Laminations,or how to build a quantum-logic-valued model of set theory

An explicit construction of the colimit of a filtered diagram in the category of topoi and logical morphisms is given and then used to construct a family of topoi with a fixed Boolean algebra of truth values but with varying amounts of cocompleteness. This same construction, when applied to the diagram of complete Boolean algebras in a quantum logic Q gives a partial topos, a noncategory which is a close to being a model of set theory with algebra of truth values Q as a noncategory can be.     

Sequence characterization of Riordan arrays

In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of the hitting-time subgroup.     

Spontaneous hydrolysis reactions of cis- and trans-beta-methyl-4-methoxystyrene oxides (Anethole oxides): buildup of trans-anethole oxide as an intermediate in the spontaneous reaction of cis-anethole oxide

Rates and products of the reactions of trans- and cis-beta-methyl-4-methoxystyrene oxides (1 and 2) (anethole oxides) and beta,beta-dimethyl-4-methoxystyrene oxide (3) in water solutions in the pH range 4-12 have been determined. In the pH range ca. 8-12, each of these epoxides reacts by a spontaneous reaction. The spontaneous reaction of trans-anethole oxide (1) yields ca. 40% of (4-methoxyphenyl)acetone and 60% of 1-(4-methoxyphenyl)-1, 2-propanediols (erythro:threo ratio ca. 3:1). The spontaneous reaction of cis-anethole oxide is more complicated. The yields of diol and ketone products vary with pH in the pH range 8-11, even though there is not a corresponding change in rate. These results are interpreted by a mechanism in which 2 undergoes isomerization in part to the more reactive trans-anethole oxide (1), which subsequently reacts by acid-catalyzed and/or spontaneous reactions, depending on the pH, to yield diol and ketone products. The buildup of the intermediate trans-anethole oxide in the spontaneous reaction of cis-anethole oxide was detected by (1)H NMR analysis of the reaction mixture. Other primary products of the spontaneous reaction of 2 are (4-methoxyphenyl)acetone (73%) and threo-1-(4-methoxyphenyl)-1,2-propanediol (ca. 3%). The rates and products of the spontaneous reaction of 2 and its beta-deuterium-labeled derivative were determined, and the lack of significant kinetic and partitioning deuterium isotope effects indicates that the isomerization of 2 to ketone and to trans-anethole oxide must occur primarily by nonintersecting reaction pathways.     

(<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.     

Multivariate Interpolation Using Carlitz's Inversion Formulas

Multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite–Fejér interpolation formulas of the Newton type are established by using Carlitz's inversion formulas. The recurrence relation for constructing Lagrange interpolation is also given. In addition, by setting q1 in the obtained formulas, we obtain the corresponding polynomial interpolation formulas with combinatorial form.     

Synthesis of homoallyl ethers via allylation of acetals in ionic liquids catalyzed by trimethylsilyl trifluoromethanesulfonate

The chemoselective allylation of acetals using allyltrimethylsilane in ionic liquids is catalyzed by TMS triflate (5.0-20.0 mol %). The reaction proceeds smoothly at room temperature to afford the corresponding homoallyl ether in good yield. Since the ionic liquids are easily recovered and recycled, they are a useful alternative to dichloromethane, which is the commonly used solvent for allylations. [reaction--see text]