The Mathematical Gazette: Volume 45 -

Some Problems in Combinatorics
H. G. Forder
Published online by Cambridge University Press:

03 November 2016, pp. 199-201
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The following problems are merely different forms of one
1. In how many ways can a convex polygon of n + 1 sides be split into triangles by non-intersecting diagonals? We represent such a splitting by a symbol. Take one side of the polygon as base and letter the other n sides in order by a, b, c,…. If a triangle with sides lettered x, y is cut off, denote the diagonal by (xy). If this diagonal and side z form the sides of another triangle of the dissection, denote the third side of that triangle by ((xy)z). Similarly for two diagonals, as the figure illustrates. Finally the symbol for the unlettered base is the symbol for the dissection.

On the D Calculus for Linear Differential Equations with Constant Coefficients
A. Robinson
Published online by Cambridge University Press:

03 November 2016, pp. 202-206
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1. Introduction. While the operational calculus for the solution o initial value problems for linear differential equations with constam coefficients is now commonly introduced in a thoroughly satisfactory manner—either by means of the Laplace transform (e.g. ref. 1) of by the method of J- Mikusinski (ref. 2)—the same cannot be said of the simpler method for the general integration of such equations which is known as the D calculus. In fact, I cannot recall any text which gives a consistent definition of the meaning of expressions of the type [F(D)/G(D)]y, where F and G are polynomials. The reason for this state of affairs seems to be that the scope of the D calculus is rather limited so that the validity of the result can be verified in each case (e.g. ref. 3). In particular, this applies to the decomposition of an operator F-1(D) into partial fractions, which is the central step: in the solution of an equation with nonvanishing right hand side of general form. Nevertheless a rational approach to the entire problem is perhaps not out of place. This is attempted in the present paper.

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