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We answer a 1975 question of G. R. Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds: (i) X is metrizable, (ii) dimX = 0 or (iii) X is a union of finitely many pairwise disjoint generalized simple closed curves. We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta. 1. Introduction. The aim of this paper is to investigate homogeneous spaces which are continuous images of ordered compacta. In 1975, G. R. Gordh proved that if a homo­ geneous and hereditarily unicoherent continuum is the continuous image of an ordered compactum, then it is metrizable, and so indecomposable (7, Theorem 3). Further, he asked if, in general, every homogeneous continuum which is the continuous image of an ordered compactum must be either metrizable or a generalized simple closed curve. Our Theorem 1 provides an affirmative answer to Gordh's question. Moreover, in Theorem 2, we prove that a homogeneous space which is not 0-dimensional and which is the continuous image of an ordered compactum is either metrizable or a union of finitely many pairwise disjoint generalized simple closed curves. Our methods of proof involve characterizations of continuous images of arcs obtained in ( 16) in terms of cyclic elements and T-sets. When dealing with the class A of all homogeneous and 0-dimensional spaces which are the continuous images of ordered compacta, the situation becomes less clear. By a recent theorem of M. Bell, each member of A is first countable. Moreover, by a result of (18), each member of A can be embedded into a dendron. We give a rather simple construction leading to a wide subclass of A. In particular, we show that not all members of A are orderable, and that there exists a strongly homogeneous space X which is the continuous image of an ordered compactum and which is not first countable. It follows that X $ A. Our investigations of the class A led to some natural questions which are stated at the end of the paper. All spaces considered in this paper are Hausdorff.

This paper is a continuation of [5]. As stated there, the problem is to explicitly decompose the space L2 = L2(G(F)\G(A)) into simpler invariant subspaces, and to deal with the associated continuous spectrum in case G is a connected reductive algebraic group defined over a global function field. In that paper the solution was begun by studying Eisenstein series associated to cusp forms on Levi components of parabolic subgroups; these Eisenstein series and the associated intertwining operators were shown to be rational functions satisfying functional equations. To go further it is necessary to consider more general Eisenstein series and intertwining operators, and to show that they have similar properties. Such Eisenstein series arise from the cuspidal ones by a residue taking process, which is detailed in a disguised form suitable for induction in the first part of this paper: the main result is a preliminary form of the spectral decomposition.

In an earlier paper [6] we considered the application of Lagrangian methods to the enumeration of plane rooted trees with given colour partition. We obtained an expression which generalised Tutte’s result [9], and a correspondence, which, when specialised, gives the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte Theorem [1]. A corollary of these results is a one-to-one correspondence [4], between trees and generalised derangements, for which no combinatorial description has yet been found.In this paper we extend these methods to the enumeration of rooted labelled trees to demonstrate how another pair of well-known and apparently unrelated theorems may be obtained as the result of a single enumerative approach. In particular, we show that a generalisation of Good's result [3], also considered by Knuth [7], and the matrix tree theorem [8] have a common origin in a single system of functional equations, and that they correspond to different coefficients in the power series solution.

In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions. Remarkably, the number of such trees which are planted is a multiple of the number of spanning arbores­ cences on a graph with the edge partition as its adjacency matrix. This result is used (Section 4) to obtain the number of rooted plane i£-chromatic trees with fixed chromatic partition, a number given by Tutte [5] for planted trees in the case K = 2. Finally, a new combinatorial correspon­ dence between two sets of trees is given (Section 5) which yields the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte (BEST) theorem as a special case. We use a familiar decomposition of rooted trees. Associated with this is a system of functional equations which may be solved by a specialisa­ tion, given in Section 2, of the Lagrange theorem in many variables. The specialisation accounts for the persistence, in combinatorial enumeration, of determinants of matrices with row and column constraints (see, for example, [6]). Throughout this paper we use the notation: the number of non-root vertices of colour i is nt = ]C^i/

It is the purpose of this paper to establish a closer connection between the logarithmic capacity of sets and double trigonometric series. In (9), closed sets of logarithmic capacity zero were established as sets of uniqueness for a particular class of double trigonometric series under circular (C, 1) summability. By slightly changing this class of series but still maintaining closed sets of logarithmic capacity zero as sets of uniqueness, it is shown in this paper that closed sets of positive logarithmic capacity form sets of multiplicity.

In this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

We characterize diametrically maximal and constant width sets inC(K), whereKis any compact Hausdorff space. These results are applied to prove that the sum of two diametrically maximal sets needs not be diametrically maximal, thus solving a question raised in a paper by Groemer. A characterization of diametrically maximal sets inis also given, providing a negative answer to Groemer's problem in finite dimensional spaces. We characterize constant width sets inc0(I), for everyI, and then we establish the connections between the Jung constant of a Banach space and the existence of constant width sets with empty interior. Porosity properties of families of sets of constant width and rotundity properties of diametrically maximal sets are also investigated. Finally, we present some results concerning non-reflexive and Hilbert spaces.

