半质环的两个交换性结果

定理1 设R是半质环,m,n是固定正整数,且n>1.如果R满足条件(x^my)ⁿ-x^my∈Z(R),?x,y∈R,则R是交换环.定理2 设R是半质环,m,n,s,t是固定正整数,且(m+n)t=s+1,mt>1.如果R满足条件[x^m,yⁿ]^t-[x,y^s]∈Z(R),?x,y∈R,则R是交换环.     

对称凸集上初等对称函数为Schur-凹的充要条件

设x,y∈Rⁿ,x被y所控制记作x(?)y。又w∈Rⁿ,令S_k(x)为第k个初等对称函数。Q_m,n为前n个自然数取m个的严格增序列的集合。对于β∈Q_m,n写w_β=(W_{^β(1),…,WB(m)}∈R^m。本文主要证明了下面的结论:(1)S_k(x)在(?)_w上是Schur-凹的充要条件是(2)S_k(x)≥0,(?)x∈(?)_w的充要条件是S_k(w)≥0且S_k(x)在(?)_w上是Schur-凹的(3)S_k(x)≥0,(?)x∈(?)的充要条件是     

矩阵方程AXAT=C的对称斜反对称解

设A∈R^m×n,C∈R^m×m给定,利用矩阵的广义奇异值分解和对称斜反对称矩阵的性质,得到了矩阵方程(1)AXA^T=C存在对称斜反对称解的充要条件和通解表达式;证明了若方程(1)有解,则一定存在唯一极小范数解,并给出了极小范数解的具体表达式和求解步骤.     

关于一类高次不定方程的解

本文得到了方程(axⁿ±c)/(ax^m±c)=y²+1,c=1,2,4适合m≡n(mod2)的全部解.     

极大高阶交换子的端点估计

该文主要确立了当b∈BMO 时, 极大高阶奇异积分算子交换子T_{b, m}* 满足如下不等式 |{y∈Rⁿ:T_{b, m}*f(y)>λ}|≤C||b||^m_BMO∫_Rⁿ|f(y)|/λ (1+log⁺|f(y)|/λ)^mdy 且T_{b, m}* 在L^p(Rⁿ)(1 < p <∞上有界.     

Mn(C)中的不可约性

对于M_n(C)(所有n×n矩阵的全体)中的不可约矩阵得到以下结果:对于任意A∈M_n(C),设λ₁, λ₂,…,λ_m为A的所有特征值,这里m≤n而且当i≠j时,λ_i≠λ_j.则A是不可约的当且仅当任意P∈A′(A),P^*=P=P²,有σ(P|ker(A-λ₁))=σ(P|ker(A-λ₂))=…=σ(P|ker(A-λ_m))为单点集.     

Heisenberg群上一类左不变LPDO的局部基本解及局部可解性

In this paper it is proved that local fundamental solution exists in some space W^m(H_n) (m∈Z), if the left invariant differential operator on the Heisenberg group H_n satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on H_n. If there exist R≥0, r,s∈R and operators {B_λ|λ∈Γ_R} ∈V^s(Γ_R, M^r) such that, for almost all λ∈Γ_R, B_λ is the right inverse of Ⅱ_λ(L), then there exists E∈W^m(H_n) (when m≥0 or m even) or E∈W^m-1(H_n) (when m<0 and odd) such that LE =δ(near the origie) Where m=min([r],-[2s]-n-2); 2. Let L(W,T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ|≥R,(?) and C_λ≥ C|λ|^x(C>0), then the same conclusion as above holds with m=min(-[2r]-n-2,[-2s]-n-2).     

非参数回归函数最近邻估计Lp收敛的速度

Let (X,Y) be a R^d×R¹-valued random vector with E(|Y|)<∞,m(x)=E(Y|X=x) be the regression funvion of Y with respect to X.Suppose that (X_i, Y_i),i=1, …,n, are iid samples drawn from (X,Y). It is desired to estimate m(x) based on these samples,everoye discussed in 1981 (see [2]) the pointwise L_p-convergence of the nearest neigthoor estimate m_n(x) (see (5) of the present paper). In this article we further study the rate of this convergence.It is shown that if there exists p≥2 such that E |Y|^p<∞,then E|m_n(x)-m(x)|^p=O(n^-p/(d+2))a.s. for suitabte choice of the weighte C_m (see(4) of the present paper).     

