1. Introduction. Many problems involving prime numbers depend on estimating sums of the form ΣΛ(n)f(n), for appropriate functions f(n), (here, as usual, Λ(n) is the von Mangoldt function). Three distinct general methods have been used to estimate such sums. The earliest is due to Vinogradov (see [13, Chapter 9]); the second involves zerodensity bounds for Dirichlet L–functions (see [8, Chapters 15 and 16] for example); and the third, due to Vaughan (see [12] for example) uses an arithmetical identity as will be explained later. The second and third methods are much simpler to apply than the first. On the other hand Vinogradov's technique is at least as powerful as Vaughan's and occasionally more so. In many cases Vaughan's identity yields better bounds than the use of zero–density estimates, but sometimes they are worse. The object of this paper is to present a simple extension of Vaughan's method which is essentially as powerful as any of the techniques mentioned above, to discuss its general implications, and to apply it to the proof of the following result of Huxley [4], which has previously only been within the scope of the zero density method.

The class of developable topological spaces, which includes the metrizable spaces, has been fundamentally involved in investigations in point set topology. One example is the remarkable edifice of theorems relating to these spaces constructed by R. L. Moore (13). Another is the role played by the developable property in several metrization theorems, including Alexandroff and Urysohn's original solution of the general metrization problem (1).

This paper presents an anslysis of the concept of developable space in terms of certain more extensive classes of spaces satisfying the first axiom of countability : spaces with a base of countable order and those having what is here called a θ-base. The analysis is given in the characterizations of Theorems 3 and 4 below.

Let S1 . . . ,Sr be r line segments, each of non-zero length, in n-dimensional euclidean space Rn. If a polytope Z is defined as the vector (Minkowski) sum

(1) Z = S1 + . . . + Sr,

then the segments Si will be called the components of Z. Since we do not wish to exclude the possibility that some of the components may be parallel, the polytope Z may be written in the form (1) in many different ways. For this reason it is convenient to define a zonotope to be the polytope Z together with some specified set of components {S1 , . . . , Sr}. Figures 1, 2 and 3 show some zonotopes of 1, 2 and 3 dimensions with 4, 5 and 6 components.

The central problem in the present paper is the construction of symmetric and of skew-symmetric ( = skew) matrices C of order v, with diagonal elements 0 and other elements + 1 or — 1, satisfying

The following necessary conditions are known: v ≡ 2 (mod 4) and

a and b integers, for symmetric matrices C (Belevitch (1, 2), Raghavarao (14)), and v = 2 or v ≡ 0 (mod 4) for skew matrices C.

Introduction. Let R be a ring and let R[x] be the ring of all polynomials in a commutative indeterminate x over R. Let J(R) denote the Jacobson radical (5) of the ring R and let L(R) be the lower radical (4) of R. The main object of the present note is to determine the radicals J(R[x]) and L(R[x]). The Jacobson radical J(R[x]) is shown to be a polynomial ring N[x] over a nil ideal N of R and the lower radical L(R[x]) is the polynomial ring L(R)[x].

Let k be a local field, with char(k) ≠ 2. A quadratic space V over k is a finite dimensional vector space together with a non-degenerate quadratic form Q: V → k.The special orthogonal group SO(V) consists of all linear maps T: V → V which satisfy:

Q(Tv) = Q(v) for all ν and det T = 1.

Mares [9] has called a projective module semiperfect if every homomorphic image has a projective cover and has shown that many of the properties of semiperfect rings can be extended to these modules. More recently Zelmanowitz [16] has called a module regular if every finitely generated submodule is a projective direct summand. In the present paper a class of semiregular modules is introduced which contains all regular and all semiperfect modules. Several characterizations of these modules are given and a structure theorem is proved. In addition several theorems about regular and semiperfect modules are extended.

