Classification of Kantowski-Sachs and Bianchi Type III Space-Times According to Their Killing Vector Fields in Teleparallel Theory of Gravitation

In this paper we classify Kantowski-Sachs and Bianchi type Ⅲ space-times according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the telepaxallel Killing vector fields are 4 or 6, which are the same in numbers as in general relativity. In case of 4 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of t. In the case of 6 Killing vector fields the metric functions become constants and the Killing vector fields in this case are exactly the same as in general relativity. Here we also discuss the Lie algebra in each case.     

Abelian Chern-Simons theory, Stokes’ theorem, and generalized connections

Generalized connections and their calculus have been developed in the context of quantum gravity. Here we apply them to abelian Chern-Simons theory. We derive the expectation values of holonomies in U(1) Chern-Simons theory using Stokes’ theorem, flux operators and generalized connections. A framing of the holonomy loops arises in our construction, and we show how, by choosing natural framings, the resulting expectation values nevertheless define a functional over gauge invariant cylindrical functions.The abelian theory considered in the present article is the test case for our method. It can also be applied to the non-abelian theory. Results will be reported in a companion article.     

半刚性连接钢框架整体结构屈曲荷载计算模型

在综合考虑梁柱任意半刚性连接及弹性支撑特性的基础上，建立了单层多跨平面钢框架结构整体失稳的线弹性屈曲荷载的解析计算模型，并同时考虑了所有柱脚的半刚性特征，使模型的适用范围更为广泛．本模型中的半刚性连接特性通过在所有梁柱连接处设置虚拟约束弹簧实现，弹性支撑则通过在柱顶引入具有任意刚度参数的水平弹簧实现．基于上述的基本模型，通过对所有构件建立失稳状态的控制方程并引入相应的连续性条件和边界条件，建立了平面单层任意多跨框架结构整体失稳的临界荷载解析计算方法，并研制了相应的计算程序．分析结果表明，半刚性连接特性和水平支撑效应对框架结构整体稳定性的影响极为显著，而且作用于多跨框架结构柱硕竖向荷载的分布特征对结构的整体稳定性有不可忽视的影响．另外，结构的支撑刚度及节点的半刚性连接特性对单层框架整体稳定性的影响具有明显的界限性和区间效应，对此需进一步开展深入研究．     

Lie algebroids,Lie groupoids and TFT

We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid one can associate a BF-like topological field theory which localizes on the space of algebroid morphisms, that can be seen as a generalization of flat connections to the groupoid case. We discuss the finite gauge transformations and discuss the corresponding moduli spaces. We consider the theories both without and with boundaries.     

Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs

This research explored students’ views of geometric objects through the implementation of a curriculum module that allowed them to explore the relationships between transformational geometry and linear algebra. The majority of the students were middle and secondary mathematics education majors enrolled in a one-semester geometry course that is aimed at prospective teachers. A preponderance of the evidence suggests that the participating students, for the most part, viewed isometries operationally and viewed geometric objects (triangle, etc.) as “perceived.” Results also suggest that these two views influenced the students’ abilities to understand and to construct geometric proofs in transformational geometry.     

相配群胚上的诱导联络

本文引入了相配群胚和相配主丛的概念,它们是一对具有相同传递函数的局部平凡李群胚（Г→_→P,α,β）和主丛（B,P,π,G）.我们首先考察了相配群胚Г的内子群胚GГ的局部平凡化,利用这个局部平凡化证明了在B和GГ之间自然存在着一个丛同构.通过这个丛同构以及B的联络日逐步得到了H在GГ和Г上的诱导联络,进而定义了Г上分别以α,β为投影的左联络和右联络,这两个联络都是在Г上整体有定义的,与以往李群胚上的联络只是定义在李代数胚上不同.     

A construction of a nonparametric quantum information manifold

We present a construction of a Banach manifold structure on the set of faithful normal states of a von Neumann algebra, where the underlying Banach space is a quantum analogue of an Orlicz space. On the manifold, we introduce the exponential and mixture connections as dual pair of affine connections.     

Nonholonomic connections with extended Galilean groups of transformations

We continue the investigation of regular representations of the extended Galilean group {ie463-01} acting on the smooth canonical fiber bundle π: R ⁿ⁺¹ → I R ⁿ . First-and second-order nonholonomic affine connections Γ ₁, Γ ₂, Γ _1,2 are constructed using the results presented in papers [3, 4, 5, 6].     

On some bidimensional denumerable chains of infinite order

We study homogeneous chains of infinite order (ξ_t)_{t∈$\text{Z}$} with the set of states taken to be $\text{X=(}\text{N}\text{{0,1})x{-1,1}}$. Our approach is to interpret the half-infinite sequence ..., ξ_?n,..., ξ_?1, ξ₀, where $\text{ξ}{}_{\mathrm{t}}\text{=(i}{}_{\mathrm{t}}\text{, ε}{}_{\mathrm{t}}\text{) ∈ X, t ∈}\text{Z}$ as the continued fraction to the nearer integer expansion (read inversely) of a y ? [?$\text{1}\text{2}$,$\text{1}\text{2}$]. Thus, we are led to study certain Y-valued Markov chains, where Y = [?$\text{1}\text{2}$, $\text{1}\text{2}$] and then by making use of their properties we establish the existence of denumerable chains of infinite order under conditions different from those given in Theorem 2.3.8 of Iosifescu-Theodorescu (1969). A (weak) variant of mixing is proved as well.     

Multivalued maps as a tool in modeling and rigorous numerics

Applications of the fixed point theory of multivalued maps can be classified into several areas: (1) Game theory and mathematical economics; (2) Discontinuous differential equations, differential inclusions, and optimal control; (3) Computing homology of maps; (4) Computer assisted proofs in dynamics; (5) Digital imaging. We give an overview of the most classical and well developed areas of applications (1) and (2), where a multivalued map is used as a generalization of a single-valued continuous map, and we survey more recent applications (3), (4), and (5), where multivalued maps play the role of a numerical tool. Dedicated to Felix Browder on his 80th birthday