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     

On the covering multiplicity of lattices

Let the lattice Λ have covering radiusR, so that closed balls of radiusR around the lattice points just cover the space. The covering multiplicityCM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity for ann-dimensional lattice isn ifn≤8, and conjecture that it exceedsn in all other cases. We determine the covering multiplicity of the Leech lattice and of the latticesI _n, A_n, D_n, E_n and their duals for small values ofn. Although it appears thatCM(I _n)=2 ⁿ⁻¹ ifn≤33, asn → ∞ we haveCM(I _n)∼2.089... ⁿ . The results have application to numerical integration.     

The antipode construction for sphere packings

Summary A construction for sphere packings is introduced that is parallel to the anticode construction for codes. This provides a simple way to view Vardy's recent 20-dimensional sphere packing, and also produces packings in dimensions 22, 44–47 that are denser than any previously known.Oblatum 18-VII-1994 & 7-XII-1994     

The essential selfcommutator of a subnormal operator

In this paper an easier proof is obtained of Alexandru Aleman's extension of an inequality of Axler and Shapiro for subnormal operators to the essential norm. The method is applied to show that a hyponormal operator whose essential spectrum has area zero must be essentially normal.

     

The evaluation and use of properties of individual ions in slution

A critical examination is given of extra-thermodynamic methods for evaluation of properties of individual ions in solution. The ways in which such data may be applied, or are required in various problems concerning specific properties of electrolyte solutions and in the fields of kinetics, electrochemistry, colloid and interfacial chemistry, are outlined. Special attention is given to the problem of evaluation of individual ionic thermodynamic functions for the hydration of ions but, in some cases, corresponding data for nonaqueous solutions are available for discussion and are of interest in a comparative way. The reliability of the various methods is assessed and estimates of numerical values for various individual ionic functions are tabulated. The question of scales of individual ionic radii, on which some of the derived data depend, is also examined. The dependence of various properties for anions and for cations on ionic radius is discussed and data are plotted comparatively. It is shown that for most, but not all, properties, the values for anions are more sensitive to increasing ionic radius than are the values for cations. This behavior, as well as the exceptions, is of theoretical interest. The significance of ion-specific behavior at interfaces is considered and the importance of interactions between ions of like sign of charge is indicated.Presented at the Symposium on Electrolytes, University of Reading, September 13, 1977.