Hopf代数的双交叉积

本文定义并详细讨论了交叉余积，考虑交叉积与交叉余积合起未成为双代数的问题，讨论了由内作用，内余作用构造的双交叉积．我们还用正合序列刻划了双交叉积．     

一列非同构的离散交叉积

本文研究了R上的一类离散交叉积的因子结构及其在小波分析中的应用问题.利用群测度构造离散交叉积的方法,我们构造了R上的一列离散交叉积,并且利用比率集的方法证明了上述交叉积是互相不同构的超有限III_γ型因子.上述结论推广了已有的一些结果.     

2- 上循环交叉积

利用 von Neumann 代数 M 和带有正规的2- 上循环 μ 的离散群 G, 定义了2- 上循环交叉积, 推广了经典的离散交叉积, 并证明了2- 上循环交叉积具有结合律.     

弱群余代数的交叉积

该文引入弱群交叉积的概念,并给出弱群交叉积代数和通常的张量积余代数构成弱半Hopf群余代数的充要条件,接着证明了弱群交叉积上的对偶定理,推广了沈和王~([7-8])的主要结果.     

六阶图Q与星图S_n的积图交叉数

目前关于积图的交叉数的研究已经推广到六阶图与星图的积图.研究得到了一个特殊六阶图Q与n个孤立点nK_1的联图交叉数,然后通过收缩的方法,得到了Q与星图S_n的积图交叉数.     

完全二部图K3,3与星Sn的积图的交叉数

确定图的交叉数是NP-完全问题．目前有关完全二部图与星图的积图的交叉数结果并不多．引入了一些新的收缩技巧,建立了积图K3,3□Sn与完全三部图K3,3□Sn之间的交叉数关系．从而,为进一步完全确定积图K3,3□Sn的交叉数提供了一条新途径．     

完全二部图K_{3,3}与星S_n的积图的交叉数

确定图的交叉数是NP-完全问题. 目前有关完全二部图与星图的积图的交叉数结果并不多. 引入了一些新的收缩技巧, 建立了积图K_{3,3}\square S_n与完全三部图K_{3,3,n}之间的交叉数关系. 从而, 为进一步完全确定积图K_{3,3}\square S_n的交叉数提供了一条新途径.     

斜Pairing,余循环变形与双交叉积

本文讨论了双代数的余循环变形和双交叉积．当双交叉积ＸＡ是ＸＡ的２ 余循环变形时，给出了ＸＡ为ＸＡ的某个斜Ｐａｉｒｉｎｇ变形ＸτＡ的一个充分条件．然后，利用函子间的自然同构，给出双交叉积ＸＡ为斜Ｐａｉｒｉｎｇ变形的一个充要条件．     

泊松图P(4,1)与路Pn的笛卡尔积的交叉数

泊松图$P(m, 1)$与路$P_n$的笛卡尔积的交叉数是一个NP-完全问题, Y.H. Peng和Y.C.Yiew 证明了$P(3,1)$与$P_n$的笛卡尔积的交叉数为$4n$, 我们证明明了$P(4,1)$与$P_n$的笛卡尔积的交叉数为$8n$.     

弱Hopf代数上的ω-交叉积

在弱Hopf代数上,定义了ω-交叉积概念,并给出了ω-交叉积是弱Hopf代数的充要条件,推广了弱Hopf代数的相应结论.     

A Barrier Function Method for the Nonconvex Quadratic Programming Problem with Box Constraints

In this paper a barrier function method is proposed for approximating a solution of the nonconvex quadratic programming problem with box constraints. The method attempts to produce a solution of good quality by following a path as the barrier parameter decreases from a sufficiently large positive number. For a given value of the barrier parameter, the method searches for a minimum point of the barrier function in a descent direction, which has a desired property that the box constraints are always satisfied automatically if the step length is a number between zero and one. When all the diagonal entries of the objective function are negative, the method converges to at least a local minimum point of the problem if it yields a local minimum point of the barrier function for a sequence of decreasing values of the barrier parameter with zero limit. Numerical results show that the method always generates a global or near global minimum point as the barrier parameter decreases at a sufficiently slow pace.     

Staircase Parameterization in Dynamical Selection

This paper deals with a variant of a dynamical selection scheme introduced by Attouch and Cominetti for ill-posed convex minimization which combines approximation with the steepest descent method by mean of a suitable parameterization of the approximation parameter as a function of the time. This variant applies to a general inclusion with a maximal monotone operator by mean of a staircase parameterization. A discrete analogue is also considered. Applications to selecting a particular zero of a maximal monotone operator or a particular fixed point of a nonexpansive mapping via regularization techniques are presented. Finally, the alternative use of well-posedness by perturbations is discussed.     

Partially isometric dilations of noncommuting -tuples of operators

Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.

     

Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure

In this paper we prove the existence of a solution of a coupled system involving a two phase incompressible flow in the ground and the mechanical deformation of the porous medium where the porosity is a function of the global pressure. The model is strongly coupled and involves a nonlinear degenerate parabolic equation. In order to show the existence of a weak solution, we consider a sequence of related uniformly parabolic problems and apply the Schauder fixed point theorem to show that they possess a classical solution. We then prove the relative compactness of sequences of solutions by means of the Fréchet-Kolmogorov theorem; this yields the convergence of a subsequence to a weak solution of the parabolic system.     

Spectrum of a homogeneous graph

We describe the spectrum of the Laplacian for a homogeneous graph acted on by a discrete group. This follows from a more general result which describes the spectrum of a convolution operator on a homogeneous space of a locally compact group. We also prove a version of Harnack inequality for a Schrödinger operator on an invariant homogeneous graph.     

Construction of a CPA contraction metric for periodic orbits using semidefinite optimization

A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. While the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit, the explicit construction of the metric is a difficult problem.In this paper, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine (CPA) function, which is affine on each simplex of a triangulation of the phase space. The contraction conditions are formulated as conditions on the values at the vertices.The paper states a semidefinite optimization problem. We prove on the one hand that a feasible solution of the optimization problem determines a CPA contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough. An objective function can be used to obtain a bound on the largest Floquet exponent of the periodic orbit.     

Kripke submodels and universal sentences

We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with (homo)morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms (including ? and ⊥) using ∧, ∨, ?, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model‐consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory Δ if and only if every rooted Kripke model of Δ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.