The domination numbers of the 5 × n and 6 × n grid graphs

The k × n grid graph is the product P_k × P_n of a path of length k ? 1 and a path of length n ? 1. We prove here formulas found by E. O. Hare for the domination numbers of P₅ × P_n and P₆ × P_n. © 1993 John Wiley & Sons, Inc.     

Cusp Forms on GL(2n) with GL(n) × GL (n) Periods,and Simple Algebras

The notion of a period of a cusp form on GL(2,D(??)), with respect to the diagonal subgroup D(??)^X × D(??)^X, is defined. Here D is a simple algebra over a global field F with a ring ?? of adeles. For D^x = GL(1), the period is the value at 1/2 of the L-function of the cusp form on GL(2, ??). A cuspidal representation is called cyclic if it contains a cusp form with a non zero period. It is investigated whether the notion of cyclicity is preserved under the Deligne - Kazhdan correspondence, relating cuspidal representations on the group and its split form, where D is a matrix algebra. A local analogue is studied too, using the global technique. The method is based on a new bi-period summation formula. Local multiplicity one statements for spherical distributions, and non - vanishing properties of bi - characters, known only in a few cases, play a key role.     

POSITIVE SOLUTIONS TO(k,n-k) NONHOMOGENEOUS BOUNDARYVALUE PROBLEM

By the Schauder fixed point theory,this paper establishes the existence of positive solutions to a(k,n k) m-point boundary value problem.We show that there exists a positive constant b such that the problem has at least one positive solution when the homogeneous boundary parameter is smaller than b,and no positive solution when this parameter is greater than b.     

Gauge Orbit Types for Theories with Gauge Group O(n), SO(n) or Sp(n)

We determine the orbit types of the action of the group of local gauge transformations on the space of connections in a principal bundle with structure group O(n), SO(n) or Sp(n) over a closed, simply connected manifold of dimension 4. On the way we derive a classification of Howe subgroups of SO(n) up to conjugacy.     

POSITIVE SOLUTIONS TO SINGULAR(k,n-k) CONJUGATE BOUNDARY VALUE PROBLEMS

By the fxed point index theory, the existence of one, two and three positive solutions to(k, n-k) conjugate boundary value problems is obtained, where n > 2, 1 ≤ k ≤ n-1, the nonlinear term may be noncontinuous and singular at any point of [0,1]. Our results extend some of the existent results.     

奇异(k,n-k)共轭边值问题的正解

对固定的 １≤ ｋ≤ｎ－１,在对 ｆ（ｔ,ｙ）更弱的条件下,本文重新建立了奇异边值问题正解的存在性．允许ｆ（ｔ,ｙ）在ｙ＝０,ｔ＝０和ｔ＝１处具有奇性,本文只用到格林函数的正性和一个锥不动点定理,并且构造了格林函数的精确表达式．     

变号(k,n-k)共轭边值问题解的存在问题

讨论了下列变号(k,n-k)共轭边值问题(SCBVP)正解的存在问题{(-1)(n-k)u(n)(t)=f(t,u(t)),0t1,u(i)(0)=0,0≤i≤k-1,u(j)=0,0≤j≤n-k-1,其中n≥2,1kn-1.对于0t1,非线性项f允许变号,即本文允许非线性项f取负值并且可以无下界.本文利用不动点指数定理,在无任何单调性假设条件下,得到了边值问题正解的存在性结论.     

完全三部图K(n-k,n,n)的色唯一性

通过对图的特征子图个数的比较,给出了图K(n-k,n,n)色唯一性的数值条件.     

The geometry of SO(n)\SO0(n, 1)

Consider the Lie group SO₀(n, 1) with the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left and right. This paper studies the geometry of the quotient space of the homogeneous submersion SO₀(n, 1) → SO(n)\SO₀(n, 1). It is a cohomogeneity one manifold, which can be expressed as a warped product. Its group of isometries, geodesics, and sectional curvatures are calculated.     

A (k,n-k) Conjugate Boundary Value Problem with Semip ositone Nonlinearity

The existence of positive solution is proved for a(k, n- k) conjugate boundary value problem in which the nonlinearity may make negative values and may be singular with respect to the time variable. The main results of Agarwal et al.(Agarwal R P, Grace S R, O'Regan D. Semipositive higher-order differential equations. Appl. Math. Letters, 2004, 14: 201–207) are extended. The basic tools are the Hammerstein integral equation and the Krasnosel'skii's cone expansion-compression technique.     

f~((k))(1≤k≤n-1)与原函数f和最高阶导数f~((n))之间的一个不等式

证明了f(k)(1≤k≤n-1)与原函数f和最高阶导数f(n)之间的一个不等式关系.     

奇异超线性(k,n-k)多点边值问题的正解

应用锥理论和不动点指数方法,在与相应的线性算子第一特征值有关的条件下,获得奇异超线性(k,n-k)多点边值问题正解的存在性结果,推广和改进了已有的主要结论.     

方程f~（n）（x）＝Af~（n－k）（（ax＋b）／（cx－a））

本文提出方程ｆ~（ｎ）（ｘ）＝Ａｆ（ｎ－ｋ）（（ａｘ＋ｂ）／（ｃｘ－ａ）），证明它是可积的．所得结论是文献［１］中结论的推广．