Sharp Growth Theorems and Coefficient Bounds for Starlike Mappings in Several Complex Variables

Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.     

同态模Hom(A,B)的同调维数

本文从研究函子(?)与Hom的联系入手,来考虑求Hom(A,B)的弱维数与投射维数。当K为域时,且条件(a)[R:K]∞,A是有限生成右R模;(b)·[R:K]<∞,S是右凝聚代数:(c)[S:K]∞,R是右Noether代数,有一成立得到1.wd_R(?)SHom(A,B)r.id_RA+1.wd_sB。     

B0,I0,B1,I1,C0-半群

利用半群的理想和双理想呈现的包含关系定义了新的半群类B0,I0,B1,I1,C0-半群,讨论其性质,证明了正则半群S是C0-半群的充要条件是S是矩形带;B0,I0,B1,I1,C0-半群各自的直积半群和关于同余的商半群保持B0,I0,B1,I1,C0性.     

The Dirichlet problem with a volume constraint

For B the open unit disk in R², let W₁(B) denote the Sobolev space of vector functions x: B→R³ such that x and its first partial derivatives are square integrable. For any y∈W₁(B), S(y) is the set of all x in W₁(B) for which x-y∈W₁₀(B), the closure in W₁(B) of C ₀ ^∞ (B). Assume that for all x ∈ S(y) the area functional A(x)>0. For a given constant K, we show that there is an x_o∈S(y) minimizing the “Dirichlet Integral” $$D(x) = \iint_B {(|x_u |^2 } + |x_v |^2 )dudv$$ in the subset of all x ∈ S(y) for which the oriented volume enclosed by y and x, V(y,x)=K. x_o is analytic on B and is a solution to the differential equation Δx=2H(x_u∧x_v) for some constant H.     

The schoenflies extension in the analytic case

Summary Let S be an (n−1)-sphere in a euclidean n-space E. Let B be the closed n-ball in E bounded by S. Let z be an arbitrary point of . A real analytic diffeomorphism f of S into E admits a homeomorphic extension F which is defined over some open neighborhood N of B and such that F | (N−z) is an analytic diffeomorphism. We give a new proof of this theorem to serve as a model for a forthcoming theory of analytic families of such extensions. To Enrico Bompiani on his scientific Jubilee.     

厄米特矩阵的迹的几点性质

应用矩阵Ａ＝（ａｉｊ）∈Ｃｎ×ｎ的弗罗伯尼范数ＡＦ和谱范数ＡＳ，研究厄米特矩阵的迹的性质，得到几个结论：Ｔｒ（ＡＢ）＝∑ｎｉ＝１λｉ∑ｎｊ＝１ｔｉｊμｊ（λｉ，μｊ分别为Ａ，Ｂ的特征值，０≤ｔｉｊ≤１，且∑ｎｉ＝１ｔｉｊ＝１，ｊ＝１，２，…，ｎ）；Ｔｒ（ＡＢ）≤Ｔｒ（Ａ）ＢＳ；Ｔｒ（ＡＢ）Ｈ（ＡＢ）］≤Ｔｒ（ＡＨＡ）［ｍａｘ１≤ｉ≤ｎλｉ］２（λｉ是Ｂ的特征值）等．     

The Zeros and the Final Value of a Polynomial Form

ABSTRACT

A polynomial form f, is a not necessarily linear map, from an infinite module over a ring 𝔷 to a finite abelian group of exponent n satisfying some additional conditions. Denote the zeros of f by Ω_f. We show it satisfies a weak closure condition. Among all 𝔷-submodules of finite index, there is a submodule B such that |f (B)| (the order of the subset f (B)) is as small as possible. f (B) is called the final value of f and D. S. Passman asks if f (B) is necessarily a subgroup of S. This paper shows that if the degree of f ≤ 2 then the final value is a subgroup and if the form f has arbitrary degree from an finitely generated infinite abelian group, then the final value is 0.     

Verallgemeinerungen eines Satzes von S. Piccard

The following result is due to S. Piccard ([12], S.30): “If A,B ?? are Baire sets of second category and if the function f: ?×?→? is defined by f(x,y):=x?y (x,y ε ?), then the interior of f(A×B) is non void”. In this note the two main results assure, that the theorem of S. Piccard remains valid, if (1) ? is replaced by topological spaces X,Y,Z, (2) f:X×Y→Z is a function, which satisfies a certain global (respectively local) solvability condition, (3) A ?X contains a Baire set of second category and (4) B ?Y is only of second category.     

单位球上Bergman空间上的紧算子

当S满足一些可积条件时,我们证明了Lap(B)(p>1)上的有界算子S是紧的当且仅当S的Berezin变换在单位球B的边界趋近于零.     

Lattice Points in Large Borel Sets and Successive Minima

Let B be a Borel set in E^d with volume V(B) = ∞. It is shown that almost all lattices L in E^d contain infinitely many pairwise disjoint d-tuples, that is sets of d linearly independent points in B. A consequence of this result is the following: let S be a star body in E^d with V(S ) = ∞. Then for almost all lattices L in E^d the successive minima λ₁(S,L),..., λ_d(S,L) of S with respect to L are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the semi-continuity of the successive minima.     

