Balancing families of sets

Given k finite sets S₁,…,S_k, to what extent is it possible to partition their union into two parts A and B in such a way that, for each j, S_j ∩ A and S_j ∩ B contain approximately the same number of elements? Bounds are found for this and similar questions.     

Coexistence of phases in Ising ferromagnets

We derive a new inequality for ferromagnetic Ising spin systems and then use it to obtain information about the number of phases which can coexist in such systems. We show in particular that for even interactions only two phases (up and down magnetization) can coexist below the critical temperature at zero magnetic field (h=0) whenever the energy is a continuous function of the temperature. We also prove that the derivatives with respect toh ath=0 of the odd correlation functions (triplet,...) diverge like the susceptibility in the vicinity of the critical temperature (at least for pair interactions). Our results also apply to higher order Ising spins (not just spin 1/2).Research supported in part by NSF Grant #MPS 75-20638 and USAFOR Grant #73-2430D.John Simon Guggenheim Fellow.     

Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit

Equivalence (allowing for convex combinations) of microcanonical, canonical and grandcanonical ensembles for states of classical systems is established under very mild assumptions on the limiting state. We introduce the notion of conditional equilibrium (C.E.), a property of states of infinite systems which characterizes convex combinations of limits of microcanonical ensembles. It is shown that C.E. states are, under quite general conditions, mixtures of Gibbs states.Supported in part by NSF Grant No. MCS 75-21684 A02Supported in part by NSF Grant No. MPS 72-04534Supported in part by NSF Grant No. Phy 77-22302     

On the electron affinity of perfluorocycloalkanes and perfluoroalkanes

There is an increasing body of evidence showing that perfluorocycloalkanes have a higher electron affinity than their open chain analogs, the perfluoroalkanes. A new molecular orbital model is presented to explain these results and compared with the electrostatic model of Mittal and Libby. Explicit experiments are suggested which would allow comparison of the two models.     

An observation on the effect of fluctuations far from equilibrium: The Lotka-Volterra model

Spontaneous fluctuations in the Lotka-Volterra model of chemical reactions are known to grow in an unbounded way when species held in excess are neglected. This result is obtained in a simple way using generalized fluctuation-dissipation principles but appears to be an artifact of ignoring fluctuations in the variables that are held fixed. When fluctuations in the other concentrations are included in the model, the spontaneous fluctuations become bounded.This work was supported by the National Science Foundation through Research Grant No. MPS 74-00483 AO1.     

The effect of disordering on the spectral properties of partial monolayers of H on Si(111)

Ultraviolet photoemission measurements are reported for a H covered Si(111) surface for which the H coverage ranged from a fraction to a full monolayer. These measurements reveal striking differences depending whether the Si(111) substrate is kept at room temperature (RT) or 150°C. In particular, the 150°C sample UPS spectral series shows monotonic growth with little line shape change while the RT series shows significant line shape modification with coverage. These results are interpreted as island growth at 150°C and disordered adsorption at RT. Theoretical model calculations are carried out of the electron density of states of a fractional monolayer of H chemisorbed to Si(111) that reproduce the essential features of the RT data and confirm the role of disordering there.     

Factorizing the complete graph into factors with large star number

The graph G has star number n if any n vertices of G belong to a subgraph which is a star. Let f(n, k) be the smallest number m such that the complete graph on m vertices can be factorized into k factors with star number n. In the present paper we prove that $\text{c}{}_1{}^{\mathrm{n}}\text{k ≤ f(n, k) < c}{}_2{}^{\mathrm{n}}\text{k}$.