一个Bargmann系统与耦合Burgers族解的对合表示

本文给出耦合Burgers族的换位表示，并通过对耦合Burgers族Lax系统的非线性化得到一个Bargmann系统，证明该系统为Liouville完全可积的，还给出耦合Burgers族解的对合表示.     

Lax表示的变形与Hamilton方程族的Lax表示

本文首先给出了构造演化方程族的Ｌａｘ表示的马文秀方法的一种变形，后对这一方法作了改进，使之适用于Ｈａｍｉｌｔｏｎ形式的方程族．作为应用，得到了具有非等谱Ｌａｘ表示的杨方程族．     

Differential-algebraic approach to constructing representations of commuting differentiations in functional spaces and its application to nonlinear integrable dynamical systems

There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only one conserved quantity is analyzed in detail, the corresponding Lax type representations of differentiations are constructed for an infinite hierarchy of nonlinear dynamical systems of the Burgers and Korteweg–de Vries type. A related infinite bi-Hamiltonian hierarchy of Lax type dynamical systems is constructed.     

A new derivation of Painlevé hierarchies

We give a new derivation of two Painlevé hierarchies. This is done by extending the accelerating-wave reductions of the Korteweg-de Vries and dispersive water wave equations to their respective hierarchies. We also consider the extension of this reduction of Burgers equation to the Burgers hierarchy.     

A dynamic-programming algorithm for hierarchical discretization of continuous attributes

Discretization techniques can be used to reduce the number of values for a given continuous attribute, and a concept hierarchy can be used to define a discretization of a given continuous attribute. Traditional methods of building a concept hierarchy from a continuous attribute are usually based on the level-wise approach. Unfortunately, this approach suffers from three weaknesses: (1) it only seeks a local optimal solution instead of a global optimal, (2) it is usually subject to the constraint that each interval can only be partitioned into a fixed number of subintervals, and (3) the constructed tree may be unbalanced. In view of these weaknesses, this paper develops a new algorithm based on dynamic-programming strategy for constructing concept hierarchies from continuous attributes. The constructed trees have three merits: (1) they are global optimal trees, (2) each interval is partitioned into the most appropriate number of subintervals, and (3) the trees are balanced. Finally, we carry out an experimental study using real data to show its efficiency and effectiveness.     

Painlevé Hierarchies and the Painlevé Test

In a series of recent papers, we derived several new hierarchies of higher-order analogues of the six Painlevé equations. Here we consider one particular example of such a hierarchy, namely, a recently derived fourth Painlevé hierarchy. We use this hierarchy to illustrate how knowing the Hamiltonian structures and Miura maps can allow finding first integrals of the ordinary differential equations derived. We also consider the implications of the second member of this hierarchy for the Painlevé test. In particular, we find that the Ablowitz–Ramani–Segur algorithm cannot be applied to this equation. This represents a significant failing in what is now a standard test of singularity structure. We present a solution of this problem.     

Detecting evolving patterns of self‐organizing networks by flow hierarchy measurement

Hierarchies occur widely in evolving self‐organizing ecological, biological, technological, and social networks, but detecting and comparing hierarchies is difficult. Here we present a metric and technique to quantitatively assess the extent to which self‐organizing directed networks exhibit a flow hierarchy. Flow hierarchy is a commonly observed but theoretically overlooked form of hierarchy in networks. We show that the ecological, neurobiological, economic, and information processing networks are generally more hierarchical than their comparable random networks. We further discovered that hierarchy degree has increased over the course of the evolution of Linux kernels. Taken together, our results suggest that hierarchy is a central organizing feature of real‐world evolving networks, and the measurement of hierarchy opens the way to understand the structural regimes and evolutionary patterns of self‐organizing networks. Our measurement technique makes it possible to objectively compare hierarchies of different networks and of different evolutionary stages of a single network, and compare evolving patterns of different networks. It can be applied to various complex systems, which can be represented as directed networks. © 2011 Wiley Periodicals, Inc. Complexity, 2011     

具有等级约束的三台机排序问题的可中断在线算法

研究具有等级约束的三台机在线排序问题．机器和工件的等级数均为1或2,工件只能在等级数不超过自身等级的机器上加工,且加工允许中断,目标是极小化最大工件完工时间．如果有两台机器等级为1,给出竞争比为3／2的在线算法,并证明算法是最好可能的;如果只有一台等级为1的机器,也给出竞争比为3／2的在线算法．     

Symmetries for the Ablowitz–Ladik Hierarchy: Part I. Four‐Potential Case

In the paper, we first investigate symmetries of isospectral and non‐isospectral four‐potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of u_n,t= L^mH⁽⁰⁾ , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero‐curvature representations of the isospectral and non‐isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non‐isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac‐Moody‐Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four‐potential and two‐potential Ablowitz–Ladik hierarchies. The even order members in the four‐potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two‐potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L² .     

