6元素集合上T0拓扑总数的计算

文中证明了有限预序集与有限偏序集的一些性质,并基于有限集上的拓扑和其上预序的一一对应关系,利用这些性质通过对极小元和极大元个数进行分类讨论,以一种有别于计算机算法而又容易理解的计算方法得出6元素集合上的T₀拓扑总数为130023.     

考虑偏差值的中立型区间交叉效率研究

在不确定性环境下,当决策单元(DMU)的投入产出数据为区间数形式时,为解决决策单元之间既不是合作也不是竞争关系时的交叉评价问题,本文提出一种中立型区间交叉效率模型。从所有被评价者的角度出发解决评价权重的选取问题,以决策单元投入得分的平均偏差与产出得分的平均偏差之和最小化为目标,建立决策单元在最佳和最差两种生产状态下的中立型区间交叉效率模型。在本文提出的中立型模型视角下,DMU的投入得分平均偏差和产出得分平均偏差之和达到最小。算例结果表明该中立型区间交叉效率模型的有效性,解决了不确定性环境下的交叉评价问题,保证评价的客观公正,更加符合现实。     

Low Mach number limit of inviscid Hookean elastodynamics

The low Mach number limit of inviscid Hookean elastodynamic equations is rigorously proved in bounded domain, whole space and periodic domain, respectively. The uniform existence of smooth solutions and convergence results as the Mach number tends to zero are obtained in three different domains.     

The Newton modified barrier method for QP problems

The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues, to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a hot start. Starting at such a hot start, at mostO(In In ^-1) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, >0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that onlyO(In ^-1)O(In In ^-1) Newton steps are required to reach an -approximation to the solution from any hot start. In order to reach the hot start, one has to perform Newton steps, wherem characterizes the size of the problem andC>0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.Partially supported by NASA Grant NAG3-1397 and National Science Foundation Grant DMS-9403218.     

Geometric indices and the Alexander polynomial of a knot

It is well-known that any Laurent polynomial satisfying and is the Alexander polynomial of a knot in . We show that can be realized by a knot which has the following properties simultaneously: (i) tunnel number 1; (ii) bridge index 3; and (iii) unknotting number 1.

     

Realization of fixed point sets

Letf:XX be a selfmap of a compact connected polyhedron, andA a nonempty closed subset ofX. In this paper, we shall deal with the question whether or not there is a mapg:XX homotopic tof such that the fixed point set Fixg ofg equalsA. We introduce a necessary condition for the existence of such a mapg. It is shown that this condition is easy to check, and hence some sufficient conditions are obtained.Partially supported by the Natural Science Foundation of Liaoning University.     

The theoretical basis of column chromatography in multicomponent separations. Part 2: Peak capacity. Column systems with good selectivity

The selectivity of a column system, S, defined by equation 28, includes a variation coefficient (β) of plate number with capacity ratio which has a large influence on the peak capacity, as shown in Fig. 18. Some typical chromatograms are given. In order to predict S for a column system from Kovat's Index, equations 40 and 41 are given to calculate the constant of the carbon number rule for squalane at different temperatures or for different stationary phases. The specific retention value of heptane on squalane at different temperatures can be calculated from equation 42. The nonpolarity index, defined by equation 43, was used to calculate the retention value of heptane on various stationary phases. In liquid chromatography, the order of elution may be reversed by changing the composition of the eluent on the same chemically bonded silica (manufactured in China). The linear relations between the log retention values of different kinds of solutes or of a single solute on the silicas of different surface areas when using the same eluent are given.     

Structure elucidation of triglycerides by chromatographic techniques. Part 1: Capillary GC of ozonized triglycerides

Structure determination of triglycerides according to their number of unsaturated fatty acids (N.U.F.A.) is more easily achieved after ozonolysis of their mixtures (fats and oils) than by capillary gas chromatographic analysis of the mixture as such. Analyses performed before and after ozonolysis of cocoa buttel, hazelnut oil, and a mixture of both fats illustrate the potential of this approach to the structure elucidation of triglycerides by chromatographic techniques.     

Dissolution of ionisable groups and lignocellulosic components during low-temperature kraft pulping of pinus radiata

A wide range of kraft pulps from radiata pine produced by low-temperature kraft pulping in flow-through reactors was assessed for carboxyl and hexenuronic acid (HexA) contents using the conductometric titrations and UV spectrophotometer, respectively. The Kappa number of pulps varied from 20.8 to 84.7 when using a cooking liquor of 1 M effective alkali as Na₂O, 25% sulfidity, and cooking time of 100–250 min. The experimental results showed that the carboxyl groups (including HexA) and HexA groups dissolved and their residual values in the pulp samples correlate linearly with Kappa number and pulp yield. The dissolving rate of all carboxyl groups is much faster than the loss of HexA. The HexA/lignin ratio decreased non-linearly with Kappa number.     

Scaling properties of ℤ k- 1 actions on the circle

We derive universal scaling properties for ^k–1 actions on the circle whose generators have rotation numbers algebraic of degreek. As fork=2 these properties can be explained for arbitraryk in terms of a renormalization group transformation. It has at least one trivial fixed point corresponding to an action whose generators are pure rotations. The spectrum of the linearized transformation in this fixed point is analyzed completely. The fixed point is hyperbolic with a (k–1)-dimensional unstable manifold. In the casek=2 the known results are therefore recovered.