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The term Geometrography is new to mathematical science, and it may be defined, in the words of its inventor, as “the art of geometrical constructions.”
We show that every simple graph of order 2r and minimum degree ≧4r/3 has the property that for any partition of its vertex set into 2-subsets, there is a cycle which contains exactly one vertex from each 2-subset. We show that the bound 4r/3 cannot be lowered to r, but conjecture that it can be lowered to r + 1.
§ 1. Extensions of Pascal's theorem are already known if we look at the matter from certain particular points of view – an extension of the theorem from the more general point of view is still a desideratum.
§ 1. This paper deals with certain formulae which, though probably not all new, have not appeared in the text-books. They were suggested to the writer while engaged in discussing the expression for the intensity of the transmitted beam in the Lummer Gehrcke Interference Spectroscope, viz.
We study, from the point of view of abelian and Kummer surfaces and their moduli, the special quintic threefold known as Nieto's quintic. It is, in some ways, analogous to the Segre cubic and the Burkhardt quartic and can be interpreted as a moduli space of certain Kummer surfaces. It contains 30 planes and has 10 singular points: we describe how some of these arise from bielliptic and product abelian surfaces and their Kummer surfaces.
A new approach to fractional integrals of distributions on a half-line is suggested. The results admit an extension to a large class of Mellin convolutions.
A corollary of the main theorem presented in this note is a generalisation of the well-known result that a self-adjoint square root of a positive self-adjoint compact linear map in a Hilbert space is itself a compact linear map. The method used here exploits the techniques developed recently in the study of k-set contractions ((1), (2)).
In a paper which appeared in the Proceedings of the Edinburgh Mathematical Society, Vol. XXXII., Session 1913–14,* I showed how the application of Laplace's transformation to certain linear differential equations enables us to solve some homogeneous integral equations of the first and second kinds, and I obtained, by an extension of this method, the solution of integral equations whose nucleus is of the form f(zt) or ef(z)f(t).