共查询到20条相似文献，搜索用时 506 毫秒
1.
Heppes 《Discrete and Computational Geometry》2003,29(3):477-481
Abstract. Kershner proved in 1939 that the density of a covering of the plane by congruent circles is at least 2π/
[3]. In 1950 L. Fejes Tóth [2] extended this result showing that the same density bound holds for coverings with congruent
ellipses which do not ``cross'. In the present paper we prove that the non-crossing assumption is not necessary if the ellipses
are sufficiently ``fat'. 相似文献
2.
3.
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
4.
Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ
Ω
and λ
Π
denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type
are considered where M
n
(z,Ω, Π) does not depend on f and represents the smallest value possible at this place. We prove that
if Δ is the unit disk and Π is a convex domain. This generalizes a result of St. Ruscheweyh.
Furthermore, we show that
holds for arbitrary simply connected domains whereas the inequality 2
n-1
≤ C
n
(Ω,Π) is proved only under some technical restrictions upon Ω and Π . 相似文献
5.
D. Junghenn 《Semigroup Forum》2008,66(2):328-336
Abstract. Let
be a semidirect product of semitopological semigroups S and T . If S and T act on topological spaces X and Y , respectively, then under suitable conditions there is a natural action of
on X × Y . In this paper we characterize the almost periodic and strongly almost periodic compactification of the flow
,
in terms of related compactifications of (S,X) and (T,Y) . 相似文献
6.
7.
Let K be a field of characteristic 0 and let p, q, G 0 , G 1 , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G n (x)) n=0 ∞ be defined by the second order linear recurring sequence
In this paper we give conditions under which the diophantine equation G n (x) = G m (P(x)) has at most exp(1018) many solutions (n, m) ε ℤ2, n, m ⩾ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to Evertse, Schlickewei and Schmidt [14]. Under the same conditions we present also bounds for the cardinality of the set
相似文献
8.
Let K be a field of characteristic 0 and let p, q, G
0
, G
1
, P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G
n
(x))
n=0
∞ be defined by the second order linear recurring sequence
In this paper we give conditions under which the diophantine equation G
n
(x) = G
m
(P(x)) has at most exp(1018) many solutions (n, m) ε ℤ2, n, m ⩾ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to Evertse, Schlickewei and Schmidt [14]. Under the same conditions we
present also bounds for the cardinality of the set
In the last part we specialize our results to certain families of orthogonal polynomials.
This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT.
The second author was supported by the Hungarian National Foundation for Scientific Research Grants No 16741 and 38225.
Received June 5, 2001; in revised form February 26, 2002
RID="a"
ID="a" Dedicated to Edmund Hlawka on the occasion of his 85th birthday 相似文献
9.
J. Schoißengeier 《Monatshefte für Mathematik》2000,131(3):227-234
For a real number x let be the fractional part of x and for any set M let c
M
be the characteristic function of M. For and a positive integer N let
be the discrepancy of the sequence modulo 1. In this paper we prove that
(Received 2 May 2000; in revised form 19 June 2000) 相似文献
10.
Abstract. Given an m × n rectangular mesh, its adjacency matrix A , having only integer entries, may be interpreted as a map between vector spaces over an arbitrary field K . We describe the kernel of A : it is a direct sum of two natural subspaces whose dimensions are equal to
and
, where c = gcd (m+1,n+1) - 1 . We show that there are bases to both vector spaces, with entries equal to 0,1 and -1 . When K = Z/(2), the kernel elements of these subspaces are described by rectangular tilings of a special kind. As a corollary, we count
the number of tilings of a rectangle of integer sides with a specified set of tiles. 相似文献
11.
J. Schoi?engeier 《Monatshefte für Mathematik》2000,74(4):227-234
For a real number x let be the fractional part of x and for any set M let c
M
be the characteristic function of M. For and a positive integer N let
be the discrepancy of the sequence modulo 1. In this paper we prove that
相似文献
12.
E.G. Coffman Jr. George S. Lueker Joel Spencer Peter M. Winkler 《Probability Theory and Related Fields》2001,120(4):585-599
A random rectangle is the product of two independent random intervals, each being the interval between two random points
drawn independently and uniformly from [0,1]. We prove that te number C
n
of items in a maximum cardinality disjoint subset of n random rectangles satisfies
where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constat K,
Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Q
n
of items in a maximum cardinality disjoint subset of the cubes satisies
Received: 1 September 1999 / Revised version: 3 November 2000 / Published online: 14 June 2001 相似文献
13.
