Studies on the ring opening reactions of 3-oxa-1-azabicyclo[3.1.0]hexan-2-ones. Synthesis of aminomethyl oxazolidinones and aziridinyl ureas

The reaction of fused ring aziridines, 3-oxa-1-azabicyclo[3.1.0]hexan-2-ones, with amine nucleophiles can provide either an aminomethyl oxazolidinone or an aziridinyl urea. The amine, reaction solvent, and aziridine substitution have been examined with the aid of computational studies to identify reaction conditions that provide a single product. Polar solvents provided only the aminomethyl oxazolidinone products. Formation of aziridinyl ureas required control of aziridine substitution, solvent, and reactant stoichiometry.     

Projection-forcing multisets of weight changes

Let F be a finite field. A multiset S of integers is projection-forcing if for every linear function ?:Fn→Fm whose multiset of weight changes is S, ? is a coordinate projection up to permutation and scaling of entries. The MacWilliams Extension Theorem from coding theory says that S={0,0,…,0} is projection-forcing. We give a (super-polynomial) algorithm to determine whether or not a given S is projection-forcing. We also give a condition that can be checked in polynomial time that implies that S is projection-forcing. This result is a generalization of the MacWilliams Extension Theorem and work by the first author.     

Bismuth(III) bromide in organic synthesis. A catalytic method for the allylation of tetrahydrofuranyl and tetrahydropyranyl ethers

A bismuth bromide-catalyzed (10.0 mol %) multicomponent reaction involving the allylation of THF- and THP-ethers, followed by in situ derivatization with acetic anhydride to generate highly functionalized esters has been developed under solvent-free conditions. To the best of our knowledge, this is the first report of a catalytic procedure for the allylation of THF- and THP-ethers to yield ring-opened products.     

(<Emphasis Type="Italic">m</Emphasis>,<Emphasis Type="Italic">r</Emphasis>)-central Riordan arrays and their applications

For integers m > r ≥ 0, Brietzke (2008) defined the (m, r)-central coefficients of an infinite lower triangular matrix G = (d, h) = (d_n,k)_n,k∈N as d_{mn+r,(m?1)n+r}, with n = 0, 1, 2,..., and the (m, r)-central coefficient triangle of G as

$${G^{\left( {m,r} \right)}} = {\left( {{d_{mn + r,\left( {m - 1} \right)n + k + r}}} \right)_{n,k \in \mathbb{N}}}.$$

It is known that the (m, r)-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = (d, h) with h(0) = 0 and d(0), h′(0) ≠ 0, we obtain the generating function of its (m, r)-central coefficients and give an explicit representation for the (m, r)-central Riordan array G^(m,r) in terms of the Riordan array G. Meanwhile, the algebraic structures of the (m, r)-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of m and r. As applications, we determine the (m, r)-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.

     

Application of Polynomial Interpolation in the Chinese Remainder Problem

This paper presents an application of polynomial interpolation in the solution of the Chinese Remainder Problem for bother integers and polynomials.     

A Unified Approach to Construct a Class of Daubechies Orthogonal Scaling Functions

We use Lorentz polynomials to give an efficient way to prove Daubechies’ results on the existence of spline type orthogonal scaling functions and to evaluate a class of Daubechies scaling functions in a unified approach.     

Environment-friendly organic synthesis using bismuth compounds. Bismuth triflate catalyzed synthesis of substituted 3,4-dihydro-2H-1-benzopyrans

A highly catalytic method for the synthesis of dihydrobenzopyrans from salicylaldehydes has been developed. An extension of this method to the synthesis of a pyrano[2,3,b]benzopyran has also been achieved. Bi(OTf)₃·xH₂O (1 < x < 4) (0.1 mol%) smoothly catalyzes the condensation of substituted salicylaldehydes with 2,2-dimethoxypropane to give the corresponding substituted 3,4-dihydro-2H-1-benzopyrans as a mixture of diastereomers (9:1) in moderate yields. The relative configuration of the methoxy groups in the two diastereomers was established by NOE experiments. The advantages of this method include the use of an easy to handle, inexpensive and relatively non-toxic catalyst.     

ON AN EXTENSION OF ABEL-GONTSCHAROFF＇S EXPANSION FORMULA

We present a constructive generalization of Abel-Gontscharoff＇s series expansion to higher dimensions. A constructive application to a problem of multivariate interpolation is also investigated. In addition, two algorithms for constructing the basis functions of the interpolants are given.     

Fourier transform of Bernstein-Bézier polynomials

Explicit formulae, in terms of Bernstein-Bézier coefficients, of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on an interval are derived in this paper. Examples are given and discussed to illustrate the general theory. Finally, this consideration is related to the study of refinement masks of spline function vectors.