孤子族的生成及换位表示的一般结构

本文通过对谱问题ψｘ＝Ｕ（ｕ，λ）ψ的直接研究，利用谱梯度提供一条获得孤子方程族的途径，进一步，我们给出了孤子方程换位表示的一般结构，同时我们还将看到同一个谱问题可产生两族不同的孤子发展方程。     

BKP 与CKP 可积系列的递归算子

借助于新引进的算子B, 本文给出了BKP 与CKP 可积系列约束条件在其Lax 算子L中的动力学变量上的具体体现, 即奇数阶动力学变量u_2k+1 能被偶数阶动力学变量u_2k 显式表达. 同时本文给出了BKP 与CKP 可积系列的流方程以及(2n + 1)- 约化下递归算子的统一公式, 揭示了BKP 可积系列和CKP 可积系列的重要区别. 作为例子, 本文给出了BKP 与CKP 可积系列在3- 约化下的递归算子的显式表示, 并验证了u₂ 的t₁ 流通过递归算子的确可以产生u₂ 的t₇ 流, 该流方程与3- 约化下产生的对应流方程是一致的.     

Conservation laws for a super G-J hierarchy with self-consistent sources

Based on a well known super Lie algebra, a super integrable system is presented. Then, the super G-J hierarchy with self-consistent sources are obtained. Furthermore, we establish the infinitely many conservation laws for the integrable super G-J hierarchy. The methods derived by us can be generalized to other nonlinear equations hierarchies with self-consistent sources.     

Collision detection for deforming necklaces

In this paper, we propose to study deformable necklaces—flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macro-molecules, muscles, ropes, and other linear objects in the physical world. We exploit this linearity to develop geometric structures associated with necklaces that are useful for collision detection in physical simulations. We show how these structures can be implemented efficiently and maintained under necklace deformation. In particular, we study a bounding volume hierarchy based on spheres which can be used for collision and self-collision detection of deforming and moving necklaces. As our theoretical and experimental results show, such a hierarchy is easy to compute and, more importantly, is also easy to maintain when the necklace deforms. Using this hierarchy, we achieve a collision detection upper bound of O(nlogn) in two dimensions and O(n^2−2/d) in d-dimensions, d3. To our knowledge, this is the first subquadratic bound proved for a collision detection algorithm using predefined hierarchies. In addition, we show that the power diagram, with the help of some additional mechanisms, can be used to detect self-collisions of a necklace in a way that is complementary to the sphere hierarchy.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

两台带服务等级的可拒绝同型机排序问题的在线算法

两台同型机M_1,M_2, 加工速度一致, 但拥有不同的加工能力,用其服务等级表示, M_1的服务等级为1, M_2的服务等级为2. 工件j按列表在线到达,每个工件带有三个参数: 长度t_j,等级g_j=1或2, 罚值p_j. 当j到达时, 可以被拒绝, 但要付出相应的罚值p_j, 也可以被接受并分配给服务等级不超过该工件等级的机器加工,事实上等级为1的工件只能分给M_1加工, 等级为2的工件可以分给M_1或M_2加工, 加工不允许中断. 目标为极小化加工工件集的最晚完工时间(makespan)和拒绝工件集的总罚值之和. 对于该问题给出了一个在线算法, 其竞争比为11/6, 以及问题一个下界5/3.     

Weight distributions and weight hierarchies of two classes of binary linear codes

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t-resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes.     

Abelian Chern-Simons vortices and holomorphic Burgers hierarchy

We consider the Abelian Chern-Simons gauge field theory in 2+1 dimensions and its relation to the holomorphic Burgers hierarchy. We show that the relation between the complex potential and the complex gauge field as in incompressible and irrotational hydrodynamics has the meaning of the analytic Cole-Hopf transformation, linearizing the Burgers hierarchy and transforming it into the holomorphic Schrödinger hierarchy. The motion of planar vortices in Chern-Simons theory, which appear as pole singularities of the gauge field, then corresponds to the motion of zeros of the hierarchy. We use boost transformations of the complex Galilei group of the hierarchy to construct a rich set of exact solutions describing the integrable dynamics of planar vortices and vortex lattices in terms of generalized Kampe de Feriet and Hermite polynomials. We apply the results to the holomorphic reduction of the Ishimori model and the corresponding hierarchy, describing the dynamics of magnetic vortices and the corresponding lattices in terms of complexified Calogero-Moser models. We find corrections (in terms of Airy functions) to the two-vortex dynamics from the Moyal space-time noncommutativity.     

一个新的非等谱可积族和一些相关对称

利用李群$M_nC$的一个子群我们引入一个线性非等谱问题,该问题的相容性条件可导出演化方程的一个非等谱可积族,该可积族可约化成一个广义非等谱可积族.这个广义非等谱可积族可进一步约化成在物理学中具有重要应用的标准非线性薛定谔方程和KdV方程.基于此,我们讨论在广义非等谱可积族等谱条件下的一个广义AKNS族$u_t=K_m(u)$的$K$对称和$\tau$对称.此外,我们还考虑非等谱AKNS族$u_t=\tau_{N+1}^l$的$K$对称和$\tau$对称.最后,我们得到这两个可积族的对称李代数,并给出这些对称和李代数的一些应用,即生成了一些变换李群和约化方程的无穷小算子.     

Positive and negative hierarchies of nonlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem

Positive and negative hierarchies of nonlinear integrable lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct three integrable coupling systems of the positive hierarchy through enlarging Lax pair method.     

Hierarchies in φ‐spaces and applications

We establish some results on the Borel and difference hierarchies in φ‐spaces. Such spaces are the topological counterpart of the algebraic directed‐complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non‐collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω‐ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)