It is assumed that the reader is familiar with the following notions: regressive function, regressive set, regressive isol, infinite series of isols, the minimum of two regressive isols, combinatorial function, and canonical extension. We shall use the slightly more general definition of a regressive function introduced in (3). The next three notions are defined in (2), the fifth in (3), and the last two in (7 and 8).

Given a function ƒ(z), continuous on C: |z| = 1 in the complex plane, there is a close analogy between approximation in the sense of least squares by polynomials on the unit circle and interpolation by polynomials in the nth roots of unity to the same function. For detailed discussion of the problem and its generalization for a suitable Jordan curve one can refer to Walsh (3) or to a recent paper by Curtiss (2). More recently, Curtiss (1) has considered the problem of interpolation by polynomials in non-equally spaced points on the unit circle and has pointed out the limitations inherent in the problem.

La caractérisation fondamentale des sous-groupes du groupe additif R des nombres réels en tant que groupes ordonnés est la suivante:

A. Un groupe abélien totalement ordonné G est isomorphe à un sous-groupe de R si et seulement si G est archimédien.

Nous nous proposons ici de donner une nouvelle caractérisation des sousgroupes de R qui nous semble présenter un certain intérêt pour plusieurs raisons. D'abord elle est très différente de la caractérisation A ci-dessus, bien que notre démonstration utilise A. Puis elle ajoute un nouveau théorème de r-groupe à ceux déjà connus. Finalement, en employant les caractérisations des anneaux de valuation généraux et discrets données dans (1), le théorème en question possède quelques corollaires parmi lesquels se trouve le théorème de s-groupe pour un domaine d'intégrité (“s-Gruppensatz” dans la terminologie de Krull (5)).

The purpose of this paper is to introduce a generalization of metric space that arises naturally out of the notion of signal function as it occurs, for example, in (5). In §§ 2-5, the basic definitions and motivation are given. In §§ 6 and 7 several elementary topological properties are proved, and in §§ 8 and 9 an important example from special relativity is developed.

Let R be a commutative ring with unit and A be a unitary commutative R-algebra. Let As be a generalized algebra of quotients of A with respect to a multiplicatively closed subset S of A. If (A) and (As) denote the categories of complexes and their homomorphisms over A and As respectively, then one easily sees that there exists a covariant functor T:(A) → (AS) such that T is onto and T(X, d) is universal over As whenever (X, d) is universal over A. Actually the category (AS) is equivalent to a subcategory of (A) where contains all those complexes (X, d) over A such that for each s in S, the module homomorphism ϕs: x → sx of Xn into itself is one-one and onto for each n ⩾ 1.

The principal theorem of this paper, a generalization of a theorem given by R. H. Cameron (2), provides a means of approximating certain Wiener integrals to any desired degree of accuracy by an (n + 1)-fold Riemann integral with sufficiently large n. The generalization is in the use of a general complete orthonormal set of functions, whereas Cameron's paper used only the odd harmonic set.

Let C′ be the class of real-valued functions x(t) defined on [0, 1] and such that x(0) = 0 and which are continuous except perhaps for one left continuous jump. Let C be the class of continuous members of C′.

A pro-p-group G is said to be a Demushkin group if

1. (1) dimFp H1(G, Z/pZ) < ∞,

2. (2) dimFp H2(G, Z/pZ) = 1,

3. (3) the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro-p-group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F9 (F, F); cf. §1.4.

Some time ago, J. G. Wendel proved that the operators on the group algebra L1(G) which commute with convolution correspond in a natural way to the measure algebra M(G) (13). One might ask if Wendel's theorem can be restated in a more general setting. It is this question that is the point of departure for our present paper. Let K be a Banach module over L1(G). Our interest is in operators from L1(G) into K, and from K into L∞(G), which commute with the module composition (where L∞(G) is thought of as a module over L1(G) also). Such operators we call (L1(G), K)- and (K, L∞(G))-homomorphisms, respectively. Investigations of various other kinds of module homomorphisms occur in A. Figà-Talamanca (6) and B. E. Johnson (9; 10).

The present paper began as a natural outgrowth of our first paper, where we characterized the module homomorphisms from group algebras into a fairly restrictive class of group algebra modules. We now investigate module homomorphisms from group algebras into a more general class of group algebra modules. Although the two papers are thus related, they can be read quite independently.

Section 2 contains our extension, Theorem 2.1, of P. J. Cohen's theorem on factorization in Banach algebras (1). Our extension is to Banach modules over Banach algebras equipped with an approximate identity. We should mention first that J.-K. Wang observed the existence of such a generalization, and secondly, that our proof requires no ideas different from those in Cohen's proof. Nevertheless, we include a proof that condenses the original proof considerably.

