Let p be a rational prime, G a non-trivial finite p group, and K the field of p elements, regarded as a trivial G-module according to context; then we define:

d(G) = dimKH1(G, K), the minimal number of generators of G,

r(G) = dimKH2(G, K),

r′(G) = the minimal number of relations required to define G,

where, in the last equation, it is sufficient to take the minimum over those presentations of G with d(G) generators. It is well known (see § 2) that the following inequalities hold:

We shall consider only finite p-groups, so that the class of groups with r = d coincides with that consisting of those groups whose Schur multiplicator is trivial.

A group P is said to be a CEF-group if, for every countable group G, there is a factor group of P which contains a subgroup isomorphic to G. It was shown by Higman, Neumann, and Neumann [5] that the free group of rank two is a CEF-group. More recently, Levin [6] proved that if P is the free product of two cyclic groups, not both of order two, then P is a CEF-group. Later, Hall [3] gave an alternative proof of Levin's result.

In this paper we give a new proof of Levin's result (Theorem 2). The proof given yields information about the factor group H of P in which a given countable group G is embedded; for example, if G is given by a recursive presentation (this concept is denned in [4]), then a recursive presentation is obtained for H, and certain decision problems (in particular, the word problem) are solvable for the recursive presentation obtained for H if and only if they are solvable for the given recursive presentation of G.

The results of Rudin in [7] show that under certain conditions, the maximum modulus principle characterizes the algebra A (G) of functions analytic on an open subset G of the plane C (see below). In [2], Birtel obtained a characterization of A(C) in terms of the Liouville theorem; he proved that every singly generated F-algebra of continuous functions on C which contains no non-constant bounded functions is isomorphic to A(C) in the compact-open topology. In this paper we show that the Montel property of the topological algebra A (G) also characterizes it. In particular, any Montel algebra A of continuous complex-valued functions on G which contains the polynomials and has continuous homomorphism space M (A) homeomorphic to G is precisely A(G).

A famous theorem of Levitzki states that in a left Noetherian ring each nil left ideal is nilpotent. Lanski [5] has extended Levitzki's theorem by proving that in a left Goldie ring each nil subring is nilpotent. Another important theorem in this area which is due to Herstein and Small [3] states that if a ring satisfies the ascending chain condition on both left and right annihilators, then each nil subring is nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields both Lanski's theorem and Herstein- Small's theorem. We make use of the ascending chain condition on principal left annihilators in order to obtain, at an intermediate step, a theorem (Theorem 1.1) which produces sufficient conditions for a nil subring to be left T-nilpotent.

Bernstein's classical integral representation theorem for completely monotonie functions can be proved most elegantly, on a commutative semigroup with identity, by the integral version of the Kreĭn-Milman theorem [2]. The key to this approach is the identification (as exponentials) of the extremal points of the normalized completely monotonie functions. Alternate proofs of this identification are given in § 1. The first (Corollary 1.3) is based on the Kreĭn-Milman theorem and the second (see remarks following Corollary 1.5) is derived from elementary analytic techniques. Other interesting facts about completely monotonie functions are mentioned in passing. For example, we observe that the normalized completely monotonie functions form a simplex (Corollary 1.4). In Corollary 1.6 we note that the product of completely monotonie functions corresponds to the convolution of their representing measures. Thus the normalized completely monotonie functions form an affine semigroup [3].

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

In what follows, we suppose that ƒ(z) = Σ0∞anzn is regular for |z| < 1. Let

and

Then (see, for example, [6, pp. 235-236]), for 0 ≦ r < ρ < 1, we have:

The following results are well known.

Our aim in this paper is to continue our investigation of the homology of deleted products of finite, contractible, 2-dimensional polyhedra. In [1], we observed that if X is such a polyhedron, then a homeomorph of X can be constructed by starting with a 2-simplex and appending n-simplexes (n = 1, 2). In this paper, we are concerned with those polyhedra which have the property that if they are constructed as above, then at some stage we are forced to add to Xi–1 a 2-simplex τ at two of its 1-faces, 〈u3, u1〉 and 〈u3, u2〉, where there is a simple closed curve S in ∂(St(u3, Xi–1)) such that u1 and u2 are not in S but every sequence of 1-simplexes in ∂(St(u3, Xi–1)) from u1 to u2 intersects S. The cone over the complete graph on five vertices and the cone over the houses-and-wells figure are examples of such spaces.

1. Definitions and notation. A generalized Steiner system (t-design, tactical configuration) with parameters t, λt, k, v is a system (T, B), where T is a set of v elements, B is a set of blocks each of which is a k-subset of T (but note that blocks bi and bj may be the same k-subset of T) and such that every set of t elements of T belongs to exactly λt of the blocks. If we put λt = u we denote by Su(t, k, v) the collection of all systems with these parameters. Thus Q ∈ Su(t, k, v) means Q = (T, B) is a system with the given parameters. If λt = u = 1, we write S(t, k, v) instead of S1(t, k, v) and refer to the system as a Steiner system. If t = 2, the system is called a balanced incomplete block design.

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.

