Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.     

The Braided product of Ockham algebras

In this paper we introduce an algebraic concept of the product of Ockham algebras called the Braided product. We show that ifL _i MS(i=1, 2, ,n) then the Braided product ofL _i(i=1, 2, ,n) exists if and only ifL ₁, ,L _n have isomorphic skeletons.     

Linear programming with entropic perturbation

In this paper, we derive an unconstrained convex programming approach to solving standard form linear programs through an entropic perturbation. The whole duality theory is established by using only one simple inequality lnz z –1 forz > 0. A curved search algorithm is also proposed for obtaining a pair of primal and dual-optimal solutions. The proposed algorithm is proven to be globally convergent with a quadratic rate of convergence. Computational results are included in support of theoretic findings.The work is partially supported by the North Carolina Supercomputing Center, the Cray Research Award, and the National Science Council of the Republic of China # NSC 81-0415-E-007-10.     

Quantum chemical definition and calculation of oxidation number

A new quantum chemical definition of oxidation number is proposed, in the present paper, as a direct generalization of the corresponding classical definition. According to the proposed general definition, the oxidation number can be calculated by use of molecular orbital data and a population analysis method or by use of other quantum chemical methods. For the practical calculation, we present a corresponding concrete calculation procedure within the framework of the maximum overlap population principle, which is very simple and very easy to use. The calculated numerical results are, on the whole, in good agreement with chemists' intuitive concepts of chemical bonding.     

Chiral Phosphinophenyloxazolines Bearing Alkoxymethyl Substituents: Synthesis and Application in the Palladium Catalyzed Allylic Substitution Reactions

The chiral phosphine‐oxazoline ligands 3 and 4 bearing 4‐alkoxymethyl substituents on the oxazoline ring with (R)‐configuration were prepared from L‐serine methyl ester in 66% and 33% yields, respectively. Along this synthetic pathway, the β‐hydroxylamides derived from L‐serine methyl ester and 2‐halobenzoyl chlorides were expediently converted to the corresponding oxazolines by using diethylaminosulfur trifluoride as the activation agent. Potassium diphenylphosphide was the reagent of choice for replacing the bromine atom on the phenyl ring, giving the desired oxazoline‐phosphine ligands 3 and 4 . Together with [Pd(η³‐allyl)Cl]₂, ligands 3 and 4 induced an enantioselective allylic substitution reaction of 1,3‐diphenyl‐2‐pro‐penyl acetate by dimethyl malonate. Although ligands 3 and 4 exhibit the (R)‐configuration, differing from the (S)‐configuration of Pfaltz‐Helmchen‐Williams phosphine‐oxazoline ligands, all these ligands led to the same enantiotopic preference in the allylic substitution reaction. To facilitate the recovery and reuse of the phosphine‐oxazoline ligand, immobilization on Merrifield resin was attempted, albeit in low loading.