We consider here the equation

1

This equation was first studied by Hammerstein [4] under the assumption that the linear operator

2

is selfadjoint and completely continuous. V. Nemytsky [5] and M. Golomb [3] dropped the assumption that A be selfadjoint and positive. M. Vainberg [6] considered (among other cases) the case in which A is a bounded operator generated by a Carleman kernel. The kernels considered in this work do not necessarily generate bounded, completely continuous or selfadjoint, operators.

Let M be an n-dimensional connected C∞ manifold with a linear connection Γ. M is said to be of recurrent curvature with respect to Γ if the corresponding curvature tensor R satisfies [1], [4]

where Δ denotes covariant derivative with respect to Γ and W is a nonzero covector called the recurrence co-vector. Let T be the torsion of Γ.

Let R be a commutative Noetherian ring with identity, and let M be a fixed ideal of R. Then, trivially, ring multiplication is continuous in the ilf-adic topology. Let S be a multiplicative system in R, and let j = js: R → S-1R, be the natural map. One can then ask whether (cf. Warner [3, p. 165]) S-1R is a topological ring in ihe j(M)-adic topology. In Proposition 1, I prove this is the case if and only if M ⊂ p(S), where

Hence S -1 R is a topological ring for all S if and only if M ⊂ p*(R), where

The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.

The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

One very interesting and important problem in ring theory is the determination of the position of the singular ideal of a ring with respect to the various radicals (Jacobson, prime, Wedderburn, etc.) of the ring. A summary of the known results can be found in Faith [3, p. 47 ff.] and Lambek [5, p. 102 ff.]. Here we use a new technique to obtain extensions of these results as well as some new ones.

Throughout we adopt the Bourbaki [2] conventions for rings and modules: all rings have 1, all modules are unital, and all ring homomorphisms preserve the 1.

Let R be a commutative ring with unity and let M(R) be the multiplicative group of 4 x 4 triangular matrices (a ij ) over R, where a 11 is a unit element of R and a ii = 1 for i = 2, 3, 4. If V(=AN 2 ∧ N 2 A) denotes the variety of groups which are both abelian-by-class-2 and class-2-by-abelian, then it is routine to verify that M(R) ∊ V. Here we prove the following,

Theorem. Let F(V) denote the free group of finite or countable infinite rank of the variety V. Then for a suitable choice of R, F(V) is embedded in M(R).

Let L(P) denote the total edge length and A(P) the total surface area of a three-dimensional convex polyhedron P. In [5] it was shown that if P belongs to the set of all polyhedra with triangular faces then for all

with equality if and only if is a regular tetrahedron.

It is not difficult to establish the inequality

The purpose of this paper is to investigate for graphs the existence of certain valuations which have some "magic" property. The question about the existence of such valuations arises from the investigation of another kind of valuations which are introduced in [1] and are related to cyclic decompositions of complete graphs into isomorphic subgraphs.

Throughout this paper the word graph will mean a finite undirected graph without loops or multiple edges having at least one edge. By G(m, n) we denote a graph having m vertices and n edges, by V(G) and E(G) the vertex-set and the edge-set of G, respectively. Both vertices and edges are called the elements of the graph.

Let L denote a semi-simple, finite dimensional Lie algebra over an algebraically closed field K of characteristic zero. If denotes a Cartan subalgebra of L and denotes the centralizer of in the universal enveloping algebra U of L, then it has been shown that each algebra homomorphism (called a "mass-function" on ) uniquely determines a linear irreducible representation of L. The technique involved in this construction is analogous to the Harish-Chandra construction [2] of dominated irreducible representations of L starting from a linear functional . The difference between the two results lies in the fact that all linear functionals on are readily obtained, whereas since is in general a noncommutative algebra the construction of mass-functions is decidedly nontrivial.

Let T—c be a Fredholm operator, where T is a bounded linear operator on a complex Banach space and c is a scalar, the set of all such scalars is called the Φ-set of T [2] and was studied by many authors. In this connection, the purpose of the present paper is to investigate some classes Φ(V) of all such operators for any subset V of the complex plane.

Let X be a Banach space over the field C of complex numbers with dim Z = ∞, unless otherwise stated, B(X) the Banach algebra of all bounded linear operators and K(X) the closed two-sided ideal of all compact operators on X.