Serge Lawrencenko Michael D. Plummer Xiaoya Zha 《Discrete and Computational Geometry》2002,28(3):313-330
Abstract. Let G be an infinite locally finite plane graph with one end and let H be a finite plane subgraph of G . Denote by a(H) the number of finite faces of H and by l(H) the number of the edges of H that are on the boundary of the infinite face or a finite face not in H . Define the isoperimetric constant h (G) to be inf
H
l(H) / a(H) and define the isoperimetric constant h (δ) to be inf
G
h (G) where the infimum is taken over all infinite locally finite plane graphs G having minimum degree δ and exactly one end. We establish the following bounds on h (δ) for δ ≥ 7 :
相似文献
14.
Goran Peskir 《Probability Theory and Related Fields》2002,124(1):100-111
Let (B
t
)
t
≥ 0) be a standard Brownian motion started at zero, let g : ℝ_+ →ℝ be an upper function for B satisfying g(0)=0, and let
be the first-passage time of B over g. Assume that g is C
1 on <0,∞>, increasing (locally at zero), and concave (locally at zero). Then the following identities hold for the density
function f of τ:
in the sense that if the second and third limit exist so does the first one and the equalities are valid (here is the standard normal density). These limits can take any value in [0,∞]. The method of proof relies upon the strong Markov
property of B and makes use of real analysis.
Received: 30 August 2001 / Revised version: 25 February 2002 / Published online: 22 August 2002 相似文献
15.
Chang 《Semigroup Forum》2008,66(1):68-80
Abstract. For the generator A of a C
0
-semigroup on a Banach space (X, ||⋅||) , we apply the perturbation of Desch-Schappacher type to solve the Volterra integordifferential equation
which can be applied to treat boundary value problems and inhomogeneous retarded differential equations. 相似文献
16.
Dan Ismailescu 《Discrete and Computational Geometry》2009,41(4):583-589
Given any plane strictly convex region K and any positive integer n≥3, there exists an inscribed 2n-gon Q
2n
and a circumscribed n-gon P
n
such that
The inequality is the best possible, as can be easily seen by letting K be an ellipse. As a corollary, it follows that for any convex region K and any n≥3, there exists a circumscribed n-gon P
n
such that
This improves the existing bounds for 5≤n≤11. 相似文献
17.
Let K, D be centrally symmetric convex bodies in
Let k < n and let dk(K, D) be the smallest Banach–Mazur distance between k-dimensional sections of K and D. Define
where the supremum is taken over all n-dimensional convex symmetric bodies K, D. We prove that, for any k < n,
where
means that
for some absolute constants C, a > 0. 相似文献
18.
We investigate the relationship between the constants K(R) and K(T), where
is the exact constant in the Kolmogorov inequality, R is the real axis, T is a unit circle,
is the set of functions x L
p(G) such that x
(r) L
s(G), q, p, s [1, ], k, r N, k < r, We prove that if
thenK(R) = K(T),but if
thenK(R) K(T); moreover, the last inequality can be an equality as well as a strict inequality. As a corollary, we obtain new exact Kolmogorov-type inequalities on the real axis. 相似文献
19.
Lewin 《Semigroup Forum》2008,66(1):43-62
Abstract. Given a set A and a function
, we study the set of all functions
that are continuous for all topologies for which f continuous.
We prove that in a sense to be made precise in the text, for any essentially infinitary function f , any non-constant such g equals f
n
, for some n∈ N. We also prove a similar result for the clone of n -ary functions from
. 相似文献
20.
Dimitrios Betsakos 《Mathematische Zeitschrift》2008,259(1):187-196
Let denote the density of the hyperbolic metric for a domain Ω in the extended complex plane . We prove the inequality
with C = 8.27. The inequality was proved by Sugawa and Vuorinen with C = 10.33. The proof uses monotonicity properties of the hyperbolic metric for the thrice punctured extended plane. Gardiner
and Lakic proved the inequality
with an unspecified constant C
1. We show that the best constant Σ1 in this inequality is between 3.25 and 8.27. We also prove a related conjecture formulated by Sugawa and Vuorinen.
The author was partially supported by the EPEAK programm Pythagoras II (Greece). 相似文献