R(L)型诱导空间的权、特征和浓度

针对F格R(L)引入权、特征和浓度的概念,证明了F格R(L),满层LF拓扑空间(LX,δ)及其所诱导的R(L)型诱导空间(R(L)X,ω(δ))三者在权、特征和浓度三方面三个重要的等式。     

关于A-拓扑的2~X一些基数函数

本文赋予超空间2X一种新的拓扑(A-拓扑),证明了此空间的一些基数函数不等式的成立,从而推广了文[1]的一些结果。     

Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T ^*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2(G,K) the maximal number of functionally independent functions from A\C. We prove that (G,K) is equal to the codimension (G,K) of maximal dimension orbits of the Borel subgroup BG ^C in the complex algebraic variety G ^C/K ^C. Moreover, if (G,K)=1, then all G-invariant Hamiltonian systems on T ^*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.     

Linear algebraic groups and countable Borel equivalence relations

If is an equivalence relation on a standard Borel space , then we say that is Borel reducible to if there is a Borel function such that . An equivalence relation on a standard Borel space is Borel if its graph is a Borel subset of . It is countable if each of its equivalence classes is countable. We investigate the complexity of Borel reducibility of countable Borel equivalence relations on standard Borel spaces. We show that it is at least as complex as the relation of inclusion on the collection of Borel subsets of the real line. We also show that Borel reducibility is -complete. The proofs make use of the ergodic theory of linear algebraic groups, and more particularly the superrigidity theory of R. Zimmer.

     

Superrigidity and countable Borel equivalence relations

We formulate a Borel version of a corollary of Furman's superrigidity theorem for orbit equivalence and present a number of applications to the theory of countable Borel equivalence relations. In particular, we prove that the orbit equivalence relations arising from the natural actions of on the projective planes over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.     

Borel extractions of converging sequences in compact sets of Borel functions

It is well known by a classical result of Bourgain–Fremlin–Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence (f_n)_nω in K one can extract a subsequence (f_nk)_kω converging to f. In the present work we prove that this extraction can be achieved in a “Borel way.” This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence.     

Absolute Borel sets and function spaces

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Cech-complete spaces. We also show that the absolute Borel class of is determined by the uniform structure of the space of continuous functions ; however the case of absolute metric spaces is still open. More precisely, we prove that, for metrizable spaces and , if is a uniformly continuous surjection and is an absolute Borel set of multiplicative (resp., additive) class , , then is also an absolute Borel set of the same class. This result is new even if is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the \v{C}ech-completeness of a metric space is determined by the linear structure of .

     

拟亚纯映射的Borel点

对于平面上的Ｋ拟亚纯映射,利用孙道椿,杨乐［１］建立的一个基本不等式,讨论了拟亚纯映射在单位圆内的奇异Ｂｏｒｅｌ点．将亚纯函数的有关结果［２］推广到Ｋ拟亚纯映射．     

Borel Subalgebras of Schur Superalgebras

It is proved that any Schur superalgebra is representable as a product of two Borel subalgebras of that superalgebra, which are symmetric w.r.t. its natural anti-isomorphism (Bruhat-Tits decomposition). This readily implies that any simple module is uniquely defined by its highest weight, and all other weights are strictly less than is the highest under the dominant ordering. It is stated that the fundamental theorem of Kempf, which is valid for all classical Schur algebras, might be true for superalgebras only if they are semisimple. Nevertheless, a weaker theorem of Grothendieck holds true for superalgebras since Borel subalgebras are quasihereditary. Also we formulate an analog of the Donkin-Mathieu theorem for Schur superalgebras, and show that it is valid in the elementary non-classical case, that is, for the algebras S(1|1, r).__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 305–334, May–June, 2005.     

Borel subrings of the reals

A Borel (or even analytic) subring of either has Hausdorff dimension or is all of . Extensions of the method of proof yield (among other things) that any analytic subring of having positive Hausdorff dimension is equal to either or .

     

非扭仿射Kac-Moody代数中Borel子代数的扩代数

决定了非扭仿射Kac-Moody代数中所有包含标准Borel子代数的子代数。     

拟亚纯映射的最大型Borel方向

作者用几何方法研究了更广泛的K-拟亚纯映射;定义了K-拟亚纯映射的最大型Borel方向;证明了有限正级K-拟亚纯映射最大型Borel方向的存在性;并导出了最大型Borel方向的一个充要条件.     

Representations of Parabolic and Borel Subgroups

We demonstrate a relationships between the representation theory of Borel subgroups and parabolic subgroups of general linear groups. In particular, we show that the representations of Borel subgroups could be computed from representations of certain maximal parabolic subgroups.     

The Borel directions of algebroidal function and its coefficient functions

In this article, the relationship between the Borel direction of algebroidal function and its coefficient functions is studied for the first time. To begin with, several theorems of algebroidal functions in unit disk are proved. By these theorems, some interesting conclusions are obtained.     

Borel complexity of the space of probability measures

Using a technique developed by Louveau and Saint Raymond, we find the complexity of the space of probability measures in the Borel hierarchy: if is any non-Polish Borel subspace of a Polish space, then , the space of probability Borel measures on with the weak topology, is always true , where is the least ordinal such that is .

     

Effective Borel measurability and reducibility of functions

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single‐valued as well as for multi‐valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya‐Srivastava in order to prove a generalization of the Representation Theorem of Kreitz‐Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach‐Hausdorff‐Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Common Borel directions of a meromorphic function with zero order and its derivative

There is a meromorphic function of zero order for which the function and its derivative have no common Borel direction.

     

On Conjectures of Mathai and Borel

Mathai has conjectured that the Cheeger–Gromov invariant ₍₂₎ = ₍₂₎ - is a homotopy invariant of closed manifolds with torsion-free fundamental group. In this paper we prove this statement for closed manifolds M when the rational Borel conjecture is known for = ₁(M), i.e. the assembly map : H _*(B, ) L_*() is an isomorphism. Our discussion evokes the theory of intersection homology and results related to the higher signature problem.