The projective dimension of three cubics is at most 5

Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the example with largest projective dimension he constructed has $\mathrm{pd}(R/I)=5$. Based on computational evidence, it had been conjectured that $\mathrm{pd}(R/I)\le 5$. In the present paper we prove this conjectured sharp bound.     

Orientation-reversing involutions on handlebodies

The observation that the quotient orbifold of an orientation- reversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the -fibers of some -bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses.

     

Melting temperatures of poly(2,6-dimethyl-1,4-phenylene oxide) in methylene chloride solutions: Thermodynamic analysis

Melting temperatures were observed visually for poly-(2,6-dimethyl-1,4-phenylene oxide) in methylene chloride at nine concentrations (polymer weight fractions ranging from 0.0042 to 0.2362). The data were analyzed upon the assumptions that ΔH_u and ΔS_u, the molar heat and entropy of fusion per polymer unit, are constant over the temperature range studied, and that a Flory-Huggins chemical potential expression with a concentration-independent pair interaction parameter, χ₁ = (0.5 + ψ₁) + ψ₁Φ/T, satisfactorily describes the polymer unit activity in the binary solutions. Computation gave ΔH_u = 1404 cal/mole of units (therefore Δh = 11.7 cal/g), ψ₁ = ?0.569₁, and Φ = 342.₄°K. The effect of using various combinations of data points upon the values of these three parameters, as determined by least-squares linear regression treatment of the melting temperature expression, is indicated.     

A free-volume-based approach to modeling thermoset cure behavior

The free-volume theory for the temperature dependence of transport properties of glass-forming polymers is extended to obtain their relationship to the extent of cure. This treatment centers on the unifying role of molecular mobility and yields a model which connects extent of reaction, viscosity, diffusivity, ionic conductivity and dipole relaxation time. The temporal dependence of these properties is expressed by coupling the extended free-volume model with a relationship for the rate of cure, which included diffusional limitations. Analyses based on this model are applied to the observed behavior of a model epoxy-amine resin system. The intrinsic kinetics of this model system are shown to be first order. It is shown that diffusional limitations strongly affected the progress of the reaction in the final stages of cure. The diffusion-modified rate expression predictions agree with extent of reaction versus time data over the range of experimental temperatures. The temporal dependence of viscous behavior of the curing resin is measured. The extended free-volume model accurately describes the evolution of resin viscosity during cure. The dielectric behavior is similarly characterized and is in close agreement with the predictions of the general free-volume expression. The results of this study indicate that the free-volume theory modified to account for molecular weight effects allows prediction of resin properties with a two-parameter model. The results show that a power-law relationship exists between viscosity and ionic conductivity. This result suggests that electrical properties may be used for on-line measurement of resin viscosity during cure.     

Micro-interferometry for measurement of thermal displacements at fiber/matrix interfaces

A micro-interferometric technique for measuring out-of-plane thermal displacements on a scale commensurate with the dimensions of the fiber/matrix unit cell is described. A scanning micro-interferometer is used to image surface displacements of samples containing a single-pitch-based carbon fiber embedded in an epoxy matrix. The interferometer design gives the necessary resolution to detect small changes in thermal displacements in the fiber/matrix interface region. The samples were heated electrically through the fiber to create radially symmetric temperature and displacement fields. Repeatable displacement measurements were obtained on a radial line across the interface region with an accuracy of ±25 Å. A sharp expansion of the matrix surrounding the fiber was observed with each heating. Overall, the experiments demonstrate the utility of micro-interferometry for measuring submicron displacements.     

Nevanlinna–Pick interpolation on distinguished varieties in the bidisk

This article treats Nevanlinna–Pick interpolation in the setting of a special class of algebraic curves called distinguished varieties. An interpolation theorem, along with additional operator theoretic results, is given using a family of reproducing kernels naturally associated to the variety. The examples of the Neil parabola and doubly connected domains are discussed.     

Caratheodory interpolation kernels

Kernels over the unit disk for which a version of Carathéodory interpolation is true are characterized in a simple computationally verifiable manner.