Chemistry of Succinylsuccinic Acid Derivatives. Part I. On the photodimerization of diethyl succinylsuccinate

Topochemically controlled photochemical dimerization of diethyl succinylsuccinate in the solid is reported. The structure evaluation by elemental and spectroscopic analysis of the product is given. The participation of chelated enols in photochemical cycloadditions in the crystalline solid state is here demonstrated for the first time.     

Ramsey's theorem—A new lower bound

This paper gives improved asymptotic lower bounds to the Ramsey function R(k, t). Section 1 considers the symmetric case k = t while the more general case is considered in Section 2.     

On the number of lattice points between two enlarged and randomly shifted,copies of an oval

Summary Let A be an oval with a nice boundary in ²,R a large positive number,c>0 some fixed number and a uniformly distributed random vector in the unit square [0,1]². We are interested in the number of lattice points in the shifted annular region consisting of the difference of the sets {(R+c/R)A–} and {(R–c/R)A–}. We prove that whenR tends to infinity, the expectation and the variance of this random variable tend to 4c times the area of the set A, i.e. to the area of the domain where we are counting the number of lattice points. This is consistent with computer studies in the case of a circle or an ellipse which indicate that the distribution of this random variable tends to the Poisson law. We also make some comments about possible generalizations.     

Discrepancy in arithmetic progressions

It is proven that there is a two-coloring of the first integers for which all arithmetic progressions have discrepancy less than . This shows that a 1964 result of K. F. Roth is, up to constants, best possible.

     

Synthesis of(±)-descarboxylquadrone

The synthesis of a biologically active anlogue of the antitumor agent quadrone, termed descarboxylquadrone ($\text{3}$), is described.