A numerically stable least squares solution to the quadratic programming problem

The strictly convex quadratic programming problem is transformed to the least distance problem — finding the solution of minimum norm to the system of linear inequalities. This problem is equivalent to the linear least squares problem on the positive orthant. It is solved using orthogonal transformations, which are memorized as products. Like in the revised simplex method, an auxiliary matrix is used for computations. Compared to the modified-simplex type methods, the presented dual algorithm QPLS requires less storage and solves ill-conditioned problems more precisely. The algorithm is illustrated by some difficult problems.      

Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes

We consider the problem of estimating the covariance of two diffusion-type processes when they are observed only at discrete times in a nonsynchronous manner. In our previous work in 2003, we proposed a new estimator which is free of any ‘synchronization’ processing of the original data and showed that it is consistent for the true covariance of the processes as the observation interval shrinks to zero; Hayashi and Yoshida (Bernoulli, 11, 359–379, 2005). This paper is its sequel. Specifically, it establishes asymptotic normality of the estimator in a general nonsynchronous sampling scheme.     

Quadratic characters in groups of odd order

We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes.     

Well-posedness of the spatially homogeneous Landau equation for soft potentials

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderately soft potentials, we assume additionally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka [H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Geb. 46 (1) (1978-1979) 67-105].     

On the reactivity of isoindolo[2,1-a]quinazoline-5-ones

Base- and acid-catalyzed nucleophilic addition of 11H-isoindolo[2,1-a]quinazoline-5-one to aromatic aldehydes and maleimides was investigated. The aldol adducts and condensation products were obtained stereoselectively. Main diastereomers of the Michael adducts were isolated in 74-89% yield, and converted by N-methylation to new stable α-substituted isoindole derivatives, for which 6-methylisoindolo[2,1-a]]quinazoline-5-one stands as the unsubstituted reference. The stability of the latter was monitored in moist aerated CDCl₃ solution, and one of the oxidative hydrolysis product was characterized by X-ray diffraction analysis as the corresponding N-arylphthalimide. The reactivity of the unsubstituted 6-methylisoindolo[2,1-a]]quinazoline-5-one was also investigated with acetylenic Michael acceptors. Fully conjugated isoindole derivatives possessing an original pull-push-pull structure were obtained. The conformations and molecular orbitals of the dibenzoylacetylene adduct were studied at the DFT level of theory. Its static quadratic hyperpolarizabilty β₀ was also calculated at the ZINDO level.     

New reformulations for probabilistically constrained quadratic programs

The mixed integer quadratic programming (MIQP) reformulation by Zheng, Sun, Li, and Cui (2012) for probabilistically constrained quadratic programs (PCQP) recently published in EJOR significantly dominates the standard MIQP formulation ( and ) which has been widely adopted in the literature. Stimulated by the dimensionality problem which Zheng et al. (2012) acknowledge themselves for their reformulations, we study further the characteristics of PCQP and develop new MIQP reformulations for PCQP with fewer variables and constraints. The results from numerical tests demonstrate that our reformulations clearly outperform the state-of-the-art MIQP in Zheng et al. (2012).     

二次Lagrangian有限元方程的几何多重网格法

为了构造快速求解二次Lagrangian有限元方程的几何多重网格法,在选择二次Lagrangian有限元空间和一系列线性Lagrangian有限元空间分别作为最细网格层和其余粗网格层以及构造一种新限制算子的基础上,提出了一种新的几何多重网格法,并对它的计算量进行了估计.数值实验结果,与通常的几何多重网格法和AMG01法相比,表明了新算法计算量少且稳健性强.     

A Petrov-Galerkin method for equal width equation

The equal width equation is solved numerically by Petrov-Galerkin method using linear hat function and quadratic B-spline function as trial and test functions respectively. Product approximation has been used in this method. A linear stability analysis of the scheme is shown to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. The Maxwellian initial condition pulse and the development of an undular bore are also studied.     

Evolving a three-world framework for solving algebraic equations in the light of what a student has met before

In this paper we consider data from a study in which students shift from linear to quadratic equations in ways that do not conform to established theoretical frameworks. In solving linear equations, the students did not exhibit the ‘didactic cut’ of Filloy and Rojano (1989) or the subtleties arising from conceiving an equation as a balance (Vlassis, 2002). Instead they used ‘procedural embodiments’, shifting terms around with added ‘rules’ to obtain the correct answer (Lima & Tall, 2008). Faced with quadratic equations, the students learn to apply the formula with little success. The interpretation of this data requires earlier theories to be seen within a more comprehensive framework that places them in an evolving context. We use the developing framework of three worlds of mathematics (Tall, 2004, Tall, 2013), based fundamentally on human perceptions and actions and their consequences, at each stage taking into account the experiences that students have ‘met-before’ (Lima and Tall, 2008, McGowen and Tall, 2010). These experiences may be supportive in new contexts, encouraging pleasurable generalization, or problematic, causing confusion and even mathematical anxiety. We consider how this framework explains and predicts the observed data, how it evolves from earlier theories, and how it gives insights that have both theoretical and practical consequences.     

Multicategory ψ-Learning and Support Vector Machine: Computational Tools

Many margin-based binary classification techniques such as support vector machine (SVM) and ψ-learning deliver high performance. An earlier article proposed a new multicategory ψ-learning methodology that shows great promise in generalization ability. However,ψ-learning is computationally difficult because it requires handling a nonconvex minimization problem. In this article, we propose two computational tools for multicategory ψ-learning. The first one is based on d.c. algorithms and solved by sequential quadratic programming, while the second one uses the outer approximation method, which yields the global minimizer via sequential concave minimization. Numerical examples show the proposed algorithms perform well.