In § 2, a fibre-type construction for graphs is given which, applied to block designs with λ = 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande [22], in which the number of points in the intersection of any pair of blocks attains only two values.

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.

For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.

In this paper, we consider the extent to which a local Noether lattice (ℒ, M) is characterized by the sub-multiplicative lattice, denoted δℒ, of M-primary elements. (Here we use the notation (ℒ, M) to indicate that M is the maximal element of ℒ.) In particular, we call ℒM-complete if, given any decreasing sequence {Ai} of elements and any n ≧ 1, it follows that Ai ≦ A V Mn for large i, where A = ΛAi And we show that, given two Mi-complete local Noether lattices (ℒ1, M1) and (ℒ2, M2), with δℒ1 ≅ δℒ2, it follows that ℒ1 ≅ ℒ2. Further, we show that any local Noether lattice (ℒ, M) is a sublattice of a local Noether lattice (ℒ*, M) which is M-complete and such that δℒ = δℒ*.

1.1. Let Σ an be an infinite series and sn its nth partial sum. Let (pn) be a sequence of positive numbers such that

If the sequence

(1)

is of bounded variation, that is, Σ |tn – tn –1| < ∞, then the series Σ an is said to be absolutely (R, pn, 1) summable or |R, pn, 1| summable.

Let ƒ be an integrable function with period 2π and let its Fourier series be

(2)

Dikshit [4] (cf. Bhatt [1] and Matsumoto [7]) has proved the following theorems.

THEOREM I. Suppose that (i) the sequence (pn/Pn) is monotone decreasing, (ii) mn > 0, (iii) the sequence (mnpn/Pn) decreases monotonically to zero, and (iv) the series Σ mnPn/Pn) is divergent.

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.

I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.

A complex function on the real line is said to be bounded in mean square if it is locally in L2 (i.e. on each finite interval) and satisfies

(1.1)

The set of all such functions clearly forms a linear space over the complex numbers and is a Banach space B under the norm ‖·‖B defined by (1.1). This space, among others, has been discussed by Beurling in [1], where it was shown to be the dual, in the Banach space sense, of a certain Banach (convolution) algebra of functions. We have used Beurling's characterization of B and others of his results throughout this paper, and indeed the essence of one or two of the proofs has been derived from his theorems.

Our aim in this paper is to extend to difference-differential rings the beautiful theorem of Kolchin [5, Theorem 3] for the differential case. The necessity portion of Kolchin's result is not obtained.

What might well be called the Ritt basis theorem states that if a commutative ring R with identity is finitely generated over a subring R0, then the ascending chain condition for radical ideals of R0 implies the ascending chain condition for radical ideals of R. (This is indeed a basis theorem. If we define a basis for a radical ideal A to be a finite set B such that then every radical ideal of a ring R has a basis if and only if the ascending chain condition for radical ideals holds in R.) It is the Ritt basis theorem rather than the Hilbert basis theorem which has appropriate generalizations in differential and difference algebra, where in fact it originated.

Let C be a full subcategory of T, the category of based topological spaces and based maps, and let Cn be the corresponding category of n-tuples. Let S, T: Tn → T be covariant functors which respect homotopy classes and let u, v: S → T be natural transformations, u and v are homotopic inC, denoted u ≃ v(C), if uX ≃ vX: SX → TX (X ∈ Cn), that is to say, for every X ∈ C, uX and vX are homotopic (all homotopies are required to respect base points), u and v are naturally homotopic inC, denoted u ≃n v; (C), if there exist morphisms

such that, for every X ∈ C, utX is a homotopy from uX to vX and such that, for every t ∈ I, ut:S → T is a natural transformation.

Let ℭ be a finite group with a representation as an irreducible group of linear transformations on a finite-dimensional complex vector space. Every choice of a basis for the space gives the representing transformations the form of a particular group of matrices. If for some choice of a basis the resulting group of matrices has entries which all lie in a subfield K of the complex field, we say that the representation can be realized in K. It is well known that every representation of ℭ can be realized in some algebraic number field, a finitedimensional extension of the rational field Q.

A Sturm-Liouville function is simply a non-trivial solution of the Sturm-Liouville differential equation

(1.1)

considered, together with everything else in this study, in the real domain. The associated quantities whose higher monotonicity properties are determined here are defined, for fixed λ > –1, to be

(1.2)

where y(x) is an arbitrary (non-trivial) solution of (1.1) and x1, x2, … is any finite or infinite sequence of consecutive zeros of any non-trivial solution z(x) of (1.1) which may or may not be linearly independent of y(x). The condition λ > –1 is required to assure convergence of the integral defining Mk, and the function W(x) is taken subject to the same restriction.

In this paper we investigate indefinite Finsler spaces in which the metric tensor has signature n — 2. These spaces are a generalization of Lorentz manifolds. Locally a partial ordering may be defined such that the reverse triangle inequality holds for this partial ordering. Consequently, the spaces we study may be made into what Busemann [3] terms locally timelike spaces. Furthermore, sufficient conditions are obtained for an indefinite Finsler space to be a doubly timelike surface (see [2; 4]). In particular, all two-dimensional pseudo-Riemannian spaces are shown to be doubly timelike surfaces.