共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
Emmanuel Preissmann 《Monatshefte für Mathematik》2007,45(1):233-239
Let X
0 be the germ at 0 of a complex variety and let
f: X0? \Bbb Cn0f:\ X_0\rightarrow {\Bbb C}^n_0
be a holomorphic germ. We say that f is pseudoimmersive if for any
g: \Bbb R0? X0g:\ {\Bbb R}_0\rightarrow X_0
such that
f °g ? C¥ f \circ g \in C^{\infty}
, we have
g ? C¥g\in C^{\infty}
. We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered. 相似文献
3.
Wolfgang M. Ruppert 《Archiv der Mathematik》1999,72(4):278-281
We give an elementary argument for the well known fact that the endomorphism algebra
End(A)?\Bbb Q {\rm {End}}(A)\otimes {\Bbb Q } of a simple complex abelian surface A can neither be an imaginary quadratic field nor a definite quaternion algebra. Another consequence of our argument is that a two-dimensional complex torus T with
\Bbb Q (?d)\hookrightarrow End\Bbb Q (T){\Bbb Q }(\sqrt {d})\hookrightarrow {\rm{End_{{\Bbb Q }}}}(T) where
\Bbb Q (?d){\Bbb Q }(\sqrt {d}) is real quadratic, is algebraic. 相似文献
4.
Given a finite subset
A{\cal A}
of an additive group
\Bbb G{\Bbb G}
such as
\Bbb Zn{\Bbb Z}^n
or
\Bbb Rn{\Bbb R}^n
, we are interested in efficient covering of
\Bbb G{\Bbb G}
by translates of
A{\cal A}
, and efficient packing of translates of
A{\cal A}
in
\Bbb G{\Bbb G}
. A set
S ì \Bbb G{\cal S} \subset {\Bbb G}
provides a covering if the translates
A + s{\cal A} + s
with
s ? Ss \in {\cal S}
cover
\Bbb G{\Bbb G}
(i.e., their union is
\Bbb G{\Bbb G}
), and the covering will be efficient if
S{\cal S}
has small density in
\Bbb G{\Bbb G}
. On the other hand, a set
S ì \Bbb G{\cal S} \subset {\Bbb G}
will provide a packing if the translated sets
A + s{\cal A} + s
with
s ? Ss \in {\cal S}
are mutually disjoint, and the packing is efficient if
S{\cal S}
has large density.
In the present part (I) we will derive some facts on these concepts when
\Bbb G = \Bbb Zn{\Bbb G} = {\Bbb Z}^n
, and give estimates for the minimal covering densities and maximal packing densities of finite sets
A ì \Bbb Zn{\cal A} \subset {\Bbb Z}^n
. In part (II) we will again deal with
\Bbb G = \Bbb Zn{\Bbb G} = {\Bbb Z}^n
, and study the behaviour of such densities under linear transformations. In part (III) we will turn to
\Bbb G = \Bbb Rn{\Bbb G} = {\Bbb R}^n
. 相似文献
5.
Franki Dillen Johan Fastenakels Joeri Van der Veken Luc Vrancken 《Monatshefte für Mathematik》2007,40(1):89-96
In this article we study surfaces in
\Bbb S2×\Bbb R {\Bbb S}^2\times {\Bbb R}
for which the unit normal makes a constant angle with the
\Bbb R {\Bbb R}
-direction. We give a complete classification for surfaces satisfying this simple geometric condition. 相似文献
6.
Wojciech Jaworski 《Monatshefte für Mathematik》2008,155(2):135-144
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
, whenever
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
. 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
and P is an n-dimensional polytope with integral vertices. Then we have
.
Moreover, in the 3-dimensional case we prove a stronger inequality, namely
.
Authors’ addresses: Martin Henk, Institut für Algebra und Geometrie, Universit?t Magdeburg, Universit?tsplatz 2, D-39106 Magdeburg,
Germany; J?rg M. Wills, Mathematisches Institut, Universit?t Siegen, ENC, D-57068 Siegen, Germany 相似文献
7.
S. Bespamyatnikh 《Discrete and Computational Geometry》2001,25(2):235-255
We explore a new approach for computing the diameter of n points in \Bbb R
3
that is based on the restriction of the furthest-point Voronoi diagram to the convex hull. We show that the restricted Voronoi
diagram has linear complexity. We present a deterministic algorithm with O(nlog
2
n) running time. The algorithm is quite simple and is a good candidate to be implemented in practice. Using our approach the
chromatic diameter and all-furthest neighbors in \Bbb R
3
can be found in the same running time.
Received February 18, 2000, and in revised form June 27, 2000. Online publication January 17, 2001. 相似文献
8.
