共查询到20条相似文献,搜索用时 31 毫秒
1.
It is shown that a Banach space E has type p if and only for some (all) d ≥ 1 the Besov space B(1/p – 1/2)d p,p (?d ; E) embeds into the space γ (L2(?d ), E) of γ ‐radonifying operators L2(?d ) → E. A similar result characterizing cotype q is obtained. These results may be viewed as E ‐valued extensions of the classical Sobolev embedding theorems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
2.
Álvaro Lozano-Robledo 《The Ramanujan Journal》2009,19(1):53-61
In this note we present examples of elliptic curves and infinite parametric families of pairs of integers (d,d′) such that, if we assume the parity conjecture, we can show that E
d
,E
d′ and E
dd′ are all of positive even rank over ℚ. As an application, we show examples where a conjecture of M. Larsen holds.
相似文献
3.
Peter M. Gruber 《Geometriae Dedicata》1980,9(1):87-90
Let K be the space of compact subsets of E
d, endowed with the Hausdorff-metric. It is shown that the isometries of K onto itself are the mappings generated by rigid motions of E
d. 相似文献
4.
Alexandr V. Kuzminykh 《Journal of Geometry》2004,79(1-2):134-145
A family of convex bodies in Ed is called neighborly if the
intersection of every two of them is (d-1)-dimensional. In the present paper we
prove that there is an infinite neighborly family of centrally symmetric convex bodies
in Ed, d 3, such that every two of them are affinely equivalent
(i.e., there is an affine transformation mapping one of them onto another), the
bodies have large groups of affine automorphisms, and the volumes of the bodies are
prescribed. We also prove that there is an infinite neighborly family of centrally
symmetric convex bodies in Ed such that the bodies have large groups of
symmetries. These two results are answers to a problem of B. Grünbaum (1963). We
prove also that there exist arbitrarily large neighborly families of similar convex
d-polytopes in Ed with prescribed diameters and with arbitrarily large
groups of symmetries of the polytopes. 相似文献
5.
Hiroshi Maehara 《Discrete and Computational Geometry》1989,4(1):15-18
The unit distance graphE
n
is the graph whose vertices are the points in Euclideann-space, and two vertices are adjacent if and only if the distance between them is 1. We prove that for anyn there is a finite bipartite graph which cannot be embedded inE
n
as an induced subgraph and that every finite graph with maximum degreed can be embedded inE
N
,N=(d
3 –d)/2, as an induced subgraph. 相似文献
6.
In this note, first, we give a very short new proof of the theorem which yields a lower bound for the surface area of Voronoi
cells of unit ball packings in E
d
and implies Rogers' upper bound for the density of unit ball packings in E
d
for all d ≥ 2. Second we sharpen locally a classical result of Gauss by finding the locally smallest surface area Voronoi cells of
lattice unit ball packings in E
3.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
7.
Let Atf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function ME f(X) = supt∈E |Atf(x)| where E is a fixed set in IR+ and f is a radial function ∈ Lp(IRd). Let Pd = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < pd, d ? 2, and (ii) p = pd, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for ME to map radial functions in Lp to the Lorentz space LP,q. 相似文献
8.
Rolf Schneider 《Israel Journal of Mathematics》1971,9(2):241-249
Ifs is a mapping from the set of all convex bodies in Euclidean spaceE
d
toE
d
which is additive (in the sense of Minkowski), equivariant with respect to proper motions, and continuous, thens(K) is the Steiner point of the convex bodyK. 相似文献
9.
J. M. Wills 《Geometriae Dedicata》1997,65(1):117-126
The packing density of large lattice packings of spheres in Euclidean E
d
measured by the parametric density depends on the parameter and on the shape of the convex hull P of the sphere centers; in particular on the isoperimetric coefficient of P and on the second term in the Ehrhart polynomial of the lattice polytope P. We show in E
d
, d 2, that flat or spherelike polytopes generate less dense packings, whereas polytopes with suitably chosen large facets generate dense packings. This indicates that large lattice packings in E
3 of high parametric density may be good models for real crystals. 相似文献
10.
Let G be a closed, additive semigroup in a Hausdorff topological vector space. Then G is a group if and only if it satisfies natural convexity conditions of algebraic or geometric-topological type. This yields
a characterization of the geometric lattices among the discrete, additive semigroups of Euclidean d-space
\mathbbEd{\mathbb{E}^{d}} and, more generally, of direct sums of subspaces and lattices in
\mathbbEd{\mathbb{E}^{d}}. 相似文献
11.
In 1964 Grünbaum conjectured that any primitive set illuminating from within a convex body in E
d
, d ≥ 3 , has at most 2
d
points. This was confirmed by V. Soltan in 1995 for the case d = 3 . Here we give a negative answer to Grünbaum's conjecture for all d ≥ 4 , by constructing a convex body K ⊂ E
d
with primitive illuminating sets of an arbitrarily large cardinality.
Received December 1, 1997, and in revised form January 21, 1999. 相似文献
12.
Franz Aurenhammer 《Discrete and Computational Geometry》1990,5(1):243-254
A new duality between order-k Voronoi diagrams inE
d and convex hulls inE
d+1 is established. It implies a reasonably simple algorithm for computing the order-k diagram forn points in the plane inO(k
2
n logn) time and optimalO(k(n–k)) space.Research was supported by the Austrian Fond zur Foerderung der wissenschaftlichen Forschung. 相似文献
13.
