截面箭图代数的Gerstenhaber括号积

本文基于截面箭图代数Λ的极小投射双模分解,利用平行路的语言清晰地刻画了截面箭图代数的Hochschild上同调空间的Gerstenhaber括号积,并由此得到了截面基本圈代数Λ的二阶上同调群中的每个元素都定义了Λ的一个非交换Poisson结构,进而定义了Λ的一个单参数形变的一阶乘法映射.     

K-Theory of Braided Monoidal Categories

We show that the commutativity constraint of a braided monoidal category gives rise to an algebraic structure on its K-theory known as a Gerstenhaber algebra. If, in addition, the braiding has a compatible balanced structure the Gerstenhaber bracket on the K-theory is generated by a Batalin–Vilkovisky differential. We use these algebraic structures to prove a generalization of the Anderson–Moore–Vafa theorem which says that the order of the twist, in a semi-simple balanced monoidal category with duals and finitely many simple objects, is finite.     

广义Hamilton系统的保结构算法

基于广义Hamilton系统微分方程解析解理论。给出了构造保持系统真解典则性的高阶显式积分格式的方法,并说明其可推广到广义Hamilton控制系统。该方法保持了原系统的几何定性特征,因而是稳定的。数值例子说明了算法的有效性。     

Metriplectic structure, Leibniz dynamics and dissipative systems

A metriplectic (or Leibniz) structure on a smooth manifold is a pair of skew-symmetric Poisson tensor P and symmetric metric tensor G. The dynamical system defined by the metriplectic structure can be expressed in terms of Leibniz bracket. This structure is used to model the geometry of the dissipative systems. The dynamics of purely dissipative systems are defined by the geometry induced on a phase space via a metric tensor. The notion of Leibniz brackets is extendable to infinite-dimensional spaces. We study metriplectic structure compatible with the Euler-Poincaré framework of the Burgers and Whitham-Burgers equations. This means metriplectic structure can be constructed via Euler-Poincaré formalism. We also study the Euler-Poincaré frame work of the Holm-Staley equation, and this exhibits different type of metriplectic structure. Finally we study the 2D Navier-Stokes using metriplectic techniques.     

r-Matrix Structure for a Restricted Flow with Bargmann Constraint

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R^2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.     

On the relation between the A-polynomial and the Jones polynomial

This paper shows that the noncommutative generalization of the A-polynomial of a knot, defined using Kauffman bracket skein modules, together with finitely many colored Jones polynomials, determines the remaining colored Jones polynomials of the knot. It also shows that under certain conditions, satisfied for example by the unknot and the trefoil knot, the noncommutative generalization of the A-polynomial determines all colored Jones polynomials of the knot.

     

Local solvability of linear differential operators with double characteristics. I. Necessary conditions

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators L, defined, say, in an open set $\Omega \subset \mathbb{R}^n.This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators L, defined, say, in an open set Suppose the principal symbol p _k of L vanishes to second order at , and denote by the Hessian form associated to p _k on . As the main result of this paper, we show (under some rank conditions and some mild additional conditions) that a necessary condition for local solvability of L at x ₀ is the existence of some such that . We apply this result in particular to operators of the formwhere the X _j are smooth real vector fields and the α _jk are smooth complex coefficients forming a symmetric matrix . We say that L is essentially dissipative at x ₀, if there is some such that e ^iθ L is dissipative at x ₀, in the sense that . For a large class of doubly characteristic operators L of this form, our main result implies that a necessary condition for local solvability at x ₀ is essential dissipativity of L at x ₀. By means of H?rmander’s classical necessary condition for local solvability, the proof of the main result can be reduced to the following question: suppose that Q _A and Q _B are two real quadratic forms on a finite dimensional symplectic vector space, and let Q _C : = {Q _A ,Q _B } be given by the Poisson bracket of Q _A and Q _B . Then Q _C is again a quadratic form, and we may ask: when can we find a common zero of Q _A and Q _B at which Q _C does not vanish? The study of this question occupies most of the paper, and the answers may be of independent interest. In the second paper of this series, building on joint work with F. Ricci, M. Peloso and others, we shall study local solvability of essentially dissipative left-invariant operators of the form (0.1) on Heisenberg groups in a fairly comprehensive way. Various examples exhibiting a kind of exceptional behaviour from previous joint works, e.g., with G. Karadzhov, have shown that there is little hope for a complete characterization of locally solvable operators on Heisenberg groups. However, the “generic” scheme of what rules local solvability of second order operators on Heisenberg groups becomes evident from our work.      

Gottlieb groups of spheres

This paper takes up the systematic study of the Gottlieb groups of spheres for k≤13 by means of the classical homotopy theory methods. We fully determine the groups for k≤13 except for the 2-primary components in the cases: k=9,n=53;k=11,n=115. In particular, we show if n=2ⁱ−7 for i≥4.