This paper arose out of an attempt to solve the following problem due to Suprunenko [5, Problem 2.77]. For which pairs of abelian groups A, B is every extension of A by B nilpotent? We obtain complete answers when A and B are p-groups and (a) A has finite exponent or (b) B is divisible or (c) A has infinite exponent, is countable and B is non-divisible. The structure of a basic subgroup of A plays a central role in cases (b) and (c).

At the outset we must say that the problem is too difficult to solve in complete generality. If G/A ≅ 2?, then the nilpotency of G depends solely on properties of the associated homomorphism θ.B → Aut A. Thus for instance if A is torsion-free and B finite, G is nilpotent if and only if the extension is a central one, and we would need detailed information on finite subgroups of the group Aut A.

One of the distinguishing features of the topological category is the following periodicity in the set of homotopy TOP structures on X.

THEOREM (Siebenmann). Let Xm, m ≦ 5, be a connected compact topological manifold with non-empty boundary. Then

It was conjectured by Siebenmann (see [3], p. 283) that the analogous periodicity should also exist for noncompact manifolds.

The purpose of this paper is to prove that this is indeed the case, namely:

THEOREM 1. Let Let Xm, m ≦ 6, be a connected noncompact topological manifold with non-empty boundary. Then

The concept of an operational logic has been developed by Randall and Foulis ([l]-[4], [10], [11]) as a part of a larger effort to obtain a formalism suitable for expressing, comparing, and evaluating various approaches to empirical science, statistics, and in particular, quantum mechanics. The structure of these logics is similar to that of an orthomodular partially ordered set which is often used as a model for quantum logics. However, the operational logic is a more general structure which, among other features, allows for the creation of a tensor product of logics to represent the coupling of physical systems. Randall and Foulis have shown that, given certain reasonable physical constraints, such a product is not possible within the category of orthomodular posets [12].

In [7] Schoenberg introduced the following variation-diminishing spline approximation methods.

Let m > 1 be an integer and let Δ = {xi} be a biinfinite sequence of real numbers with xi ≧ xi + l < xi+m. To a function f associate the spline function Vf of order m with knots Δ defined by

(1.1)

where

and the Nj(x) are B-splines with support xj < x < xj+m normalized so that ΣjNj(x) = 1. See, e.g., [2] for a precise definition of the Nj(x) and a discussion of the properties of Vf.

A compact bordered Klein surface of (algebraic) genus g ≦ 2 is said to have maximal symmetry [5] if its automorphism group is of order 12(g – 1), the largest possible. An M*-group acts as the automorphism group of a bordered surface with maximal symmetry. M*-groups were first studied in [6], and additional results about these groups are in [5, 7, 8].

Here we construct a new, interesting family of M*-groups. Each group G in the family is an extension of a cyclic group by the automorphism group of a torus T with holes that has maximal symmetry. Furthermore, G acts on a bordered Klein surface X that is a fully wound covering [7] of T, that is, an especially nice covering in which X has the same number of boundary components as T. The construction we use for the new family of M*-groups is a standard one that employs group automorphisms to define extensions of groups.

Let L be a very ample line bundle on a smooth, connected, projective, ruled not rational surface X. We have considered the problem of classifying biholomorphically smooth, connected, projected, ruled, non rational surfaces X with smooth hyperplane section C such that the genus g = g(C) is less than or equal to six and dim where is the map associated to . L. Roth in [10] had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see [8].

If g = 2 ≠ hl,0(X) by [12, Lemma (2.2.2) ] it follows that X is a rational surface. Thus we can assume g ≦ 3.

Since X is ruled, h2,0(X) = 0 and

see [4] and [12, p. 390].

Given a lattice L and a convex sublattice K of L, it is well-known that the map Con L → Con K from the congruence lattice of L to that of K determined by restriction is a lattice homomorphism preserving 0 and 1. It is a classical result (first discovered by R. P. Dilworth, unpublished, then by G. Grätzer and E. T. Schmidt [2], see also [1], Theorem II.3.17, p. 81) that any finite distributive lattice is isomorphic to the congruence lattice of some finite lattice. Although it has been conjectured that any algebraic distributive lattice is the congruence lattice of some lattice, this has not yet been proved in its full generality. The best result is in [4]. The conjecture is true for ideal lattices of lattices with 0; see also [3].

Putnam showed in [5] that the spectrum of the real part of a bounded subnormal operator on a Hilbert space is precisely the projection of the spectrum of the operator onto the real line. (In fact he proved this more generally for bounded hyponormal operators.) We will show that this result can be extended to the class of unbounded subnormal operators with bounded real parts.

