On Prime and Semiprime Near-Rings with Generalized Derivation

Abstract

Let N be a left near-ring and S be a nonempty subset of N. A mapping F from N to N is called commuting on S if [F(x),x] = 0 for all x € S. The mapping F is called strong commutativity preserving (SCP) on S if [F(x),F(y)] = [x,y] for all x, y € S. In the present paper, firstly we generalize the well known result of Posner which is commuting derivations on prime rings to generalized derivations of semiprime near-rings. Secondly, we investigate SCP-generalized derivations of prime near-rings.     

Semicircular-Like Laws and the Semicircular Law Induced by Orthogonal Projections

The main purpose of this paper is to construct semicircular elements from orthogonal projections. The construction, itself, would be the main result of this paper. Free distributional data and operator-theoretic properties of certain semicircular-like elements, and semicircular elements will be considered.     

投影的约化极小模和空间的夹角

Let M and N be nonzero subspaces of a Hilbert space H, and PM and PN denote the orthogonal projections on M and N, respectively. In this note, an exact representation of the angle and the minimum gap of M and N is obtained. In addition, we study relations between the angle, the minimum gap of two subspaces M and N, and the reduced minimum modulus of （I - PN）PM,     

Generalized Bregman Projections in Convex Feasibility Problems

We present a method for finding common points of finitely many closed convex sets in Euclidean space. The Bregman extension of the classical method of cyclic orthogonal projections employs nonorthogonal projections induced by a convex Bregman function, whereas the Bauschke and Borwein method uses Bregman/Legendre functions. Our method works with generalized Bregman functions (B-functions) and inexact projections, which are easier to compute than the exact ones employed in other methods. We also discuss subgradient algorithms with Bregman projections.     

A NOTE ON GENERALIZED DERIVATION RANGES

This paper is devoted to the generalized derivations determined by compact operatorsacting on Banach spaces.The author introduces the concept of “family of approximatelylinearly indepedent vectors” and employs this concept to prove the following result:Let X,Y be Banach spaces,A∈B(X),B∈B(Y)be compact operators.Then _(AB) has closed rangeif and only if both A and B have closed range.     

Optimal Rates of Linear Convergence of Relaxed Alternating Projections and Generalized Douglas-Rachford Methods for Two Subspaces

We systematically study the optimal linear convergence rates for several relaxed alternating projection methods and the generalized Douglas-Rachford splitting methods for finding the projection on the intersection of two subspaces. Our analysis is based on a study on the linear convergence rates of the powers of matrices. We show that the optimal linear convergence rate of powers of matrices is attained if and only if all subdominant eigenvalues of the matrix are semisimple. For the convenience of computation, a nonlinear approach to the partially relaxed alternating projection method with at least the same optimal convergence rate is also provided. Numerical experiments validate our convergence analysis     

正交射影在广义框架理论中的应用

本文研究了可分的Hilbert空间H中的广义框架,运用算子理论方法,研究了可分的 Hilbert空间H中广义框架的性质,给出了广义框架的对偶广义框架的一些刻画,并且证明了两个广义框架是强非交的一个充分必要条件．     

Sobolev Mapping of the Bergman Projections on Generalized Hartogs Triangles

In this paper,we investigate Sobolev mapping properties of the Bergman projection.The domain we focus on is defined by■ where m=(m₁,...,m_N) ∈(Z⁺)^N,n=(N₁,...,nN) ∈(Z⁺)^N,N∈Z⁺.Sobolev irregularity of the Bergman projections on Ω_n/m is shown.We also prove some Sobolev regularity results of the Bergman projections on Ω_n/m for m=(1,...,1).     

Free Mutual Information for Two Projections

The present paper provides a proof of \(i^*(\mathbb {C}P+\mathbb {C}(I-P); \mathbb {C}Q+\mathbb {C}(I-Q))=-\chi _{{ orb}}(P,Q)\) for two projections P, Q without any extra assumptions. An analytic approach is adopted to the proof, based on a subordination result for the liberation process of symmetries associated with P, Q.     

