Operators that are points of spectral continuity,II

This paper continues an earlier study of those bounded operators on a Hilbert space at which the spectrum is continuous, where the spectrum is considered as a function whose domain is the set of all bounded operators furnished with the norm topology and whose range is the collection of compact subsets of the complex plane furnished with the Hausdorff metric. In this paper the points of continuity of the essential spectrum, the approximate point spectrum, and certain related subsets of the spectrum are characterized.The first author was supported by National Science Foundation Grant MCS 77-28396.     

On the enumeration of self-dual codes

We give the complete classification of all binary, self-dual, doubly-even (32, 16) codes. There are 85 non-equivalent, self-dual, doubly-even (32, 16) codes. Five of these have minimum weight 8, namely, a quadratic residue code and a Reed-Muller code, and three new codes. A set of generators is given for a code in each equivalence class together with its entire weight distribution and the order of its entire group with other information facilitating the computation of permutation generators. From this list it is possible to identify all self-dual codes of length less than 32 and the numbers of these are included.     

Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds

A substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.     

Iridium(III) Complexes Bearing Pyrene‐Functionalized 1,10‐Phenanthroline Ligands as Highly Efficient Sensitizers for Triplet–Triplet Annihilation Upconversion

“Chemistry‐on‐the‐complex” synthetic methods have allowed the selective addition of 1‐ethynylpyrene appendages to the 3‐, 5‐, 3,8‐ and 5,6‐positions of Ir^III‐coordinated 1,10‐phenanthroline via Sonogashira cross‐coupling. The resulting suite of complexes has given rise to the first rationalization of their absorption and emission properties as a function of the number and position of the pyrene moieties. Strong absorption in the visible region (e.g. 3,8‐substituted Ir‐3 : λ_abs=481 nm, ?=52 400 m ^?1 cm^?1) and long‐lived triplet excited states (e.g. 5‐substituted Ir‐2 : τ_T=367.7 μs) were observed for the complexes in deaerated CH₂Cl₂. On testing the series as triplet sensitizers for triplet–triplet annihilation upconversion, those Ir^III complexes bearing pyrenyl appendages at the 3‐ and 3,8‐positions ( Ir‐1 , Ir‐3 ) were found to give optimal upconversion quantum yields (30.2 % and 31.6 % respectively).     

Biology-enabling inositol phosphates, phosphatidylinositol phosphates and derivatives

This review highlights inositol polyphosphate- and phosphatidylinositol-based small molecule probes that have advanced our understanding of intracellular signalling.     

On Angles Whose Squared Trigonometric Functions Are Rational

We consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call ``geodetic.' We construct a convenient basis for the vector space over Q generated by these angles. Geodetic angles and rational linear combinations of geodetic angles appear naturally in Euclidean geometry; for illustration we apply our results to equidecomposability of polyhedra. Received April 7, 1998, and in revised form September 2, 1998.     

Solubility of vinylidene fluoride polymers in supercritical CO2 and halogenated solvents

The cloud‐point behaviors of poly(vinylidene fluoride) (PVDF) and poly(vinylidene fluoride‐co‐22 mol % hexafluoropropylene) (VDF–HFP₂₂) are reported at temperatures up to 250 °C and pressures up to 3000 bar in supercritical CO₂, CHF₃, CH₂F₂, CHClF₂, CClF₃, CH₃CHF₂, CH₂FCF₃, CHF₂CF₃, and CH₃CClF₂. The molecular weight of PVDF has a smaller effect on the cloud point than the solvent quality. Cloud‐point pressures for both fluoropolymers decrease as the solvent polarizability, polar moment per molar volume, and density increases. However, it is extremely difficult to dissolve either fluoropolymer in CClF₃, which has a large polarizability and a small dipole moment. CO₂ is an effective solvent because it complexes with the C F dipole at low temperatures where energetic interactions fix the phase behavior. In addition, polymer architecture has a strong impact on the cloud‐point pressure. VDF–HFP₂₂ has lower cloud‐point pressures than PVDF in all solvents because it has a larger free volume that promotes facile interactions between the solvent and the polymer segments. Cloud‐point data are also reported for amorphous poly(tetrafluoroethylene‐co‐x mol % 2,2‐bistrifluoromethyl‐4,5‐difluoro‐1,3‐dioxole) (TFE–PDD_x ; x = 65 and 85) in CO₂. These data provide an interesting comparison to the PVDF–CO₂ and VDF–HFP₂₂–CO₂ systems because TFE–PDD₆₅ and TFE–PDD₈₇ have very high glass‐transition temperatures of 160 and 240 °C, respectively. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2832–2840, 2000     

Genetic algorithms applied to the solution of hybrid optimal control problems in astrodynamics

Many space mission planning problems may be formulated as hybrid optimal control problems, i.e. problems that include both continuous-valued variables and categorical (binary) variables. There may be thousands to millions of possible solutions; a current practice is to pre-prune the categorical state space to limit the number of possible missions to a number that may be evaluated via total enumeration. Of course this risks pruning away the optimal solution. The method developed here avoids the need for pre-pruning by incorporating a new solution approach using nested genetic algorithms; an outer-loop genetic algorithm that optimizes the categorical variable sequence and an inner-loop genetic algorithm that can use either a shape-based approximation or a Lambert problem solver to quickly locate near-optimal solutions and return the cost to the outer-loop genetic algorithm. This solution technique is tested on three asteroid tour missions of increasing complexity and is shown to yield near-optimal, and possibly optimal, missions in many fewer evaluations than total enumeration would require.