The Wave Function of the Lyman-Alpha Photon Part II. The Shape,Position, and Trajectory of the Photon

In Part I of the present paper we derived the wave function of the Lyman-a \alpha photon in both the linear and angular momentum bases using relativistic concepts for the photon wave function. In the present paper, Part II, we derive two [(X)\vec] \vec{X} -representations. In the first we assume one-particle theory for the photon wave function and the usual commutation rules for the position operators X_i X_i and linear momentum operators P_i P_i . The second representation employs the quantized photon field to derive an [(X)\vec] \vec{X} -representation. The stress, energy tensor density is used to provide a probability density in [(x)\vec] \vec{x} -space which is relativistic. The two methods of defining [(x)\vec] \vec{x} -space are compared.¶It is found in the present case that, despite the use of particle operators, the photon resembles a field far more than it does a particle.     

The Approximation Property and Locally Convex Spaces Defined by the Ideal of Approximable Operators

The connection between the approximation property and certain classes of locally convex spaces associated with the ideal ?? of approximable operators will be discussed. It will be shown that a Frechet Montel space has the approximation property iff it is a mixed ??-space, or equivalently, iff its strong dual is a ??-space. Similarly, a Silva space has the approximation property iff it is a mixed ??-space or equivalently iff it is a ??-space. We shall also show that a Frechet Schwartz space with the bounded approximation property is always a ??-space.     

Some generalizations of the concept of a p-space

In this article, some generalizations of the concept of a p-space are introduced and studied. The notion of a source of a space in a larger space and the concepts of partial plumage, s-embedding, p-embedding, p^?-embedding, s-space, and p^?-space are defined and studied in depth (see Theorems 2.6, 2.7, 3.2, 4.3, 4.4, 4.10 and their corollaries). An example of a hereditarily p^?-space which is not a p-space and is a perfect image of a hereditarily p-space is indicated (Example 2.9). Among the main results, we establish that if a paracompact space X is p-embedded in a pseudocompact space as a dense subspace, then X is a p-space (Corollary 4.8), and that if X has a countable network and is p^?-embedded in a pseudocompact space, then X is metrizable (Corollary 4.11). The following problem is posed: is every paracompact Gδ-subspace of a pseudocompact space ?ech-complete?     

D-spaces and covering properties

We study the D-space property and its generalizations, the notions of an aD-space and a weak aD-space in connection with covering properties. A brief survey on D-spaces is presented in Section 1.Among new results, it is proved that if a linearly ordered space is an aD-space, then it is paracompact. The statement further extends the list of equivalences in [Proc. Amer. Math. Soc. 125 (1997) 1237]. We also establish some sufficient conditions for the free topological group of a Tychonoff space to be a D-space. In particular, the free topological group of a semi-stratifiable space is shown to be a D-space, while it need not be semi-stratifiable. A similar result is established for the free topological group of a space with a point-countable base. Some new interesting open problems on D-spaces and on spaces close to them are formulated. In particular, we discuss several such questions in connection with the sum theorems.     

Equilibrium description of a particle system in a heat bath

AnN particle system evolving in 3-space under the action of a friction force, an external potential and a fluctuating force is studied, using the Langevin equation. Necessary and sufficient conditions are given for both (a) the existence and (b) the Maxwell-Boltzmann character of an invariant measure. Both probabilistic and system theoretical tools play an important part in this study. In addition, the general results mentioned above are applied to the study of a system ofn oscillators coupled through velocity, with a first-neighbour-only type of interaction.     

On the modulus of L- and M-weakly compact operators

In the present paper it is shown that every L-weakly compact operator from an AL-space into a KB-space has an L-weakly compact modulus and that every M-weakly compact operator from a Banach lattice into an order complete AM-space with unit has an M-weakly compact modulus.     

On hereditarily strong Σ-spaces

In this paper the structure of hereditarily strong Σ-spaces (hsΣ-spaces, for short) is dealt with. The main result asserts that an hsΣ-space is the disjoint union of two σ subspaces one of which is an F_σ, the other a G_δ subset. Examples are given that in many ways, this decomposition cannot be improved. Then we investigate the question when an hsΣ-space is a σ-space. It is shown that a GO-space (or a first countable compactum) is metrizable iff it is an hsΣ-space, thereby proving a conjecture of J. van Wouwe. σ-spaces are characterized as being identical with perfect hsΣ-spaces. The question whether a Lindelöf, first countable hsΣ-space is a σ-space is shown to be independent of set theory. A characterization of hsΣ-spaces with no compact subsets of cardinality >2^ω is given.     

三维欧氏Steiner最小树的Delaunay四面体网格混合智能算法

Steiner最小树问题是组合优化中经典的NP难题,在许多实际问题中有着广泛的应用,而三维欧氏Steiner最小树问题是对二维欧氏Steiner最小树问题的推广。由于三维欧氏Steiner树问题的求解非常困难,至今为止的相关成果较为少见。本文针对该问题,利用Delaunay四面体网格剖分技术,提出了一种混合型智能求解方法,不仅可以尽量避免拓扑结构陷入局部最优,且对较大规模的问题求解亦有良好的效果。算法在Matlab环境下编程实现,经实例测试,获得了满意的效果。     

Discreteness Criterions of Isometric Subgroups for Quaternionic Hyperbolic Space

In this paper, we give analogue of Jörgensen's inequality for nonelementary groups of isometries of complex hyperbolic 2-space generated by two elements, one of which is either loxodromic or boundary elliptic for the same group of isometries of quaternionic hyperbolic 2-space. And we give a sufficient condition for a nonelementary subgroup of isometries of quaternionic hyperbolic 2-space generated by two elements one of which is parabolic not to be discrete.     