Given a set E of v elements, and given positive integers k, l (l ≤ k ≤ v), and λ, we understand by a tactical configurationC[k, l, λ, v] (briefly, configuration) a system of subsets of E, having k elements each, such that every subset of E having l elements is contained in exactly λ sets of the system.

A necessary condition for the existence of a configuration C[k, l, λ, v] is known (6) to be

A (group) divisible design is a tactical configuration for which the v points are split into m classes of n each, such that points have joining number λ (resp. λ2) if and only if they are in the same (resp. in different) classes. We are interested in such designs with a nice automorphism group. We first investigate divisible designs with equally many points and blocks admitting an automorphism group acting regularly on all points and on all blocks, i.e., with a Singer group (Singer [50] obtained the first result in this direction for the finite projective spaces).

As in the case of block designs, one may expect a divisible design with a Singer group to be equivalent to some sort of difference set; as it turns out, one here obtains a generalisation of the relative difference sets of Butson and Elliott [11] and [20].

Let Mn denote the algebra of n-square matrices over the complex numbers; and let Un, Hn, and Rk denote respectively the unimodular group, the set of Hermitian matrices, and the set of matrices of rank k, in Mn. Let ev(A) be the set of n eigenvalues of A counting multiplicities. We consider the problem of determining the structure of any linear transformation (l.t.) T of Mn into Mn having one or more of the following properties:

• (a) T(Rk) ⊆ for k = 1, …, n.

• (b) T(Un) ⊆ Un

• (c) det T(A) = det A for all A ∈ Hn.

• (d) ev(T(A)) = ev(A) for all A ∈ Hn.

We remark that we are not in general assuming that T is a multiplicative homomorphism; more precisely, T is a mapping of Mn into itself, satisfying

T(aA + bB) = aT(A) + bT(B)

for all A, B in Mn and all complex numbers a, b.

The integer part of a non-negative real number p will be denoted by [p]. For any integer n, n* will denote the greatest even integer less than or equal to n, that is, n* = n or n — 1 according as n is even or odd respectively.

The order of a set A, denoted by |A|, is the number of elements in A. The set whose elements are a1, a2, … , an will be denoted by {a1, a2 … , an. The empty set will be denoted by Λ. A set will be said to include each of its elements. A set separates two elements if it includes one but not both of them.

An unoriented graph U consists of two disjoint sets V(U), E(U), the elements of V(U) being called vertices of U and the elements of V(U) being called edges of U, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices which the edge is said to join.

The problem of determining the possible morphological types of convex polyhedra in three-dimensional Euclidean space E3 is well known to be quite hopeless. We lack not only any general way of determining whether there exists a convex polyhedron having as faces ƒ3 triangles, ƒ4 quadrangles, . . . , and ƒnn-gons, but even much more special questions of this kind seem to be rather elusive.

Soit C un espace de Banach (nous ne parlerons que d'espaces sur le corps des nombres réels, pour simplifier). On dit que C possède la propriété de Nachbin ou est un espace (N), si pour toute application linéaire continue u dans C d'un sous-espace vectoriel F d'un espace de Banach E, il existe une application linéaire de même norme de E dans C qui prolonge u.

In classical two-valued logic there is a three way relationship among formal systems, Boolean algebras and set theory. In the case of infinite-valued logic we have a similar relationship among formal systems, MV-algebras and what is called Bold fuzzy set theory. The relationship, in the latter case, between formal systems and MV-algebras has been known for many years while the relationship between MV-algebras and fuzzy set theory has hardly been studied. This is not surprising. MV-algebras were invented by C. C. Chang [1] in order to provide an algebraic proof of the completeness theorem of the infinitevalued logic of Lukasiewicz and Tarski. Having served this purpose (see [2]), the study of these algebras has been minimal, see for example [6], [7]. Fuzzy set theory was also being born around the same time and only in recent years has its connection with infinite-valued logic been made, see e.g. [3], [4], [5]. It seems appropriate then, to take a further look at the structure of MV-algebras and their relation to fuzzy set theory.

