由x_0=(x_1 λx_2)/(1 λ)所想到的

众所周知 ,解析几何中有向线段的定比分点公式是 x0 =x1 λx21 λ ,在各类数学问题中有与此相类似的结构 .命题 1　梯形的上底长为 l1,下底长为 l2 ,过腰上一点 P作底的平行线 ,交另一腰于 Q.且 APPB= λ( λ≠ - 1 ) .设 PQ长为 l0 ,那么 l0 =l1 λl21 λ.证明 :设 BA 延长线交CD延长线于 E,如图 1 .由△AED∽△ PEQ 可得 :AEAE λPB=l1l0( 1 )由△ AED∽△ BEC得 :AEAE λPB PB=l1l2( 2 )由 ( 1 ) ,( 2 )可得　l0 =l1 λl21 λ.特殊地 :当λ=1时 ,即可得到梯形的中位线定理 .　　命题 2　棱台上底面积为 S1,下底面积…     

二次曲线系Ax~2 Bxy Cy~2 Dx Ey λ=0

二次曲线系Ax~2 Bxy cy~2 Dx Ey λ=0(其中A、B、C不全为零,λ是参变数,下同)有一些重要性质,值得研究。定理1 二次曲线系Ax~2 Bxy Cy~2 Dx Ey λ=0……(1)中的非退化二次曲线是下列三种情形之 1°当B~2-4AC<0时,是一簇同轴(指对称轴)位似椭圆; 2°当B~2-4AC>0时,是一簇共渐近线双曲线; 3°当B~2-4AC=0时,是一簇同轴(指对称轴)同p(焦点到准线的距离)抛物线。证明1°当B~2-4AC<0时,曲线系(1)中的所有非退化二次曲线均是椭圆,它们经过适当的坐标变换,总可以化成最简椭圆方程;又因为经过坐标变换得到的新方程的二次项系数和一次项系数只与原方程的二次     

Enumeration of 2-(9, 3, λ) Designs and Their Resolutions

We consider 2-(9, 3, ) designs, which are known to exist for all 1, andenumerate such designs for = 5 and their resolutions for 3 5, the smallestopen cases. The number of nonisomorphic such structures obtained is 5 862 121 434, 426, 149 041, and 203 047732, respectively. The designs are obtained by an orderly algorithm, and the resolutions by two approaches:either by starting from the enumerated designs and applying a clique-finding algorithm on two levels or by anorderly algorithm.     

Nested Optimal λ-Packings and λ-Coverings of Pairs with Triples

We say that an optimal -packing ( -covering) of pairs with triples can be nested if it is possible to add one point to each block such that an optimal 2 -packing (2 -covering) of pairs with quadruples is obtained. A packing (covering) of order v can exist only if v 4 and v / 3 (v - 1) / 2 = v / 4 2(v - 1) / 3 ( v / 3 (v - 1) / 2 = v / 4 2 (v - 1) / 3 ). It is shown in this paper that this condition is also sufficient, except possibly for v = 20.     

On blocking sets in symmetric BIBD's with λ⩾2

Blocking sets in symmetric BIBD's with 2 are defined and bounds for their sizes are found. Thus, some results known in projective planes are generalized to symmetric BIBD's.     

On λ-harmonic graphs

A λ harmonic graph G, a λ-Hgraph G for short, means that there exists a constant A such that the equality λd（ui） = ∑（vi,vj）∈E（G） d（vj） holds for all i = 1, 2,…, ｜V（G）｜, where d（vi） denotes the degree of vertex vi. In this paper, some harmonic properties of the complement and line graph are given, and some algebraic properties for the λ-Hgraphs are obtained.     

On λ-harmonic graphs

A λ harmonic graph G, a λ-Hgraph G for short, means that there exists a constant denotes the degree of vertex vi. In this paper, some harmonic properties of the complement and line graph are given, and some algebraic properties for the λ-Hgraphs are obtained.     

