The discriminant of an algebraic torus

For a torus T defined over a global field K, we revisit an analytic class number formula obtained by Shyr in the 1970s as a generalization of Dirichlet?s class number formula. We prove a local-global presentation of the quasi-discriminant of T, which enters into this formula, in terms of cocharacters of T. This presentation can serve as a more natural definition of this invariant.     

Lie Tori of Type BC 2 and Structurable Quasitori

Structurable quasitori are introduced and shown to give all Lie tori of type BC ₂ via the Kantor construction. Structurable quasitori, including Jordan quasitori, are classified. The methods include a use of generalized quadrangles of order (2,t).     

Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmu¨ller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.     

基于臭氧分子体系的动力学研究

在李代数动力学群得到Hamiltonian的基础上,通过求解正则运动方程.对固定键角近似下臭氧分子体系相空间的庞加莱截面、KAM环面以及振动体系的能量属性进行了研究,获得了有关臭氧分子振动体系丰富的动力学信息.     

Knotted pictures of the GHZ states on the surface of trivial torus

By means of the method of torus knot theory, this paper presents the complete process of obtaining the knotted pictures of eight GHZ states on the surface of trivial torus from the knotted pictures of eight basic three-qubit states on the surface of trivial torus. Thus, we obtain eight knotted pictures 12₁ linkage on the ordinary plane.     

Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing

This work focuses on one-dimensional (1D) quasi-periodically forced nonlinear wave equations. This means studying with Dirichlet boundary conditions, where ε is a small positive parameter, (t) is a real analytic quasi-periodic function in t with frequency vector ω=(ω₁,ω₂…,ω_m) and the nonlinearity h is a real analytic odd function of the form It is shown that, under a suitable hypothesis on (t) and h, there are many quasi-periodic solutions for the above equation via KAM theory.     

Integrable structure of melting crystal model with two -parameters

This paper explores integrable structures of a generalized melting crystal model that has two q-parameters q₁,q₂. This model, like the ordinary one with a single q-parameter, is formulated as a model of random plane partitions (or, equivalently, random 3D Young diagrams). The Boltzmann weight contains an infinite number of external potentials that depend on the shape of the diagonal slice of plane partitions. The partition function is thereby a function of an infinite number of coupling constants t₁,t₂,… and an extra one Q. There is a compact expression of this partition function in the language of a 2D complex free fermion system, from which one can see the presence of a quantum torus algebra behind this model. The partition function turns out to be a tau function (times a simple factor) of two integrable structures simultaneously. The first integrable structure is the bigraded Toda hierarchy, which determines the dependence on t₁,t₂,…. This integrable structure emerges when the q-parameters q₁,q₂ take special values. The second integrable structure is a q-difference analogue of the 1D Toda equation. The partition function satisfies thisq-difference equation with respect to Q. Unlike the bigraded Toda hierarchy, this integrable structure exists for any values of q₁,q₂.     

From Metacyclophanes to Cyclacenes: Synthesis and Properties of [6.8]3Cyclacene

Conjugated belts : [6.8]₃cyclacene as the first hydrocarbon cyclacene was synthesized in a de novo strategy. Various [2₃]metacyclophanes are described as intermediates. The synthetic approach was extended to larger cyclacenes, and [2₄]metacyclophanes as precursors of [6.8]₄cyclacene were synthesized.