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. 相似文献
Structurable quasitori are introduced and shown to give all Lie tori of type BC2 via the Kantor construction. Structurable quasitori, including Jordan quasitori, are classified. The methods include a use of generalized quadrangles of order (2,t). 相似文献
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. 相似文献
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. 相似文献
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…,ωm) 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. 相似文献
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. 相似文献
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.