共查询到19条相似文献,搜索用时 140 毫秒
1.
Wu-Yi HSIANG 《数学年刊B辑(英文版)》2006,27(1)
In the study of n-dimensional spherical or hyperbolic geometry, n ≥3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in Sn(l) and Hn(-1). 相似文献
2.
We shall prove the equivalences of a non-degenerate circle-preserving map and a Mobius transformation in Rn, of a non-degenerate geodesic-preserving map and an isometry in Hn, of a non-degenerate line-preserving map and an affine transformation in Rn. That a map is non-degenerate means that the image of the whole space under the map is not a circle, or geodesic or line respectively. These results hold without either injective or surjective, or even continuous assumptions, which are new and of a fundamental nature in geometry. 相似文献
3.
ON THE TOPOLOGY,VOLUME,DIAMETER AND GAUSS MAP IMAGE OF SUBMANIFOLDS IN A SPHERE 总被引:1,自引:0,他引:1 下载免费PDF全文
WU Bingye 《数学年刊B辑(英文版)》2004,25(2):207-212
In this paper, the author uses Gauss map to study the topology, volume and diameter of submanifolds in a sphere. It is proved that if there exist ε, 1≥ε > 0 and a fixed unit simple p-vector a such that the Gauss map g of an n-dimensional complete and connected submanifold M in Sn p satisfies (g, a) ≥ε, then M is diffeomorphic to Sn, and the volume and diameter of M satisfy εnvol(Sn) ≤vol(M) ≤ vol(Sn)/ε and επ ≤diam(M) ≤ π/ε, respectively. The author also characterizes the case where these inequalities become equalities. As an application, a differential sphere theorem for compact submanifolds in a sphere is obtained. 相似文献
4.
Let Mnbe an n-dimensional submanifold without umbilical points in the(n + 1)-dimensional unit sphere Sn+1.Four basic invariants of Mnunder the Moebius transformation group of Sn+1are a 1-form Φ called moebius form,a symmetric(0,2) tensor A called Blaschke tensor,a symmetric(0,2) tensor B called Moebius second fundamental form and a positive definite(0,2) tensor g called Moebius metric.A symmetric(0,2) tensor D = A + μB called para-Blaschke tensor,where μ is constant,is also an Moebius invariant.We call the para-Blaschke tensor is isotropic if there exists a function λ such that D = λg.One of the basic questions in Moebius geometry is to classify the hypersurfaces with isotropic para-Blaschke tensor.When λ is not constant,all hypersurfaces with isotropic para-Blaschke tensor are explicitly expressed in this paper. 相似文献
5.
Leng Yan Xu Hongwei 《高校应用数学学报(英文版)》2007,22(2):153-162
A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n p)-dimensional manifold Nn p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H > 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn p is isometric to the hyperbolic space Hn p(-1). As a consequence, this submanifold M is congruent to Sn(1/ H2-1) or theVeronese surface in S4(1/√H2-1). 相似文献
6.
In this paper we generalized the uniqueness theorem of. Alexadroff-Fenchel-Jessen, the cohn-Vossen theorem and the Hilbert-Liebmann-Hsiung theorem to hypersurfaces in a sphere S~(n+1) or a hyperbolic space H~(n+1) 相似文献
7.
The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations 下载免费PDF全文
Let {Xn;n≥1} be a sequence of independent random variables on a probability space(Ω,F,P) and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained. 相似文献
8.
A nonlinear parabolic system is derived to describe incompressible nuclear waste-disposal contamination in porous media. A sequential implicit tirne-stepping is defined, in which the pressure and Darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the brine, radionuclid and heat are treated by a combination of a Galerkin finite element method and the method of characteristics. Optimal-order convergence in L2 is proved. Time-truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the brine, radionuclide and heal equalios, temporal and spatial error are lossened by direct compulation of the velocity in the mixed method, as opposed to differentiation of the pressure. 相似文献
9.
Here we consider the numerical approximations of the 2D simplified Ericksen-Leslie system.We first rewrite the system and get a new system.For the new system,we propose an easy-to-implement time discretization scheme which preserves the sphere constraint at each node,enjoys a discrete energy law,and leads to linear and decoupled elliptic equations to be solved at each time step.A discrete maximum principle of the schemc in the finite element form is also proved.Some numerical simulations are performed to validate the scheme and simulate the dynamic motion of liquid crystals. 相似文献
10.
We consider the following problem: for a collection of points in an n-dimensional space, find a linear projection mapping the points to the ground field such that different points are mapped to different values. Such projections are called normal and are useful for making algebraic varieties into normal positions. The points may be given explicitly or implicitly and the coefficients of the projection come from a subset S of the ground field. If the subset S is small, this problem may be hard. This paper deals with relatively large S, a deterministic algorithm is given when the points are given explicitly, and a lower bound for success probability is given for a probabilistic algorithm from in the literature. 相似文献
11.
12.
