In a survey of methods in enumerative map theory [14], W. T. Tutte pointed out that little has been done towards enumerating unrooted maps other than plane trees. A notable exception is to be found in the work of Brown, who took the initial step in this direction by enumerating non-separable maps up to sense-preserving homeomorphisms of the plane [2]. He then took a further step, allowing sense-reversing homeomorphisms, by counting triangulations and quad-rangulations of the disc [3, 4]. In all these problems, however, there is a fixed outer region of the plane. This can be considered as a certain type of rooting of a planar map, which is normally regarded as lying on the sphere or closed plane. It is our object here to find an expression for the number of unrooted planar maps in a given set, in terms of the numbers of maps in that set which have been rooted in a special way.

A K-coloured rooted tree t is said to have colour partitionL if L is a K × ∞ matrix with elements lij equal to the number of non-root vertices of colour i and degree j. If adjacent vertices are of different colours then t is called a chromatic tree and L a chromatic partition. The tree has edge partitionD where D is a K × K matrix with elements dij equal to the number of edges, directed away from the root, from a vertex of colour i to a vertex of colour j.

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.

This paper extends several graph-theoretic results to matroids. The main result of Tutte's paper [10] which introduced the theory of n-connection for matroids was a generalization of an earlier result of his [9] for 3-connected graphs. The latter has since been strengthened by Halin [3] and in Section 3 of this paper we prove a matroid analogue of Halin's result. Tutte used his result for 3-connected graphs to deduce a recursive construction of all simple 3-connected graphs having at least four vertices. In Section 4 we generalize this by giving a recursive construction of all 3-connected matroids of rank at least three. Section 2 contains a generalization to minimally n-connected matroids of a result of Dirac [2] for minimally 2-connected graphs.

The width of a lattice L is the maximum number of pairwise noncomparable elements in L.

It has been known for some time ([5] ; see also [4]) that there is just one subdirectly irreducible lattice of width twro, namely the five-element nonmodular lattice N5. It follows that every lattice of width two is in the variety of N5, and that every finitely generated lattice of width two is finite.

Beginning a study of lattices of width three, W. Poguntke [6] showed that there are infinitely many finite simple lattices of width three. Further studies on width three lattices were made in [3], where it was asked whether every finitely generated simple lattice of width three is finite. In this paper we will show that, in fact, more is true:

THEOREM 1.1. Every finitely generated subdirectly irreducible lattice of width three is finite.

Consider the following formula due to Young [7] for the calculation of the homogeneous product sum, hλ, in terms of Schur functions;

where the operation Srs is defined as follows:

Y1: Srs, where r < s, “represents the operation of moving one letter from the s-th row up to the r-th row; and the resulting term is regarded as zero, when any row becomes less than a row below it, or when letters from the same row overlap; as, for instance, happens when λ1 = λ2 in the case of S13S23.“

The following example of the above is given by Robinson [4].

Calculation by other means shows that the above analysis of h(3,2,1) is correct; however, it will be noticed that the operator S123S23 does not appear in the above yet it is not specifically excluded by the rule Y1.

Let p be an integer and let H(p) be the class-number of the field

where ζp is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4).

In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4).

We designate by h(p) the class-number of the real quadratic field

Let l < p < ∞, p ≠ 2 and α > 0. In what follows sp(α) will denote the space of all real or complex sequences for which

(1.1)

In this paper we show that the spaces sp(α) are Banach spaces under the natural norm and in fact share many properties that the usual lp spaces have. Our main results give characterizations of the surjective isometries of sp(α). These turn out to be quite different than the results for lp. For example, we show that for α ≠ 1, an operator T is a surjective isometry if and only if T is a modulus one multiple of the identity. The methods used are valid for both real and complex scalars. They involve the use of a disjoint support condition together with a property of semi inner products. In the complex case the information on isometries allows us to give complete descriptions of the Hermitian operators as well as the adjoint abelian operators.

Let A be a quasi-central complex Banach algebra with a bounded approximate identity and Prim A the structure space of A. In [15], we have shown that every central double centralizer T on A can be represented as a bounded continuous complex-valued function ΦT on Prim A such that Tx + P = ΦT(P)(x + P) for all x ∈ A and P ∈ Primal when the center Z(A) of A is completely regular. Here x + P for P ∈ Prim A denotes the canonical image of x in A/P. In particular, in the case of quasi-central C*-algebras, this result is equivalent to the Dixmier's representation theorem of central double centralizers on C*-algebras (see [3, Section 2] and [9, Theorem 5]).

In this paper, it is shown that if Z(A) is completely regular then the space Prim A is locally quasi-compact and for each element z of Z(A), ΦLz vanishes at infinity, where Lz for z ∊ Z(i) is the central double centralizer on A defined by Lz(x) = zx for all x ∊ A.

At the Georgia conference in 1961 Fox presented a paper, “Knots and periodic transformations”, in which he asked which knots may be fixed by a periodic transformation of the 3-sphere. He distinguished eight cases according to the type of fixed point set of T and its relationship to the knot. Except for case a), all these cases have since received some attention and conditions have been given for knots to fall into each of these classes. In fact, the problem of deciding which knots fall into category d) (that is “periodic knots“; I will refer to them as cyclically periodic knots) has been the subject of at least six papers [2], [3], [13], [15], [17], [22], but measured by their effectiveness at determining the periods of the knots to nine crossings, the theorems contained in these papers are not entirely satisfactory.

