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.

The purpose of this note is to characterize "block-graphs", a collection of graphs defined by a construction involving certain subgraphs called "blocks". A related operation on a graph leads to the study of "cut-point-graphs". The precise relationship between these two operations is made explicit. In order that this characterization be self-contained, we include the necessary definitions.

Let A: a1 < a2 < … be a sequence of positive integers and let f(A:n) = f(n) denote the number of representations of n as the sum of one or more consecutive a's. Although f(n) will usually oscillate rapidly, an asymptotic estimate for its average value

can be obtained for many sequences A.

In [3] K. Goldberg showed that if A is a 0-1 matrix that satisfies

(1.1)

then for some permutation matrix P, PAP* is a direct sum of matrices each of which is either zero or consists only of ones. More recently J. L. Brenner [1] proved that if A ∦ 0 (i.e. A has non-negative entries) and satisfies (1) then there exists a permutation matrix P such that PAP* = A1⊕ … ⊕An in in which each Ai is either 0 or all positive, Ai > 0, and satisfies (1) as well.

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)).

Let A: {a1, ≤ a2, ≤ ߪ ≤ at} be a set of non-negative integers not exceeding n, and let rn(A;m) = rr(m) denote the numbers of representations of m as the sum of k a's, order taken into account, i. e.

1

Since 0 ≤ ai1 + ai2 + … + aik ≤ kn, we have

2

A number of interesting Diophantine equations are of the a form f(n) = f(a) f(b). Thus the case f(a) = aa has been studied by C. Ko [2] and W. H. Mills [3] and a class of non-trivial solutions has been found, though whether these give all the solutions is still unsettled. The case f(n) = n! has been mentioned by W. Sierpinski. Here the situation is that besides the trivial solutions m! = m! 1! and (m!-1)! m! = (m!)! and the special solution 10! =7! 6! no other solutions are known, nor are they known not to exist. In the present note we show that the equation in the title has no solutions. A sketch of a somewhat different proof that this equation has at most a finite number of solutions was recently communicated to the author by P. Erdös.

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.

Marcus [2] has proved the following theorem.

Suppose A is a non-negative normal matrix satisfying p(A) = 0 in which p(λ) is a monic polynomial no two of whose non-zero roots have the same modulus. Then there exists a permutation matrix P such that PAP* is a direct sum, PAP* = A1 ⊕ A2 ⊕ … ⊕ Am, in which each Ai is either O or primitive.

This note gives a generalisation of this result, dropping the non-negative assumption and weakening the normality assumption.

The eigenvalues of a second order self-adjoint elliptic differential operator on Riemannian n-space R will be considered. Our purpose is to obtain asymptotic variational formulae for the eigenvalues under the topological deformations of (i) removing an ɛ -cell (and adjoining an additional boundary condition on the boundary component thereby introduced); and (ii) attaching an ɛ -handle, valid on a half-open interval 0 < ɛ ≤ ɛo. In particular the formulae will exhibit the non-analytic nature of the variation. Similar variational problems for singular ordinary differential operators have been considered by the writer in [3] and [4].

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.

A sequence of non-negative numbers, μ0 μ1 … μn, is called completely monotonic [5, p. 108] if (-1)n Δn μk ≧ 0 for n, k = 0, 1, 2, …. Such sequences occur in many connexions, such as the Hausdorff moment problem and Hausdorff summability [1, Chapter XI, 5, Chapters III and IV].

It is natural to inquire as to the circumstances under which the inequality "≧" above can be strengthened to ">". As it happens, this can be done always, except for sequences all of whose terms past the first are identical. The formal statement follows. (In it, as above, Δnμk is the n-th forward difference, i. e. Δ0μk = μk; Δnμk = Δn-1 μk+1 - Δn-1μk.)

Let f(n) denote the number of dissections of a regular n-gon into n-2 triangles by n-3 non-intersecting diagonals. It is known that

and that

1

for n = 3, 4, . . . , where f(2) = 1 by definition. (For pertinent references on this and related problems see, e. g. , Motzkin [2].) The object of this note is to obtain a simple expression for g(n) , the number of such dissections remaining when those which differ only by a rotation, reflection, or both are not considered as being different. For convenience we shall let g(2) = 1 and f(k) = 0 when k is not an integer.

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 Theorem 2.20 of his Geometric Algebra, Artin shows that any ordering of a plane geometry is equivalent to a weak ordering of its skew field. Referring to his Theorem 1. 16 that every weakly ordered field with more than two elements is ordered, he deduces his Theorem 2.21 that any ordering of a Desarguian plane with more than four points is (canonically) equivalent to an ordering of its field. We should like to present another proof of this theorem stimulated by Lipman's paper [this Bulletin, vol.4, 3, pp. 265-278]. Our proof seems to bypass Artin's Theorem 1. 16; cf. the postscript.

In this note we give a characterisation of ordered skew fields by certain properties of the negative elements.

Properties of negative elements of a skew field K are then interpreted as statements about "betweenness" in any affine geometry over K, or as statements about "separation" in any projective geometry over K.

In this way, our axioms for ordered fields permit of immediate translation into postulates of order for affine or projective geometry (over a field). The postulates so obtained seem simple enough to be of interest, (cf. [4], p. 22)

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

The (γ, r) summation method, 0 < r < 1, is the "circle method" employed by G. H. Hardy and J. E. Littlewood. It is also known as the Taylor method. Its Lebesgue constants, say L(Tr, n), n = 1, 2, …, were studied by K. Ishiguro [1] in the notation L*(n;1-r). He noted that

1

where Im{z} denotes the imaginary part of the complex number z, and proved that

2

Here

3

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.

The object of this paper is to present results concerning the structure of 3-connected graphs and of 5-chromatic and 6-chromatic graphs and also a theorem on contraction and a theorem of Turan type. The Axiom of Choice is assumed.