正则环的内射理想

设R为正则环, A∈ M_n(R). 如果M_n(R)AM_n(R)为M_n(R) 的内射理想,该文证明了有U, V∈ GL_n(R)使得UAV为对角矩阵.     

齐次乘子算子的交换子

记H^l={w∈C^∞(R^k\{0}):w是l次齐次函数),R^_(-a)(m)是Taylor级数余项算子的n重叠合:m=(m₁,…,m_n)∈Zⁿ,Z记非负整数的集,α∈(R^k)ⁿ,定义 其中a=(a₁,…,a_n),a_i,f∈(R^k), 主要结果如下: 1.证明了几个介于算子T_^R(-a)^{(m)_w(ξ)),(a,f})的类与多线性奇异积分算子的类之间的对等定理; 2.作为应用,算子及 的某些有界性结果被给出,其中Ω∈H⁰,|β|≤|m|,且,m_i≥1。     

On the regular Sylow p-subgroups of Chevalley groups over \mathbb{Z}_{p^m }

We prove that a Sylow p-subgroup of the general linear group of dimension n over the residue ring modulo p ^m is regular for n ² < p and powerful if and only if n = 2 and m = 1. We obtain similar results for the Sylow p-subgroups of normal types over the same ring.     

A gallai-type problem

Two disjoint sets in R^m are said to be n-incident if every flat spanned by n distinct points of one set contains a point of the other. We obtain bounds for the dimension of the flat spanned by the union of n-incident sets and consider a related problem.     

A remark on full rank perfect codes

Any full rank perfect 1-error correcting binary code of length n=2^k-1 and with a kernel of dimension n-log(n+1)-m, where m is sufficiently large, may be used to construct a full rank perfect 1-error correcting binary code of length 2^m-1 and with a kernel of dimension n-log(n+1)-k. Especially we may construct full rank perfect 1-error correcting binary codes of length n=2^m-1 and with a kernel of dimension n-log(n+1)-4 for m=6,7,…,10.This result extends known results on the possibilities for the size of a kernel of a full rank perfect code.     

Two Commuting Involutions Fixing F n ∪ F n-1

Consider G=Z ₂² as the group generated by two commuting involutions, and let be a smooth G-action on a smooth and closed manifold M. Suppose that the fixed point set of Φ consists of two connected components, F ⁿ and F ^n-1, with dimensions n and n−1, respectively. In this paper we prove that, if in the fixed data of Φ at least two eigenbundles over F ⁿ have dimension greater than n, and at least one eigenbundle over F ^n-1 has dimension greater than n−1, then the action (M,Φ) bounds equivariantly.It is well known that, if is a smooth involution on a smooth and closed m-dimensional manifold M ^m such that the fixed point set of T has constant dimension n, and if m > 2n, then (M ^m ,T) bounds equivariantly; this fact was proved by R. E. Stong and C. Kosniowski 27 years ago. As a consequence of our result, we will see that the same fact is true when, besides n-dimensional components, the fixed point set contains additional (n−1)-dimensional components.     

Affine geometries and sets of equiorthogonal frequency hypercubes

Equiorthogonal frequency hypercubes are one particular generalization of orthogonal latin squares. A complete set of mutually equiorthogonal frequency hypercubes (MEFH) of order n and dimension d, using m distinct symbols, has (n − 1)^d/(m − 1) hypercubes. In this article, we prove that an affine geometry of dimension dh over 𝔽_m can always be used to construct a complete set of MEFH of order m^h and dimension d, using m distinct symbols. We also provide necessary and sufficient conditions for a complete set of MEFH to be equivalent to an affine geometry. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 435–441, 2000     

Integrable Submanifolds in Almost Complex Manifolds

Let (M,J) be a germ of an almost complex manifold of real dimension 2m and let n (n<m) be an integer. We study a necessary and sufficient condition for M to admit an integrable submanifold N of complex dimension n. If n=m−1, we find defining functions of N explicitly from the coefficients of the torsion tensor. For J obtained by small perturbation of the standard complex structure of ℂ ^m this condition is given as an overdetermined system of second order PDEs on the perturbation. The proof is based on the rank conditions of the Nijenhuis tensor and application of the Newlander-Nirenberg theorem. We give examples of almost complex structures on ℂ³: the ones with a single complex submanifold of dimension 2 and the ones with 1-parameter or 2-parameter families of complex submanifolds of dimension 2.     