关于压缩算子的不变子空间

证明关于压缩算子的如下不变子空间定理：如果T是Hilbert空间H上的压缩算子,且集合Z’={λ∈D;存在z∈H,使得‖z‖=1,且‖（λ-T）z‖＜1/3(1-‖λ‖}是开单位圆D的控制集,那么T有非平凡的不变子空间,这个定理包含了S.Brown,B.Chevreau,C.fPearcy和B.Beauzamy的两个重要结果作为特殊情况,特别是,为个定理包含了S.Brown等人的Hilbert空间上的每个具有厚谱的压缩算子都有平凡的不变子空间这个重要结果作为特殊情况。     

Nest代数上的在零点广义可导映射

设A为B（H）的子代数,　是A到B（H）的线性映射,我们说　在0点广义可导（广义双边可导）,如果对任意的S,T∈A且ST=0（ST=0或TS=0）,有　（ST）=　（S）T+S　（T）-S　（I）T.本文主要得到如下结果:（1）有限Nest代数上的每个范数拓扑连续的在0点广义可导的线性映射是广义内导子;（2）若N是完备Nest且H_　 H,则algN上的每个范数拓扑连续的在0点广义双边可导的线性映射是广义内导子.     

Self-maps of p-local infinite projective spaces

Denote by Z (p) (resp.Z p ) the p localization (resp.p completion) of Z.Then we have the canonical inclusion Z (p) → Z p .Let S 2n-1 (p) be the p-local (2n-1)-sphere and let B 2n (p) be a connected p-local space satisfying S 2n-1 (p) ～= ΩB 2n (p) ;then H - (B 2n (p) ,Z (p) ) = Z (p) [u] with |u| = 2n.Define the degree of a self-map f of B 2n (p) to be k ∈ Z (p) such that f *(u) = ku.Using the theory of integer-valued polynomials we show that there exists a self-map of B 2n (p) of degree k if and only if k is an n-th power in Z p .     

广义正则平面曲线的顶点

<正> 引言.四顶点定理.自印度加尔各答大学教授 Mukhopadhyaya 证明后,继而论之者颇多.Blaschke 证明:一正则卵形线若与任一圆之交点至多有四点时,则此卵形线至多有四顶点.另一人证明:一正则卵形线若与一圆有2n个交点时,则此卵形线至少有2n个顶点.Fog 及 Graustein 曾推广四顶点定理至简单闭曲线.Jackson 从研究二顶点曲线的特性出发,以研究四顶点曲线,指出了更广泛的四顶点曲线族.因不具二顶点曲线的特性就为四顶点曲线.以上这些曲线是指正则曲线,即曲线上的曲率不仅存     

The uniqueness of the adjoint operation I

Let H be a complex, infinite-dimensional Hilbert space. Let B(H) denote the set of bounded linear operators on H. This paper contains a nonlinear characterization of the adjoint operation on B(H). The statement of this result is:THEOREM:Let h: B(H) B(H)be a function such that h(I)0.Then h(ST)=h(T)h(S)and h(S)S0for all elements Sand Tof B(H)if and only if h(S)=S^* for all S B(H).     

初基演算

<正> 命题演算的构成,通常有三步骤的说法,即从 Johanson 的“极小演算”到 Heyting的构造论命题演算再到二值演算.此外,Lewis 从模态或严格蕴涵出发,也分别了许多步骤,以达到二值演算为其极限;特别值得注意的是最后三个步骤,即从 S4 到 S5 到二值演算.这两个三步骤就某意义说乃是通常命题演算的构成中最本质的步骤.综合这两个三步骤,会带来许多便利,而本文所提出的也就是作为二者共同基础的初基演算.     

和一类积分算子相关的单叶函数的子类

在Ruscheweyh定义了解析函数的Ruscheweyh导数[1]之后,许多学者相继研究了与Ruscheweyh导数有关的单叶或者多叶解析函数类.近来,Jung,Ki m和Srivastava[5]引入了下面的单参数积分算子类:Iσf(z)=zΓ2(σσ)∫0zlogtzσ-1f(t)dt,σ0,f∈Α.算子Iσ和Flett[6]研究的乘数变换密切相关.本文利用算子Iσ定义了两个函数类.首先研究在单位圆内解析的单叶函数类Rσ(A,B),给出函数类的包含关系Rσ(A,B)Rσ+1(A,B),同时也考虑了在积分算子Fλ的作用下的函数类的包含关系以及当λ取特殊值1时的特殊情况.其次研究了函数类Rσ(A,B)中系数为正实数的函数类Sσ(A,B),给出函数f(z)属于类Sσ(A,B)的充分必要条件.     

Über die Wertannahme der Ahlfors Funktion in beliebigen Gebieten

Let B be a domain ?? that supports nonconstant bounded analytic functions, let pe B. In [1] S.Fisher posed the question: Does the Ahlforsfunction for B and p map a maximal B onto the unit disc D? We give an example of a maximal B such that in B F(z) omits two values of D.     

A non-normal topology generated by a two-point selection

We construct a two-point selection , where is the set of the irrational numbers, such that the space is not normal and it is not collectionwise Hausdorff either. Here, τ_f denotes the topology generated by the two-point selection f. This example answers a question posed by V. Gutev and T. Nogura. We also show that if is a two-point selection such that the topology τ_f has countable pseudocharacter, then τ_f is a Tychonoff topology.     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.