Maintaining the positive definiteness of the matrices in reduced secant methods for equality constrained optimization

We propose an algorithm for minimizing a functionf on ⁿ in the presence ofm equality constraintsc that locally is a reduced secant method. The local method is globalized using a nondifferentiable augmented Lagrangian whose decrease is obtained by both a longitudinal search that decreases mainlyf and a transversal search that decreases mainly c. Our main objective is to show that the longitudinal path can be designed to maintain the positive definiteness of the reduced matrices by means of the positivity of _k ^T _k , where _k is the change in the reduced gradient and _k is the reduced longitudinal displacement.Work supported by the FNRS (Fonds National de la Recherche Scientifique) of Belgium.     

A twist on chiral Potts

We show that the chiral Potts model may be formulated so that the rapidity lines carry a second integer variable-an increment or twist in each bond crossing it. This modification does not affect those properties of the chiral Potts model which lead to integrability, since it is equivalent to one of the automorphisms allowed for in the theory. In particular, transfer matrices still form commuting families and still satisfy hierarchies of functional equations. Surprisingly, the superintegrable case with twists retains the special algebraic properties which lead to its Ising-like spectra. The formalism should be useful for considering systems with twisted boundary conditions or embedded interfaces.     

Exact molecular mass determination of various forms of native and de-N-glycosylated human plasma-derived antithrombin by means of electrospray ionization ion trap mass spectrometry

Human plasma-derived antithrombin was characterized in both the native and de-N-glycosylated forms (without separation of isoforms) by means of electrospray ionization ion trap mass spectrometry (ESI-ITMS). In order to determine the limits of the instrument set-up, the molecular mass precision and accuracy of the ESI-ITMS analysis was evaluated with the standard protein enolase and some instrumental data acquisition parameters were optimized. Mass precision was determined as a function of the number of averaged mass spectra (= scans) and data acquisition time. For this study, 20 and 50 scans were averaged and the data acquisition time was chosen to be between 0.5 and 5 min. It turned out that data acquisition times longer than approximately 2 min show no significant differences of the standard deviation of the determined molecular mass. Furthermore, the ion trap scan rate was varied at constant acquisition time of 2 min and the number of averaged scans was set to 20. At the scan rate of 13,000 u s(-1) a mass precision of +/-1.8 Da and a mass accuracy of +0.026% were determined. On reducing the scan rate to 5500 u s(-1), better agreement with the theoretical molecular mass was obtained, showing a mass accuracy of +0.012% but with a decrease in the mass precision to +/-3.0 Da. Using the optimized scan rate of 13,000 u s(-1) and a data acquisition time of 2 min, the exact molecular mass was determined of the three forms of antithrombin, namely the alpha-form, the beta-form and the natural mixture (present in human plasma) containing both forms. The protonated molecular masses were found to be 57,854 and 55,664 Da for the affinity chromatography-isolated alpha-and beta-form, respectively. The mass difference of 2190 Da is attributed to the known difference in carbohydrate content at one specific site. The protonated molecular mass of the dominating species of the natural mixture in human plasma was shown to be 57,850 Da, corresponding to the alpha-form, the major component in native plasma. In this mixture the beta-form was also detected, exhibiting a protonated molecular mass of 55,655 Da, but showing a much lower abundance, as expected. To obtain a complete release of the N-glycan residues by means of PNGase F, a denaturation, reduction and alkylation step of the glycoproteins was performed before the enzymatic reaction. After enzymatic removal of all N-glycans, the protonated molecular masses obtained were 49,399, 49,380 and 49,391 Da for the alpha-form, the beta-form and the unseparated natural mixture, respectively. These values are in good agreement (+0.026% for the alpha-form, -0.012% for the beta-form and +0.010% for the unseparated mixture) with the calculated molecular mass based on the SwissProt data. The determined molecular masses after reduction/alkylation and de-N-glycosylation of the alpha-and beta-forms are almost equal, indicating that no major differences exist between the three preparations on the amino acid level.     

Construction of positive exact (2+1)-dimensional shock wave solutions for two discrete Boltzmann models

It is proved that (2+1)-dimensional (spacex, y; timet) positive exact shock wave solutions of two discrete Boltzmann models exist. For each densityN _i, these solutions are linear combinations of three similarity shock waves,N _i =n _0i + _j n _ji /[1+d _j exp( _j y+y _j x+ _j t)],j=1,2,3. Two models with four independent densities are investigated: the square discrete-velocity Boltzmann model and the model with eight velocities oriented toward the eight corners of a cube.The positivity problem for the densities is nontrivial. Two classes of solutions are considered for which the two first similarity shock wave components depend on only one spatial dimension, _j=const· _j ,j=1,2. For the positivity, if ₁₂>0, it is sufficient to prove that the 16 asymptotic shock limitsn _0i ,n _0i +n _3i , _j=0 ² n _ji , _j=0 ³ n _ji are positive. The density solutions are built up with five arbitrary parameters and we prove that there exist subdomains of the arbitrary parameter space in which the 16 shock limits are positive. We study numerically two explicit shock wave solutions. We are interested in the movement of the shock front when the time is growing and in the possible appearance of bumps. In the space, at intermediate times, these bumps represent populations of particles which are larger than at initial time or at equilibrium time.     

An exact penalty function algorithm for optimal control problems with control and terminal equality constraints,part 2

In Part 1 of this paper, implementable and conceptual versions of an algorithm for optimal control problems with control constraints and terminal equality constraints were presented. It was shown that anyL accumulation points of control sequences generated by the algorithms satisfy necessary conditions of optimality. Since such accumulation points need not exist, it is shown in this paper that control sequences generated by the algorithms always have accumulation points in the sense of control measure, and these accumulation points satisfy optimality conditions for the corresponding relaxed control problem.This work was supported by the United Kingdom Science Research Council, by the US Army Research Office, Contract No. DAA-29-73-C-0025, and by the National Science Foundation, Grant No. ENG-73-08214-A01.     

Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations

In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665‐669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth‐order Lax pairs instead of the fifth‐order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto‐Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd.     

一维搜索的收敛性比较

精确一维搜索与不精确一维搜索是一维搜索的两种主要形式,为了进一步探讨有关其收敛性的内容,本文将分别运用精确一维搜索中的0．618黄金分割法与不精确一维搜索中的Armijo—Goldstdn搜索法进行比较．     

一个Boussinesq系统的精确解

利用三角函数方法、双曲正切函数方法、双曲正割函数方法和Jacobi椭圆函数方法获得Boussinesq系统的精确行波解．     

高值与低值空间聚类的精确检测

提出了两个调整数据的新方法来弥补通常空间聚类检测的不足,从而能够对高值或低值聚类进行精确检测.两个方法被分别在模拟的空间类型中应用于Moran'I检测.结果表明,当不存在空间聚类时,该方法对原检测毫无影响;当有聚类趋势时,经该方法调整数据后,可以显著地检测出高值或低值聚类的存在.