Towards relative invariants of real symplectic four-manifolds

Let (X, ω, c_X) be a real symplectic four-manifold with real part . Let be a smooth curve such that We construct invariants under deformation of the quadruple (X, ω, c_X, L) by counting the number of real rational J-holomorphic curves which realize a given homology class d, pass through an appropriate number of points and are tangent to L. As an application, we prove a relation between the count of real rational J-holomorphic curves done in [W2] and the count of reducible real rational curves done in [W3]. Finally, we show how these techniques also allow us to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics. Received: March 2005; Revision: September 2005; Accepted: September 2005     

Poisson structures on complex flag manifolds associated with real forms

For a complex semisimple Lie group and a real form we define a Poisson structure on the variety of Borel subgroups of with the property that all -orbits in as well as all Bruhat cells (for a suitable choice of a Borel subgroup of ) are Poisson submanifolds. In particular, we show that every non-empty intersection of a -orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.

     

On a Weyl-Titchmarsh theory for discrete symplectic systems on a half line

Recently, Bohner and Sun [9] introduced basic elements of a Weyl-Titchmarsh theory into the study of discrete symplectic systems. We extend this development through the introduction of Weyl-Titchmarsh functions together with a preliminary study of their properties. A limit point criterion is described and characterized. Green’s function for the half-line is introduced as a limit of such functions in the regular case and half-line solutions obtained are seen to satisfy λ-dependent boundary conditions at infinity.     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

辛体系下含对称破缺因素动力学系统的近似守恒律

自冯康先生创立Hamilton系统辛几何算法以来,诸如辛结构和能量守恒等守恒律逐渐成为动力学系统数值分析方法有效性的检验标准之一。然而,诸如阻尼耗散、外部激励与控制和变参数等对称破缺因素是实际力学系统本质特征,影响着系统的对称性与守恒量。因此,本文在辛体系下讨论含有对称破缺因素的动力学系统的近似守恒律。针对有限维随机激励Hamilton系统,讨论其辛结构;针对无限维非保守动力学系统、无限维变参数动力学系统、Hamilton函数时空依赖的无限维动力学系统和无限维随机激励动力学系统,重点讨论了对称破缺因素对系统局部动量耗散的影响。上述结果为含有对称破缺因素的动力学系统的辛分析方法奠定数学基础。     

On the Complex Difference Equation of Hypergeometric Type on Non-uniform Lattices

In this article,we obtain a new fundamental theorems for Nikiforov-Uvarov-Suslov complex difference equation of hypergeometric type by the method of Euler integral transformation,its expression is different from Suslov’s Theorem.We also establish the adjoint equation for Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on non-uniform lattices,and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well.The particular solutions of the adjoint equation are then obtained.As an appliction of these particular solutions,we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices and other important results.     

Decomposing a new nonlinear differential-difference system under a Bargmann implicit symmetry constraint

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.     

一类具有三种状态的可修排队系统研究

研究一类具有三种状态的可修排队模型主算子的豫解集.通过研究该主算子的共轭算子的豫解集得到此主算子的豫解集.     

Embeddings of Un(q2) and symmetric strongly regular graphs

In the geometric setting of the embedding of the unitary group U_n(q²) inside an orthogonal or a symplectic group over the subfield GF(q) of GF(q²), q odd, we show the existence of infinite families of transitive two‐character sets with respect to hyperplanes that in turn define new symmetric strongly regular graphs and two‐weight codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 248–253, 2010     

Finite group action and Gromov-Witten invariants in symplectic four-manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.