Suppose that

1.1

is a transcendental integral function. In this article we develop the theory initiated by Wiman [22, 23] and deepened by other writers including Valiron [18, 19, 20], Saxer [15], Clunie [4, 5] and Kövari [10, 11], which describes the local behaviour of f(z), near a point where | f(z) | is large, in terms of the power seriesf of f(z).

W. Burnside [3, p. 343] showed that a cyclic group of order pm (p prime, m > l) cannot occur as a regular subgroup of a simply transitive primitive group. (For definitions and notation see [9].) Groups which are contained regularly in a primitive group G only when G is doubly transitive are therefore called B-groups [9, p. 64].

To give a geometric interpretation to the inverted incidence relation between the flats of a geometry has for years been a tempting idea in combinatorial geometries [1]. If G is a geometric lattice, the inverted lattice G' is not necessarily geometric.

The goal of this paper is to relate plane cubic graphs which are not bipartite to plane bipartite cubic graphs which have properties (face-coloring with three colors, existence of two-factors whose cycles are boundaries of faces) characterizing this class of graphs.

Let C by a Cayley-Dickson algebra over an algebraically closed field F of characteristic 0. A multiplication table for a basis of this 8-dimensional alternative algebra can be found in [3], page 137, where we take α = β = γ = — 1.

Consider the 1st order linear system

1

when An(t) is real valued and continuous on I, ai, i+1(t) ≠ 0, aij(t) ≡ 0 for j ≥ i + 2, t ∊ I and i = 1, …, n — l. In what follows, components cj of a vector c will be identifiable by the superscript. The purpose of this note is to give a simple induction proof of the following theorem of Frobenius [1].

The concept of standard representations of simple Lie algebras was introduced by I. Z. Bouwer [1], One of the difficulties was that of existence. The order zero standard representations are simply those having a dominant weight vector and these have been completely characterized, for example in [2].

Z. Opial [11] proved in 1960 the following theorem:

Theorem 1. If u is a continuously differentiable function on [0, b], and if u(0)= u(b)=0 and u(x) > 0 for x ∊ (0, b), then

1

where the constant b/4 is the best possible.

In [5] N. D. Lane and P. Scherk discuss arcs in the conformai (inversive) plane which are met by every circle at not more than three points; i.e., arcs of cyclic order three. This paper is concerned with the analysis of normal arcs of cyclic order four in the conformai plane.

This paper gives several monotonicity properties of all oscillatory solutions of equations with separable and nonseparable nonlinearities which are more general than the Emden- Fowler equations

*

Principally, if x(t) is an oscillatory solution, conditions are given such that; if a(t)↑ ∞ as t → ∞, then x(t) → 0; and, if a(t) ↓ 0 as t → ∞, then lim sup | x(t) | = ∞.

In his paper [1] on homogeneous spaces W. L. Chow states that “ Any one-toone adjacency preserving transformation of the Grassmann space of all the [r] of Sn(0 < r < n— 1) onto itself is a transformation of the basic group of the space.” In the proof both the transformation and its inverse are assumed to be adjacency preserving. See also Dieudonne [2] p. 81.

Let E be a real Banach space with norm ρ. Let S={x ∊ E: ρ(x) = 1}. A norm on E is admissible if it generates the same topology as ρ.

The normρ is Gateaux differ-entiable if for each x ∊ S and u ∊ E

exists.

Throughout this note we shall use the following conventions and notations: All matrices have entries in the field of complex numbers. I denotes the identity matrix with compatible dimensions. A* is the conjugate transpose of a matrix A. A being self adjoint means A = A*.

Theorem. “If a crunode of a real, irreducible, plane, algebraic curve changes into an acnode via the intermediary stage of a real cusp, two real inflexions are introduced in a neighborhood of the double point.”

We show that any Gaussian process can be derived in a simple manner from a Markov process if it has zero mean and covariance identical to the covariance of a real valued function of a temporally homogeneous Markov process.

Generalized Sverdrup's lemma, Kabe [5], is used here to give a more direct treatment of less than full rank regression model.

In this note it is shown that a contractive matrix selfmap T of l∞ has T2 a contraction; hence the techniques used in showing ITNC for lp(1 ≤ p < ∞), c0, and c do not apply to l∞.