Embedding EnG in Euclidean space

It is shown that if G is an arbitrary upper semicontinuous decomposition of Eⁿ for which π(N_G embeds in S^m for some m?3, then the decomposition space $\text{E}{}^{\mathrm{n}}\text{G}$ embeds as a closed subset of E^n+m+1. The proof consists of constructing a cell-like upper semicontinuous decomposition G? of E^n+m+1 which intersects Eⁿ to yield precisely G and using Edwards' Cell-Like Approximation Theorem to show that G? is shrinkable. As an immediate corollary, $\text{E}{}^{\mathrm{n}}\text{G}$ embeds in E^n+2k+2 whenever G is an arbitrary k-dimensional upper semicontinuous decomposition of Eⁿ. This is an improvement of (n?1)-dimensions over the corresponding dimension theoretic result and examples due to Daverman show that this result is sharp in case n is odd and off by no more than one dimension in case n is even.     

On commuting varieties of upper triangular matrices

It is known that the variety of the pairs of n×n commuting upper triangular matrices is not a complete intersection for infinitely many values of n; we show that there exists m such that this happens if and only if n>m. We also show that m<18 and that m could be found by determining the dimension of the variety of the pairs of commuting strictly upper triangular matrices. Then, we define an embedding of any commuting variety into a grassmannian of subspaces of codimension 2.     

Compact complex homogeneous manifolds with large automorphism groups

Let X be a compact complex homogeneous manifold and let Aut(X) be the complex Lie group of holomorphic automorphisms of X. It is well-known that the dimension of Aut(X) is bounded by an integer that depends only on n=dim X. Moreover, if X is K?hler then dimAut (X)≤n(n+2) with equality only when X is complex projective space. In this article examples of non-K?hler compact complex homogeneous manifolds X are given that demonstrate dimAut(X) can depend exponentially on n. Let X be a connected compact complex manifold of dimension n. The group of holomorphic automorphisms of X, Aut(X), is a complex Lie group [3]. For a fixed n>1, the dimension of Aut(X) can be arbitrarily large compared to n. Simple examples are provided by the Hirzebruch surfaces F _m , m∈N, for which dimAut(F _m )=m+5, see, e.g. [2, Example 2.4.2]. If X is homogeneous, that is, any point of X can be mapped to any other point of X under a holomorphic automorphism, then the dimension of the automorphism group of X is bounded by an integer that depends only on n, see [1, 2, 6]. The estimate given in [2, Theorem 3.8.2] is roughly dimAut(X)≤(n+2) ⁿ . For many classes of manifolds, however, the dimension of the automorphism group never exceeds n(n+2). For example, it follows directly from the classification given by Borel and Remmert [4], that if X is a compact homogeneous K?hler manifold, then dimAut(X)≤n(n+2) with equality only when X is complex projective space P ⁿ . It is an old question raised by Remmert, see [2, p. 99], [6], whether this same bound applies to all compact complex homogeneous manifolds. In this note we show that this is not the case by constructing non-K?hler compact complex homogeneous manifolds whose automorphism group has a dimension that depends exponentially on n. The simplest case among these examples has n=3m+1 and dimAut(X)=3m+3 ^m , so the above conjectured bound is exceeded when n≥19. These manifolds have the structure of non-trivial fiber bundles over products of flag manifolds with parallelizable fibers given as the quotient of a solvable group by a discrete subgroup. They are constructed using the original ideas of Otte [6, 7] and are surprisingly similar to examples found there. Generally, a product of manifolds does not result in an automorphism group with a large dimension relative to n. Nevertheless, products are used in an essential way in the construction given here, and it is perhaps this feature that caused such examples to be previously overlooked. Oblatum 13-X-97 & 24-X-1997     

All spectral dominant norms are stable

A vector norm |·|on the space of n×n complex valued matrices is called stable if for some constant K&>0, not depending upon A or m, we have |A^m|?K|A|^m We show that such a norm is stable if and only if it dominates the spectralradius.