On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

Optimality Conditions and Duality for?Nondifferentiable Multiobjective Fractional Programming Problems with?(C,α,ρ,d)-convexity

The purpose of this paper is to consider a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function contains a term involving the support function of a compact convex set. Based on the (C,α,ρ,d)-convexity, sufficient optimality conditions and duality results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problem are established. The results extend and improve the corresponding results in the literature.     

Connected subgroups of SO(2, n) acting irreducibly on ?2,n

We classify all connected subgroups of SO(2, n) that act irreducibly on ℝ^{2, n} . Apart from SO ₀(2, n) itself these are U(1, n/2), SU(1, n/2), if n even, S ¹ · SO(1, n/2) if n even and n ≥ 2, and SO ₀(1, 2) for n = 3. Our proof is based on the Karpelevich Theorem and uses the classification of totally geodesic submanifolds of complex hyperbolic space and of the Lie ball. As an application we obtain a list of possible irreducible holonomy groups of Lorentzian conformal structures, namely SO ₀(2, n), SU(1, n), and SO ₀(1, 2).     

Cauchy problem F(t, x(t), x′(t)) = 0, x(0) = 0: solvability and asymptotics of solutions

For the singular Cauchy problem, the authors find some sufficient conditions for the existence of continuously differentiable solutions x: (0, ρ] → ℝ (ρ > 0 is sufficiently small) of the form

矩阵方程AX=B,XC=D的定秩解

本文研究了一类矩阵方程组解的秩的范围.利用矩阵的奇异值分解以及Frobenius范数的特征,得到了解的极值秩以及解的通式,并就这些问题的特殊情况进行了讨论,得到了一些结果.     

Subexponential ultracontractivity and L p ? L q functional calculus

 We prove that if a symmetric submarkovian semigroup (T _t ) _t>0 satisfies an estimate of the form

Lie Symmetries of the Equation ut(x, t)?+?g(u)ux(x, t) = 0

In this paper the invariance criterion is applied for the nonlinear equation

Maximum principles for real (p, p)-forms on K?hler manifolds

In this paper, we extend the maximum principle for (1, 1)-Hermitian symmetric tensor to a complete K?hler manifold with bounded holomorphic bisectional curvature and nonnegative orthogonal bisectional curvature. We also achieve a maximum principle for real (p, p)-forms on a compact K?hler manifold with nonnegative holomorphic sectional curvature and vanishing Bochner tensor.     

Nonexistence of [n, 5, d]q Codes Attaining the Griesmer Bound for q4 ?2q2 ?2q+1 ≤ d≤ q4?2q2 ?q

We prove that there does not exist a [q⁴+q³−q²−3q−1, 5, q⁴−2q²−2q+1]_q code over the finite field for q≥ 5. Using this, we prove that there does not exist a [g_q(5, d), 5, d]_q code with q⁴ −2q² −2q +1 ≤ d ≤ q⁴ −2q² −q for q≥ 5, where g_q(k,d) denotes the Griesmer bound.MSC 2000: 94B65, 94B05, 51E20, 05B25     

Complete description of invariant Einstein metrics on the generalized flag manifold SO(2n)/U(p)?× U(n ? p)

We find the precise number of non-K?hler SO(2n)-invariant Einstein metrics on the generalized flag manifold M = SO(2n)/U(p)×U(n−p) with n ≥ 4 and 2 ≤ p ≤ n−2. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems. We also examine the isometric problem for these Einstein metrics.     

On solutions of functional-differential equations f ′( x ) = a ( x ) f ( g ( x )) + b ( x ) f ( x ) + c ( x ) in the large

We consider solutions of functional-differential equations

On the Diophantine Equation G n ( x )?= G m ( P ( x ))

 Let K be a field of characteristic 0 and let p, q, G ₀ , G ₁ , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G _n (x)) _n=0 ^∞ be defined by the second order linear recurring sequence

On the Diophantine Equation G n ( x )?= G m ( P ( x ))

 Let K be a field of characteristic 0 and let p, q, G ₀ , G ₁ , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G _n (x)) _n=0 ^∞ be defined by the second order linear recurring sequence

Sum of sequence spaces and matrix transformations mapping in s α 0 ((Δ ? λ I ) h ) + s β ( c ) ((Δ ? μ I ) l )

We deal with the sum of sequence spaces. Then we apply these results to characterize matrix transformations mapping between s ^h,l (λ, μ) = s _α ⁰((Δ − λI) ^h ) + s _β ^(c)((Δ − μI) ^l ) and s _γ . Among other things the aim of this paper is to reduce the set (s ^h,l (λ, μ), s _γ to a set of the form S _τ,γ .      

Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

The generalized bisymmetric solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 + ? + A l X l B l = C and its optimal approximation

For fixed generalized reflection matrix P, i.e. P ^T  = P, P ² = I, then matrix X is said to be generalized bisymmetric, if X = X ^T  = PXP. In this paper, an iterative method is constructed to find the generalized bisymmetric solutions of the matrix equation A ₁ X ₁ B ₁ + A ₂ X ₂ B ₂ + ⋯ + A _l X _l B _l  = C where [X ₁,X ₂, ⋯ ,X _l ] is real matrices group. By this iterative method, the solvability of the matrix equation can be judged automatically. When the matrix equation is consistent, for any initial generalized bisymmetric matrix group , a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm generalized bisymmetric solution group can be obtained by choosing a special kind of initial generalized bisymmetric matrix group. In addition, the optimal approximation generalized bisymmetric solution group to a given generalized bisymmetric matrix group in Frobenius norm can be obtained by finding the least norm generalized bisymmetric solution group of the new matrix equation , where . Given numerical examples show that the algorithm is efficient. Research supported by: (1) the National Natural Science Foundation of China (10571047) and (10771058), (2) Natural Science Foundation of Hunan Province (06JJ2053), (3) Scientific Research Fund of Hunan Provincial Education Department(06A017).     

Complete ( q 2?+ q +?8)/2-caps in the spaces PG (3, q ), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG (3, 5)

An infinite family of complete (q ² + q + 8)/2-caps is constructed in PG(3, q) where q is an odd prime ≡ 2 (mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q ² + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails for q = 17.      

Spherical Harmonic Analysis on Buildings of Type ? n

 To any locally finite thick building of type there is naturally associated a commutative algebra of operators. When is constructed from a local field F with local ring , and , then is isomorphic to the convolution algebra of compactly supported bi-K-invariant functions on PGL(n+1,F). We give a proof, valid for any , that the multiplicative functionals on may all be expressed in terms of Hall–Littlewood polynomials. Regarding as a subalgebra of the C ^*-algebra of bounded operators on the space of square summable functions on the vertex set of , we find the spectrum of the C ^*-algebra , the closure of . This generalizes results obtained in [3] when n = 1 and in [5] when n = 2.