The BKK root count in

The root count developed by Bernshtein, Kushnirenko and Khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus . In this paper, we modify this bound slightly so that it counts the number of isolated zeros in . Our bound is, apparently, significantly sharper than the recent root counts found by Rojas and in many cases easier to compute. As a consequence of our result, the Huber-Sturmfels homotopy for finding all the isolated zeros of a polynomial system in can be slightly modified to obtain all the isolated zeros in .

     

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.     

Synthesis and performance of HPTLC plates modified with cyanopropyl- and octadecyl-trichlorosilane

Chemical modification of commercial high performance thin-layer chromatography plates with various mixtures of cyano-propyltrichlorosilane and octadecyltrichlorosilane is described. Surface coverage by different treatments is demonstrated as well as the variations in chromatographic performance. With regard to the development in aqueous media the utility of CN/ ODS plates compared to ODS plates is also shown.     

Characterization of OV-1/FFAP-mixture coated glass capillary columns used for the separation of free fatty acids

As an alternative to acid pretreated UCON and FFAP capillaries for the analysis of wide-boiling free fatty acid mixtures, OV-1 : FFAP phase mixtures were used on high-temperature silylated inert glass capillary columns. The HETP-carrier gas velocity curves, peak asymmetry factors, coating efficiency, gas phase and stationary phase contributions to the mass transfer resistance were determined for various OV-1 : FFAP ratios. Mixed-phase capillaries showed optimum performance at a 2 : 1 OV-1 : FFAP ratio. The thermal and long-term stability of OV-1-stabilized FFAP columns surpassed those of the UCON and pure FFAP reference column used.     

Ionization constants for deuterium oxide in deuterium oxide-organic mixtures

A potentiometric method using a glass electrode has been applied to determination of ionization constants for deuterium oxide (D₂O) in binary mixtures of D₂O with dioxane, tetrahydrofuran, acetone, dimethylsulfoxide, CH₃CH₂OD, and CH₃OD at 25°C. The results are compared with values of ionization constants for H₂O obtained previously in the corresponding H₂O-organic mixtures, and the isotope effect is shown to be small. Further calculations with the D₂O results show that the first five solvents mentioned above are neither appreciably acidic nor basic in D₂O solution, but that CH₃OD shows slightly acidic behavior (pK_a=16.0±0.3).     

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.