In this paper we describe the outlines of the theory of analytic sets from the point of view of recent work on the subject. Our aim is to present the concepts and some of the principal results in a setting useful to workers in analysis, especially those workers not familiar with the field or its current developments. No attempt has been made to include all the results concerning analytic sets—not even in a particular category. There are some excellent monographs (12; 18) as well as chapters in books (2, 8, 9, 15, 17) where the subject is treated extensively. These, however, do not contain the recent results and consider only metric spaces. Our emphasis is on a general topological setting.

Let f be a function mapping the open unit disk D into the extended complex plane. A point ζ on the unit circle C is called an ambiguous point of f if there exist two Jordan arcs J1 and J2, each having an endpoint at ζ and lying, except for ζ, in D, such that

In Bernstein's proof of the Weierstrass Approximation Theorem, the polynomials

are constructed in correspondence with a function f ∊ C [0, 1] and are shown to converge uniformly to f. These Bernstein polynomials have been the starting point of many investigations, and a number of generalizations of them have appeared. It is our purpose here to consider several generalizations in the form of infinite series and to establish some of their properties.

The letter k denotes an infinite cardinal. A space is a compact Hausdorff space unless otherwise indicated. A space is called extremally disconnected (k-extremally disconnected) if it is the Stone space for a complete (k-complete) Boolean algebra. A map is a continuous function from one space into another. A map f:X —> Y is called minimal if f is onto, but f(M) is properly contained in Y for each closed proper subset M of X. A space F is called free if F has a dense subset X such that every space-valued function on X extends to a map on all of F or, equivalently, if F is the Stone-Cech compactification of some discrete topological space X.

In his paper (3) on finite H-planes, Kleinfeld has defined invariants s and t for an H-plane π as follows: let P and k be any point and line of π such that P is incident with k, then let s be the number of non-neighbour points to P on k and let t be the number of neighbour points to P on k. He has shown that s and t are independent of the choice of P and k, t divides s, π has s2 + st + t2 points (lines), π* has order s/t, and, if t ≠ 1, s ≤ t2. In the case s = t2, π is called uniform and has the property that each pair of neighbour lines (points) has exactly t points (lines) in common.

The theorems of the present paper deal with conditions which are necessary and sufficient in order that a solvable or nilpotent infinite group should have a completely reducible matrix representation over a given algebraically closed field.

It is known (17) that a locally nilpotent group of matrices is always solvable. Thus the first theorem of the present paper is a partial generalization of Theorem 1 of (16), which states:

If G is a locally nilpotent subgroup of the full linear group GL(n, P) over a perfect field P, then G is completely reducible if and only if each matrix of G is diagonizable (by a similarity transformation over some extension field of P).

Throughout this paper X denotes a fixed Hausdorff locally compact group with left Haar measure dx. Various spaces of functions and measures on X will recur in the discussion, so we name and describe them forthwith. All functions and measures on X will be scalarvalued, though it matters little whether the scalars are real or complex.

C = C(X) is the space of all continuous functions on X, Cc = Cc(X) its subspace formed of functions with compact supports. M = M(X) denotes the space of all (Radon) measures on X, Mc = MC(X) the subspace formed of those measures with compact supports. In general we denote the support of a function or a measure ξ by [ξ].

In the classical packing problem unit spheres are placed without overlapping in D-dimensional space. When D = 2, the densest packing is a familiar regular arrangement of circles, each circle touching six others. In this packing, circles cover a fraction π/√12 = 0.9069 . . . of the area of the plane. The densest packing is not known for D ≥ 3.

Most of the packings to be considered here use spheres of many different sizes. In this way greater densities are obtainable; small spheres can fill up some of the space left over after large spheres have been packed.

Let G be a discrete group with group algebra C[G] over the complex numbers C. In (5) Kaplansky essentially proves that if G has a normal abelian subgroup of finite index n, then all irreducible representations of C[G] have degree ≤n. Our main theorem is a converse of Kaplansky's result. In fact we show that if all irreducible representations of C[G] have degree ≤n, then G has an abelian subgroup of index not greater than some function of n. (The degree of a representation of C[G] for arbitrary G is defined precisely in § 3.)

