Reaction of α,β-unsaturated acid chlorides with β-diketones. Synthesis of 4,6,7,8-tetrahydro-2H-1-benzopyran-2,5(3H)-diones

Reaction of an α,β-unsaturated acid chloride with the anion of 5,5-dimethyl-1,3-cyclohexane-dione ( 2 ) gives either the enol ester ( 4 ) or a cyclized product ( 5 ), depending upon the reaction conditions. When the acid chloride is in excess, the ester is obtained. When the anion is in excess and the acid chloride is acryloyl or crotonyl, the cyclized product is obtained. A similar cyclized product ( 9 ) was formed from acryloyl chloride and an excess of the anion of 4-hydroxy-6-methyl-2-pyrone ( 8 ).     

PHOTOCHEMICAL ESTIMATION OF OXYGEN SOLUBILITY

Abstract— A photochemical technique for estimating the solubility of oxygen in a solvent has been developed and used to estimate the solubility of oxygen in chloroform. From a measurement of the change in rubrene concentration and rubrene fluorescence lifetime as a sealed rubrene solution is irradiated and from the fluorescence lifetime of rubrene in nitrogen, air and oxygen-saturated solvent the oxygen solubility constant and rate constant for oxygen quenching of rubrene fluorescence can be measured. For chloroform these values are 9.8 m M /atm and 7.9 10₉ M _-1 s_-1 respectively.     

The GHS and other correlation inequalities for a class of even ferromagnets

We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin –1/2 Ising models, ⁴ field theories, and other continuous spin models. The proofs are based on the properties of a classG ^– of probability measures which contains all measures of the form const exp(–V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ). A new proof of the GKS inequalities using similar ideas is also given.Supported in part by National Science Foundation Grant MPS 71-02838 A 04.Supported by National Science Foundation Grant MPS 74-24696.Supported in part by National Science Foundation Grant MPS 74-04870.     

Optimal crossing-free Hamiltonian circuit drawings of Kn

The number of crossing-free Hamiltonian circuits in planar drawings of K_n is studied. In particular, it is shown that for planar drawings of K_n, (1) there are drawings having as many as $\text{(}\text{3}\text{20}\text{)·10}{}^{\mathrm{<mtext>[</mtext><mtext>n</mtext><mtext>3</mtext><mtext>]</mtext>}}$ such circuits and (2) no drawing contains more than $\text{2·6}{}^{\mathrm{n?2}}\text{([}\text{n}\text{2}\text{])!}$