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.

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.

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].

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 an earlier paper [3] we considered an inversion formula for the Gauss transformation G defined by

1.1

We noted there that formally G is inverted by,

1.2

and we showed that if e-D2 is interpreted via the power series for the exponential function, that is if

1.3

then under certain conditions on φ,

1.4

The concept of a Boolean ring, as a ring A in which every element is idempotent (i. e., a2 = a for all a in A), was first introduced by Stone [4]. Boolean algebras and Boolean rings, though historically and conceptually different, were shown by Stone to be equationally interdefinable. Indeed, let (A, +, x) be a Boolean ring with unit 1, and let (A, ∪, ∩, ') be a Boolean algebra, where ∩, ∪, ', denote "union", " intersection", and "complement". The equations which convert the Boolean ring into a Boolean algebra are:

1. I

   Conversely, the equations which convert the Boolean algebra into a Boolean ring are:

2. II

The orthogonal latin squares displayed in [1] and [2] have the property that their row permutations are transformed amongst themselves by a permutation of order 7. In this note I present three examples of orthogonal latin squares of order 10 whose row permutations are transformed amongst themselves by a permutation of order 9.

Throughout this note let be a polynomial of degree n. The following results are immediate.

For R ≧ 1

1

If p(z) has no zeros in |z| < 1, then

2

Linear differential equations both ordinary and partial are often studied by means of Green's functions. One reason for this is that linearity permits superposition of solutions. A Green's function describes the "effect" of a point source, and the description of line, surface, or volume sources is achieved by superposing, that is to say, integrating, this function over the source distribution.

For equations with constant coefficients the use of integral transforms permits the calculation of such source functions in the form of integrals. Only in the simplest cases is explicit evaluation by elementary functions possible, and this has perforce led to the use of asymptotic estimates, which so thoroughly pervade the domain of applied mathematics.

A note by N. Kimura gives the sums

explicitly linearly interms of the sums for positive integers p.

We note here that by a similar simple method, the Bernoulli numbers and Euler numbers may be related similarly to these sums.