Fixed point theorems for mixed monotone operators with PPF dependence

In this paper, fixed point theorems for mixed monotone operators are investigated by weakening the requirements in the contractive assumption and strengthening the metric space utilized with a partial order, taking the domain space C[[a,b],E]$C[[a,b],E]$ different from the range E$E$ and using the method of reflection operators.     

Bivariate interpolatory rational splines

Smoothing conditions in terms of Bézier coefficients of piecewise rational functions on an arbitrary triangulation are derived. This facilitates the solution of the problem of bivariate rational spline interpolation, with or without convexity constraints, particularly on the three and four-directional meshes. For such a triangulation, we also derive the conformality condition that a bivariate rationale spline function must satisfy, and we demonstrate the interpolation scheme with a low-degree example.The research of this author was supported by NSF Grant # DMS-92-06928.     

Fixed-point theorems in partially ordered metric spaces for operators with PPF dependence

In this paper, fixed point theorems of a nonlinear operator are developed, where the domain space C[[a,b],E]$C[[a,b],E]$ is different from the range, E$E$, which is a metric space with partial order. An application of the fixed point theorem to a periodic boundary value problem with delay is also given.     

Discriminant Bounds for Spinor Regular Ternary Quadratic Lattices

The main goals of the paper are to establish a priori boundsfor the prime power divisors of the discriminants of spinorregular positive definite primitive integral ternary quadraticlattices, and to describe a procedure for determining all suchlattices.     

An example of a nonregular semimonotoneQ-matrix

In their paper, Aganagic and Cottle gave necessary and sufficient conditions for aP ₀-matrix to be aQ-matrix. In his paper, Pang showed that the same characterization holds for anL-matrix.In this paper, we show that a similar characterization for anE ₀-matrix (semimonotone orL ₁-matrix) is not possible. This is done by providing a counterexample to the inclusionE ₀ Q R ₀, thus answering in the negative a question first posed by Pang.     

A study of epoxyolefin cyclizations catalyzed by bismuth trifluoromethanesulfonate and other metal triflates

Epoxyolefin cyclizations have attracted considerable interest due to their importance in biosynthetic pathways. Bismuth trifluoromethanesulfonate as well as several other metal triflates are shown to be highly effective (0.1 mol %) catalysts for the cyclization of geraniolene oxide. The product composition is found to be more dependent on solvent and substrate concentration than on the nature of the metal triflate. Cyclization products are favored in CH₂Cl₂ and under high dilution conditions. Ether solvents favored acyclic products.     

Reactivity of ionic liquids

Ionic liquids are becoming widely used in synthetic organic chemistry and yet relatively little attention has been paid to the intrinsic reactivity of these low temperature molten salts. Clues to the non-innocent nature of many ionic liquids are contained in the reports of altered reactivity of dissolved substrates, unexpected catalytic activity and unforeseen by-product formation. In this review, we focus on the reactivity of ionic liquids, as opposed to reactivity in ionic liquids (although discussion of the latter is often included where it aids understanding of the former).     

Quasilinearization for functional differential equations with retardation and anticipation

The technique of generalised quasilinearization is developed, using the method of lower and upper solutions and the monotone iterative technique (MIT), to prove the existence of unique solutions for functional differential equations (FDE) with retardation and anticipation.     

Riordan arrays associated with Laurent series and generalized Sheffer-type groups

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined by using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different type of power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type of inverse relations of various series is constructed by using pairs of Riordan arrays. Finally, several applications involving various arrays, polynomial sequences, special formulas and identities are also presented as illustrative examples.     

Positivity and boundedness of solutions of impulsive reaction-diffusion equations

Employing the generalized quasilinearization for nonlinear reaction-diffusion equations, existence of positive bounded solution is proved.