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1.
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
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
10.
Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an (n + 1)-dimensional Riemannian manifold N^n+1 whose (n - 1)th Ricci curvature satisfying Ric^N(n-1) (n - 1)c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If |φ|^2 - (n- 2)√n(n- 1)|H||φ|+ n(2n - 1)(H^2+ c) 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c + H^2 〉 0, then M must have only one end. 相似文献
11.
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
12.
We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation. 相似文献
13.
David Kalaj 《Israel Journal of Mathematics》2011,182(1):123-147
Let ρ
Σ = h(|z|2) be a metric in a Riemann surface Σ, where h is a positive real function. Let H
r
1 = {w = f(z)} be the family of a univalent ρ
Σ harmonic mapping of the Euclidean annulus A(r
1, 1):= {z: r
1 < |z| < 1} onto a proper annulus A
Σ of the Riemann surface Σ, which is subject to some geometric restrictions. It is shown that if A
Σ is fixed, then sup{r
1: ℋ
r
1 ≠ ∅} < 1. This generalizes similar results from the Euclidean case. The cases of Riemann and of hyperbolic harmonic mappings
are treated in detail. Using the fact that the Gauss map of a surface with constant mean curvature (CMC) is a Riemann harmonic
mapping, an application to the CMC surfaces is given (see Corollary 3.2). In addition, some new examples of hyperbolic and
Riemann radial harmonic diffeomorphisms are given, which have inspired some new J. C. C. Nitsche-type conjectures for the
class of these mappings. 相似文献
14.
Bart De Bruyn 《Annals of Combinatorics》2010,14(3):307-318
Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space
D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into
D¢ = DQ(2n, \mathbb K¢){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let
e¢: D¢? S¢ @ PG(2n - 1, \mathbb K¢){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H¢{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry
S @ PG(2n - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e¢°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e¢°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ. 相似文献
15.
Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H
n
, n > 2. For ν > 0, the Brownian bridge B
(ν) of length ν on H is the process B
t
, 0 ≤t≤ν, conditioned by B
0 = B
ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge
on ℝ3). The same result holds for the simple random walk on an homogeneous tree.
Received: 4 December 1998 / Revised version: 22 January 1999 相似文献
16.
S Thangavelu 《Proceedings Mathematical Sciences》1990,100(2):147-156
The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH
n is considered. It is proved thatS
R
α
are uniformly bounded onL
p(Hn) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)]. 相似文献
17.
Subhash J. Bhatt 《Proceedings Mathematical Sciences》2006,116(2):161-173
Given anm-tempered strongly continuous action α of ℝ by continuous*-automorphisms of a Frechet*-algebraA, it is shown that the enveloping ↡-C
*-algebraE(S(ℝ, A∞, α)) of the smooth Schwartz crossed productS(ℝ,A
∞, α) of the Frechet algebra A∞ of C∞-elements ofA is isomorphic to the Σ-C
*-crossed productC
*(ℝ,E(A), α) of the enveloping Σ-C
*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK
*(S(ℝ, A∞, α)) =K
*(C
*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC
*-algebra defined by densely defined differential seminorms is given. 相似文献
18.
Michel Talagrand 《Israel Journal of Mathematics》1992,79(2-3):207-224
Consider a setA of symmetricn×n matricesa=(a
i,j)
i,j≤n
. Consider an independent sequence (g
i)
i≤n
of standard normal random variables, and letM=Esupa∈A|Σi,j⪯nai,jgigj|. Denote byN
2(A, α) (resp.N
t(A, α)) the smallest number of balls of radiusα for thel
2 norm ofR
n
2 (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN
2(A, α))1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN
t≤K(δ)M.
Work partially supported by an N.S.F. grant. 相似文献
19.
Ji-pu Ma Tseng Yaun Rong Functional Analysis Research Center Harbin Normal University Harbin China 《中国科学A辑(英文版)》2007,50(9):1233-1239
Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. < ∞ or codim R(T) = const.< ∞} and {T ∈ R(E, F) RankT =const.<∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry. 相似文献
20.
Joseph Weier 《Annali dell'Universita di Ferrara》1958,8(1):29-37
Riassunto SeM edN sono varietà poliedriche chiuse connesse ed orientate di dimensioni rispettivem edn, conm≥n>2, edf∶M→N è una trasformazione continua, allora per ognir, minore din e non inferiore a 2, si definisce un omomorfismo indotto ϕrπ:r
(N)→H
m-n+r
(M) dal quale si ricavano certi invarianti topologici.
Résumé Soientm≥n>r≥2 des entiers etM, N des variétés polyédrales closes connexes orientées satisfaisant dimM=m et dimN=n, de plusH i(M) le groupe de Betti à i dimensions deM,M,π i (N) le groupe de Hurewicz ài dimensions deN, etf∶M→N une application continue. Alorsf définit, pour,r=2, 3, …n−1, un homomorphisme réciproque ϕrπ:r (N)→H m-n+r (M) comme il suit. Etant donné un élément α du groupe πr (N) et uner-sphère continue orientéeS de α, on peut supposer quef −1(S) soit un polyèdre finiA àm−n+r dimensions. Parf est induit dansA un (m−n+r)-cyclez à coefficients entiers, et la classe d'homologie dez est justement l'image ϕr(α) de α par ϕr. Pourr=1, on obtient un homomorphisme réciproque ϕrπ:r (N)→H m-n+r (M) du groupe fondamentalF(N) deN dans le groupe d'homologie àm−n+1 dimensions deM. A l'aide des homomorphismes ϕ,,ϕ2,ϕ,3...,ϕn-i, on parvient à certaines expressions caractéristiques dépendantes seulement de la classe d'homotopie def, en particulier on obtient des constantes pour les images des bases de Betti deM, pour Fimage du groupe de torsion deM, et pour l'image réciproque du groupe fondamental deN.相似文献