The cell growth problem is a combinatorial problem which may be stated as follows: A plane animal is made up of cells, each of which is a square of unit area. It starts as a single cell, and grows by adding cells one at a time in such a way that the new cell has at least one side in contact with a side of a cell already present in the animal. The problem is to find the number of different animals of area n, it being understood that animals which can be transformed into each other by reflections or rotations of the plane will be regarded as the same animal.

Let P be a closed region in the plane bounded by a simple closed curve, and let S be a simplicial dissection of P. We may say that S is a dissection of P into a finite number α of triangles so that no vertex of any one triangle is an interior point of an edge of another. The triangles are ‘'topological” triangles and their edges are closed arcs which need not be straight segments. No two distinct edges of the dissection join the same two vertices, and no two triangles have more than two vertices in common.

In (2), John von Neumann introduced the concept of a continuous ring as a generalization to the infinite limiting case of the total matric algebras over a division ring. Von Neumann sketched a theory of arithmetic for such continuous rings and asserted :

(⋆) every continuous ring contains purely transcendental elements c.

This means: for every polynomial p(t) = tm + Z1tm-1 + . . . + zm {m ≥ 1) which has all coefficients zi in the centre of , the element p(c) has a reciprocal in , that is, (p(c))-l exists such that p(c).(p(c))-l = (p(c))-l.p(c) = 1.

Let C(M, Q) be the Clifford algebra of an even dimensional vector space M relative to a quadratic form Q. When Q is non-degenerate, it is well known that there exists an isomorphism of the orthogonal group O(Q) onto the group of those automorphisms of C(M, Q) which leave invariant the space M ⊂ C(M, Q). These automorphisms are inner and the group of invertible elements of C(M, Q) which define such inner automorphisms is called the Clifford group.

If instead of the group O(Q) we take the group of similitudes γ(Q) or even the group of semi-similitudes Γγ(Q), it is possible to associate in a natural way with any element of these groups an automorphism or semi-automorphism, respectively, of the subalgebra of even elements C+(M, Q) ⊂ C(M, Q). Each one of the automorphisms of C+(M, Q) so defined can be extended, as it is shown here (Theorem 2), to an inner automorphism of C(M, Q), although the extension is not unique.

One important point in the determination of the automorphisms of the classical groups is the study of group-theoretic properties of the elements of order 2, that is, the involutions. The group of similitudes and the projective group of similitudes of a non-degenerate quadratic form Q are extensions of the orthogonal group and the projective orthogonal group, respectively. These extended groups may contain involutions which do not belong to the orthogonal group or the projective orthogonal group. To study the automorphisms of such groups, group-theoretic properties of some of these new involutions should be established.

The aim of this paper is to give some properties of a similitude T of ratio ρ whose square T2 is equal to the scalar multiplication by — p (it is always assumed that we are dealing with a field of characteristic ≠2). If ρ = — 1, T is an involution in the group of similitudes and for any ρ the coset of T in the projective group of similitudes is an involution.

Consider the differential equation

1.1

where

1.2

is regular in |z| < R.

The indicial equation associated with (1.1) is of the form

We shall denote the two roots of this equation by α and β, where . Corresponding to the root α there exists a unique solution of (1.1) of the following form

1.3

In those cases for which there exists a unique second solution of (1.1) of the form

1.4

Let ω be a continuous differential r-form defined in a bounded domain R of Euclidean n-space, En, where n ≧ 1 and 0 ≦ r ≦ n — 1. ω is called a flat form in R, (3, p. 263), if there exists a constant N such that for every (r + 1)-simplex σ contained in R, where |σ| designates the (r + 1)-volume of σ. For n = 1 and ω a zero form, flatness is the same thing as the usual Lip 1 condition. As is well known, a necessary and sufficient condition that a continuous real-valued function of one variable f(x) satisfy a Lip 1 condition over an interval (a, b) is that its upper and lower symmetric derivatives be bounded in (a, b) (4, pp. 22, 327). We intend to prove the analogue of this result for r-forms.

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.

Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

A finite sequence of open sets L1 L2, … , Ln is called a linear chain if each Li intersects only the L's adjacent to it in the sequence. The finite sequence is a circular chain if we also insist that the first and last links intersect each other. The 1-skeleton of the covering is an arc for a linear chain and a simple closed curve for a linear chain.

A compact metric continuum X is called snake-like if for each ∈ > 0, X can be covered by a linear chain of mesh less than ∈. Likewise, X is called circle-like if for each ∈ > 0, X can be irreducibly covered with a circular chain of mesh less than ∈. This definition is more restrictive than that given in (3, p. 210) for there a pseudo-arc is not circle-like but here it is. The present usage is in keeping with definitions of Burgess.

In a paper (5) published in the Proceedings of the Cambridge Philosophical Society, Primrose obtained the formulae for the number of points contained in a non-degenerate quadric in PG(n, s), the finite projective geometry of n dimensions based on a Galois field GF(s). In § 3 of the present paper the formulae for the number of p-flats contained in a non-degenerate quadric in PG(n, s) are obtained. In § 4 an interesting property of a non-degenerate quadric in PG(2k, 2m) is proved. These properties of a quadric will be used in solving some combinatorial problems of statistical interest in a later paper.