关于三维Minkowski空间中直线汇的一些注记

本文研究了三维Minkowski空间中直线汇的一些性质,特别是关于类时线汇的性质．讨论了线汇基本元素的存在性,并证明了关于三维Minkowski空间中类时线汇的配分参数的一个结果,推广了苏步青1927年的—个成果．     

Curvature of $$C_5\oplus C_{12}$$ C 5 ⊕ C 12 -Manifolds

Mediterranean Journal of Mathematics - In this paper, we consider spacelike curves in the light-cone 3-space that is canonically embedded in the light-cone 4-space and the de Sitter 4-space in...     

On the total mean curvature of a nonrigid surface

Using the Green’s theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field, which immediately yields the following well-known theorem: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.     

On non-orientable surfaces in 4-space which are projected with at most one triple point

We show that if a non-orientable surface embedded in 4-space has a projection into 3-space with at most one triple point, then it is ambient isotopic to a connected sum of some unknotted projective planes and an embedded surface in 4-space with vanishing normal Euler number.

     

积空间是D -空间的空间类

人们知道每个C-似空间是 D -空间, 且每个正则弱θ -可加细 C-散布空间也是D -空间。上述空间类的积空间还是D -空间吗?在这篇文章中作者讨论了该问题, 得到如下结论：正则弱θ -可加细空间的有限积是D -空间; 正则Lindel\"of C-散布空间的可数积是D -空间     

Group-like structures and the whitehead product in pro-homotopy and shape

The notion of ‘H-space’ is of considerable importance in the homotopy theory of CW-complexes. This paper studies a similar notion in the framework of pro-homotopy and shape theories. This is achieved by following the general plan set forth by Eckmann and Hilton. Examples of shape H-space are also given; it is observed that every compact connected topological monoid is a shape H-space. The Whitehead product is defined and studied in the pro-homotopy and shape categories; and, it is shown that this Whitehead product vanishes on an H-object in pro-homotopy. These results are the natural extension of some well-known classical results in the homotopy theory of CW-complexes.     

On the completion of (LF)-spaces

Once the existence of metrizable (LF)-spaces was discovered, the problem whether the completion of an (LF)-space is or is not an (LF)-space was answered in the negative, because no (LF)-space can be a Fréchet space. However, some (non-metrizable) (LF)-spaces are complete, e.g. the classical Köthe's strict (LF)-spaces. In this paper we will carry out a thorough study of the completeness of (LF)-spaces stressing upon the stable completion properties of (LB)-spaces. A basic tool for handling this problem is an Open Mapping Theorem for completions of (LF)-spaces, which is also proved in the present paper.This research, supported by a Fulbright-MEC fellowship, was carried out during the author's visit at the University of Maryland while on leave from the University of Extremadura. The author is grateful to Professor S. A. Saxon for very helpful conversation and, especially, for calling his attention to the problem of completions of (LF)-spaces.     

L~p空间中神经网络逼近的几何速度

该文主要研究了L~p空间中神经网络逼近的几何速度.将凸贪婪迭代法应用于L~p空间中满足"δ-角度的条件"的一类函数.作者将文献的结论推广到L~p空间中,得到当0q1时人工神经网络逼近的速度是o(q~n).     

The β-space property in monotonically normal spaces and GO-spaces

In this paper we examine the role of the β-space property (equivalently of the MCM-property) in generalized ordered (GO-)spaces and, more generally, in monotonically normal spaces. We show that a GO-space is metrizable iff it is a β-space with a Gδ-diagonal and iff it is a quasi-developable β-space. That last assertion is a corollary of a general theorem that any β-space with a σ-point-finite base must be developable. We use a theorem of Balogh and Rudin to show that any monotonically normal space that is hereditarily monotonically countably metacompact (equivalently, hereditarily a β-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a β-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos.     

A note on biduals of strict (LF)-spaces

Grothendieck asked in 1954 in [1] the following questions. (1) Is the bidual of a strict inductive limit of a sequence of locally convex spaces the inductive limit of the biduals? (2) Is the bidual of a strict (LF)-space again an (LF)-space? (3) Is the bidual of a strict (LF)-space complete? M. Valdivia gave a (negative) answer to the first question in 1979 in [5]. Since his counterexample is not an (LF)-space, problem (2) remained open. The aim of this note is to present a negative solution to questions (2) and (3). The answer to question (2) is negative even if every step of the (LF)-space is distinguished, in which case the strong bidual is complete by a result of Grothendieck. Moreover, we show that the strong dual of a strict (LF)-space need not be countably barrelled.     

The 2-twist-spun trefoil has the triple point number four

The triple point number of an embedded surface in 4-space is the minimal number of the triple points on all the projection images into 3-space. We show that the 2-twist-spun trefoil has the triple point number four.