The main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelianmod p representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer seems to require developments in transcendence theory in characteristic p.

A model subspace Kϴ of the Hardy space H2 = H2(ℂ+) for the upper half plane ℂ+ is H2(ℂ+) ϴ ϴH2(ℂ+) where ϴ is an inner function in ℂ+. A function ω: ⟼ [0,∞) is called an admissible majorant for Kϴ if there exists an f ∈ Kϴ, f ≢ 0, |f(x)| ≤ ω(x) almost everywhere on ℝ. For some (mainly meromorphic) ϴ's some parts of Adm ϴ (the set of all admissible majorants for Kϴ) are explicitly described. These descriptions depend on the rate of growth of argϴ along ℝ. This paper is about slowly growing arguments (slower than x). Our results exhibit the dependence of Adm B on the geometry of the zeros of the Blaschke product B. A complete description of Adm B is obtained for B's with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.

This paper is concernedwith the structure of the arithmetic sum of a finite number of central Cantor sets. The technique used to study this consists of a duality between central Cantor sets and sets of subsums of certain infinite series. One consequence is that the sum of a finite number of central Cantor sets is one of the following: a finite union of closed intervals, homeomorphic to the Cantor ternary set or an M-Cantorval.

Let denote the set of integers representable as a sum of two squares. Since can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that hasmany properties in common with the set of prime numbers. In this paper we exhibit “unexpected irregularities” in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of than expected, and infinitely many intervals containing considerably fewer than expected.

A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about tn by establishing new connections between themodular jinvariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbert class field of . This shows that tn is a new class invariant according to H. Weber’s definition of class invariants.

This paper studies closed subspaces L of the Hardy spaces Hp which are g-invariant (i.e., g. L ⊆ L) where g is inner, g ≠ 1. If p = 2, theWold decomposition theorem implies that there is a countable “g-basis” f1, f2, . . . of L in the sense that L is a direct sum of spaces fj . H2[g] where H2[g] = {f o g | f ∈ H2}. The basis elements fj satisfy the additional property that ∫T |fj|2gk = 0, k = 1, 2, . . . . We call such functions g-2-inner. It also follows that any f ∈ H2 can be factored f = hf ,2 . (F2 o g) where hf,2 is g-2-inner and F is outer, generalizing the classical Riesz factorization. Using Lp estimates for the canonical decomposition of H2,we find a factorization f = hf ,p.(Fpog) for f ∈ Hp. If p ≤ 1 and g is a finite Blaschke product we obtain, for any g-invariant L ⊆ Hp, a finite g-basis of g-p-inner functions.

Let X be a countable fundamental set in a Hilbert space H, and let T be the operator Whenever T is well-defined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is injective is a stable basis (also known as a Riesz basis). This paper considers the above three properties for subspaces H of L2(ℝd), and for sets X of the form with Φ either a singleton, a finite set, or, more generally, a countable set. The analysis is performed on the Fourier domain, where the two operators TT* and T* T are decomposed into a collection of simpler "fiber" operators. The main theme of the entire analysis is the characterization of each of the above three properties in terms of the analogous property of these simpler operators.

Let F be a field of characteristic different from p containing a primitive p-th root of unity. This paper studies the cup product pairing Hl(F, p) x Hl(F, p) → H2(F, p) and its relationship to valuation theory and Galois theory. Sufficient conditions on the pairing which guarantee the existence of a valuation on the field are described. In the non p-adic case these results provide a converse to the well-known structure theory in this situation. In the p-adic case, the pairing is described using the notion of "relative rigidity". These results are analogues of results in quadratic form theory developed in the past decade, which cover the special case p = 2. Applications to the maximal pro-p Galois group of F are also described.

Finite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

Introduction. In this paper, we study the following question: How long a cycle must there be in a 3-connected 3-regular graph on n vertices? For planar graphs this question goes back to Tait [6], who conjectured that any planar 3-connected 3-regular graph is hamiltonian. Tutte [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tutte's example, Grunbaum and Motzkin [3] constructed an infinite family of 3-connected 3-regular planar graphs such that the length of a longest cycle in each member of the family is at most nc, where c = 1 — 2~17 and n is the number of vertices. The exponent c was sub­ sequently reduced by Walther [8, 9] and by Grunbaum and Walther [4]. It is natural to ask what one can say when the planarity condition is dropped. For 2-connected 3-regular graphs, Bondy and Entringer [2] proved that the length of a longest cycle is at least 4 log 2?z — 4 log 2log2w — 20, and an example due to Lang and Walther [5] shows that this result is essentially best possible. Let f(n) denote the largest integer k such that every 3-connected 3-regular graph on n vertices contains a cycle of length at least k. For planar graphs, Barnette [1] proved that

Recently, H. M. Srivastava extended certain interesting generating functions of L. Carlitz to the forms:andwhere are general oncand many-parameter sequences of functions. In the present paper some general addition formulas for analogous sequences of functions are derived, and a number of interesting applications of the main results are given.