Before proceeding we establish some notation. If T is a (not necessarily bounded) operator on a Hilbert space, then D(T) will denote its domain, and σ(T) its spectrum. For K a subspace of D(T), T|K will denote the restriction of T to K. Norms of bounded operators and elements in Hilbert spaces will be indicated by ‖ ‖. All Hilbert space inner products will be written 〈,〉. If W is a set in C, the closure of W will be written clos W, the topological boundary will be written bdy W, and the projection of W onto the real line will be written π(W),

In recent years, a considerable amount of research has been devoted to questions involving set porosity, particularly as it relates to differentiation theory. We may express the type of question in which we are interested by using the language of path derivatives and sequential derivatives. A path derivative of a function/is defined by writing

where at each point x a set Ex is given. A special case of the path derivative is the sequential derivative, defined by writing

where hn is a fixed sequence of nonzero numbers converging to zero.

Let ∧ be a bounded, non-empty, open subset of Rn and given any x in Rn, let

let k ∊ N and suppose that p ∞ (1, ∞). It is known (c.f. e.g. [4]) that if u belongs to the Sobolev space WKp(∧) and u/dk ∊ Lp(∧), then . Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out that

Under appropriate smoothness conditions on the boundary of ∧, m and d are equivalent, and thus may in this case be characterized as the subspace of W1,2(∧) consisting of all functions u ∊ W1,2(∧) such that u/d ∊ L2(∧). Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out that

Besides stating the problem and the results, we shall give in this section a brief overview of the classical necessary and sufficient conditions in the calculus of variations, in order to clearly situate the contribution of this article.

1.1 The problem. We are given an interval [a, b], two points xa, xb in Rn, and a function L (the Lagrangian) mapping [a, b] × Rn × Rn to R. The basic problem in the calculus of variations, labeled (P), is that of minimizing the functional

over some class X of functions x and subject to the constraints x(a) = xa, x(b) = xb. Let us take for now the class X of functions to be the continuously differentiable mappings from [a, b] to Rn; we call such functions smooth arcs.

Given a triangular matrix A whose nth row consists of the n points

(1.1)

Turán et al. ([12], [1], [2], [3]) considered the problem of existence, uniqueness, representation, convergence, etc. of polynomials f2n – 1 of degree ≧2n – 1 where the values of f2n – 1 and those of its second derivative are prescribed at the points (1.1), i.e.,

(1.2)

The choice of the points (1.1) is important. They found the zeros

(1.3)

of the polynomial

(1.1)

where Pn – 1 is the (n − 1) Legendre polynomial with the normalization Pn – 1(l) = 1 to be the most convenient.

The ideal generation conjecture has recently been proved for general points in , (k a field) [1], [6]. The proof in [1] is by induction. One of the starting points for the induction (called A(5) in [1]) is proved in [6]. The theoretical proof of A(5) in [6] seems to be very difficult, apparently even more difficult than the induction. Because of this, and also because [6] is not publically available, I feel it is worth knowing that A(5) can be proved numerically with modest readily available computing facilities. In this note I discuss the computation involved, and give a few explicit examples. In the course of working out these examples I found 26 points in , that satisfy the ideal generation conjecture, but which cannot be extended to 27 or 28 points satisfying the ideal generation conjecture. This phenomenon can be interpreted combinatorially, leading to an infinite number of similar examples.

A net {aα} of positive, norm one elements of a C*-algebra A excises a state f of A if

This notion has been used explicitly by the second author [4, 5, 6] for pure states, but the present paper will explore it more fully. The name is motivated by the following example. Let K be the unit disk in the complex plane, A = C(K) and f(a) = a(0). Define an(reiθ) = ϕn(r), where

Note that the sets {t ∊ K:an(t) ∊ 0} form rings about 0 with radii tending to 0. In this sense the sequence {an} “cuts out” the state f and, in the limit, isolates it from all other states.

This paper is devoted to the study of sets of four unequal integers ni such that the twelve quotients ni/nj(i ≠ j) of pairs of distinct members sum to zero. (Without the restriction i ≠ j the condition would be equivalent to (Σ ni)(Σ l/ni) = 0, of no great interest.) Constructions for these sets are given, and relations between them are studied. It is found that each set belongs to an orbit of six related sets, and that each such orbit is related to four neighbours, each of which is another orbit of six sets. A study is made of the graph formed by assigning a node to each orbit of six solutions and joining it to the nodes assigned to its four neighbours. This appears to comprise one component containing an infinity of cycles together with an infinite forest of infinite trees.