In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space
\Bbb Rd{\Bbb R}^d
containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the
only extremal bodies, which is proved for all three dimensional and several prominent lattices. 相似文献
9.
P. Biran 《Geometric And Functional Analysis》2001,11(3):407-464
We prove that every symplectic Kähler manifold (M;W) (M;\Omega) with integral [W] [\Omega] decomposes into a disjoint union (M,W) = (E,w0) \coprod D (M,\Omega) = (E,\omega_0) \coprod \Delta , where (E,w0) (E,\omega_0) is a disc bundle endowed with a standard symplectic form w0 \omega_0 and D \Delta is an isotropic CW-complex. We perform explicit computations of this decomposition on several examples.¶As an application we establish the following symplectic intersection phenomenon: There exist symplectically irremovable intersections between contractible domains and Lagrangian submanifolds. For example, we prove that every symplectic embedding j:B2n(l) ? \Bbb CPn \varphi:B^{2n}(\lambda) \to {\Bbb C}P^n of a ball of radius l2 3 1/2 \lambda^2 \ge 1/2 must intersect the standard Lagrangian real projective space \Bbb RPn ì \Bbb CPn {\Bbb R}P^n \subset {\Bbb C}P^n . 相似文献
10.
11.
Min Ho Lee 《Monatshefte für Mathematik》2004,78(4):187-196
Let
t: D ?D¢\tau: {\cal D} \rightarrow{\cal D}^\prime
be an equivariant holomorphic map of symmetric domains associated to a homomorphism
r: \Bbb G ?\Bbb G¢{\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime
of semisimple algebraic groups defined over
\Bbb Q{\Bbb Q}
. If
G ì \Bbb G (\Bbb Q)\Gamma\subset {\Bbb G} ({\Bbb Q})
and
G¢ ì \Bbb G¢(\Bbb Q)\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q})
are torsion-free arithmetic subgroups with
r (G) ì G¢{\bf\rho} (\Gamma) \subset \Gamma^\prime
, the map G\D ?G¢\D¢\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime
of arithmetic varieties and the rationality of D{\cal D}
and
D¢{\cal D}^\prime
as well as the commensurability groups of
s ? Aut (\Bbb C)\sigma \in {\rm Aut} ({\Bbb C})
determines a conjugate equivariant holomorphic map
ts: Ds ?D¢s\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma}
of fs: (G\D)s ?(G¢\D¢)s\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma
of . We prove that is rational if is rational. 相似文献
12.
Vicente Miquel 《Archiv der Mathematik》1999,72(5):376-384
Given a compact Kähler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let q\cal q (resp. \Bbb R Pn{\Bbb R} P^n) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \Bbb C Pn{\Bbb C} P^n of constant holomorphic sectional curvature 4l \lambda . We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) £\leq volume (q\cal q)/ volume (\Bbb C Pn)({\Bbb C} P^n) (resp. £\leq volume (\Bbb R Pn)({\Bbb R} P^n) / volume (\Bbb C Pn)({\Bbb C} P^n)), then there is a holomorphic isometry between M and \Bbb C Pn{\Bbb C} P^n taking P isometrically onto q\cal q (resp. \Bbb R Pn{\Bbb R} P^n). We also classify the Kähler manifolds with boundary which are tubes of radius r around totally real and totally geodesic submanifolds of half dimension, have the holomorphic sectional and some (n-1)-Ricci curvatures bounded from below by those of the tube \Bbb R Pnr{\Bbb R} P^n_r of radius r around \Bbb R Pn{\Bbb R} P^n in \Bbb C Pn{\Bbb C} P^n and have the first Dirichlet eigenvalue not lower than that of \Bbb R Pnr{\Bbb R} P^n_r. 相似文献
13.
Bruno Franchi Maria Carla Tesi 《NoDEA : Nonlinear Differential Equations and Applications》2001,8(4):363-387
In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1, a: \Bbb Rn ×\Bbb Rn ? \Bbb R, a(y,x) ? áA(y)x,x?p/2-1, A ? Mn ×n(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that At(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l2/p(x) |x|2 £ áA(x)x,x? £ L 2/p(x) |x|2, \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces. 相似文献
14.
Min Ho Lee 《Monatshefte für Mathematik》2007,338(2):321-336
We introduce vector-valued Jacobi-like forms associated to a representation
r: G? GL(n,\Bbb C)\rho: \Gamma \rightarrow GL(n,{\Bbb C})
of a discrete subgroup
G ì SL(2,\Bbb C)\Gamma \subset SL(2,{\Bbb C})
in
\Bbb Cn{\Bbb C}^n
and establish a correspondence between such vector-valued Jacobi-like forms and sequences of vector-valued modular forms
of different weights with respect to ρ. We determine a lifting of vector-valued modular forms to vector-valued Jacobi-like
forms as well as a lifting of scalar-valued Jacobi-like forms to vector-valued Jacobi-like forms. We also construct Rankin-Cohen
brackets for vector-valued modular forms. 相似文献
15.