A ring R with identity element is n-finite if for any pair A, B of n × n matrices over R, AB = In implies BA = In . In module theoretic terms, R is n-finite if and only if in a free R-module of rank n any generating set of n elements is free. If R is n-finite for all positive integers n then R is said to be strongly finite. It is known that all commutative rings, all Artinian rings and all Noetherian rings are strongly finite. These and many other interesting results appear in a paper of P. M. Cohn [1].

Let R be a ring with 1. All modules considered are to be unital left R-modules unless otherwise noted.

Definition. A σ-set for R is a nonempty set Σ of left ideals of R satisfying the following conditions:

1. (σ1) If I ∊ Σ, J is a left ideal of I, and J ⊇ I, then J ∊ Σ.

2. (σ2) If I ∊ Σ and r ∊ R, then Ir-1 = {s ∊ R | sr ∊ I} ∊ Σ.

3. (σ3) If I is a left ideal of R, J ∊ Σ, and It-1 ∊ Σ for each t ∊ J, then I ∊ Σ.

In this note we shall present a result about incidence functions on a locally finite partially ordered set, a result which is related to theorems of Lambek [2] and Subbarao [6]. Our terminology and notation will be that of Smith [4, 5] and Rota [7].

Let (L, ≤) be a partially ordered set which is locally finite in the sense that for all x, y ∊ L the interval [x, y] = {z | x ≤ z ≤ y} is finite. Denote by A(L, ≤) the set of functions f from L × L into some field, which is fixed once and for all, such that f(x, y) = 0 whenever x ≰ y.

Greenleaf states the following conjecture in [1, p. 69]. Let G be a (connected, separable) amenable locally compact group with left Haar measure, μ, and let U be a compact symmetric neighbourhood of the unit. Then the sets, {U m}, have the following property: given ɛ > 0 and compact K ⊂ G, ∃ m 0 = m 0(ɛ, K) such that

It is well known that if a module M is expressible as a direct sum of modules with local endomorphism rings, then such a decomposition is essentially unique. That is, if M = ⊕i∊I Mi = ⊕j∊J Nj then there is a bijection f: I → J such that M i is isomorphic to Nf(i) for all i∊I (see [1]). On the other hand, a nonprincipal ideal in a Dedekind domain provides an example where such a theorem fails in the absence of the local hypothesis. Group algebras of certain groups over rings R of algebraic integers is another such example, where even the rank as R-modules of indecomposable summands of a module is not uniquely determined (see [2]). Both of these examples yield modules which are expressible as direct sums of two indecomposable modules in distinct ways. In this note we construct a family of rings which show that the number of summands in a representation of a module M as a direct sum of indecomposable modules is also not unique unless one has additional hypotheses.

We are concerned with the nth-order linear differential equation

1

where the coefficients are continuous. Aliev [1, 2] showed, in papers unavailable to the author that for n = 4

(see Definition 2). Theorems 1 and 5 give respectively nth-order generalizations of these two results.

Let G be a finite connected graph without loops or multiple edges. A maximal tree subgraph T of G is called a spanning tree of G. Denote by k(G) the number of all trees spanning the graph G. A. Rosa formulated the following problem (private communication): Let x(≠2) be a given positive integer and denote by α(x) the smallest positive integer y having the following property: There exists a graph G on y vertices with x spanning trees. Investigate the behavior of the function α(x).

Let Q denote the field of rational numbers. If m, n are distinct squarefree integers the field formed by adjoining √m and √n to Q is denoted by Q(√m, √n). Since Q(√m, √n) = Q(√m, √n) and √m + √n has for its unique minimal polynomial x 4 —2(m + n)x 2 + (m - n)2, Q(√m, √n) is a biquadratic field over Q. The elements of Q(√m, √n) are of the form a0 + a1√m + a2√n + a3√mn, where a 1, a 2, a 3 ∊ Q. Any element of Q(√m, √n) which satisfies a monic equation of degree ≥ 1 with rational integral coefficients is called an integer of Q(√m, √n). The set of all these integers is an integral domain. In this paper we determine the explicit form of the integers of Q(√m, √n) (Theorem 1), an integral basis for Q(√m, √n) (Theorem 2), and the discriminant of Q(√m, √n) (Theorem 3).