It is the purpose of this paper to show that many of the enumerative techniques available for counting rooted plane trees may be extended to tree-rooted maps, that is, rooted maps in which a spanning tree is distinguished as root tree. For example, tree-rooted maps are enumerated by partition, and the average number of trees in a rooted map with n edges is determined. An enumerative similarity between Hamiltonian rooted maps (that is, rooted maps with a distinguished Hamiltonian polygon) and tree-rooted maps is discussed. A 1-1 correspondence is established between treerooted maps with n edges and Hamiltonian rooted trivalent maps with 2n + 1 vertices in which the root vertex is exceptional, being divalent, both of which are in 1-1 correspondence with non-separable Hamiltonian-rooted triangularized digons with n internal vertices, where both the latter are as defined in (2).

We shall be concerned at first with some properties of the solutions of the matrix differential equation

1.1

where

is an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.

A (v, k, λ) difference set D is a set of k distinct residues {a1, a2, … , ak} modulo v such that every residue b ≢ 0 (mod v) can be expressed in exactly λ ways in the form b ≡ ai — aj (mod v). With each difference set we may associate a binary periodic sequence (s1, s2, …) with si = 1 if i (mod v) is in D, and si = 0 otherwise. Since this sequence is periodic of period v, we need only consider one cycle from the sequence. Such cycles we agree to call (binary) difference cycles. Difference cycles (equivalently, difference sets) have been studied intensively (2, 4). They have important applications to digital communications, mainly because they have 2-level autocorrelation. In this paper we shall point out certain other (equivalent) properties of difference cycles which seem susceptible to immediate generalization, but show that these generalizations are vacuous.

Throughout this paper E denotes a compact Hausdorff space, which, to avoid trivial complications, is assumed to contain at least two points. C(E), with the uniform norm, is the Banach algebra of all continuous real-valued functions defined on E; C+(E) is the set of those functions in C(E) which take only non-negative values. A subset of C(E) is a wedge if and only if it is closed under addition and multiplication by nonnegative scalars; a semi-algebra is a wedge closed under (pointwise) multiplication. The set C+(E) is a semi-algebra, and all semi-algebras considered in this paper are contained in C+(E). For a subset K of C+(E), the closed wedge (semi-algebra) generated by K is the least closed wedge (semi-algebra) containing K.

In this paper we consider measures determined by arbitrary functions G(x) for which finite right and left limits exist everywhere and indicate how some of these measures permit the definition of generalized integrals of constructive or Denjoy type. These definitions are related to corresponding descriptive definitions based on the Perron approach as given by Ward (6) and Henstock (2). An exposition of the introductory theory is given in (1).

In this paper it is proved that if a circle-like continuum M cannot be embedded in the plane, then M is not a continuous image of any plane continuum (Theorem 5).

Suppose that (S, ρ) is a metric space. A finite sequence of domains L1, L2, … , Ln is called a linear chain provided Li intersects Lj if and only if |i — j| ⩽ 1. If, in addition, there is a positive number ∊ such that, for each i, the diameter of Li is less than ∊, then the linear chain is called a linear ∊-chain. If for each positive number ∊ the continuum M can be covered by a linear ∊-chain, then M is said to be chainable (or snake-like) (2).

These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.

The principal results of these notes are the following. New sphere packings are given in [2m], m ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2m], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.

In (4) G. Lewittes proved some theorems connecting automorphisms of a compact Riemann surface with the Weierstrass points of the surface, and in (5) he applied these results to elliptic modular functions. We refer the reader to these papers for definitions and details. It is our purpose in this note to point out that these results are of a purely algebraic nature, valid in arbitrary algebraic function fields of one variable over algebraically closed ground fields (with an obvious restriction on the characteristic). We shall also make use of the calculation carried out in (5) to obtain a rather easy extension of a theorem proved in (6, p. 312).

Let ∑ be a projective space PG(3, q) of dimension 3 and finite order q. Then ∑ contains (q + 1)(q2 + 1) points and an equal number of planes, and (q2 + 1) (q2 + q + 1) lines. It will be convenient to consider lines and planes as sets of points and to treat the incidence relation as set inclusion. Each plane contains q2 + q + 1 points and an equal number of lines. Each line contains q + 1 points and is contained in an equal number of planes. Each point is contained in q2 + q + 1 planes and an equal number of lines.

A spread of lines of ∑ is a set of q2 + 1 lines of ∑ which are pairwise disjoint, or skew; it can also be defined as a set of lines such that each point (or each plane) is incident with exactly one of the lines.

Let Mm(F) be the vector space of m-square matrices

where F is a field; let f be a function on Mm(F) to some set R. It is of interest to determine the linear maps T: Mm(F) → Mm(F) which preserve the values of the function ƒ; i.e., ƒ(T(X)) = ƒ(X) for all X. For example, if we take ƒ(X) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank. Other classical invariants that may be taken for f are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues. Dieudonné (1), Hua (2), Jacobs (3), Marcus (4, 6, 8), Mori ta (9), and Moyls (6) have conducted extensive research in this area. A class of matrix functions that have recently aroused considerable interest (4; 7) is the generalized matrix functions in the sense of I. Schur (10).