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 .     

SOME PROPERTIES OF ROSEN’S MLE FOR GENERAL DISTRIBUTIONS

1 IntroductionConsider the lnultivariate linear model (MLM) as follows:mX = Z AiBiC E (1)i= 1where X, Ai, Bi and C are p x nfp x qi(qi 5 p), qi x ki and ki x n matrices respectively, Z is ap x p definite positive matrix with p(C1) p 5 n and R(CL) G R(Cfu--,) g' g R(CI), p(.)and R(.) stand for the rank and the colunu spanned linear space Of a matriX respbctively.e = (e1,'2,... f e.), e1le21',f n are iid. p--variate random vectors with D(e1) = Z > 0,E(El) = 0, A: aild C: are …     

Upper Bounds on Character Sums with Rational Function Entries

We obtain formulac and estimates for character sums of the type S(x,f,p^m)=∑x=1^p^mx(f(f)),where p^m is a prime power with m≥2,x is a multiplicative character(mod p^m),and f=f1/f2 is a rational function over Z.In particular,if p is odd,d=deg(f1) deg(f2) and d^*=max(deg(f1),deg(f2)) then we obtain |S(x,f,p^m)|≤(d-1)p^m(1-1/d^*) for any non-constant f (mod p) and primitive character x.For p=2 an extra factor of 2√2 is needed.     

INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/（p^e）

Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence (a) over Z/(pe), there is a unique p-adic decomposition (a) = (a)0 (a)1·p … (a)e-1 ·pe-1, where each (a)i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and G' (f(x), pe) the set of all primitive sequences generated by f(x) over Z/(pe). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 deg(μ(x)),p- 1) = 1,set ψe-1 (x0, x1,…, xe-1) = xe-1·[ μ(xe-2) ηe-3 (x0, x1,…, xe-3)] ηe-2 (x0, x1,…, xe-2),which is a function of e variables over Z/(p). Then the compressing map ψe-1: G'(f(x),pe) → (Z/(p))∞,(a) (→)ψe-1((a)0, (a)1,… ,(a)e-1) is injective. That is, for (a), (b) ∈ G' (f(x), pe), (a) = (b) if and only if ψe - 1 ((a)0, (a)1,… , (a)e - 1) =ψe - 1 ((b)0,(b)1,… ,(b)e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions ψe-1 and ψe-1 over Z/(p) are both of the above form and satisfy ψe-1((a)0,(a)1,… ,(a)e-1) = ψe-1((b)0,(b)1,… ,(b)e-1) for (a),(b) ∈ G'(f(x),pe), the relations between (a) and (b), ψe-1 and ψe-1 are discussed.     

环Z/(p~e)上本原序列压缩映射的新结果

令Z/(pe)表示整数剩余类环,其中p为素数且e 2为正整数.令f(x)表示Z/(pe)上的n次本原多项式,G′(f(x),pe)表示Z/(pe)上所有由f(x)生成的本原序列构成的集合.设序列a∈G′(f(x),pe),它有唯一的p进制展开a=a0+a1p+···+ae-1pe-1.令φ(x0,x1,...,xe-1)=g(xe-1)+μ(x0,x1,...,xe-2)表示由Fe p到Fp的一个e变元多项式.那么,φ可以诱导出一个从G′(f(x),pe)到F∞p的压缩映射.在p为奇素数且f(x)为强本原多项式的条件下,人们已经证明该压缩映射是保熵的.而本文证明该压缩映射在f(x)为本原多项式的条件下仍然是保熵的.当deg(g(x))2时,我们还要求deg(g(x))为奇数,或者g(x)=xk+∑k-2i=0cixi.     

Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality

It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,Z Rn Z Rn f(x)g(y)|x||x.y||y|dxdy6 B(p,q,,,,n)kfkLp(Rn)kgkLq(Rn).The main purpose of this paper is to give the sharp constants B(p,q,,,,n)for the above inequality for three cases:(i)p=1 and q=1;(ii)p=1 and 1q 6∞,or 1p 6∞and q=1;(iii)1p,q∞and 1p+1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1p+1q1.     

