Characterizations of Classical Orthogonal Polynomials

We give a simple unified proof and an extension of some of the characterization theorems of classical orthogonal polynomials of Jacobi, Bessel, Laguerre, and Hermite. In particular, we prove that the only orthogonal polynomials whose derivatives form a weak orthogonal polynomial set are the classical orthogonal polynomials.     

A Dual-Based Add Heuristic for Uncapacitated Facility Location

This paper presents a heuristic method for solving the uncapacitated facility-location problem (UFLP), which is similar to Erlenkotter's ‘dual ascent’ procedure. The heuristic is of the ‘add’ type, which progressively selects facilities to open according to a certain criterion derived from the analysis of the linear programming dual. Computational experience with both (static) UFLPs and dynamic UFLPs reveals that the heuristic method yields solutions in most cases superior in quality to those achieved by the dual-ascent procedure, with barely noticeable additional computation time.     

Gibbsianness of fermion random point fields

We consider fermion (or determinantal) random point fields on Euclidean space ℝ^d. Given a bounded, translation invariant, and positive definite integral operator J on L²(ℝ^d), we introduce a determinantal interaction for a system of particles moving on ℝ^d as follows: the n points located at x₁,· · ·,x_n ∈ ℝ^d have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)⁻¹ is a Gibbs measure admitted to the specification.     

Sequential Fourier-Feynman transform, convolution and first variation

Cameron and Storvick introduced the concept of a sequential Fourier-Feynman transform and established the existence of this transform for functionals in a Banach algebra of bounded functionals on classical Wiener space. In this paper we investigate various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or continuous. Also we study the relationships involving this transform and the first variation.

     

Multiple Degree Reduction and Elevation of Bézier Curves Using Jacobi–Bernstein Basis Transformations

In this article, we find the optimal r times degree reduction of Bézier curves with respect to the Jacobi-weighted L ₂-norm on the interval [0, 1]. This method describes a simple and efficient algorithm based on matrix computations. Also, our method includes many previous results for the best approximation with L ₁, L ₂, and L _∞-norms. We give some examples and figures to demonstrate these methods.     

Pricing Asian options of discretely monitored geometric average in the regime‐switching model

This paper provides analytic pricing formulas of discretely monitored geometric Asian options under the regime‐switching model. We derive the joint Laplace transform of the discount factor, the log return of the underlying asset price at maturity, and the logarithm of the geometric mean of the asset price. Then using the change of measures and the inversion of the transform, the prices and deltas of a fixed‐strike and a floating‐strike geometric Asian option are obtained. As the numerical results, we calculate the price of a fixed‐strike and a floating‐strike discrete geometric Asian call option using our formulas and compare with the results of the Monte Carlo simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Practical method for crystalline-liquid resolution of chrysanthemic acids utilizing chiral 1,1′-binaphthol monoethyl ethers directed for process chemistry

We have developed an efficient practical resolution method for (1R,3R)-trans-chrysanthemic acid 1 and (1R,3S)-trans-2,2-dimethyl-3-(2,2-dichloroethenyl)cyclopropanecarboxylic acid 2, based on the preliminary results of the simpler analogues, (1R)-2,2-dichlorocyclopropanecarboxylic acid 3 and (1R)-2,2-dimethylcyclopropanecarboxylic acid 4, using a crystalline-liquid separation procedure (without column chromatography) with chiral 1,1′-binaphthol monoethyl ethers (R)-5b as the key auxiliary. Direct esterifications of 1, 2, 3, and 4 with (R)-5b gave four sets of (1R)- and (1S)-diastereomeric esters 8, 9, 6, and 7, respectively, with markedly different melting points. All of these diastereomers were easily obtained using a simple and one-step crystalline-liquid separation. The separated diastereomers 8 and 9 were easily hydrolyzed to the desired enantiopure acids 1 (>98%) and 2 (>99%), respectively, with recovery of (R)-5b (>90%).     

Primal-Dual Active-Set Algorithm for Chemical Equilibrium Problems Related to the Modeling of Atmospheric Inorganic Aerosols

A general equilibrium model for multiphase multicomponent inorganic atmospheric aerosols is proposed. The thermodynamic equilibrium is given by the minimum of the Gibbs free energy for a system involving an aqueous phase, a gas phase, and solid salts. A primal-dual algorithm solving the Karush-Kuhn-Tucker conditions is detailed. An active set/Newton method permits to compute the minimum of the energy and tracks the presence or not of solid salts at the equilibrium. Numerical results show the efficiency of our algorithm for the prediction of multiphase multireaction chemical equilibria.Communicated by R. GlowinskiThis work has been partially supported by the United States Environmental Protection Agency through Cooperative Agreement X-83234201 to the University of Houston. The second author was supported by the Swiss National Science Foundation, Grant PBEL2-103152.