Π r P1-bundle from which a surjective morphism to Π m P1 exists

Let : X Y be a morphism of smooth projective varieties over an algebraically closed field k of characteristic not equal to 2 whose closed fibres are all isomorphic to ^r P¹ and let ': X ^r P¹ be a surjective morphism. This article gives a sufficient condition concerning ' and Y under which X is isomorphic to Y× ^r P¹.     

Spaces which contain a copy of Sω or S2 and their applications

Let S_ω and S₂ denote the sequential fan and Arens' space, respectively. In this paper, we shall prove the following main results. (1) If Π^∞_i=1 X_i contains a copy of S_ω (S₂), then some Πⁿ_i=1 X_i contains a copy of S_ω (S_ω or S₂, respectively). (2) Let f: X → Y be a closed map such that any f^-1(y) contains no closed copy of S_ω (S₂). If X contains a closed copy of S_ω (S₂), then Y contains a closed copy of S_ω (S_ω or S₂, respectively).As applications of (1) and (2), we shall consider the Fréchet or strongly Fréchetness, or sequentiality of products of finitely or countably many spaces.     

On (2, 4) complete intersection threefolds that contain an Enriques surface

We study nodal complete intersection threefolds of type (2, 4) in \({\mathbb {P}}^5\) which contain an Enriques surface in its Fano embedding. We completely determine Calabi–Yau birational models of a generic such threefold. These models have Hodge numbers \(h^{11}=2,h^{12}=32\). We also describe Calabi–Yau varieties with Hodge numbers \((h^{11},h^{12})\) equal to (2, 26), (23, 5) and (31, 1). The last two pairs of Hodge numbers are, to the best of our knowledge, new.     

Example of a Smooth Homeomorphism Violating the Luzin N−1 Property

This paper discusses some results concerning the validity of the Luzin N and N^?1 properties for homeomorphisms. Also, we present a counterexample in the form of a C^∞-homeomorphism that violates the N^?1 property.

     

Conditions for the spectrum associated with an asymptotically straight leaky wire to contain an interval [− α2/4, ∞)

The method of singular sequences is used to provide a simple and, in some respects, a more general proof of a known spectral result for leaky wires. The method introduces a new concept of asymptotic straightness. Received: 4 October 2007     

Brouwer’s Fan Theorem as an axiom and as a contrast to Kleene’s alternative

The paper is a contribution to intuitionistic reverse mathematics. We introduce a formal system called Basic Intuitionistic Mathematics BIM, and then search for statements that are, over BIM, equivalent to Brouwer’s Fan Theorem or to its positive denial, Kleene’s Alternative to the Fan Theorem. The Fan Theorem is true under the intended intuitionistic interpretation and Kleene’s Alternative is true in the model of BIM consisting of the Turing-computable functions. The task of finding equivalents of Kleene’s Alternative is, intuitionistically, a nontrivial extension of the task of finding equivalents of the Fan Theorem, although there is a certain symmetry in the arguments that we shall try to make transparent. We introduce closed-and-separable subsets of Baire space \({\mathcal{N}}\) and of the set \({\mathcal{R}}\) of the real numbers. Such sets may be compact and also positively noncompact. The Fan Theorem is the statement that Cantor space \({\mathcal{C}}\) , or, equivalently, the unit interval [0, 1], is compact and Kleene’s Alternative is the statement that \({\mathcal{C}}\) , or, equivalently, [0, 1], is positively noncompact. The class of the compact closed-and-separable sets and also the class of the closed-and-separable sets that are positively noncompact are characterized in many different ways and a host of equivalents of both the Fan Theorem and Kleene’s Alternative is found.     

The universality of ? 1 as a dual space

Let X be a Banach space with a separable dual. We prove that X embeds isomorphically into a ${{\mathcal L}_\infty}$ space Z whose dual is isomorphic to ? ₁. If, moreover, U is a space with separable dual, so that U and X are totally incomparable, then we construct such a Z, so that Z and U are totally incomparable. If X is separable and reflexive, we show that Z can be made to be somewhat reflexive.     

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P4

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in P ⁴ such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronesé variety V ²⊂P ⁵ and the theory of associativity equations of two-dimensional topological field theory. Received: 15 August 2001     

A PλM-policy for an M/G/1 queueing system

We introduce P_λ^M-service policy for an M/G/1 queueing system. The stationary distribution of the workload under this policy is explicitly obtained through a decomposition technique, renewal reward theorem, and level crossing argument.     

Nontangential Weighted Limit of the Infinitesimal Carathéodory Metric in an h-Extendible Boundary Point of a Smooth Bounded Pseudoconvex Domain in Cn

A boundary point of a domain in Cⁿ is said to be h-extendible if its Catlin's multitype coincides with its D'Angelo's type. The main purpose of this paper is to study the existence of nontangential weighted limit of the infinitesimal Carathéodory metric in such a point of a smooth bounded pseudoconvex domain in Cⁿ.     

