Figure 1: An Apollonian Circle Packing Apollonius’s Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles to fill all remaining spaces. A remarkable consequence of Descartes’ Theorem is if the initial four tangent circles have integr...

The goal of this paper is to describe how to apply a rigorous framework for modular speciication and validation to a real-time critical system. We are especially interested in generation of functional test sets from a structured speciication of the system. We adopt a theoretical framework for black-box testing from formal spec-iications. Thanks to our modular speciication framework, we can take...

A combinatorial in situ three-component chiral oxazoline-Schiff base copper(II) complex catalyst formation method was developed. This simple combinatorial chiral catalyst approach provided a modular library of chiral oxazoline-Schiff base copper(II) complex catalysts. The catalytic activity of these in situ generated catalysts can be rapidly and conveniently evaluated in the asymmetric Henry re...

We transpose a conservative extension theorem from structured operational semantics to conditional term rewriting. The result is useful for the development of software renovation factories, and for modular speci cation of abstract data types.

Abstraet---A new development in the field of internals in packed columns is the use of structured packing types. Recently, a new structured packing type coated with a thin alumina layer (KATAPAK rM) has been developed, in this report, the results of an experimental and theoretical study concerning the possible applicability of this new packing material for hydrogenation processes in a trickle-b...

The modular catalysts were fabricated via the combination of the Ni-MgO-Al2O3 mixed oxide nanoplates and the mesoporous SiO2 coating. Due to the dual confinements, the catalysts show high catalytic activity with enhanced coke- and sintering-resistance in the dry reforming of methane reaction.

Simple peptides and their analogues having a primary amino group as the catalytic residue mediate the direct asymmetric intermolecular aldol reaction with high stereoselectivity and furnish the corresponding aldol products with up to 99% ee; this intrinsic ability of highly modular peptides may explain the initial molecular evolution of aldolase enzymes.

The title compound, C(11)H(14)Cl(2)O, was synthesized by the reaction of a dichloro-methane solution of (R)-carvone and potassium tert-butano-late in the presence of a catalytic amount of benzyl-triethyl-ammonium chloride in chloro-form. The cyclo-hexene ring adopts a half-boat conformation. The cyclo-propyl ring is unsymmetrical, the shortest C-C bond being distal to the alkyl-substituted C at...