Rates of convergence, saturation theorems and the socalled “inverse problems” for Bernstein polynomials have been intensively studied (see, e.g., [1 ; 4; 8; 14; 17]). The same problems for some other positive operators have also been investigated by many authors. In this paper, we shall use a uniform approach to study the saturation and inverse problems for a class of linear combinations of operators including Bernstein polynomials, and Szâsz, Post-Widder, Gauss-Weierstrass and Baskakov operators.

In trying to extend the concept of torsion to rings more general than commutative integral domains the first thing that we notice is that if the definition is carried over word for word, integral domains are the only rings with torsion-free modules. Thus, if m is an element of any right module M over a ring containing a pair of non-zero elements x and y such that xy = 0, then either mx = 0 or (mx)y = 0. A second difficulty arises in the non-commutative case: Does the set of torsion elements of M form a submodule? The answer to this question will not even be "yes" for arbitrary non-commutative integral domains.

In the theory of enumeration, the part devoted to the counting of planar maps gives rather surprising results. Of special interest to the combinatorialists is the conspicuous feature of counting numbers associated with families of maps as discussed in the papers of Tutte, Brown and Mullin. These formulas are by no means easy to prove; this is also a part of their charm. We are convinced therefore that these maps possess deep combinatorial properties. We discuss such properties in this paper.

The main result of this article is the construction of a bisection between planar maps and a special family of trees: their vertices are labeled with numbers that differ by at most 1 on adjacent vertices. These trees are said to be well labeled. Combinatorialists usually claim that theorems specifying the “geometry” of objects lead to enumerating formulas. Our result obeys this general rule: we can indeed obtain a new proof of the formula counting the number of rooted planar maps with m edges. Four different proofs of this formula are known to date.

A band is a closed connected set in the 2-sphere, bounded by one or more disjoint simple closed curves.

Consider a band B with bounding curves J1, J2, … , Jk. On each curve Ji let there be chosen mi ≥ 0 points to be called vertices, with the restriction that the sum of the k integers mi is to be even. Write

(1)

Next consider a set of n disjoint open arcs in the interior of B which join the 2n vertices in pairs and partition the remainder of the interior of B into simply connected domains. We call the resulting dissection of B a slicing with respect to the given set of vertices. The arcs are the internal edges of the slicing and the simply connected domains are its internal faces, or slices.

We consider the problem

$$\left\{ \begin{matrix} {{\varepsilon }^{2}}\Delta u-u+f(u)=0,u>0 & \text{in}\,\Omega \\ \frac{\partial u}{\partial v}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{on}\,\partial \Omega , \\\end{matrix} \right.$$

where $\Omega $ is a bounded smooth domain in ${{R}^{N}},\varepsilon >0$ is a small parameter and $f$ is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as $\varepsilon $ approaches zero, at multiple critical points of the mean curvature function $H(P),P\in \partial \Omega $. It is also proved that this equation has multiple interior spike solutions which concentrate, as $\varepsilon \to 0$, at sphere packing points in $\Omega $.

In this paper, we prove the existence of solutions with multiple spikes both on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.

Let G be the group of R-rational points on a reductive, Zariskiconnected, algebraic group defined over R, let K be a maximal compact subgroup, and let g be the corresponding complexified Lie algebra of G. It is a curious fault of the current representation theory of G that for technical reasons one very rarely works with representations of G itself, but rather with a certain category of simultaneous representations of g and K. The reasons for this are, roughly speaking, that for a given (g,K)-module of finite length there are clearly any number of overlying rather distinct continuous G-representations, whose ‘essence’ is captured by the (g, K)-module alone. At any rate, this paper will propose a remedy for this inconvenience, and define a category of smooth representations of G of finite length which will, I hope, turn out to be as easy to work with as representations of (g, K) and occasionally much more convenient. It is to be considered a report on what has been to a great extent joint work with Nolan Wallach, and is essentially a sequel to [38].