度量投影的收敛性（英文）

讨论了赋范空间中度量投影的收敛性．得到了在局部紧集控制下，Chebyshev凸集序列的度量投影的收敛性与K－M收敛，Wijsman收敛和Kuratowski收敛都等价．本文的结论完善了M．Tsukada在[1]和[2]结果．     

On the Projections of Multivalued Maps

This paper considers analogues of the Helmholtz projections of the set of selections of a piecewise smooth multivalued map , n2. It is shown that, for mn–1 (m=1), the closure of the projection of on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map which ensure that the closure of the projection of contains the zero element. Possible applications to optimal control problems are discussed.     

On Sums of Projections

In the paper, for all n, we describe the set _n of all real numbers admitting a collection of projections P ₁,...,P _n on a Hilbert space H such that _k=1 ⁿ P _k= I (I is the identity operator on H) and study the problem to find all collections of this kind for a given _n .     

Automorphisms on the Poset of Products of Two Projections

Let ℌ be a complex Hilbert space with dimℌ ≥ 3 and $${\cal     

On the Norm of the Metric Projections

LetXbe a Banach space. GivenMa subspace ofXwe denote withP_Mthe metric projection ontoM. We defineπ(X) sup{P_M: Ma proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=L_p, we obtain the inequality P_M2^|2/p−1|, for every subspaceMofL_p. We also show thatπ(X)=π(X*).     

On the Determination of Convex Bodies by Translates of Their Projections

It is a well-known fact that a three-dimensional convex body is, up to translations, uniquely determined by the translates of its orthogonal projections onto all planes. Simple examples show that this is no longer true if only lateral projections are permitted, that is orthogonal projections onto all planes that contain a given line. In this article large classes of convex bodies are specified that are essentially determined by translates or homothetic images of their lateral projections. The problem is considered for all dimensions , and corresponding stability results are proved. Finally, it is investigated to which degree of precision a convex body can be determined by a finite number of translates of its projections. Various corollaries concern characterizations and corresponding stability statements for convex bodies of constant width and spheres.     

Weighted Generalized Inverses,Oblique Projections,and Least-Squares Problems

A generalization with singular weights of Moore–Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.     

广义n参数Wiener过程和广义OUPn的导出过程的注记

记为非负单增函数且一阶偏导数均存．本文证明：（ｉ）广义ｎ参数Ｗｉｅｎｅｒ过程（ＧＢｎ）和广义ｎ参数Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ过程（广义ＯＵＰｎ）在Ｄ中导出的过程是马氏过程；（ｉｉ）如果ＧＢｎ（或广义ＯＵＰｎ）在Ｄ中导出的过程成为某个ＧＢｋ（或某个广义ＯＵＰｋ），则ｆｊ（ｔ１…，ｔｋ）未必恒为正常数．（ｊ＝ｋ＋１，…，ｎ）．     

Phase Retrieval with One or Two Diffraction Patterns by Alternating Projections with the Null Initialization

Alternating projection (AP) of various forms, including the parallel AP (PAP), real-constrained AP (RAP) and the serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null initialization is proposed to produce good-quality initial guess. Numerical experiments show that the null initialization is more accurate than the spectral initialization and that AP converges faster to the true object than other iterative schemes such as the Wirtinger flow (WF). In numerical experiments AP with the null initialization converges globally to the true object.     

On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S⊂ℝ ^k+m be a compact semi-algebraic set defined by P ₁≥0,…,P _ℓ ≥0, where P _i ∈ℝ[X ₁,…,X _k ,Y ₁,…,Y _m ], and deg (P _i )≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ ^k+m onto ℝ ^m . We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m) ^{O(q ℓ)}. We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is . For fixed q and ℓ, both the bounds are polynomial in k+m. The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship.     

On Fourier Orthogonal Projections in the Rotation Algebra

Projections are constructed in the rotation algebra that areorthogonal to their Fourier transform and which are fixed underthe flip automorphism. Such projections are expected in a constructionof an inductive limit structure for the irrational rotationalgebra that is invariant under the Fourier transform (namely,as two circle algebras of the same dimension, which are swappedby the Fourier transform, plus a few points). The calculationis based on Rieffel's construction of the Schwartz space asan equivalence bimodule of rotation algebras as well as on thetheory of theta functions.