For a positive integer k and a prime p ≡ 1 (mod k), there is a proper subgroup, R, of the multiplicative group (mod p) consisting of the kth power residues (mod p). A necessary and sufficient condition that an integer t be an element of R is that the congruence xk ≡ t (mod p) be solvable. The cosets, not R, formed with respect to R are called classes of kth power nonresidues, and form with R a cyclic group of order k. Let ρ be a primitive kth root of unity and let S be a class of non-residues that is a generator of this cyclic group. There is a kth power character X (mod p) such that

Let Mn denote the space of all n-square matrices over an algebraically closed field F. For A, B ∊ Mn, let

define the iterated commutators of A and B. Recently several research papers (1, 2, 4, and 5) have appeared on these commutators. In (1), Kato and Taussky have proved that for n = 2 the iterated commutators of A and B satisfy the linear relation

This paper is concerned with some of the properties of arcs in the real affine plane which are met by every parabola at not more than four points. Many of the properties of arcs of parabolic order four which we consider here are analogous to the corresponding properties of arcs of cyclic order three in the conformai plane which are described in (1). The paper (2), on parabolic differentiation, provides the background for the present discussion.

In Section 2, general tangent, osculating, and superosculating parabolas are introduced. The concept of strong differentiability is introduced in Section 3; cf. Theorem 1. Section 4 deals with arcs of finite parabolic order, and it is proved (Theorem 2) that an end point p of an arc A of finite parabolic order is twice parabolically differentiable.

The generalized Lototsky transform (or the [F, dn] transform) of a sequence {sn} into a sequence {tn) was defined in (2) in the following way: Let {dn) (dn ≠ —1) be a fixed complex sequence and let

then the sequence {tn} is defined by

In a recent paper by V. F. Cowling and C. L. Miracle (1) on these transformations, the following two problems were left open.

Lower case italics will denote rational integers, while lower case Greek letters will denote algebraic integers. The Law of Quadratic Reciprocity can be formulated as follows:

If p and a are distinct odd primes, then a is a quadratic residue (mod p) if and only if ( — 1)(p-1)/2p is a quadratic residue (mod q).

Let Xp denote the non-principal quadratic character (mod p) and let ζp be a primitive pth. root of unity. Then

Let K denote a connected finite 1-dimensional cell complex (1, p. 95), G(K) its group of homeomorphisms, and D(K) the group of homeomorphisms of K which are isotopic to the identity. The group (K) = G(K)/D(K) is a topological invariant of K and is called the homeotopy group ofK (4). K may be thought of as a linear graph (connected finite 1- dimensional simplicial complex) extended to admit loops and multiple edges and (K) as the topological analogue of the automorphism group A(L), (the permutations of vertices which preserve edge incidence relations) of a linear graph L.

The Hardy spaces for right half-planes, , σ real, 1 ≤ p ≤ ∞, are defined to consist of all those functions f(s), holomorphic for Re s > σ, for which μp(f, x) exists and is bounded for x > σ, where

These spaces have been studied extensively (see, for example, 3, Chapter 8, and 2, §19.1).

In this paper we consider anti-commutative algebras with an invariant form, that is, an algebra A over a field F such that

and A possesses a symmetric bilinear form f(x, y) such that

Lie and Malcev algebras (2, 3) are examples of such algebras and we shall consider generalizations of these algebras obtained by introducing commutation, x o y = xy — yx, as a new multiplicative operation in the non-commutative Jordan algebras of (1).

In 1945, L. I. Kopejkina (4), at the suggestion of A. G. Kurosh, began a programme of studying the properties of free projective planes and the analogies between free planes and free groups. In this paper, this study will be extended by proving a tool theorem and several of its consequences. The theorem deals with the existence of "minimal free generators" for subplanes of free planes.

A set of points and lines and an incidence relation are said to form a projective plane if the following three axioms are satisfied.

In a recent paper (5) Newman proved the following theorem: if D is a difference set in a cyclic group G and n = q is prime, then q is a multiplier of D. If n = 2q and (v, 7) = 1, then q is a multiplier of D. The purpose of this note is to point out that a stronger statement than the first part was proved in (1), to remove the restriction (v, 7) = 1 in the second part, and to give again and make some comments about the proof of the theorem which asserts that a prime divisor of n is a multiplier of D if prime to v.

A class of problems in the geometry of numbers, for which there are but fragmentary results, may be expressed in general terms, as follows. Let 5 be a given point-set and let G be a given discrete point-set, both in Euclidean n-space. Suppose that Λ is a lattice which contains G but no point of S not in G. Such lattices, if they exist, will be said to be admissible for S with respect to G, and the general problem is to investigate their properties and, if possible, classify them.