整数集上四连贯n次多项式的构造

连贯、m (m∈ N,m≥ 3)连贯的定义见[1]或 [2 ].约定 :本文中表示数的字母均表整数 .定理　当an-i =p1 q1 ki-1 (pq1 p1 q) ki pqki 1 ,(i=0 ,1,… ,n- 1,n∈ N,n≥ 2 ,k-1 =k0 =0 )kn =± 1,pq1 - p1 q =± 1,a0 =p1 (q1 kn-1 qkn)时 ,多项式 f (x) =∑n-1i=0an-ixn-i a0 在整数集 Z上连贯 ,且 f(x) j (j =0 ,1)分别有因式px p1 ,qx q1 .证明　这是因为由题设可证得 :f(x) =(px p1 ) ∑n-1i=0(q1 ki qki 1 ) xn-i-1 ,f(x) 1=(qx q1 ) ∑n-1i=0(p1 ki pki 1 ) xn-i-1 .在定理中可选 :(1) kn=1,q1 =rp1 1,p …     

NEUMANN PROBLEM FOR THE LANDAU-LIFSHITZ-MAXWELL SYSTEM IN TWO DIMENSIONS

91. IntroductionIn 1935, LandauLifshitz[1] proposed the fOllowing coupled system of the nonlinear evo-lution equationZr = --a,t x (2 x (b f H)) a,E x (b f A), (1.1)- 8E7 x H = -- aE, (1.2)0t- 0H 0ZV x E = ---- -- pfZ0t p7' (1'3)v. A gv. 2 = 0, v. E = 0, (l.4)where a1, a2, a, g are constants, cr1 2 0, a 2 0, Z(x,t) = (Z1(x,t), Z2(x,t), Z3(x,t))denotes the microscopic magnetization field, H = (H1 (x, f), H2(x, t), H3(x, t)) the magneticfield, E(x, t) = (E1(x, t), E2(x, t), E3(…     

p- 局部化无限射影空间的自映射

分别记Z（p）和Zp为整数环Z的p-局部化和p-完备化,那么我们有自然的含入映射Z（p）→Zp.令S2n-1（p）为p-局部化的2n-1维球面,令B2n（p）为一个p-局部化空间,满足S2n-1（p）=～ΩB2n（p）,那么我们有H＊（B2n（p）,Z（p））=Z（p）[u],其中u的度数为2n.对于B2n（p）的任意一个自映射f,我们定义f的度数为k∈Z（p）满足f＊（u）=ku.运用整值多项式理论,我们证明存在B2n（p）的一个度数为k的自映射当且仅当k在Zp中是一个n次幂.     

借用向量分解定理巧求f(x0)的范围

已知函数 f ( xi) ( i =1,2 ,3 ,… )的范围 ,求 f( x0 )的范围 .笔者在同行们研究的基础上 ,借用向量分解定理 ,使这类问题的解决更加简单、明了 ,可操作性强 ,便于实施 .例 1　已知一次函数 f( x) ,1≤ f ( 1)≤2 ,3≤ f ( 2 )≤ 4,试确定 f( 5 )的范围 .解　设一次函数为 f( x) =ax + b,则　 f( 1) =a+ b,f( 2 ) =2 a+ b,f( 5 ) =5 a+ b.记　 p1→ =a+ b,p2→ =2 a+ b,p=5 a+ b显然 p1→ ,p2→ 不共线 ,根据向量分解定理p=λ1 p1→ +λ2 p2→ 　 (λ1 ,λ2 为实数 ) ,即　 5 a+ b=λ1 ( a+ b) +λ2 ( 2 a+ b…     

Adjoints of linear fractional composition operators on the Dirichlet space

The adjoint of a linear fractional composition operator acting on the classical Dirichlet space is expressed as another linear fractional composition operator plus a two rank operator. The key point is that, in the Dirichlet space modulo constant functions, many linear fractional composition operators are similar to multiplication operators and, thus, normal. As a particular application, we can easily deduce the spectrum of each linear fractional composition operator acting on such spaces. Even the norm of each linear fractional composition operator is computed on the Dirichlet space modulo constant functions. It is also shown that all this work can be carried out in the Hardy space of the upper half plane.This work was partially supported by Plan Nacional I+D Ref. BFM2000-0360 and Junta de Andalucía Ref. FQM-260. The first named author was also supported by Plan Propio de la Universidad de Cádiz.     