The concept of homotopy for homomorphisms of modules, suggested by analogy with fundamental properties of the topological homotopy, was developed by Eckmann and Hilton (3; 9; 10). In the present paper, this concept of homotopy is generalized to additive categories with an additional structure, and the theory of homotopy, including in particular various exact homotopy sequences, is established in full detail. It may be helpful to the reader to realize that most of our theorems, concepts, and constructions have their analogues in the homotopy theory of topological spaces. Many of these analogues may be found in a recent paper by Eckmann and Hilton (4). Explicit references will be given at various places in the paper.

The purpose of this paper is to characterize weak compactness in Orlicz spaces. Though an Orlicz space is a Banach space, it will be viewed from the standpoint of the theory of Köthe spaces. Considering that a norm-bounded subset is not weakly compact in general, we shall give some criteria for weak compactness in terms of the functional defining an Orlicz space. Because weak compactness is closely connected with the continuity of the semi-norms on the conjugate space, at the same time some properties of continuous semi-norms on Orlicz spaces will be brought to light.

The first characterization (Theorem 1) is concerned with degree of smoothness of the functional at the origin. In Theorem 2 a connection between weak compactness and boundedness (by another functional) is obtained. In Theorem 3 the result in Theorem 2 is stated as a proposition about continuous seminorms.

Let X be an n-square matrix with elements in a field F. The permanent of X is defined by

1.1

where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.

Let Mn be the linear space of n-square matrices with real elements. For a matrix X = (xij) ∈ Mn the permanent is defined by

where σ runs over all permutations of 1, 2, …, n. In (2) Marcus and May determine the nature of all linear transformations T of Mn into itself such that per T(X) = per X for all X ∈ Mn. For such a permanent preserver T, and for n < 3, there exist permutation matrices P, Q, and diagonal matrices D, L in Mn, such that per DL = 1 and either

or

Here X′ denotes the transpose of X. In the case n = 2, a different type of transformation is also possible.

The class of finite groups G of the form ABA, where A and B are subgroups of G, is of interest since it includes the finite doubly transitive groups, which admit such a representation with A the subgroup fixing a letter and B of order 2. It is natural to ask for conditions on A and B which will imply the solvability of G. It is known that a group of the form AB is solvable if A and B are nilpotent. However, no such general result can be expected for ABA -groups, since the simple groups PSL(2,2n) admit such a representation with A cyclic of order 2n + 1 and B elementary abelian of order 2n. Thus G need not be solvable even if A and B are abelian.

In (3) Herstein and the author have shown that G is solvable if A and B are cyclic of relatively prime orders; and in (2) we have shown that G is solvable if A and B are cyclic and A possesses a normal complement in G.

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

Let G and H be locally compact abelian groups with character groups G*, H*, and let < . , . > denote the pairing between a group and its dual.

In 1952 Kaplansky proved the following result, using the structure of locally compact abelian groups and category arguments.

Theorem 1.1. Let τ: G → H be an algebraic homomorphism for which there is a dual τ* : H* → G* (so that < rg, h* > = < g, τ*h* > for all g in G, h* in H*). Then τ is continuous.

The result bears a striking similarity to a well-known fact about Banach spaces which is a consequence of uniform boundedness; the present note is devoted to an analogous “uniform boundedness” for groups, which yields a non-structural proof of Kaplansky's theorem.

The list of known finite simple groups other than the cyclic, alternating, and Mathieu groups consists of the classical groups which are (projective) unimodular, orthogonal, symplectic, and unitary groups, the exceptional groups which are the direct analogues of the exceptional Lie groups, and certain twisted types which are constructed with the aid of Lie theory (see §§3 and 4 below). In this article, it is proved that each of these groups is generated by two of its elements. It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group (1), or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.

The purpose of this note is to establish an identity involving the cyclotomic polynomial and a function of the Ramanujan sums. Some consequences are then derived from this identity.

For the reader desiring a background in cyclotomy, (2) is mentioned. Also, (4) is intimately connected with the following discussion and should be consulted.

The cyclotomic polynomial Fn(x) is defined as the monic polynomial whose roots are the primitive nth roots of unity. It is well known that

2.1

For the proof of Corollary 3.2 it is mentioned that Fn(0) = 1 if n > 1 and that Fn(x) > 0 if |x| < 1 and 1 < n.

The Ramanujan sums are defined by

2.2

where the sum is taken over all positive integers r less than or equal to n and relatively prime to n. It is also well known that

2.3

where the sum is taken over all positive divisors d common to n and k.

Several studies have been made to obtain larger classes of non-associative algebras from classes of algebras with a known structure. Thus, we have right alternative algebras (2)* and non-commutative Jordan algebras (6), (7), (8), and (9). These algebras are defined by a subset of the set of identities of the algebras from which they derive their names. Also, Albert (1), among others has studied Jordan admissible algebras. This paper is concerned with algebras which are related to Lie algebras in that they satisfy some of the identities of a Lie algebra and are Lie admissible. Theorem 2 answers a question raised by Albert in (1).