Clairaut relation for geodesics of Hopf tubes

Summary In this note we use the Hopf map π: S³ → S² to construct an interesting family of Riemannian metrics h^fon the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S² is the complete lift π^-1(γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S³ of the surfaces of revolution in R^3. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S³. Finally, if we consider the sphere S³ equipped with a family h^f of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.     

一类反应扩散方程奇摄动Robin问题的广义解

讨论了一类奇摄动反应扩散方程R ob in问题.在适当的条件下,研究了问题广义解的渐近性态.     

An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

This paper gives an explicit formula for calculating the Webster pseudo torsion for a strictly pseudoconvex pseudo-hermitian hypersurface. As applications, we are able to classify some pseudo torsion-free hypersurfaces, which include real ellipsoids.     

Is combining meta‐GGA correlation functionals with the OPTX exchange functional useful?

The recent generalized gradient approximation (GGA) density functional OCS1 of Handy and Cohen is implemented in the deMon code and tested on a carefully selected set of problems. OCS1 is found to be accurate for molecular atomization energies, transition metal–ligand bonds, and systems with intramolecular hydrogen bonds. However, OCS1 encounters problems for systems with intermolecular hydrogen bonds. It also tends to elongate bond lengths systematically, and sometimes significantly. The OPTX exchange is combined with three meta‐GGA correlation functionals, Lap3, τ1, and τ2, the latter reported for the first time. The new meta‐GGA scheme OPTX exchange plus τ2 correlation called Oτ2 yields improved molecular geometries, NMR shielding constants, and an improved barrier height for the H+H₂ reaction. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Hopf Algebras of Dimension 12

We conclude the classification of Hopf algebras of dimension 12 over an algebraically closed field of characteristic zero.     

Lines,Trees, and Branch Spaces

In this paper we examine the interactions between the topology of certain linearly ordered topological spaces (LOTS) and the properties of trees in whose branch spaces they embed. As one example of the interaction between ordered spaces and trees, we characterize hereditary ultraparacompactness in a LOTS (or GO-space) X in terms of the possibility of embedding the space X in the branch space of a certain kind of tree.     

一般非均匀凸介质的迁移算子广义本征函数系统的完整性

该文讨论一般非均匀凸介质所确定的迁移算子的广义本征函数系统的完整性问题，利用我们探索的相对收敛方法，完整地刻划了一般非均匀凸介质中迁移问题的广义本征函数系统的完整性问题，我们证明了，对迁移半群E（t），当0∈Pσ（E（t）），迁移算子广义本征函数系统不完整，当0∈Pσ（E（t））时，仅当满足相对收敛条件时，迁移算子的广义本征函数系统完整．     

Polynomial Shift States of a Chaotic Map

We present a piecewise- linear map of the unit interval in which the resolvent of the Frobenius- Perron operator, considered in a polynomial basis, has an essen-tial singularity at the origin. Associated with the essential singularity are polynomial shift states, which are obtained from creation and annihilation operators in non- self- dual function spaces. Correlation functions of general polynomial observables have decay components that vanish in a finite time.     

广义路余代数和广义对偶Gabriel定理

通过将箭图的每个顶点放置一个k-余代数,首先引进了广义路余代数的概念,其次给出了广义路余代数的一些基本性质,还讨论了同构问题.证明了两个正规广义路余代数是同构的当且仅当他们的箭图及对应顶点上的单余代数是同构的.对于满足Codim C_0■1余代数C,证明了对偶Wedderburn-Malcev定理成立.作为广义路余代数的一个应用,推广了点余代数的对偶Gabriel定理.     

Trees, set compositions and the twisted descent algebra

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). Among others, a number of enveloping algebra structures are introduced and studied in detail. For example, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.