Smooth 4-folds which contain a P1-bundle¶as an ample divisor

We show that if a smooth projective 4-fold M contains an ample divisor A which is P ¹-bundle π :A→S over a smooth projective surface S, π is extended to a P ²-bundle π :S→S, unless $A$ is isomorphic to P ²×P ¹. Received: 28 September 1998 / Revised version: 16 August 1999     

Differential as a harmonic morphism with respect to Cheeger–Gromoll-type metrics

We investigate horizontal conformality of a differential of a map between Riemannian manifolds, where the tangent bundles are equipped with Cheeger–Gromoll-type metrics. As a corollary, we characterize the differential of a map as a harmonic morphism.     

sum from r=1 to (k-1)(r~2)的行列式算法

散见于各种杂志上,有许多求法．本文利用行列式和线性方程组的有关结果给出这一问题的一种算法,所得结论在形式上是非常简洁的．本文中表示n中取k的组合数．在恒等式中,令X一1,2,3,…,h－1并将所得各式相加,可得上式中用n－1,n－2,…,2,1代替n,可得以下n个等式上述n个等式,可以看成以人（k）,又（k）为未知县的线性方程组并且此线性方程组的系数行列式为,由Crammer法则‘的行列式表达式．sum from r=1 to (k-1)(r~2)的行列式算法@贾利新$解放军电子技术学院     

On the Diophantine Equation sum from k=1 to m((k~n)=(m+1)~n)

P. Erds has conjectured [1] that the Diophantine equation 1~n+2~n+…+m~n=(m+1)~n (1) has no positive integer solutions except that n=1, m=2. It is true when m≤10~(10) [3]. A generalized form of (1) has been investigated in [1] [2], and various     

sum from n=1 to +∞(1/n~(2m))的初等解法

本文用初等的方法研究sum from n=1 to(1/n~(2m))(m∈N)的求和问题。这个问题最先由Euler[8]解决。文献[1][6]给出了另两种求解方法。特别地,对于m=1的情形,即sum from n=1 to ∞(1/n~2)=((π~2)/6),已有许多不同的证明方法,可见文献[2][3][4][5]以及那里的参考文献。本文的想法,主要受文献[5][6]的启发而来的。     

A characterization for a sequence to be potentially K r+1 − e-graphic

Let n 〉 r, let lr --- （dl,d2,-,dn） be a non-increasing sequence of nonnegative integers and let Kr＋l - e be the graph obtained from Kr＋l by deleting one edge. If zr has a realization G containing Kr＋l - e as a subgraph, then r is said to be potentially Kr＋l - e-graphic. In this paper, we give a characterization for a sequence π to be potentially Kr＋l - e-graphic.     

On the Construction of Some New Harmonic Maps from ℝ m to ℍ m

In this note, we construct some new harmonic maps from ℝ ^m to ℍ ^m via symmetric methods. Supportedb y NSF Postdoctoral Research Fellowship, China, and MCSEC.     

How hard is it to tell which is a Condorcet committee?

This paper establishes the computational complexity status for a problem of deciding on the quality of a committee. Starting with individual preferences over alternatives, we analyse when it can be determined efficiently if a given committee C$C$ satisfies a weak (resp. strong) Condorcet criterion–i.e., if C$C$ is at least as good as (resp. better than) every other committee in a pairwise majority comparison. Scoring functions used in classic voting rules are adapted for these comparisons. In particular, we draw the sharp separation line between computationally tractable and intractable instances with respect to different voting rules. Finally, we show that deciding if there exists a committee which satisfies the weak (resp. strong) Condorcet criterion is computationally hard.     

sum from n=1 to ∞(1/n~(2m))的一种求和法

本文通过求出函数 F (x) =x2 m 的 Fourier级数展开式 ,得出了 ∑∞n=11n2 m =Amπ2 m 中 Am 的递推关系式 .     

线性矩阵方程sum from i=1 to P(A_iXB_i)=C的求解

<正> 文[1]在第二章以例介绍了矩阵方程AXB=C (1)的解法。对于(1)的推广形式A_1XB_1+A_2XB_2+…+A_QXB_Q=C (2)及AX+XB=C (3) 在工程技术中,特别对自动控制是有着重要应用的一类方程。如果能把它们与已经讨论过的矩阵向量方程AX=6相联系,那末,根据线性方程组的一般理论,就可以顺利地懈决矩阵     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.     

Extension of a theorem of Fejér to double Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth partial sum of its formally differentiated Fourier series divided by n converges to π^-1[F(x+0)-F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of (nonperiodic) functions of bounded variation is also well known. The aim of the present article is to extend these results to the (m, n)th rectangular partial sum of double Fourier or Fourier-Stieltjes series of a function F(x, y) of bounded variation over the closed square [0, 2π]×[0, 2π] in the sense of Hardy and Krause. As corollaries, we also obtain the following results:

Is it a function? Generalizing from single- to multivariable settings

Generalizing is a hallmark of mathematical thinking. The term ‘generalization’ is used to mean both the process of generalizing and the product of that process. This paper reports on five calculus students’ generalizing activity and what they generalized about multivariable functions. The study makes two contributions. The first is a fine-grained, actor-oriented characterization of the ways undergraduates generalized. This adds to knowledge in two areas: the use of the actor-oriented perspective and generalization in advanced mathematics. The second contribution is the products of students’ generalizing: what they generalized about what it means for a multivariable relation to represent a function). This adds to the literature about student reasoning regarding multivariable topics by characterizing the powerful ways of reasoning students possess pre-instruction.     

Involution Fixing∪from (i=1) to r(RP(1))~m_i×(CP(1))~s_i*)

§０．ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｗｒｉｔｅＦＰ（ｎ）ｆｏｒＣＰ（ｎ）（ｎ－ｄｉｍｅｎｓｉｏｎａｌｃｏｍｐｌｅｘｐｒｏｊｅｃｔｉｖｅｓｐａｃｅ）ｏｒＨＰ（ｎ）（ｎ－ｄｉ－ｍｅｎｓｉｏｎａｌｑｕａｔｅｒｎｉｏｎｉｃｐｒｏｊｅｃｔｉｖｅｓｐａｃｅ）．Ｍｎ...     

On a Generalization from Ruin to Default in a Lévy Insurance Risk Model

In a variety of insurance risk models, ruin-related quantities in the class of expected discounted penalty function (EDPF) were known to satisfy defective renewal equations that lead to explicit solutions. Recent development in the ruin literature has shown that similar defective renewal equations exist for a more general class of quantities than that of EDPF. This paper further extends the analysis of this new class of functions in the context of a spectrally negative Lévy risk model. In particular, we present an operator-based approach as an alternative analytical tool in comparison with fluctuation theoretic methods used for similar quantities in the current literature. The paper also identifies a sufficient and necessary condition under which the classical results from defective renewal equation and those from fluctuation theory are interchangeable. As a by-product, we present a series representation of scale function as well as potential measure in terms of compound geometric distribution.