Diameter graphs of polygons and the proof of a conjecture of Graham

We show that for an n-gon with unit diameter to have maximum area, its diameter graph must contain a cycle, and we derive an isodiametric theorem for such n-gons in terms of the length of the cycle. We then apply this theorem to prove Graham's 1975 conjecture that the diameter graph of a maximal 2m-gon (m?3) must be a cycle of length 2m−1 with one additional edge attached to it.     

The effect of rapid rotation on a vertical bridgman furnace at large Rayleigh number

In quasi-steady operation, convection currents in a Bridgmandevice, used for producing a semi-conductor crystal, createinhomogeneities that may make the crystal unusable. It has oftenbeen suggested that additional forces due to rotation or magnetismmight be efficacious in reducing the segregation of the elementsof the alloy. It has been found that, over a wide range of rotationrates, there is no improvement in performance due to rotationabout the vertical axis. However, numerical results that havebeen obtained previously (Lee & Pearlstein, J. Crys. Growth240, 2002) indicate that, when effects of centrifugal buoyancyare introduced, a substantial reduction in segregation is achieved.In the work reported here, by contrast, in which we extend previouslarge-Rayleigh-number asymptotic analysis to include centrifugalbuoyancy, we find no improvement in radial segregation, butrather increasing segregation with increasing rotation rate.     

Zero-temperature properties of randomly self-interacting polymers

We study the zero-temperature behavior of several simple models for randomly self-interacting polymers in one and 1+1 dimensions. Results are based on exact enumeration and closed-form expressions.     

Divinylsiloxane‐bisbenzocyclobutene‐based polymer modified with polystyrene–polybutadiene–polystyrene triblock copolymers

Divinylsiloxane‐bisbenzocyclobutene (DVS‐bisBCB) polymer has very low dielectric constant and dissipation factor, good thermal stability, and high chemical resistance. The fracture toughness of the thermoset polymer is moderate due to its high crosslink density. A thermoplastic elastomer, polystyrene–polybutadiene–polystyrene triblock copolymer, was incorporated into the matrix to enhance its toughness. The cured thermoset matrix showed different morphology when the elastomer was added to the B‐staged prepolymer or when the elastomer was B‐staged with the DVS‐bisBCB monomer. Small and uniformly distributed elastomer domains were detected by transmission electron micrographs (TEM) in the former case, but TEM did not detect a separate domain in the latter case. A high percentage of the polystyrene–polybutadiene–polystyrene triblock copolymer could be incorporated into the DVS‐bisBCB thermoset matrix by B‐staging the triblock copolymer with the BCB monomer. The elastomer increased the fracture toughness of DVS‐bisBCB polymer as indicated by enhanced elongation at break and increased K1c values obtained by the modified edge‐lift‐off test. Elastomer modified DVS‐bisBCB maintained excellent electrical properties, high T_g and good thermal stability, but showed higher coefficient of linear thermal expansion values. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1591–1599, 2006     

Effects of osmotic manipulation of intracellular hydration of HeLa S-3 cells on their proton NMR relaxation times

Pellets of HeLa from suspension cultured cells in isotonic medium (300 mosmolar) were introduced into a Bruker CXP100 NMR spectrophotometer at 80 mHz within 5 min of the start of centrifugation. T1 and T2 times were measured within a total elapsed time of 20-25 min at 80 mHz and 37 degrees C, and averaged 1430 msec and 120 msec, respectively. Extrapolation to zero extracellular space gave a corrected T1 of 1370 msec. For cells collected after 10 min in hypotonic medium (down to 30 mosmolar) increased proton density correlated well with increased cell water content, but relaxation times did not rise in proportion to that predicted for the entry of "bulk" water (T1 of 4700 msec), except when swelling approached lysis point. Cells partially dehydrated by 10 min in hypertonic medium of up to 1500 mosmolar have also been analyzed, but once again the shortening of T1 was not proportional to the loss of "free" (bulk phase) water. At the upper limit of hypertonic treatment, lacunae or vacuoles of a watery nature separated within the cytomatrix, preventing maximum dehydration. The relationship of cell water to T1 is complex over the whole range of tonicity that HeLa S-3 cells tolerate. The data indicate, however, that hypotonically induced water probably has an average T1 time considerably lower than bulk phase water. In contrast, raising the total extracellular volume with medium had precisely the predicted effect on T1 time, further strengthening the case that water taken up by cell acquires a shorter T1 time. Cells adapting to hypotonic conditions oscillated in size and water content over 2-3 hr before returning to near their initial volume. Under these circumstances, T1 oscillated in the same way but with a reduced amplitude, consistent with the above findings.