Henryka Siejka 《Israel Journal of Mathematics》1986,54(3):291-300
The class Σb is defined to consist of meromorphic univalent functionsH omitting a disc with the radiusb:H(z)=z+ Σ
0
∞
A
n
z
−n
,z>1,H(b)>b ∈ (0, 1). By aid of FitzGerald inequalities the inverse coefficients of odd Σb-functions are maximized. The result extends the corresponding estimation, due to Netanyahu and Schober, fromb=0 to the whole interval (0, 1).
The author wishes to express her gratitude to Professor O. Tammi for valuable discussions connected with the problem.
This work was supported by a grant from the Finnish Ministry of Education. 相似文献
13.
Wu-Yi HSIANG 《数学年刊B辑(英文版)》2006,27(1):1-30
Abstract In the study of n-dimensional spherical or hyperbolic geometry, n≥ 3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with.
In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas
of cones and orthogonal multiple cones in Sn(1) and Hn(—1).
(Dedicated to the memory of Shiing-Shen Chern) 相似文献
14.
Timothy A. Schroeder 《Geometriae Dedicata》2009,140(1):163-174
Associated to any Coxeter system (W, S), there is a labeled simplicial complex L and a contractible CW-complex Σ
L
(the Davis complex) on which W acts properly and cocompactly. Σ
L
admits a cellulation under which the nerve of each vertex is L. It follows that if L is a triangulation of , then Σ
L
is a contractible n-manifold. In this case, the orbit space, K
L
:= Σ
L
/W, is a Coxeter orbifold. We prove a result analogous to the JSJ-decomposition for 3-dimensional manifolds: Every 3-dimensional Coxeter orbifold splits
along Euclidean suborbifolds into the characteristic suborbifold and simple (hyperbolic) pieces. It follows that every 3-dimensional Coxeter orbifold has a decomposition into pieces which
have hyperbolic, Euclidean, or the geometry of . (We leave out the case of spherical Coxeter orbifolds.) A version of Singer’s conjecture in dimension 3 follows: That the
reduced ℓ
2-homology of Σ
L
vanishes.
相似文献
15.
Summary For P∈ F2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σn≧0 p(A,n)zn ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d|n, d∈A d, namely, the periodicity of the sequences (σ(A,2kn) mod 2k+1)n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula
to σ(A,2kn) mod 2k+1 will be established, from which it will be shown that the weight σ(A1,2kzi) mod 2k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$
is moved on some other orbit zj when A1 is replaced by A2 with A1= A(P1) and A2= A(P2) P1 and P2 being irreducible in F2[z] of the same odd order. 相似文献
16.
Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ
r
the set of all operators of finite rank r in B(E,F), and Σ
r
# the number of path connected components of Σ
r
. It is known that Σ
r
is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ
r
. In this paper,the equality Σ
r
# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ
r
is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T
A
Σ
r
= {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ
r
if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property
of Σ
r
the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝ
n
and F = ℝ
m
, then Σ
r
is a smooth and path connected submanifold of B(ℝ
n
, ℝ
m
) and its dimension is dimΣ
r
= (m+n)r−r
2 for each r, 0 <- r < min {n,m}.
Supported by the National Science Foundation of China (Grant No.10671049 and 10771101). 相似文献
17.
For a domainU on a certaink-dimensional minimal submanifold ofS
n orH
n, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k
k
ω
k
M (D)
k-1
≤Vol(∂D)
k
, where ω
k
is the volume of the unit ball ofR
k
. Also, we prove that ifD is any domain on a minimal surface inS
+
n
(orH
n, respectively), thenD satisfies an isoperimetric inequality2π A≤L
2+A2 (2π A≤L2−A2 respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH
n, then(k−1) Vol(U)≤Vol(∂U).
Supported in part by KME and GARC 相似文献
18.
We obtain a new upper bound for the sum Σ
h≤H
Δ
k
(N, h) when 1 ≤ H ≤ N, k ∈ ℕ, k ≥ 3, where Δ
k
(N, h) is the (expected) error term in the asymptotic formula for Σ
N<n≤2N
d
k
(n)d
k
(n + h), and d
k
(n) is the divisor function generated by ζ(s)
k
. When k = 3, the result improves, for H ≥ N
1/2, the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case k = 3. 相似文献
19.
TieXin Guo 《中国科学A辑(英文版)》2008,51(9):1651-1663
Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A: there exists an element p in S such that X
p
(ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit
ball S
*(1) = {f ∈ S
*: X
*
f
⩽ 1} of the random conjugate space (S
*,X
*) of (S,X) is compact under the random weak star topology on (S
*,X
*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {A
n
: n ∈ N} of at most countably many μ-atoms from E ∩ A such that E = ∪
n=1∞
A
n
and for each element F in E ∩ A, there is an H in the σ-algebra generated by {A
n
: n ∈ N} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding
classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established
as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S: X
p
⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E ∩ A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary
complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary
almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that
the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they
possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James
theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous
classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent
in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another
in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely
simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of
random metric theory. 相似文献