ON VARMA''s LACUNARY INTERPOLATION BY SPLINES OHEN TIANPING

In articles [1], [2], A. K. Varma investigated some lacunary interpolations by Spline. In this article we point out a mistake in the proof of Theorem 2 in [1] and give a new proof. Moreover, the results obtained in [1], [2] are improved.     

ON VARMA''s LACUNARY INTERPOLATION BY SPLINES OHEN TIANPING

In articles [1], [2], A. K. Varma investigated some lacunary interpolations by Spline. In this article we point out a mistake in the proof of Theorem 2 in [1] and give a new proof. Moreover, the results obtained in [1], [2] are improved.     

BOUND ON THE MAXIMUM NUMBER OF CLEAR TWO-FACTOR INTERACTIONS FOR 2n-(n-k) DESIGNS

Clear effects criterion is an important criterion for selecting fractional factorial designs [1]. Tang et al. [2] derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi's) in 2n-(n-k) designs of resolution Ⅲ and Ⅳ by constructing 2n-(n-k) designs. But the method in [2] does not perform well sometimes when the resolution is III. This article modifies the construction method for 2n-(n-k) designs of resolution Ⅲ in [2]. The modified method is a great improvement on that used in [2].     

Linearization of holomorphic operator functions II

This paper is a continuation of [2]. It contains the proof of the main linearization theorem announced in [2], some modifications of the construction and some unsolved problems for the case of several variables. Without further explanation we shall use the notations of [2], and we shall continue the numbering of formulae, sections, theorems and lemmas as begun in [2].     

BRILL－NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Lei X be an arbitrary smooth irreducible complex projective curve, E (?) X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1,D2,f}, where D1 , D2 are two effective divisors with d = deg(D1) + deg(D2), and f ∈ H0(X, [D1] |D2) is a collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using f and the Brill-Noether matrix of D1 + D2, the author constructs a 2g X d matrix WE whose zero space gives Im{H0(X,[D1]) (?) H0(X, [D1] |D1)}(?)Im{H0(X, E) (?) H0(X,[D2]) (?) H0(X,[D2] |D2)}. From this and H0(X,E) = H0(X, [D1]) (?) Im{H0(X, E) (?) H0(X, [D2])}, it is got in particular that dimH0(X, E) = deg(E) - rank(WE) + 2.     

用DEA估计生产力进步的探讨

有关生产力进步的概念与计算,文［１］、［２］基于规模报酬不变的生产前沿面来研究,本文推广到基于一般生产前沿面来研究．因此本文在讨论中用面向输出的Ｃ２ＧＳ２模型而不是文［１］、［２］中所用的面向输入的Ｃ２Ｒ模型．     

循环矩阵环的局部化

本文的主要结果如下：(1)环R关于其乘法封闭子集S满足左Ore条件当且仅当R[σ1,σ2,…，σt]关于其相应乘法封闭子集S[σ1,σ2,…，σt]满足左Ore条件．(2)若R关于其乘法封闭子集S满足左Ore条件，S^-1 R是R关于S的左分式环，其自然同态为φ：R→S^-1R，则存在环同态φ：R[σ1,σ2,…,σt]→S[σ1,σ2,…,σt]^-1 R[σ1,σ2,…σt]使得(S-1R)[φ（σ1),φ-(σ2),…φ（σt)]≌S[σl，σ2，…，σt]^-1R[σ1,σ2,…σt]。     

Corrigendum and addendum to “The strong approximation property and the weak bounded approximation property” [J. Funct. Anal. 266 (2014) 5439–5447]

We correct a mistake in the proof of [2, Theorem 1.1] and a proof of the main result in [2] is presented without depending on [2, Theorem 1.1].     

可除的四元数代数上多项式环的Grobner基

设F是一个特征不等于2的域,A是,上的一个可除代数。本文研究了A上多项式环A[x1,X2,…,xn]中理想是有限生成的,以及它的Grobner基;也表明F[x1,x2,…,xn]中有限子集G是F[x1,x2,…,xn]的Griobner基当且仅当G是A[x1,x2,…,xn]中的Grobner基。     

Almost-regular factorization of graphs

For integers a and b, 0 ? a ? a ? b, an [a, b]-graph G satisties a ? deg(x, G) ? b for every vertex x of G, and an [a, b]-factor is a spanning subgraph F such that a ? deg(x, F) ? b for every vertex x of F. An [a, b]-factor is almost-regular if b = a + 1. A graph is [a, b]-factorable if its edges can be decomposed into [a, b]-factors. When both K and t are positive integers and s is a nonnegative integer, we prove that every [(12K + 2)t + 2ks, (12k + 4)t + 2ks]-graph is [2k,2k + 1]-factorable. As its corollary, we prove that every [r.r + 1]-graph with r ? 12k² + 2k is [2k + 1]-factorable, which is a partial extension of the two results, one by Thomassen and the other by Era.     