On non-Kählerianity of nonuniform lattices¶in SO(n, 1)(n≥ 4)

In this note, we will show that no nonuniform lattice of SO(n, 1)(n≥ 4) is the fundamental group of a quasi-compact K?hler manifold Received: 18 July 2000 / Revised version: 4 September 2000     

不等式1/a+1/b≥4/(a+b)的应用

由不等式a2+b2≥2ab,我们可以得到(a+b)2≥4ab.当a,b∈R+时,容易得到a+b/ab≥4a+b,即1a+1b≥4a+b.这又是一个非常有用的基本不等式,下面我们用这个不等式来处理几个问题,看看它的威力。     

How f ar is $${\left( 1 + \fr ac{ a}{n}\right) ^n} $$1+ ann from $$ e^ a $$e a when a is an element of a B an ach algebr a?

We present effective upper and lower bounds for the distance from \(\displaystyle \left( 1 + \frac{a}{n}\right) ^n \) to \(\displaystyle e^a \) for an element a of a complex unital Banach algebra and positive integer n. Specifically

$$\begin{aligned} \frac{1}{2n} \sup \left\{ \left| \mathfrak {R}(z^2) \right| e^{\mathfrak {R}(z)} : z \in \sigma (a) \right\} \lesssim _{(2)} \left\| e^a - \left( 1 + \frac{a}{n}\right) ^n \right\| \le \frac{ \left\| a \right\| ^2}{2n} \ e^{ \left\| a \right\| }, \end{aligned}$$

where \(\sigma (a)\) is the spectrum of a. The symbol \(\lesssim _{(p)}\) means “less than or equal to, up to a term of order \(n^{-p}\)”as discussed below. Following some technical preliminaries (Sect. 1) we treat the real case (Sect.  2), extend to the complex case (Sect.  3), and then generalise to the case of a norm-unital Banach algebra (Sect.  4).

     

L 1 (μ, X) as a constrained subspace of its bidual

In this note we consider the property of being constrained in the bidual, for the space of Bochner integrable functions. For a Banach spaceX having the Radon-Nikodym property and constrained in its bidual and forY ⊂ X, under a natural assumption onY, we show thatL ¹ (μ, X/Y) is constrained in its bidual andL ¹ (μ, Y) is a proximinal subspace ofL ¹(μ, X). As an application of these results, we show that, ifL ¹(μ, X) admits generalized centers for finite sets and ifY ⊂ X is reflexive, thenL ¹ μ, X/Y) also admits generalized centers for finite sets.     

The stochastic fluid–fluid model: A stochastic fluid model driven by an uncountable-state process,which is a stochastic fluid model itself

We introduce the Stochastic Fluid–Fluid Model, which offers powerful modeling ability for a wide range of real-life systems of significance. We first derive the infinitesimal generator, with respect to time, of the driving stochastic fluid model. We then use this to derive the infinitesimal generator of a particular Laplace–Stieltjes transform of the model, which is the foundation of our analysis. We develop expressions for the Laplace–Stieltjes transforms of various performance measures for the transient and limiting analysis of the model. This work is the first direct analysis of a stochastic fluid model that is Markovian on a continuous state space.     

Children’s conceptions on the structure of an array: Using quick images as a gateway to multiplicative ideas

In this paper we report the conceptions about arrays that came to the fore as one class of second-grade students participated in whole classroom discussions and activities focused on the structure of arrays presented as a Quick Images routine. Before the intervention, students were not introduced to formal multiplication but had completed a unit on arrays. A constant comparative method was used to identify numeric and spatial structuring strategies that allowed for students’ conceptions about the structure of the array to emerge. Results indicated that not all students automatically use arrays as a composite of rows. We found that the use of Quick Images with larger arrays and non-arrays within the whole classroom discussion was successful at eliciting and directing students’ attention towards the spatial features of an array, including seeing an array as made of a composite of rows (or columns).     

关于指数丢番图方程a2x+(3a2-1)y=(4a2-1)z

设a＞3是一个整数,应用Bilu,Hanrot和Voutier关于本原素除子的深刻理论以及二次数域类数的一些结果,证明了指数丢番图方程a2x＋（3a2-1）y=（4a2-1）z仅有正整数解x=y=1.     

Zur graphischen Integration der linearen Differentialgleichung n. Ordnung (an D + 1) (an−1 D + 1) ..... (a1 D + 1) y = s (x)(a1 D + 1) y = s (x)

Ohne Zusammenfassung     

具有性质cr+1=3,ar+1=3a+1或 4a的(a+1,3)型距离正则图

设г是直径为d且型为(a 1,3)的距离正则图,其中a≥2.用l(c,a,b)表示交叉阵列ι(г)中列(c,a,6)t的个数,记r=r(г)=l(c1,a1,b1),8=8(г)=l(Cr 1,ar 1,br 1)及t=t(г)=l(cr s 1,ar s 1,br s 1).那末,若Cr 1=3,ar 1=4a或3a 1,则d=r t 2.