Invariants of Lagrangians on Weil Bundles and Their Classifications

Let Q:BA be an algebra epimorphism, where A, B are Weil algebras. Let Q M:T B M T AM denote the canonical extension of Q, from the bundle of B-velocities onto the bundle of A-velocities, over a manifold M. Classifications of all natural operators transforming real-valued functions on T AM into real-valued functions on T BM or into 1-forms on T BM of finite order with respect to Q M are given, provided the dimension of M is sufficiently large.     

Alexandroff's Double Arrow Compact Space and Approximation Theory

A compact space Q similar to the compact space known as Alexandroff's double arrow space is constructed. It is shown that the real space C(Q) has no Chebyshev subspaces of codimension >1, but the complex space C(Q) has such subspaces.     

An Example of a Positive Semidefinite Double Sequence Which is Not a Moment Sequence

The first explicit example of a positive semidefinite double sequence which is not a moment sequence was given by Friedrich. We present an example with a simpler definition and more moderate growth as (m, n) .     

Invariant Lagrangians and a new method of integration of nonlinear equations

A method for solving the inverse variational problem for differential equations admitting a Lie group is presented. The method is used for determining invariant Lagrangians and integration of second-order nonlinear differential equations admitting two-dimensional noncommutative Lie algebras. The method of integration suggested here is quite different from Lie's classical method of integration of second-order ordinary differential equations based on canonical forms of two-dimensional Lie algebras. The new method reveals existence and significance of one-parameter families of singular solutions to nonlinear equations of second order.     

Detection of Some Elements in the Stable Homotopy Groups of Spheres

Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres π＊S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres πp^nq＋2pq＋q-3S which isof order p and is represented by kohn ∈ ExtA^3,P^nq＋2pq＋q（Zp,Zp） in the Adams spectral sequence, wherep 〉 5 is an odd prime, n ≥3 and q = 2（p-1）. In the course of the proof, a new family of homotopy elements in πp^nq＋（p＋1）q-1V（1） which is represented by β＊i＇＊i＊（hn） ∈ ExtA^2,pnq＋（p＋1）q＋1 （H^＊V（1）, Zp） in the Adams sequence is detected.     

使用对称性计算一类有限元基函数的二重积分

本文使用对称性计算一类有限元基函数在非规则区域的二重积分.通过两个算例验证对称性技巧在重积分计算上所带来的极大便利性.本文的内容进一步说明了使用对称性进行重积分计算在其他学科的应用价值.     

Nontrivial Critical Networks. Singularities of Lagrangians and a Criterion for Criticality

We single out the class of so-called quasiregular Lagrangians, which have singularities on the zero section of the cotangent bundle to the manifold on which extremal networks are considered. A criterion for a network to be extremal is proved for such Lagrangians: the Euler--Lagrange equations must be satisfied on each edge, and some matching conditions must be valid at the vertices.     

中心对称三次系统的一类双纽线有界周期环域的poincare分支

证明中心对称三次系统的一类双纽线有界周期环域的 poincare分支至少可以出现作对称 (3,3)分布的六个极限环 .     

超广义Burgers方程族的非线性可积耦合及其Bargmann对称约束

基于李超代数,构造了超广义Burgers方程族的非线性可积耦合,并且利用超级恒等式得到了它的超Hamilton结构.此外,该文计算出超广义Burgers方程族的非线性可积耦合的Bargmann对称约束.     

Quasiconvex functions and null Lagrangians in the stability problems of classes of mappings

We obtain stability theorems for classes of solutions to the differential equations constructed by means of quasiconvex functions and null Lagrangians.     

正交双复数空间变换与流体问题的边界元

在确立特殊的正交双复数空间及正交双复变函数的概念上,建立其相应的运算规则及空间保角变换的概念,并用这些概念,探索了它在流体边界元中的应用前景.研究表明,所建立的概念和特殊表达算符,能够有效地将平面复数的概念扩展至三维空间,并可构成数理应用领域的一个有效工具.     

Symmetries and Their Lie Algebra of a Variable Coefficient Korteweg-de Vries Hierarchy

Isospectral and non-isospectral hierarchies related to a variable coefficient Painlev′e integrable Korteweg-de Vries(Kd V for short) equation are derived. The hierarchies share a formal recursion operator which is not a rigorous recursion operator and contains t explicitly. By the hereditary strong symmetry property of the formal recursion operator, the authors construct two sets of symmetries and their Lie algebra for the isospectral variable coefficient Korteweg-de Vries(vc Kd V for short) hierarchy.     

