Levi子群Lα(n(α)=1)在Chevalley群G(Ф,F)中的扩群

设L是复数域上单李代数,具有不可约根系Ф,固定基П.设F是一个特征不为2的域,且不是三元域,G(Ф,F)是F上Ф型的Chevalley群.设α∈П,Фα表示Ф的一种类型子根系.当n(α)=1,且Ф是Bl(l≥3),Dl(l ≥ 4),E6,E7,或E8之一时,本文决定了Levi子群Lα在G(Ф,F)中的所有扩群.     

Three classes of smooth Banach submanifolds in B(E,F)

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.     

Generalized Inverse Analysis on the Domain Ω(A,A+) in B(E,F)

Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B~+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B~+(E, F). In this paper we introduce an unbounded domain ?(A, A~+) in B(E, F) for A ∈ B~+(E, F) and A~+∈GI(A), and provide a necessary and sufficient condition for T ∈ ?(A, A~+). Then several conditions equivalent to the following property are proved: B = A+(IF+(T-A)A~+)~(-1) is the generalized inverse of T with R(B)=R(A~+) and N(B)=N(A~+), for T∈?(A, A~+), where IF is the identity on F. Also we obtain the smooth(C~∞) diffeomorphism M_A(A~+,T) from ?(A,A~+) onto itself with the fixed point A. Let S = {T ∈ ?(A, A~+) : R(T)∩ N(A~+) ={0}}, M(X) = {T ∈ B(E,F) : TN(X) ? R(X)} for X ∈ B(E,F)}, and F = {M(X) : ?X ∈B(E, F)}. Using the diffeomorphism M_A(A~+,T) we prove the following theorem: S is a smooth submanifold in B(E,F) and tangent to M(X) at any X ∈ S. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain ?(A, A~+). It seems to be useful for developing global analysis and geomatrical method in differential equations.     

SMOOTH AND PATH CONNECTED BANACH SUBMANIFOLD Σ_r OF B(E,F) AND A DIMENSION FORMULA IN B(R~n,R~m)

Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear oper- ators from E into F, Σr the set of all operators of finite rank r in B(E,F), and Σ#r the number of path connected components of Σr . It is known that Σr is a smooth Banach submani- fold in B(E,F) with given expression of its tangent space at each A ∈Σr. In this paper,the equality Σ#r = 1 is proved. Consequently, the following theorem is obtained: for any non- negative integer r, Σr is a smooth and path connected Banach submanifol...     

Local automorphisms of semisimple algebras and group algebras

Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to be an algebraically closed field of characteristic zero, K a finite group, F K the group algebra of K over F , then all local automorphisms of F K are also characterized.     

SMOOTH AND PATH CONNECTED BANACH SUBMANIFOLD ∑r OF B(E,F) AND A DIMENSION FORMULA IN B(IRn,IRm)

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear oper-ators from E into F, ∑r the set of all operators of finite rank r in B(E, F), and ∑#r the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submani-fold in B(E, F) with given expression of its tangent space at each A ∈∑r. In this paper, the equality ∑#r = 1 is proved. Consequently, the following theorem is obtained: for any non-negative integer r, ∑r is a smooth and path connected Banach submanifold in B(E, F) with the tangent space TA∑r={B∈B(E,F): BN(A) R(A)} at each A∈∑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 ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = IRn and F = IRm, then ∑r is a smooth and path connected submanifold of B(IRn,IRm) and its dimension is dim ∑r = (m + n)r- r2 for each r, 0 ≤ r < min{n,m}.     

COMPLETENESS OF THE PARABOLIC SUBGROUPS OF PROJECTIVE ORTHOGONAL GROUPS

All parabolic subgroups and Borel subgroups of PΩ(2m 1, F) over a linear-able field F of characteristic 0 are shown to be complete groups, provided m > 3.     

A SORT OF POLYNOMIAL IDENTITIES OF M_n(F)WITH CHAR F≠0

Let F denote a field,finite or infinite,with characteristic p≠0.In this paper,theauthor obtains the following result:The symmetric polynomial on t lettersS_(sym(t))(x_1,x_2,…,x_t))X_(π1)X_(π2)…X_(πt)is a polynomial identity of M_n(F)when t≥pn,and this is sharp in the sense that if t≤pn-1,it is not a polynomial identity of M_n(F).     

Linear Operators Preserving Symplectic Group over a Field Consisting of at Least Four Elements

Suppose F is a field consisting of at least four elements. Let Mn(F) and SP2n(F) be the linear space of all n×n matrices and the group of all 2n×2n symplectic matrices over F, respectively. A linear operator L:M2n(F)→M2n(F) is said to preserve the symplectic group if L(SP2n(F))=SP2n(F). It is shown that L is an invertible preserver of the symplectic group if and only if L takes the form (i) L(X)=QPXP-1 for any X∈M2n(F) or (ii) L(X)=QPXTP-1 for any X∈M2n(F), where Q∈SP2n(F) and P is a generalized symplectic matrix. This generalizes the result derived by Pierce in Canad J. Math., 3(1975), 715-724.     

