In a previous paper (1) the notion of the α-width ∈A(α) of a (0, 1)-matrix A was introduced, and a formula for the minimal α-width taken over the class of all (0, 1)-matrices having the same row and column sums as A, was obtained. The main tool in arriving at this formula was a block decomposition theorem (1, Theorem 2.1; repeated below as Theorem 2.1) that established the existence, in the class generated by A, of certain matrices having a simple block structure. The block decomposition theorem does not itself directly involve the notion of minimal α-width, but rather centres around a related class concept, that of multiplicity. We review both of these notions in § 2, together with some other pertinent definitions and results.

In n-dimensional Euclidean space En, where we shall throughout assume that n ≥ 2, the maximum number of (n — 1)-dimensional spheres which can be mutually orthogonal is n + 2, and it is well known that the sum of the squares of the reciprocals of their radii is zero, so that the spheres cannot be all real. The maximum number of such spheres which can be mutually tangent is also n + 2, and in 1936 Soddy (7) indicated the beautiful relation connecting their radii. These two formulae are the particular cases γ = 0, γ - 1 of Theorem 1 below, which gives the relation connecting the radii of a set of n + 2 such spheres when every pair is inclined at a given non-zero angle 0, where γ is written for cos θ.

Projectors, spectral functions, carriers, and collections of these objects are some of the tools which have been used to study vector lattices. One of our objectives in this paper is to show that these various approaches are not essentially different. We do this by proving that each of the above-mentioned objects can be identified with a collection of prime ideals.

Corresponding to every graph, bipartite graph, or directed bipartite graph there exists a directed graph which is connected if and only if the original graph is connected.

In this paper, it is shown that for every directed graph there exists a certain bipartite graph such that the directed graph is connected if and only if the bipartite graph is irreducible. Other connections between reducibility and connectivity are established.

Let Σ represent the class of analytic functions

(1)

which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion

(2).

In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:

Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

The Bernoulli polynomials of order k, where k is a positive integer, are defined by

Bm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

In 1947 Einstein and Strauss (2) proposed a unified field theory based on a four-dimensional manifold characterized by a nonsymmetric tensor gij and a non-symmetric connection , where

(1)

Using a variational principle in which gij and are independently varied, the above authors obtain the equivalent of the following field equations:

(2a)

(2b)

In these equations a comma denotes partial differentiation with respect to the co-ordinates of the manifold, Wij is the Ricci tensor formed from and the notation

for the symmetric and skew-symmetric parts of geometric objects Q is employed.

The usual development of the Lebesgue integral starts with a measure that may have been derived from some simpler set function, its associated class of measurable sets, the corresponding set of measurable functions, and operations which ultimately define the integral of any given function of this class, except for certain ones which are unbounded above and below. Here we propose to define a process of integration with respect to a set function more general than a measure. This process allows us to integrate virtually all functions real-valued on our space. The integrals thus obtained are all completely additive on a certain completely additive class of sets. Under rather mild hypotheses, we are able to delineate this class of sets explicitly.

The main object of this paper is to prove the following:

Theorem. Let f1, … ,fk be linearly independent continuous functions on a compact space. Then for 1 ≤ s ≤ k there exist real numbers aij, 1 ≤ i ≤ s, 1 ≤ j ≤ k, with {aij, 1 ≤ i, j ≤ s} n-singular, and a discrete probability measure ε*on, such that

(a) the functions gi = Σj=1kaijfj 1 ≤ i ≤ s, are orthonormal (ε*) to the fj for s < j ≤ k;

(b)

The result in the case s = k was first proved in (2). The result when s < k, which because of the orthogonality condition of (a) is more general than that when s = k, was proved in (1) under a restriction which will be discussed in § 3. The present proof does not require this ad hoc restriction, and is more direct in approach than the method of (2) (although involving as much technical detail as the latter in the case when the latter applies).

The groups in question are generated by elements A and B subject to the relations

in which α and β are fixed integers. We prove:

Theorem. Each group of the class just presented is finite when neither α nor β equals 1, and is nilpotent. Its order is a factor of 27 (α — 1) (β — l)∈8where ∈ is the greatest common divisor of α — 1 and β — 1, and its nilpotency class is at most 8.

We denote the commutator X-1Y-1XY by [X, Y], and Y-1XY by XY. If X1, X2, … , Xr are elements of some group then {X1, X2, … , Xr} will mean the subgroup which they generate.

