Let us define a planar partition of a graph G as a pair {H1, H2} of subgraphs of G with the following properties

1. (i) Each of H1 and H2 includes all the vertices of G.

2. (ii) Each edge of G belongs to just one of H1 and H2.

3. (iii) H1 and H2 are planar graphs.

It is not required that H1 and H2 are connected. Moreover either of these graphs may have isolated vertices, incident with none of its edges.

We describe a graph having a planar partition as biplanar.

Let ∧ denote the lattice of points X = (x1, …, xn) with integral coordinates. A basis of ∧ is a set of n points X1, …, Xn of ∧ such that every point of ∧ is expressible in the form where ui are integers. It is easy to see that points X1, …, Xn of ∧ form a basis if, and only if

(1)

where Xr = (x1(r), …, x1(r)).

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

The equation of motion of a charged particle in a combined gravitational and electromagnetic field is cast in the classical Hamilton-Jacobi form and then applied to the special case of a Schwarzschild metric, leading to the well established equation of planetary motion.

In 1948 Gabor Szegő [9] gave four proofs of a remarkable inequality communicated to him by Paul Turán, who later published an original proof [10]. The Turán theorem states that if Pn(x) is the Le gendre polynomial, then

1.1

with equality holding only when |x| = 1.

Since then many similar inequalities have been found for various special functions, particularly for the Legendre and Hermite polynomials. Reference may be had to the recent work of Danese [2] and Chatterjea [1]. Danese gives an extensive bibliography.

In this paper we will construct an example showing that the problem posed in [1] has a negative answer. Two more theorems on the subject treated in [1] will be included.

Let Io = [0, 1], R the reals, and let, for A ⊂ R, Ao be the interior of A. Let {xn} be a sequence in [0, 1> such that 0 = x1 < x2 < … and lim xn = 1. For each n, let In be closed interval having x as its midpoint (except for n = 1 in which case x1 is the left endpoint of I1) such that In ∩ Im= ϕ, and the metric density relative to Io of at 1 is zero. Let Jn be a closed interval in In concentric with In (except for n = 1, where J1 has x1 as its left endpoint) whose length is half that of In

Dans un mémoire paru en 1960 [1], G. Thierrin a étudié la structure d'une classe d'anneaux, celle des anneaux bipotent à droite. Ce travail fait suite à ce mémoire et a pour objet d' étudier, dans un première partie, la structure d'une classe particulière d' anneaux bipotents a droite, celle des anneaux complètement bipotents à droite. Dans une deuxième partie, nous nous proposons d' étudier la structure df une autre classe dr anneaux, celle des anneaux d-bipotents à droite.

In this paper an attempt is made to generalize the rather special role played by certain "Young subgroups" of Sn. In Theorem 1, the substitutional equation involving Young's raising operators for Sn is generalized to apply to an arbitrary subgroup H of a finite group G. Applying this to Sn, it is shown how the multiplicities mλH can be calculated in terms of the raising operators in Theorem 2, which calculations are illustrated by a simple example.

When two uniform functions , are given, each with a finite radius of absolute convergence R1 R2 respectively, and {λn}, {μν} are real positive increasing sequences tending to infinity, a theorem due to Eggleston [1], which is a generalisation of Hurwitz1 s composition theorem, gives information about the position of the singularities of a composition function h(z), which is assumed to be uniform, in terms of the position of the singularities of f(z) and g(z). This result can be extended to Dirichlet series with real exponents by use of the transformation z = es.

All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).

A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.

It is well known that sets of constant width share several properties with spheres. In this note we consider a simple property of the circle and we show that it is shared by every plane set of constant width. As an application we derive a stronger form of the following theorem of D. Gale, [1]: every plane set of diameter 1 is a union of three sets of diameters not exceeding √3/2, and this constant is best possible. We shall make free use of the more elementary properties of convex sets and of sets of constant width; for these properties and for the terminology see the standard reference [2], or [3].