On some low separation axioms via λ-open and λ-closure operator

In this paper, we introduce and study two new low separation axioms called λ-R ₀ and λ-R ₁ by using the notions of λ-open sets and λ-closure operator.     

Spectral properties of λ-matrices

One investigates the spectral problem for polynomial -matrices of the general form (regular and singular). One establishes a relation between the elements of the complete spectral structure of the -matrix (i.e., its elementary divisors, its Jordan chains of vectors, its minimal indices and polynomial solutions) and of the matrices which occur in its transformation to the Smith canonical form. One establishes a correspondence between the complete spectral structures of a -matrix and of three linear accompanying matrix pencils. One notes the possibility of reducing the solution of the spectral problem for a -matrix to the solution of a similar problem for its accompanying pencil. One gives a factorization of a -matrix which allows to represent it by any of its considered accompanying pencils or by its Kronecker canonical form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 180–194, 1981.     

Cardinal Invariants of λ-Topologies

We study the basic cardinal-valued invariants of C (X) such as weight, density, network weight, i-weight, and tightness, where C (X) is the space of all continuous real functions on X in the -topology.     

Embedding an Edge-colored K(a (p); λ, μ) into a Hamiltonian Decomposition of K(a (p+r); λ, μ)

Let K(a ^(p); λ, μ) be a graph with p parts, each part having size a, in which the multiplicity of each pair of vertices in the same part (in different parts) is λ (μ, respectively). In this paper we consider the following embedding problem: When can a graph decomposition of K(a ^(p); λ, μ) be extended to a Hamiltonian decomposition of K(a ^(p+r); λ, μ) for r > 0? A general result is proved, which is then used to solve the embedding problem for all ${r \geq \frac{\lambda}{\mu a}+ \frac{p-1}{a-1}}$ . The problem is also solved when r is as small as possible in two different senses, namely when r = 1 and when ${r = \frac{\lambda}{\mu a} - p+1}$ .     

Some Results About λ-Property

ＳｏｍｅＲｅｓｕｌｔｓＡｂｏｕｔλ－Ｐｒｏｐｅｒｔｙ马玉梅，张宜滨ＳｏｍｅＲｅｓｕｌｔｓＡｂｏｕｔλ－Ｐｒｏｐｅｒｔｙ￥ＭａＹｕｍｅｉ；ＺｈａｎｇＹｉｂｉｎ（Ｄｅｐｔ．ｏｆｍａｔｈ．，ＮｏｒｍａｌＣｏｌｌｅｇｅＳｈｉｊｉａｚｈｕｎａｇ，Ｈｅｂｅｉ，０...     

Elementarily λ-homogeneous binary functions

Characterizations of compact Hausdorff topological MV-algebras, StoneMV-algebras, and MV-algebras that are isomorphic to their profinite completionsare established. It is proved that compact Hausdorff topological MV-algebras areproducts (both topological and algebraic) of copies [0, 1] with the interval topologyand finite ?ukasiewicz chains with the discrete topology. Going one step further, wealso prove that Stone MV-algebras are products (both topological and algebraic) of finite ?ukasiewicz chains with the discrete topology. Finally, it is proved that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite andeach of its maximal ideals of finite rank is principal.     

L(d1, d2,..., dt)-Number λ(Cn; d1, d2,...,dt) of Cycles

An L(d1,d2,...,dt)-labeling of a graph G is a function f from its vertex set V(G) to the set {0, 1,..., k} for some positive integer k such that {f(x) - f(y)| ≥ di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L(d1,d2,...,dt)-number λ(G;d1,d2,... ,dt) of G is the smallest integer number k such that G has an L(d1,d2,... ,dt)labeling with max{f(x)|x ∈ V(G)} = k. In this paper, we obtain the exact values for λ(Cn; 2, 2,1) and λ(Cn; 3, 2, 1), and present lower and upper bounds for λ(Cn; 2,..., 2,1,..., 1)