A difference set is a set D = {d1, d2, … , dk] of k distinct residues modulo v such that each non-zero residue occurs the same number of times among the k(k — 1) differences di — dj, i ≠ j. If λ is the number of times each difference occurs, then

(1)

When we wish to emphasize the particular values of v, k, and λ involved we will call such a set a (v, k, λ) difference set. Another (v, k, λ) difference set E = {e1, e2, … ek} is said to be equivalent to the original one if there exist a and t such that (t, v) = 1 and E = {a + td1, … , a + tdk}. If t = 1 we will call the set E a slide of the set D. If D = E, then t is called a multiplier of D.

Let there be given a parabolic differential operator

where A is a second order linear elliptic (<0) differential operator in an open set Ω ⊂ Rn, having coefficients depending on x ∈ Ω and t ∈ [0, ∞]. Recently, Protter (1) investigated the asymptotic behaviour of functions u(x, t) that satisfy the differential inequality

(1.1)

Under suitable restrictions on the functions ci(t) and the coefficients of A, he proved that any solution of (1.1), subject to certain homogeneous boundary conditions, that vanishes sufficiently fast, as t → ∞, must be identically zero in Ω × [0, ∞). For example, conditions are given under which no solution of (1.1) can vanish faster than e-λt, ∀ λ > 0, unless identically zero.

A lattice ∧ in euclidean n-space, En is a group of vectors under vector addition generated by n independent vectors, X1, X2 … , Xn, called a basis for the lattice. The absolute value of the n × n determinant the rows of which are the co-ordinates of a basis is called the determinant of the lattice and is denoted by d(∧). For any lattice ∧ there is a unique minimal positive number r such that, if spheres of radius r are placed with centres at all points of ∧, the entire space is covered. The density of this covering may be defined as (Jnrn)/(d(∧)) where Jn is the volume of the unit sphere in n-dimensional euclidean space. This density will be denoted by θn(∧). The density of the most efficient lattice covering of n-space by spheres, θn is the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.

If A is an H*-algebra, then the orthogonal complement of a closed right (left) ideal I is a closed right (left) ideal P. Saworotnow (7) considered Banach algebras which are Hilbert spaces and in which the closed right ideals satisfy the complementation property of an H*-algebra. In our right complemented Banach algebras we drop the requirement of the existence of an inner product and only assume that for every closed right ideal I there is a closed right ideal IP which behaves like the orthogonal complement in a Hilbert space (Definition 1). Thus our algebras may be considered as a generalization of Saworotnow's right complemented algebras.

Let us begin with a simple result.

Proposition. Let X be a non-orientable closed surface differentiably imbedded into the euclidean 4-space R4. Then there is a line in R4 which intersects X at more than two points.

Proof. Let c be any point of R4 and let r be the smallest number such that X is contained in the closed 4-spheroid W of centre c and radius r. Clearly the boundary 3-sphere S of W intersects X.

In this paper an attempt is made to prove some of the basic theorems on the osculating spaces of a curve under minimum assumptions. The natural approach seems to be the projective one. A duality yields the corresponding results for the characteristic spaces of a family of hyperplanes. A duality theorem for such a family and its characteristic curve also is proved. Finally the results are applied to osculating hyperspheres of curves in a conformai space.

The analytical tools are collected in the first three sections. Some of them may be of independent interest.

In the analysis of mixed boundary value problems by the use of Hankel transforms we often encounter pairs of dual integral equations which can be written in the symmetrical form

(1.1)

Equations of this type seem to have been formulated first by Weber in his paper (1) in which he derives (by inspection) the solution for the case in which α — β = ½, v = 0, F ≡ 1, G ≡ 0.

The first direct solution of a pair of equations of this type was given by Beltrami (2) for the same values of α— β and v with G(p) ≡ 0 but with F(ρ) arbitrary.

Let A1 denote the class of all complex functions on the unit circle which are sums of absolutely convergent trigonometric series; that is to say, the class of all f of the form

Similarly, if 1 < p < ∞, let Ap be the class of all complex functions f on the circle whose Fourier coefficients

satisfy the condition

A graph G is called complete if any two of its vertices are connected by an edge; a set of vertices of G are said to be independent if no two of them are connected by an edge. It follows from a well-known theorem of Ramsay (1) that for each pair of positive integers k, l there is an integer f(k, l), which we take to be minimal, such that every graph with f(k, l) vertices either contains a complete graph of k vertices or a set of l independent points.