Niveau hermitien de Certaines algèbres de quaternions

Let D [d] =(a,b/F) a quaternion divisior algebra over a field F of characteristic ? 2. Denote 1, i, j , k the basis of D, such that i²[d] n, j²[d] b, ij [d] -ji [d] k and A :D → D the involution given by i [d] -i, j [d] j (and k [d] k). In [LE] D. LEWIS asks the following question :Does there exist a quadratic Pfister form [S p. 721 [d] such that the hermitian form [d] [d] D is isotropic over (D, [d]) but not hyperbolic &; In this note, we show that the answer of this question is negative, so that the hermitien level [§I], when it is finite, of (D, A) is a power of two. This result holds for quaternion algebras with standard involution [LE].     

Partially-2-Homogeneous Monounary Algebras

This paper is a continuation of [5], where k-homogeneous and k-set-homogeneous algebras were defined. The definitions are analogous to those introduced by Fraïssé [2] and Droste, Giraudet, Macpherson, Sauer [1] for relational structures. In [5] we found all 2-homogeneous and all 2-set-homogeneous monounary algebras when the homogenity is considered with respect to subalgebras, to connected subalgebras and with respect to connected partial subalgebras, respectively. The results of [3], where all homogeneous monounary algebras were characterized, were applied in [4] for 1-homogeneity.The aim of the present paper is to describe all monounary algebras which are 2-homogeneous and 2-set-homogeneous with respect to partial subalgebras, respectively; we will say that they are partially-2-homogeneous and partially-2-set-homogeneous.     

Higher holonomies: Comparing two constructions

We compare two different constructions of higher-dimensional parallel transport. On the one hand, there is the two-dimensional parallel transport associated with 2-connections on 2-bundles studied by Baez–Schreiber [2], Faria Martins–Picken [11] and Schreiber–Waldorf [12]. On the other hand, there are the higher holonomies associated with flat superconnections as studied by Igusa [7], Block–Smith [3] and Arias Abad–Schätz [1]. We first explain how by truncating the latter construction one obtains examples of the former. Then we prove that the two-dimensional holonomies provided by the two approaches coincide.     

关于“四元数自共轭矩阵与行列式的几个定理”的注记

本刊1985年第4期发表郝稚传的“四元数自共轭矩与行列式的几个定理”(称为〔1〕)的文章的主要工作分为两部分。〔1〕在英文摘要中写到:“(i)This essay has improvedthe conclusion of theorem 8 and theorem 9 in 〔2〕”即〔1〕之第一部分工作在于改进了〔2〕的定理8、9的结果,其根据就是〔1〕的定理1、2.〔1〕称定理1—若     

一个结构简单的约束变尺度算法

本文我们考虑具有线性约束凹函数的最优化问题，利用我们的算法和变尺度修正公式，提出了一个结构简单的组合算法，并在「２」，「３」和「４」同样的假设条件下，证明了该算法的收敛性和超线性收敛速度，从而使该算法比原有各算法更具实用性。     

关于挠自由群分次环的注记

本文推广ＣＮａｓｔａｓｅｃｕ，Ｆ．ＶａｎＯｙｓｔａｅｙｅｎ的有关结果，并在一定条件下回答了Ｋ．Ｒ．Ｆｕｌｌｅｒ等人的公开问题．     

Tauberian theorems for weak almost convergent functions

The almost convergent function which was introduced by Raimi [6] and discussed by Ho [4], Das and Nanda [2, 3], is the continuous analogue of almost convergent sequences (see [5]). In this paper, we establish the Tauberian conditions and the Cauchy criteria for weak almost convergent functions on R~2+ .     

BOUND ON THE MAXIMUM NUMBER OF CLEAR TWO-FACTOR INTERACTIONS FOR 2^n-（n-k） DESIGNS

Clear effects criterion is an important criterion for selecting fractional factorial designs[1].Tang et al.[2]derived upper and lower bounds on the maximum number of clear two-factor interactions（2fi＇s）in 2^n-（n-k）designs of resolution Ⅲ and Ⅳ by constructing 2^n-（n-k）designs.But the method in[2]does not perform well sometimes when the resolution is Ⅲ.This article modifies the construction method for 2^n-（n-k） designs of resolution Ⅲ in[2].The modified method is a great improvement on that used in[2].