dimensional polytope with integral vertices. Then we have
#((t+P)?\Bbb Zn) £ n! vol(P)\#((t+P)\cap{\Bbb Z}^n)\le n! {\rm vol}(P)
.
Moreover, in the 3-dimensional case we prove a stronger inequality, namely
#(K?\Bbb Zn) < vol(K) + 2 F(K)\#(K\cap{\Bbb Z}^n)< {\rm vol}(K) + 2 {\rm F}(K)
. 相似文献
52.
Peter Wills 《Physics letters. [Part B]》1985,150(4):277-281
The one loop, large Higgs mass corrections to the ? parameter are calculated in a manifestly supersymmetric way, in a model with soft breaking in which Yukawa couplings are neglected. We find that the Higgs decouples from ? and, in a phenomenologically viable class of models, which are a subset of the above model, the ? parameter is independent of Higgs mass. 相似文献
53.
本文报导了采用氩离子激光器来泵浦Pr3+:YLF晶体,应用声光调制器实现了主动锁模;同时应用振动─高反射平面镜也实现了被动锁模,两种锁模均得到了ps光脉冲.据作者了解这是这种晶体材料的第一次锁模运转. 相似文献
54.
55.
We present a general method for the linear least-squares solutionof overdetermined and underdetermined systems. The method isparticularly efficient when the coefficient matrix is quasi-square,that is when the number of rows and number of columns is almostthe same. The numerical methods for linear least-squares problemsand minimum-norm solutions do not generally take account ofthis special characteristic. The proposed method is based onLU factorization of the original quasi-square matrix A, assumingthat A has full rank. In the overdetermined case, the LU factorsare used to compute a basis for the null space of AT. The right-handside vector b is then projected onto this subspace and the least-squaressolution is obtained from the solution of this reduced problem.In the case of underdetermined systems, the desired solutionis again obtained through the solution of a reduced system.The use of this method may lead to important savings in computationaltime for both dense and sparse matrices. It is also shown inthe paper that, even in cases where the matrices are quite small,sparse solvers perform better than dense solvers. Some practicalexamples that illustrate the use of the method are included. 相似文献
56.
J. M. Wills 《Periodica Mathematica Hungarica》1991,22(3):139-146
We consider finite packings of unit-balls in Euclidean 3-spaceE
3 where the centres of the balls are the lattice points of a lattice polyhedronP of a given latticeL
3E3. In particular we show that the facets ofP induced by densest sublattices ofL
3 are not too close to the next parallel layers of centres of balls. We further show that the Dirichlet-Voronoi-cells are comparatively small in this direction. The paper was stimulated by the fact that real crystals in general grow slowly in the directions normal to these dense facets.The results support, to some extent, the hypothesis that real crystals grow preferably such that they need little volume, i.e that they are locally dense.Dedicated to A. Florian on the occasion of this 60th birthday 相似文献
57.
58.
Two isoperimetric inequalities with lattice constraints for arbitrary lattices in the euclidean plane are proved. They generalize previous results by Hadwiger et al. for the special lattice
d
(but all dimensionsd) to general lattices. 相似文献
59.
Ray L. Frost Daria L. Wain Rachael-Anne Wills Anthony Musemeci Wayde Martens 《Thermochimica Acta》2006,443(1):56-61
Thermogravimetry in tandem with mass spectrometry has been used to characterise the thermal decomposition of synthetic alunites of potassium, sodium and ammonium. Three mechanisms of decomposition are observed (a) dehydration, (b) dehydroxylation and (c) desulphation. The thermal decomposition of the three alunites is different. For NH4-alunite, an additional process of de-ammoniation is observed which occurs simultaneously with dehydration. Dehydroxylation takes place in a series of four steps. De-sulphation occurs for K-alunite at 680 °C in a single step in comparison with Na and NH4 alunites where de-sulphation is observed in a series of four steps. The temperature of desulphation is cation dependent. The thermal decomposition is not completed until around 800 °C. 相似文献
60.