We define (as in [7]) integer sequences one for each positive integer k ≧ 2, by1.1where are the kth roots of unity and (E(k))n is replaced by after multiplying out. We note that (1.1) implies , n ≠ 0 (mod k).In [7], we considered some special properties of these number sequences, proved several congruences and conjectured several others. This paper is a continuation of the work presented in [7].In Section 2 we demonstrate the asymptotic rate of growth of the numbers by showing thatIn Section 3 we present a large number of congruences (modulo 2048), some of which are proved or can be proved by the techniques presented herein, and other congruences which appear to be true on the basis of numerical evidence.

It is a classical conjecture of E. Artin that any integer a > 1 which is not a perfect square generates the co-prime residue classes (mod p) for infinitely many primes p. Let E be the set of a > 1, a not a perfect square, for which Artin's conjecture is false. Set E(x) = card(? E E: e 1, which is not a perfect square, is a primitive root (mod p) for infinitely many primes p. We shall abbreviate this conjecture of Artin as AC. Artin's conjecture was proved to be correct by Hooley (5) provided one assumes the generalized Riemann hypothesis for certain Dedekind zeta functions. The first unconditional result was obtained by Gupta and Ram Murty in (2), where it was shown that there is a finite set S, consisting of thirteen elements, such that for some a E S, AC is true for a. Subsequently, S was replaced by another finite set of seven elements in (3). In this paper, we consider the exceptional set for Artin's conjecture. More precisely, let

Let be a Hilbert space with inner product (•, •) and let E(•) and E0(•) be spectral measures in corresponding to self-adjoint operators and . In this paper we consider the set function ƒ(I × J) = E(I)E0(J) defined on the semiring of bounded rectangles, and obtain an integral representation for this set function for disjoint I, J under the hypotheses that H — H0 is a type of Carleman operator.

In this paper we prove a sharpening and generalization of the following Theorem of Khintchine (4):Let ψ1(q), …, ψnq) be n non-negative junctions of the positive integer q and assumeis monotonically decreasing. Then the set of inequalities1has an infinity of integer solutions q > 0 and p1, … , pn for almost all or no sets of numbers θ1, … , θ2, according as Σψ(q) diverges or converges.

The purpose of this paper is to exploit an analogy between algebraic extensions of fields and essential extensions of modules, in which the role of the algebraic closure of a field F is played by the injective hull H(M) of a unitary left R-module M. (The notion of * ‘algebraic’ extensions of general algebraic systems has been studied by Shoda; see, for example [5].)In this analogy, the role of a polynomial p(x) is played by a homomorphism of R-modules(1)which will be called an ideal homomorphism into M. The process of solving the equation p(x) = 0 in F, or in an algebraic extension of F, will be replaced by the process of extending an ideal homomorphism (1) to a homomorphism F* from R into M, or into an essential extension of M.

While the contents of the author's doctoral thesis (4) have, owing to their lengthy nature, been published only in small part (5, §2; 6; 7), the absence from the literature of graph theory of any characterization of infinite directed graphs with Euler lines seems to constitute a definite gap that prompts the publication in the present paper of some further material from (4). The main results characterizing such directed graphs will be obtained in §§2 and 3. In §4, we shall indicate an alternative (and perhaps better) formulation of one of these results, some extensions obtained in (4), and some comparisons between parallel results for undirected and directed graphs. A familiarity with the definitions and results of (7) will be assumed in §4, but not before.

McShane (5, 6) has introduced the concept of “Dedekind completeness” for partially ordered sets, which seems to be a natural generalization of the usual concept of completeness for lattices. It is the purpose of this paper to discuss some of the properties of Dedekind completeness, particularly with respect to a rather natural class of partially ordered sets which we call “uniform.”

1. Notation. Let s(x) be a function integrable in every finite interval of x ≥ 0. Then the Riemann-Liouville integral of s(x), of order a > 0, is defined for x ≥ 0 by(1).The object of this note is to prove a Tauberian theorem for sα(x) in the case in which α is a positive integer p, employing certain difference formulae due to Karamata (4, Lemma 2) and Bosanquet (1, Theorem 1) used already for a broadly similar purpose in an earlier paper (12) where a is any positive number.

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

The word matrix will, in this paper, always connote a matrix with non-negative elements, having in general m rows and n columns. In conformity with the definition in (4), two matrices will be said to have the same pattern if the entry in any row or column is zero or not according as the corresponding entry of the other is zero or not. The symbols ρi and σj will stand for the respective phrases “ith row-sum” and “jth column-sum” of the matrix under consideration. r1 … , rm; c1, … , cn is a set of positive numbers. It will be said to be consistent for the pattern of an m × n matrix, if there exists a matrix of that pattern for which ρi = ri and σj = cj for all i and j.