The Hamiltonian structures of 3D ODE with time-independent invariants

We have proved that any 3-dimensional dynamical system of ordinary differential equations (in short, 3D ODE) with time-independent invariants can be rewritten as Hamiltonian systems with respect to generalized Poisson brackets and the Hamiltonians are these invariants. As an example, we discuss the Kermack-Mckendrick model for epidemics in detail. The results we obtained are generalization of those obtained by Y. Nutku. First Received Nov. 22, 1993     

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 R2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.     

Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.     

Skein modules and the noncommutative torus

We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the -th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.

     

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.     

Knots and the Bracket Calculus

In this paper, we introduce a way of encoding links (long links). This ways leads to a combinatorial representation of links by words in a given finite alphabet. We prove that the link semigroup is isomorphic to some algebraically defined semigroup with a simple system of relations. Thus, knot theory is represented as a bracket calculus: the link recognition problem is reduced to a recognition problem in this semigroup.     

Lie Superalgebra Structures in <Emphasis Type="Italic">H</Emphasis><Superscript>⋅</Superscript> (<Emphasis Type="Italic">g</Emphasis>;<Emphasis Type="Italic">g</Emphasis>)

Let g=vect(M) be the Lie (super)algebra of vector fields on any connected (super)manifold M; let - be the change of parity functor, C ⁱ and H ⁱ the space of i-chains and i-cohomology. The Nijenhuis bracket makes into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in L ^* and for any Lie superalgebra g. We use a Mathematica-based package SuperLie (already proven useful in various problems) to explicitly describe the algebras l ^* for some simple finite dimensional Lie superalgebras g and their relatives - the nontrivial central extensions or derivation algebras of the considered simple ones.     

Accessibility of the matrix Riccati differential equation

A known result on the classification of transitive Lie group actions on complex Grassmann manifolds is exploited to derive a necessary and sufficient accessibility criterion for the complex matrix differential Riccati equation. We treat both cases, the symmetric as well as the non-symmetric Riccati equation. Corresponding accessibility results for the real Riccati equation are also available, but not stated here. An application to the accessibility of generalized double bracket flows is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

战车用惯性定位定向系统的支架误差分析及补偿

战车用惯性定位定向系统由于采用双轴陀螺平台结构,不可避免地存在着支架误差,从而影响系统航向精度。应用球面三角法和方向余弦矩阵法分别推导出双轴陀螺平台的支架误差公式,发现由这两种方法推导出来的公式计算结果是一致的。最后通过数学仿真和分析,给出了支架误差与载体的姿态角之间的定量关系。根据此误差公式,对惯性定位定向系统的航向输出能够进行误差补偿,从而保证了动态条件下定位定向系统的航向输出精度。