The relations among the three kinds of conditional risk measures

Let(Ω,E,P)be a probability space,F a sub-σ-algebra of E,Lp(E)(1 p+∞)the classical function space and Lp F(E)the L0(F)-module generated by Lp(E),which can be made into a random normed module in a natural way.Up to the present time,there are three kinds of conditional risk measures,whose model spaces are L∞(E),Lp(E)(1 p+∞)and Lp F(E)(1 p+∞)respectively,and a conditional convex dual representation theorem has been established for each kind.The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems.We first establish the relation between Lp(E)and Lp F(E),namely Lp F(E)=Hcc(Lp(E)),which shows that Lp F(E)is exactly the countable concatenation hull of Lp(E).Based on the precise relation,we then prove that every L0(F)-convex Lp(E)-conditional risk measure(1 p+∞)can be uniquely extended to an L0(F)-convex Lp F(E)-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter,which shows that the study of Lp-conditional risk measures can be incorporated into that of Lp F(E)-conditional risk measures.In particular,in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L0-convex conditional risk measures.∞     

Elliptic curves and their torsion subgroups over number fields of type (2, 2,

Suppose that E：y2=x(x+M)(x+N) is an elliptic curve, where M相似文献   

拟半(E,F)}——凸函数及拟半(E,F)——凸规划的性质

简、胡及唐等(Int. J. Pure Appl. Math., 2004, 14(4): 439-454)于2004年提出了拟半(E,F)--凸函数的概念. 本文进一步深入讨论拟半(E,F)--凸函数及与之相应的拟半(E,F)--凸规划的一些重要性质, 建立了此类广义凸性的若干重要结论.     

B-半-（E，F）-凸函数和规划

本文介绍了一组新的广义凸函数: b-半-(E,F)-凸函数(拟b-半-(E,F)-凸函数,伪b-半-(E,F)-凸函数),并讨论了它们的一些基本性质.研究了带不等式约束的非线性b-半-(E,F)-凸规划的最优性充分条件和对偶定理.     

线性群的自同构群的一类子群

设 E为任意域 ,F为 E的子域 ,分别以 T=GL( n,E) ,S=GL( n,F )表示域 E、F上的 n阶一般线性群 ( n≥ 2 ) ,则 S为 T的子群 .本文确定 T的自同构群 Aut T中保持 S中每个元不动的自同构全体形成的群 Gal( T/ S) .     

Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Nonexcellence

A field extension L / F is called excellent if, for any quadratic form over F, the anisotropic part (L)an of over L is defined over F; L / F is called universally excellent if L E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras.     

局部凸代数L(E)的乘法连续性

本文讨论了(E,σ(E,E′))上连续自同态代数L(E)上在几种常见拓扑下的乘法连续性,得到相应的等价条件．     

Gel''fand三元组上Lévy过程的随机积分

本文研究了算子值过程关于Gel’fand三元组EHE*上Lévy过程的随机积分.利用再生核Hilbert空间上柱Lévy过程的随机积分,定义一类算子值过程关于E*-值Lévy过程的随机积分。     

Elliptic curves and their torsion subgroups over number fields of type (2, 2, ..., 2)

Suppose thatE: y ² =x(x + M) (x + N) is an elliptic curve, whereM N are rational numbers (#0, ±1), and are relatively prime. LetK be a number field of type (2,...,2) with degree 2′. For arbitrary n, the structure of the torsion subgroup E(K) _tors of theK-rational points (Mordell group) ofE is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E(K) _tors themselves. The order of E( K)_tors is also proved to be a power of 2 for anyn. Besides, for any elliptic curveE over any number field F, it is shown that E( L)_tors = E( F) _tors holds for almost all extensionsL/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.     

Intertwining operators for L 2(E)

Let (E,Q) be a finite dimensional quadratic vector space over a finite field. For the natural representation -π of the isometry group G of (E,Q) in the space L ²(E) of all complex valued functions on E, we analyse when the intertwining algebra of π is generated by just one averaging operator.     

从特殊线性群到一般射影线性群的同态的一个性质

设F,K为域,GLn（F）,SLn（F）分别表示F上的n级一般线生群和n级特殊线性群．PGLn（F）,PSLn（F）分别表示F上的n级射影一般线性群和n级射影特殊线性群．φ：SLn（F）→PGLn（K）,n≥3为非平凡同态．本文确定了当K的持征为2时η的—个性质．