THE PLANCHEREL FORMULA FOR THE LINE BUNDLES ON SL(n + 1, R)/S(GL(1, R) × GL(n,R))

In this paper we obtain the Plancherel formula for the spaces of L~2-sections of the line bundles over the pseudo-Riemannian symmetric space G/H where G = SL(n + 1, R)and H = S(GL(1, R) × GL(n, R)). The Plancherel formula is given in an explicit form by means of spherical distributions associated with the character χ——λof the subgroup H. We follow the method of Faraut, Kosters and van Dijk.     

A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.     

具有对称自同态的环及其扩张

Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetricα-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α(1) = 1, then R is a symmetric α-ring if and only if R[x]/(x n) is a symmetric ˉα-ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q(R) the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q(R) is a symmetric ˉα-ring. Among others we also show that if a ring R is weakly 2-primal and(α, δ)-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric ˉα-ring.     

对称正交对称矩阵反问题的最小二乘解

Let P ∈ Rn×n be a symmetric orthogonal matrix. A∈Rn×n is called a symmetric orthogonal symmetric matrix if AT = A and (PA) T = PA. The set of all n × n symmetric orthogonal symmetric matrices is denoted by SRnxnp. This paper discusses the following problems: Problem I. Given X,B∈ Rn×m, find A ∈SRn×np such that||AX - B|| = min Problem II. Given A∈ Rn×n, find A∈SL such thatwhere ||·|| is the Frobenius norm, and SL is the solution set of Problem I.The general form of SL is given. The solvability conditions for the inverseproblem AX = B in SRn×nP are obtained. The expression of the solution toProblem II is presented.     

Estimates for Fourier Coefficients of Cusp Forms in Weight Aspect

Let f be a holomorphic Hecke eigenform of weight k for the modular groupΓ = SL2(Z) and let λf(n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate 1λf(n21) x2 k2(log(x + k))6n≤x is established, which improves the previous result.     

Effective condition numbers and small sample statistical condition estimation for the generalized Sylvester equation

In this paper,we investigate the effective condition numbers for the generalized Sylvester equation(AX-YB,DX-YE)=(C,F),where A,D∈R m×m,B,E∈R n×n and C,F ∈ R m×n.We apply the small sample statistical method for the fast condition estimation of the generalized Sylvester equation,which requires O(m2n+mn2) flops,comparing with O(m3+n3) flops for the generalized Schur and generalized HessenbergSchur methods for solving the generalized Sylvester equation.Numerical examples illustrate the sharpness of our perturbation bounds.     

Least-Squares Solutions of the Matrix Equation ATXA = B Over Bisymmetric Matrices and its Optimal Approximation

A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n 1-j,n 1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A~TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.     

MSPRK Methods for the Korteweg-de Vries Equation

We consider the Korteweg-de Vries (KdV) equation in the form ut+uux+uxxx=0(1) which is a nonlinear hyperbolic equation and has smooth solutions for all the time. There are a vast of results can be found in the literature for this equation, both theoretical and numerical. However, several good reasons account for needs of another numerical study of this equation are listed in[1]. Among them, the most convincing one might be that the wave equations have the multi-symplectic structure (cf. [2]), and the KdV equation is therefore a     

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS

The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation -△u=φ(r)up-1,u>0 in RN, u ∈ D1,2(RN),where N≥ 3,x = (x',z)∈ RK×RN-K,2≤K≤N,r =|x'|.It is proved that for 2(N -s)/(N-2) < p < 2* = 2N/(N -2),0 < s < 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p=2*, the above equation does not have a ground state solution but a cylindrically symmetric-solution, and when p close to 2*, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2*, the ground state solution up has a unique maximum point xp = (x'p,zp) and as p→2*, |x'p|→r0 which attains the maximum of φ on RN.The asymptotic behavior of ground state solution up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2*.     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

On the values of representation functions

For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be the set of nonnegative integers. Given n0 > 0, it is known that there exist A,A′■ N such that R2(A′,n) = R2(N \ A′,n) and R3(A,n) = R3(N \ A,n) for all n n0. We obtain several related results. For example, we prove that: If A ■ N such that R3(A,n) = R3(N \ A,n) for all n n0, then (1) for any n n0 we have...     

SUPERPOSITION FORMULAE OF A FIFTH ORDER KdV EQUATION AND ITS MODIFIED EQUATION

The Backlund transformation (BT) for a fifth order KdV equation is presented in the bilinear form.Furthermore,a nonlinear superposition formula related to the BT obtained above is proved rigorously.By the way,a nonlinear superposition formula of a modified fifth order KdV equation is also given.     

Multiple Non-Axially Solutions to a Mean Field Equation on S~2

We study the following mean field equation ■,where ρ is a real parameter. We obtain the existence of multiple non-axially symmetric solutions bifurcating from u = 0 at the values ρ = 4 n(n + 1)π for any odd integer n ≥ 3.     

中心代数上一矩阵方程的中心对称与中心斜对称解

Let Ω be a finite dimensional central algebra and chart Ω≠2 .The matrix equation AXB-CXD=E over Ω is considered.Necessary and sufficient conditions for the existence of centro(skew)symmetric solutions of the matrix equation are given.As a particular case ,the matrix equation X-AXB=C over Ω is also considered.     

ON THE MODIFIED NONLINEAR SCHRDINGER EQUATION IN THE SEMICLASSICAL LIMIT:SUPERSONIC,SUBSONIC,AND TRANSSONIC BEHAVIOR

The purpose of this paper is to present a comparison between the modified nonlinear Schro¨dinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr¨odinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.     

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

1. IntroductionConsider the large sparse system of linear equationsAx = b, (1.1)where, for a fixed positive integer cr, A e L(R") is a symmetric positive definite (SPD) matrir,having the bloCked formx,b E R" are the uDknwn and the known vectors, respectively, having the correspondingblocked formsni(ni S n, i = 1, 2,', a) are a given positthe integers, satisfying Z ni = n. This systemi= 1of linear equations often arises in sultable finite element discretizations of many secondorderseifad…     

On the Matrix Equation A~2=J

An unsolved problem is to enumerate all solutions for the matrix equation A~2=Jwhere A is an n×n (0.1)-matrix and J is the n×n matr trix with every entry being 1. Here we present a family of n×n generalized circulant (0.1)-matrices each of whose square is J. Moreover. the existence of this family is unique up to permutational similarity.     

Maximal subalgebras of the general linear Lie algebra containing Cartan subalgebras

Let gln(R) be the general linear Lie algebra of all n × n matrices over a unital commutative ring R with 2 invertible, dn(R) be the Cartan subalgebra of gln(R) of all diagonal matrices. The maximal subalgebras of gln(R) that contain dn(R) are classified completely.     

THE PATCH PROPERTY OF SELF-DUAL YANG-MILLS FIELD

SDYM-field is a holomorphic vector bandles over twistor Euclidean Space covered by two patches U_1 and U_2. In this paper, the property of the field on each patch and relationship between them are found. We point out that if the field on one patch correspond to the left SDYM-J equation, then on the other patch will be right SDYM-J equation. The Lagrangian form of the field is also found, which has relation to H_n=SL(n, c)/SU(n) nonlinear sigma model with Wess-Zumino terms.     

交换环上上三角矩阵的李三导子

Let T(n,R) be the Lie algebra consisting of all n × n upper triangular matrices over a commutative ring R with identity 1 and M be a 2-torsion free unital T(n,R)-bimodule.In this paper,we prove that every Lie triple derivation d : T(n,R) → M is the sum of a Jordan derivation and a central Lie triple derivation.