L 2-Topology and Lagrangians in the Space of Connections Over a Riemann Surface

We examine the L ²-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl lemma of harmonic analysis, and deduce local pathwise connectedness of the gauge orbits. Based on a quantitative version of the connectivity, we generalize compactness results for anti-self-dual instantons with Lagrangian boundary conditions to general gauge-invariant Lagrangian submanifolds. This provides the foundation for the construction of instanton Floer homology for pairs of a 3-manifold with boundary and a Lagrangian in the configuration space over the boundary.     

图的倍图与补倍图

计算机科学数据库的关系中遇到了可归为倍图或补倍图的参数和哈密顿圈的问题．对简单图C,如果V（D（G））：V（G）∪V（G′）E（D（G））=E（C）∪E（C″）U{vivj′｜vi∈V（G）,Vj′∈V（G′）且vivj∈E（G））那么,称D（C）是C的倍图,如果V（D（G））=V（C）∪V（G′）,E（D（C））：E（C）∪E（G′）∪{vivj′}vi∈V（G）,vj′∈V（G’）and vivj∈（G））,称D（C）是G的补倍图,这里G′是G的拷贝．本文研究了D（G）和D的色数,边色数,欧拉性,哈密顿性和提出了D（G） 的边色数是D（G）的最大度等公开问题．     

Geometrical Interpretation of General Complex Algebra and its Application in Relativistic Mechanics

In a generalized formulation of the relativistic dynamics with internal conformation an important role is played by a quadratic polynomial, the coefficients and eigenvalues of which are generated by outer and inner momenta of the relativistic particle. This polynomial induces the general complex algebra, GC. In this paper we explore the geometrical and physical aspects of the evolution generated by the algebraic operations of the GC-algebra. It is shown that the geometrical image of the GC-number is given by a straight line passing through two given points in an euclidean plane. In this representation the straight line is characterized by a norm and an argument. The motions of the straight line are described by hyperbolic trigonometry which brings a correspondence between the Euclidean geometry and the hyperbolic one. It is proved that the evolution equation governed by the generator of the GC-algebra describes the energy conservation law of the relativistic particle. This evolution is depicted on the Euclidean plane as a rotational motion of the straight line, tangent to the circle with radius equal to the mass of the particle. In this way we come to new representation for the momenta in relativistic dynamics.     

Hilbert空间中有效序列的刻画与扰动

本文研究了Hilbert空间中的有效序列,通过引入序列的伴随序列,给出了有效序列的一个等价刻画,接着讨论了保持向量序列有效性的线性算子的性质,证明了一个线性算子将任意有效序列映射为有效序列当且仅当它是酉算子;最后,给出了正规正交基中添加一个单位向量后所得到的序列是有效序列的一个充要条件.     

BRST-Invariant Algebra of Constraints in Terms of Commutators and Quantum Antibrackets

We establish the general structure of the BRST-invariant algebra of constraints in its commutator and antibracket forms via the formulation of algebra-generating equations in a supplementally extended phase space. New ghost-type variables behave as fields and antifields with respect to quantum antibrackets. The explicit form of the BRST-invariant gauge algebra is given in detail for rank-one theories with a Weyl- and a Wick-ordered ghost sector. We construct a fixed-gauge unitarizing Hamiltonian and show that the formalism is physically equivalent to the standard BRST–BFV approach.     

<Emphasis Type="Italic">β</Emphasis>-character Algebra and a Commuting Pair in Hopf Algebras

Let K be a field. Let H be a finite-dimensional semisimple and cosemisimple K-Hopf algebra. In this paper, we introduce a notion of β-character algebra C _β (H) for each group-like element β in H ^∗. We prove that Radford’s action of the Drinfel’d double D(H) on H _β (see Radford, J. Algebra, 270:670–695, 2003) and the right hit action of the β-character algebra C _β (H) on H _β form a commuting pair. This generalizes an earlier result of Zhu (Proc. Amer. Math. Soc., 125(10):2847–2851, 1997). A K-basis of C _β (H) is given when H is split semisimple. Finally, as an example, we explicitly construct all the simple modules for the Drinfel’d double of the unique 8-dimensional non-commutative and non-cocommutative semisimple Hopf algebra. Presented by S. Montgomery.