非线性Lipschitz算子的Lipschitz对偶算子及其应用

在文山中我们对非线性Ｌｉｐｓｃｈｉｔｚ算子定义了其Ｌｉｐｓｃｈｉｔｚ对偶算子，并证明了任意非线性Ｌｉｐｓｃｈｉｔｚ算子的Ｌｉｐｓｃｈｉｔｚ对偶算子是一个定义在Ｌｉｐｓｃｈｉｔｚ对偶空间上的有界线性算子．本文还进一步证明：设Ｃ为 Ｂａｎａｃｈ空间 Ｘ的闭子集，Ｃ＊Ｌ为Ｃ的 Ｌｉｐｓｃｈｉｔｚ对偶空间，Ｕ为 Ｃ＊Ｌ上的有界线性算子，则当且仅当 Ｕ为 ｗ＊－ｗ＊连续的同态变换时，存在Ｌｉｐｓｃｈｉｔｚ连续算子Ｔ，使Ｕ为Ｔ的Ｌｉｐｓｃｈｉｔｚ对偶算子．这一结论的理论意义在于：它表明一个非线性Ｌｉｐｓｃｈｉｔｚ算子的可逆性问题可转化为有界线性算子的可逆性问题．作为应用，通过引入一个新概念──ＰＸ－对偶算子，在一般框架下给出了非线性算子半群的生成定理．     

A Non-Separable Reflexive Banach Space on Which there are Few Operators

It is shown that there exists a non-separable reflexive Banachspace on which every bounded linear operator is the sum of ascalar multiple of the identity operator and an operator ofseparable range. There is a strong sense that such a Banachspace has as few operators as its linear and topological propertiesallow.     

Random commutation

We investigate the commutation between a continuous linear random operator and a continuous linear deterministic operator on a Banach space. From this we obtain probabilistic versions of theorems by Fuglede and Putnam, both of them dealing with the commutation between continuous linear operators with continuous normal operators on a Hilbert space.     

Cosine of angle and center of mass of an operator

We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.     

Perturbed projection methods for split equations of the first kind

The paper investigates the approximate solution of linear operator equations of type (J-K)u=v, with J assumed invertible, obtained by applying a linear projection on both sides of the equation, together with a linear perturbation operator on the left side and a perturbation element on the right side.     

Banach空间的Lipschitz对偶及其应用

本文引进Banach空间E的一个全新对偶空间概念—Lipschitz对偶空间,并证明：任何Banach空间的Lipschitz对偶空间是某个包含E的Banach空间的线性对偶空间,以所引进的新对偶空间为框架,本文定义了非线性Lipschitz算子的Lipshitz对偶算子,证明：任何非线性Lipschitz算子的Lipschitz对偶算子是有界线性算子．所获结果为推广线性算子理论到非线性情形（特别,运用线性算子理论研究非线性算子的特性）开辟了一条新的途径．作为例证,我们应用所建立的理论证明了若干新的非线性一致Lipschitz映象遍历收敛性定理.     

算子概率范数与共鸣定理

提出概率赋范线性空间上集合有界性的简化定义,利用算子概率范数概念·进一步研究概率赋范线性空间上的线性算子理论,并在算子概率赋范空间上,建立了概率有界、概率半有界、非概率无界意义下的共鸣定理·     

Convergence results of a modified regularized gradient method for nonlinear ill-posed problems

A modified iteratively regularized gradient method and its continuous version are proposed for nonlinear ill-posed problems, in which the Tikhonov regularization term is generated by a linear operator. The linear operator may have some physical meaning. And by employing the linear operator, scaling the problem can be avoided in the case that the nonlinear operator in the problem has larger gradient. Adopting a posteriori and a priori stopping rule respectively, we establish the convergence results by using a modified approximate source condition. The numerical results show that the linear operator effects the performance greatly.     

Elementary rotations of operators on regular Banach spaces

In this paper, we prove the existence of an elementary rotation (a Julia operator) for any continuous, linear, adjointable operator in a regular Banach space with inner product. The proof is based on a more general theorem of the author on the existence of an elementary rotation for any linear operator in a category with quadratic splitting. This result is a generalization of a well-known result about the existence of an elementary rotation for any continuous linear operator in a Krein space. The result can be useful for constructing isometric and unitary dilations as well as characteristic functions of continuous linear operators acting in regular Banach spaces with inner product. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 175–192, 2006.