In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if
{eimbxg(x-na): m,n ? \Bbb Z}\{e^{imbx}g(x-na): m,n\in{\Bbb Z}\}
is a Gabor frame for
L2(\Bbb R)L^2({\Bbb R})
with frame bounds A and B, then the following two inequalities hold:
A £ \frac2pb?n ? \Bbb Z|g(x-na)|2 £ B, a.e.A\le \frac{2\pi}{b}\sum_{n\in{\Bbb Z}}\vert g(x-na)\vert^2\le B, \quad a.e.
and
A £ \frac1a?m ? \Bbb Z|[^(g)](w-mb)|2 £ B, a.e.A\le \frac{1}{a}\sum_{m\in{\Bbb Z}}\vert \hat{g}(\omega-mb)\vert^2\le B, \quad a.e.
. In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form
è1 £ k £ r{eiáx, l?gk(x-m): m ? Dk, l ? Lk }\bigcup_{1\le k\le r}\{e^{i\langle x, \lambda\rangle}g_{k}(x-\mu):\, \mu\in \Delta_k, \lambda\in\Lambda_k \}
, where Δ
k
and Λ
k
are arbitrary sequences of points in
\Bbb Rd{\Bbb R}^d
and
gk ? L2(\Bbb Rd)g_k\in{L^2{(\Bbb R}^d)}
, 1 ≤ k ≤ r. 相似文献
16.
Michael Ruzhansky 《Archiv der Mathematik》1999,72(1):68-76
We will show that the factorization condition for the Fourier integral operators Ir m (X,Y;L )I_\rho ^\mu (X,Y;\it\Lambda ) leads to a parametrized parabolic Monge-Ampère equation. For an analytic operator, the fibration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases, by considering a more general continuation problem for the level sets of a holomorphic mapping. The results are applied to obtain Lp-continuity for translation invariant operators in \Bbb Rn{\Bbb R}^n with n £ 4n\leq 4 and for arbitrary \Bbb Rn{\Bbb R}^n with dpX×Y|L £ n+2d\pi _{X\times Y}|_\Lambda \leq n+2. 相似文献
17.
We study the problem of best approximations of a vector
a ? \Bbb Rn\alpha\in{\Bbb R}^n
by rational vectors of a lattice
L ì \Bbb Rn\Lambda\subset{\Bbb R}^n
whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend
some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem
which generalize and improve former results. 相似文献
18.
Richard Bödi 《Archiv der Mathematik》1999,73(1):73-80
We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers \Bbb C \Bbb C , the quaternions \Bbb H \Bbb H , and the Caley numbers \Bbb O \Bbb O . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension d 3 17d \geq 17 is isomorphic to the quaternion projective plane P 2\Bbb H {\cal P _2\Bbb H }. For topological compact projective planes this is true if d 3 19d \geq 19, and this bound is sharp. 相似文献
19.
Straightening and bounded cohomology of hyperbolic groups 总被引:2,自引:0,他引:2
I. Mineyev 《Geometric And Functional Analysis》2001,11(4):807-839
It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology Hnb(G,\Bbb R) H^n_b(G,{\Bbb R}) to Hn(G,\Bbb R) H^n(G,{\Bbb R}) induced by inclusion is surjective for n 3 2 n \ge 2 . We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map Hnb(G,V) ? Hn(G,V) H^n_b(G,V) \to H^n(G,V) is surjective for n 3 2 n \ge 2 when V is any bounded \Bbb QG {\Bbb Q}G -module and when V is any finitely generated abelian group. 相似文献
20.
B. Green 《Geometric And Functional Analysis》2002,12(3):584-597
We prove several results concerning arithmetic progressions in sets of integers. Suppose, for example, that a \alpha and b \beta are positive reals, that N is a large prime and that C,D í \Bbb Z/N\Bbb Z C,D \subseteq {\Bbb Z}/N{\Bbb Z} have sizes gN \gamma N and dN \delta N respectively. Then the sumset C + D contains an AP of length at least ec ?{log} N e^{c \sqrt{\rm log} N} , where c > 0 depends only on g \gamma and d \delta . In deriving these results we introduce the concept of hereditary non-uniformity (HNU) for subsets of \Bbb Z/N\Bbb Z {\Bbb Z}/N{\Bbb Z} , and prove a structural result for sets with this property. 相似文献