CO-ISOMETRIC SOLUTIONS OF EQUATION CU + U*C = 2D

This note discusses the co-isometric solutions of the operator equation CU + U*C = 2D, establishes a correspondence between such solutions and the self-adjoint solutions of the algebraic Riccati equation X2 - iDX + iXD + D2 - C2 = 0, and gives all possible co-isometric solutions parametrically. Some mistakes of Dobovivsek's results are corrected.

The Class Number of Derived Subgroups and the Structure of Camina Groups

A finite group G is called a Camina group if G has a proper normal subgroup N such that gN is precisely a conjugacy class of G for any g ∈E G - N. In this paper, the structure of a Camina group G is determined when N is a union of 2, 3 or 4 conjugacy classes of G.

A NOVEL FIXED POINT THEOREM AND ITS APPLICATIONS

In this article, a novel fixed point theorem in C[0, 1] space is established by using the properties of fixed point index. This theorem is then applied to prove the existence of positive solutions for three-point boundary value problems and generalizes some previous results.

具广义边界条件单能中子迁移方程的离散纵标法

This paper is devoted to discussing the discrete-ordinates method for the monoenergetic neutron transport equation in a slab with generalized boundary conditions. For homogeneous medium with isotropic scattering and fission, the convergence theorems for discrete-ordinates approximations are given respectively for critical eigenvalue problem and dominant eigenvalue problems: for inhomogeneous medium with anisotropic scattering and fission, a similar discussion and an estimation for the convergence rate are given for critical eigenvalue problems. Finally, some numerical results are given by use of this method.

Extreme Matroid Graphs

Let G be a simple graph and T={S :S is extreme in G}. If M(V(G), T) is a matroid, then G is called an extreme matroid graph. In this paper, we study the properties of extreme matroid graph.

Sufficient and necessary conditions of separability for bipartite pure states in infinite-dimensional systems

The PHC criterion and the realignment criterion for pure states in infinite-dimensional bipartite quantum systems are given. Furthermore, several equivalent conditions for pure states to be separable are generalized to infinite-dimensional systems.

Characterization of Lie multiplicative isomorphisms between nest algebras

Let N and M be nests on Banach spaces X and Y over the real or complex field F,respectively,with the property that if M∈M such that M-=M,then M is complemented in Y.Let AlgN and AlgM be the associated nest algebras.Assume that Φ:AlgN→AlgM is a bijective map.It is proved that,if dim X=∞ and if there is a nontrivial element in N which is complemented in X,then Φ is Lie multiplicative (i.e.Φ([A,B])=[Φ(A),Φ(B)] for all A,B∈AlgN) if and only if Φ has the form Φ(A)=-TA*T-1+τ(A) for all A∈AlgN or Φ(A)=TAT-1+τ(A) for all A∈AlgN,where T is an invertible linear or conjugate linear operator and τ:AlgN→FI is a map with τ([A,B])=0 for all A,B∈AlgN.The Lie multiplicative maps are also characterized for the case dim X<∞.

On p-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K ofG such that G = HK and H ∩K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotentif G is S4-free and every minimal subgroup of P ∩ GN is c-supplemented in NG(P), and when p = 2 P isquaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G.As some applications of this result, some known results are generalized.

CHARACTERIZATIONS OF JORDAN (?)-SKEW MULTIPLICATIVE MAPS ON OPERATOR ALGEBRAS OF INDEFINITE INNER PRODUCT SPACES

Let H and K be indefinite inner product spaces. This paper shows that a bijective map Φ: B(H) → B(K) satisfies Φ(AB+ + B+A) = Φ(A)Φ(B)+ + Φ(B)+Φ(A) for every pair A,B ∈ B(H) if and only if either Φ(A) = cUAU+ for all A or Φ(A) = cUA+U+ for all A; Φ satisfies Φ(AB+A) = Φ(A)Φ(B)+Φ(A) for every pair A, B ∈ B(H) if and only if either Φ(A) = UAV for all A or Φ(A) = UA+V for all A, where A+ denotes the indefinite conjugate of A, U and V are bounded invertible linear or conjugate linear operators with U+U = c-1I and V+V = cI for some nonzero real number c.

Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Operators

Let X1 and X2 be complex Banach spaces with dimension at least three, A1 and A2 be standard operator algebras on X1 and X2, respectively. For k ≥ 2, let （i1, i2, . . . , im） be a finite sequence such that {i1, i2, . . . , im} = {1, 2, . . . , k} and assume that at least one of the terms in （i1, . . . , im） appears exactly once. Define the generalized Jordan productT1 o T2 o··· o Tk = Ti1Ti2··· Tim ＋ Tim··· Ti2Ti1 on elements in Ai. This includes the usual Jordan product A1A2 ＋ A2A1, and the Jordan triple A1A2A3 ＋ A3A2A1. Let Φ ： A1 → A2 be a map with range containing all operators of rank at most three. It is shown that Φ satisfies that σπ（Φ（A1） o··· o Φ（Ak）） = σπ（A1 o··· o Ak） for all A1, . . . , Ak, where σπ（A） stands for the peripheral spectrum of A, if and only if Φ is a Jordan isomorphism multiplied by an m-th root of unity.

Maps Preserving Numerical Radius or Cross Norms of Products of Self-adjoint Operators

Let H be a complex Hilbert space with dimH ≥3, Bs（H） the （real） Jordan algebra of all self-adjoint operators on H. Every surjective map Ф ： Bs（H）→13s（H） preserving numerical radius of operator products （respectively, Jordan triple products） is characterized. A characterization of surjective maps on Bs （H） preserving a cross operator norm of operator products （resp. Jordan triple products of operators） is also given.

Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F(= R or C) with dimension greater than 2. Let N(X) be the set of all nilpotent operators and B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A^(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A^(-1)+ φ(T)I for all T ∈ FI + B_0(X), where c is a nonzero scalar,A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional.In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.

Fidelity of states in infinite-dimensional quantum systems

We discuss the fidelity of states in the infinite-dimensional systems and give an elementary proof of the infinite-dimensional version of Uhlmann's theorem.This theorem is used to generalize several properties of the fidelity of the finite-dimensional case to the infinite-dimensional case.These are somewhat different from those for the finite-dimensional case.

Characterization of Lie Higher Derivations on Triangular Algebras

Let A and B be unital rings, and M be an （A, B）-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri（A,M, B） be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on U.

Semi-Cover-Avoiding Properties and the Structure of Finite Groups

In this paper, the so-called π-cover-avoiding properties of subgroups are defined and investigated. In terms of this property, we characterize the π-solvability of finite groups. Some other new results are also obtained based on the assumption that some subgroups have the semi cover-avoiding properties in a finite group.

Physical origins of ruled surfaces on the reduced density matrices geometry

The reduced density matrices(RDMs) of many-body quantum states form a convex set. The boundary of low dimensional projections of this convex set may exhibit nontrivial geometry such as ruled surfaces. In this paper, we study the physical origins of these ruled surfaces for bosonic systems. The emergence of ruled surfaces was recently proposed as signatures of symmetrybreaking phase. We show that, apart from being signatures of symmetry-breaking, ruled surfaces can also be the consequence of gapless quantum systems by demonstrating an explicit example in terms of a two-mode Ising model. Our analysis was largely simplified by the quantum de Finetti's theorem—in the limit of large system size, these RDMs are the convex set of all the symmetric separable states. To distinguish ruled surfaces originated from gapless systems from those caused by symmetrybreaking, we propose to use the finite size scaling method for the corresponding geometry. This method is then applied to the two-mode XY model, successfully identifying a ruled surface as the consequence of gapless systems.

Characterizing Centralizers and Generalized Derivations on Triangular Algebras by Acting on Zero Product

Let U = Tri（fit, M, B） be a triangular ring, where A and B are unital rings, and M is a faithful （A, B）-bimodule. It is shown that an additive map φ on U is centralized at zero point （i.e., ,φ（A）B = A,φ（B） = 0 whenever AB = 0） if and only if it is a centralizer. Let 5 ： U →U be an additive map. It is also shown that the following four conditions are equivalent： （1） 5 is specially generalized derivable at zero point, i.e., 5（AB） = δ（A）B ＋ AS（B） - Aδ（I）B whenever AB = 0; （2） 5 is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ（AB） = δ（A）B ＋ Aτ1（B） = τ2（A）B ＋ Aδ（B） whenever AB = 0; （3） δ is a special generalized derivation; （4） δ is a generalized derivation. These results are then applied to nest algebras of Banach space

Observability inequalities for thin shells

We consider the exact controllability problem from boundary for thin shells. Under some check-able geometric assumptions on the middle surface, we establish the observability inequalities via the Bochnertechnique for the Dirichlet control and the Neumann control problems. We also give several examples to verifythe geometric assumptions.

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive （generalized） ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive （generalized） E-Lie derivation with ξ -if and only if it is an additive （generalized） derivation satisfying L（ξA） =- ξL（A） for all A. These results are then used to characterize additive （generalized） ξ-Lie derivations on several operator Mgebras such as Banach space standard operator algebras and yon Neumman algebras.