C?H Bond dissociation of acetylene: Local density functional calculations

The C? H bond dissociation energy of acetylene was computed by both ab initio approaches and density functional theory in a local density approximation (DFT–LDA ). Structures and energies for acetylene and its dissociation products (the ethynyl and hydrogen radicals) are presented and compared. Using directly computed HCCH and HCC· energies and the exact H· value, the DFT–LDA calculations are found to yield C? H dissociation energies ranging from 129 to 131 kcal/mol, in good agreement with recent experimental and the highest level theoretical results. The DFT–LDA results show little dependence upon the computational procedure used to obtain geometries.     

A delay-dependent model with HIV drug resistance during therapy

The use of combination antiretroviral therapy has proven remarkably effective in controlling HIV disease progression and prolonging survival. However, the emergence of drug resistance can occur. It is necessary that we gain a greater understanding of the evolution of drug resistance. Here, we consider an HIV viral dynamical model with general form of target cell density, drug resistance and intracellular delay incorporating antiretroviral therapy. The model includes two strains: wild-type and drug-resistant. The basic reproductive ratio for each strain is obtained for the existence of steady states. Qualitative analysis of the model such as the well-posedness of the solutions and the equilibrium stability is provided. Global asymptotic stability of the disease-free and drug-resistant steady states is shown by constructing Lyapunov functions. Furthermore, sufficient conditions related to the properties of the target cell density are obtained for the local asymptotic stability of the positive steady state. Numerical simulations are conducted to study the impact of target cell density and intracellular delay focusing on the stability of the positive steady state. The occurrence of Hopf bifurcation of periodic solutions is shown to depend on the target cell density.     

Boundary of Anosov dynamics and evolution equations for surfaces

We show that a compact surface of genus greater than one, without focal points and a finite number of bubbles (“good” shaped regions of positive curvature) is in the closure of Anosov metrics. Compact surfaces of nonpositive curvature and genus greater than one are in the closure of Anosov metrics, by Hamilton's work about the Ricci flow. We generalize this fact to the above surfaces without focal points admitting regions of positive curvature using a “magnetic” version of the Ricci flow, the so‐called Ricci Yang‐Mills flow.     

Two-Way Tables: Issues at the Heart of Statistics and Probability for Students and Teachers

Some problems exist at the intersection of statistics and probability, creating a dilemma in relation to the best approach to assist student understanding. Such is the case with problems presented in two-way tables representing conditional information. The difficulty can be confounded if the context within which the problem is set is one where students have preconceived opinions on the direction of the potential association present. This article considers school students’ responses to two problems of association, with data presented in 2 × 2 tables. A hierarchical rubric is presented to document students’ understandings. Teachers’ pedagogical content knowledge is also considered in relation to the same two problems. Findings include a surprising relationship of outcomes for students across the problem contexts and some concern about teachers’ pedagogical content knowledge in this area of the curriculum.     

Solving semi-infinite programs by smoothing projected gradient method

In this paper, we study a semi-infinite programming (SIP) problem with a convex set constraint. Using the value function of the lower level problem, we reformulate SIP problem as a nonsmooth optimization problem. Using the theory of nonsmooth Lagrange multiplier rules and Danskin’s theorem, we present constraint qualifications and necessary optimality conditions. We propose a new numerical method for solving the problem. The novelty of our numerical method is to use the integral entropy function to approximate the value function and then solve SIP by the smoothing projected gradient method. Moreover we study the relationships between the approximating problems and the original SIP problem. We derive error bounds between the integral entropy function and the value function, and between locally optimal solutions of the smoothing problem and those for the original problem. Using certain second order sufficient conditions, we derive some estimates for locally optimal solutions of problem. Numerical experiments show that the algorithm is efficient for solving SIP.     

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.     

Enhanced Karush–Kuhn–Tucker Conditions for Mathematical Programs with Equilibrium Constraints

In this paper, we study necessary optimality conditions for nonsmooth mathematical programs with equilibrium constraints. We first show that, unlike the smooth case, the mathematical program with equilibrium constraints linear independent constraint qualification is not a constraint qualification for the strong stationary condition when the objective function is nonsmooth. We then focus on the study of the enhanced version of the Mordukhovich stationary condition, which is a weaker optimality condition than the strong stationary condition. We introduce the quasi-normality and several other new constraint qualifications and show that the enhanced Mordukhovich stationary condition holds under them. Finally, we prove that quasi-normality with regularity implies the existence of a local error bound.     

Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow-Hurwicz-Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn-Tucker constraint qualification, and MPEC Abadie constraint qualification.     

A Class of Quadratic Programs with Linear Complementarity Constraints

We consider a class of quadratic programs with linear complementarity constraints (QPLCC) which belong to mathematical programs with equilibrium constraints (MPEC). We investigate various stationary conditions and present new and strong necessary and sufficient conditions for global and local optimality. Furthermore, we propose a Newton-like method to find an M-stationary point in finite steps without MEPC linear independence constraint qualification. The research of this author is partially supported by NSERC, and Research Grand Council of Hong Kong.