Bezdek 《Discrete and Computational Geometry》2002,28(1):75-106
Abstract. The sphere packing problem asks for the densest packing of unit balls in E
d
. This problem has its roots in geometry, number theory and information theory and it is part of Hilbert's 18th problem.
One of the most attractive results on the sphere packing problem was proved by Rogers in 1958. It can be phrased as follows.
Take a regular d -dimensional simplex of edge length 2 in E
d
and then draw a d -dimensional unit ball around each vertex of the simplex. Let σ
d
denote the ratio of the volume of the portion of the simplex covered by balls to the volume of the simplex. Then the volume
of any Voronoi cell in a packing of unit balls in E
d
is at least ω
d
/σ
d
, where ω
d
denotes the volume of a d -dimensional unit ball. This has the immediate corollary that the density of any unit ball packing in E
d
is at most σ
d
. In 1978 Kabatjanskii and Levenštein improved this bound for large d . In fact, Rogers' bound is the presently known best bound for 4≤ d≤ 42 , and above that the Kabatjanskii—Levenštein bound takes over. In this paper we improve Rogers' upper bound for the density
of unit ball packings in Euclidean d -space for all d≥ 8 and improve the Kabatjanskii—Levenštein upper bound in small dimensions. Namely, we show that the volume of any Voronoi
cell in a packing of unit balls in E
d
, d≥ 8 , is at least ω
d
/
d
and so the density of any unit ball packing in E
d
, d≥ 8, is at most
d
, where
d
is a geometrically well-defined quantity satisfying the inequality
d
<σ
d
for all d≥ 8 . We prove this by showing that the surface area of any Voronoi cell in a packing of unit balls in E
d
, d≥ 8 , is at least (d⋅ω
d
)/
d
. 相似文献
14.
A (p, q)-sigraph S is an ordered pair (G, s) where G = (V, E) is a (p, q)-graph and s is a function which assigns to each edge of G a positive or a negative sign. Let the sets E
+ and E
– consist of m positive and n negative edges of G, respectively, where m + n = q. Given positive integers k and d, S is said to be (k, d)-graceful if the vertices of G can be labeled with distinct integers from the set {0, 1, ..., k + (q – 1)d such that when each edge uv of G is assigned the product of its sign and the absolute difference of the integers assigned to u and v the edges in E
+ and E
– are labeled k, k + d, k + 2d, ..., k + (m – 1)d and –k, – (k + d), – (k + 2d), ..., – (k + (n – 1)d), respectively.In this paper, we report results of our preliminary investigation on the above new notion, which indeed generalises the well-known concept of (k, d)-graceful graphs due to B. D. Acharya and S. M. Hegde. 相似文献
15.
Pierre Fraisse 《Journal of Graph Theory》1986,10(4):553-557
Let G = (V, E) be a digraph of order n, satisfying Woodall's condition ? x, y ∈ V, if (x, y) ? E, then d+(x) + d?(y) ≥ n. Let S be a subset of V of cardinality s. Then there exists a circuit including S and of length at most Min(n, 2s). In the case of oriented graphs we obtain the same result under the weaker condition d+(x) + d?(y) ≥ n – 2 (which implies hamiltonism). 相似文献
16.
Daniel M. Oberlin 《Journal of Geometric Analysis》2010,20(2):422-438
We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two sets E,K⊂ℝ
d
. 相似文献
17.
Let Z ⊂ E
d
be ad-dimensional zonoid, whered≥3. Boltjanskii and Soltan recently proved that if Z is not a parallelotope, then Z can be illuminated by 3·2
d−2 points disjoint from Z. In the present paper we prove a related result. Namely, we show that ifd+1=2
p
, then Z can be illuminated by 2
d
/d+1 lines lying outside Z.
The work was supported by Hung. Nat. Found. for Sci. Research No. 326-0213 and 326-0113. 相似文献
18.
Wolfgang Kreitmeier 《分析论及其应用》2010,26(1):84-100
Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R
d
. Denote by D the Hausdorff dimension, by H
D
(E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H
D
(E) = diam(E)
D
if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present
a parametrised UIFS, where ω depends on c and show the inequatily H
D
(E) < diam(E)
D
, if c is small enough. 相似文献
19.
A. Shchuplev A. Tsikh A. Yger 《Proceedings of the Steklov Institute of Mathematics》2006,253(1):256-274
A finite collection of planes {E
v
} in ℂd is called an atomic family if the top de Rham cohomology group of its complement is generated by a single element. A closed
differential form generating this group is called a residual kernel for the atomic family. We construct new residual kernels
in the case when E
v
are coordinate planes such that the complement ℂd/∪ E
v
admits a toric action with the orbit space being homeomorphic to a compact projective toric variety. They generalize the
well-known Bochner-Martinelli and Sorani differential forms. The kernels obtained are used to establish a new formula of integral
representations for functions holomorphic in Reinhardt polyhedra. 相似文献
20.
In 1971, Branko Grünbaum noted that the projectived-arrangements formed by including with the facet hyperplanes of a regular polytope in Ed some of its hyperplanes of mirror symmetry and possibly the hyperplane at infinity might be expected to be simplicial. In this paper we show that none of the 4-arrangements so associated with either the 120-cell or the 600-cell inE
4 is simplicial. 相似文献