Throughout this paper, R will be a commutative integral domain with identity and x an indeterminate. If ƒ ∈ R[x], let CR(ƒ) denote the ideal of R generated by the coefficients of ƒ. Define SR = {ƒ ∈ R[x]: cR(ƒ) = R} and UR = {ƒ ∈ R(x): cR(ƒ)– 1 = R}. For a,b ∈ R, write . When no confusion may result, we will write c(ƒ), S, U, and (a:b). It follows that both S and U are multiplicatively closed sets in R[x] [7, Proposition 33.1], [17, Theorem F], and that R[x]s ⊆ R[x]U.

The ring R[x]s, denoted by R(x), has been the object of study of several authors (see for example [1], [2], [3], [12]). An especially interesting paper concerning R(x) is that of Arnold's [3], where he, among other things, characterizes when R(x) is a Priifer domain. We shall make special use of his results in our work.

Let be a valuation ring with unit element, i.e., is a commutative ring such that for any a and b in , either a divides b or b divides a. We assume 2 is a unit of . V is an n-ary nonsingular quadratic module over , O(V) or On(V) is the orthogonal group on V, and S is the set of symmetries in O(V). We define l(σ) to be the minimal number of factors in the expression of a of O(V) as a product of symmetries on V. For the case where is a field, l(σ) has been determined by P. Scherk [6] and J. Dieudonné [1]. In [3] I have generalized the results of Scherk to orthogonal groups over valuation domains. In the present paper I generalize my results of [3] to orthogonal groups over valuation rings.

Since is a valuation ring, it is a local ring with the maximal ideal A which consists of all nonunits of .

In this paper we consider the problem of determining when an algebra of formal power series over a commutative ring R is the homomorphic image of a reduced incidence algebra P(R, ∽). The question of when two such algebras are isomorphic is answered in Section 8 of [1]. A slight generalization of their notion of full binomial type is introduced here.

Section 1 contains background material together writh a summary of the results of [1]. In Section 2 we present the desired characterization, and to conclude an application appears in Section 3. In Section 3 the tools of Section 2 are used to derive an equation of R. W. Robinson and R. P. Stanley which counts labelled, acyclic digraphs.

The concept of θ-closure was introduced by Velicko to study H-closed spaces and to generalize Taimanov's extension theorem [11], [12]. More recently, this notion has been used by Dickman and Porter [1] to characterize those Hausdorff spaces in which the Fomin H-closed extension operator commutes with the projective cover (absolute) operator and [2] to study extentions of functions. If X is a topological space and A ⊂ X, we let Σ(A) and Γ(A) represent, respectively, the family of open subsets which contain A and closed subsets which contain some element of Σ(A). The θ-closure of A ⊂ X, denoted by clθ(A) (clθ(v) if A = {v}), is {x ∈ X: each V ∈ Γ(x) satisfies V ∩ A ≠ ∅} and A is called θ-closed in case clθ(A) = A.

This paper addresses some flatness properties of an (n – 1)-sphere Σ in Euclidean n-space En resulting from the presence of round balls in En tangent to Σ. The notion of tangency used here is geometric rather than differentiable, for a round n-cell Bp (that is, the set of points whose distance, in the standard metric, from some center point is less than or equal to a fixed positive number) is said to be tangent to the (n – 1)-sphere Σ in En at a point p ∈ Σ if p ∈ Bp and Int Bp ⊂ Σ = ∅. The ball Bp is called an interior tangent ball at p if Int Bp ⊂ Int Σ; otherwise, it is called an exterior tangent ball at p.

1. Introduction. The reduction of a matrix to its Jordan normal form is an unstable operation in that both the normal form itself and the reducing mapping depend discontinuously on the elements of the original matrix. For example, the matrix trivially reduces to itself in Jordan form, but there are arbitrarily small perturbations of this matrix that reduce to the form which is certainly not a small perturbation of the original matrix, and moreover the reducing mapping is not a small perturbation of the identity. In [1], Arnol'd derives the simplest possible normal forms to which families of matrices may be linearly reduced in a ‘stable’ manner. In this paper, we consider a ‘topological' version of the problem, using the classification of matrices up to topological conjugacy given in [8] and the classification of linear dynamical systems up to orbital equivalence given in [9].

Although the theory of complex Banach algebras is by now classical, the first systematic exposition of the theory of real Banach algebras was given by Ingelstam [5] as late as 1965. More recently, further attention to real Banach algebras was paid in 1970 [1], where, among other things, the (real) standard algebras on finite open Klein surfaces were introduced. Generalizing these considerations, real uniform algebras were studied in [7] and [6].

In the present paper, an attempt is made to develop the theory of real function algebras (see Section 1 for the definition) along the lines of the complex function algebras. Although the real function algebras are not structurally different from the real uniform algebras introduced in [7], they are easier to deal with since their elements are actually (complex-valued) functions.

In 1889, A. A. Markov proved the following inequality:

INEQUALITY 1. (Markov [4]). If pn is any algebraic polynomial of degree at most n then

where ‖ ‖A denotes the supremum norm on A.

In 1912, S. N. Bernstein established

INEQUALITY 2. (Bernstein [2]). If pn is any algebraic polynomial of degree at most n then

for x ∈ (a, b).

In this paper we extend these inequalities to sets of the form [a, b] ∪ [c, d]. Let Πn denote the set of algebraic polynomials with real coefficients of degree at most n.

THEOREM 1. Let a < b ≦ c < d and let pn ∈ Πn. Then

for x ∈ (a, b).

We shall consider a multiparameter eigenvalue problem of the form

(1.1)

where λ ∈ Rk while Tn and Vn(λ) are self-adjoint linear operators on a Hilbert space Hn. If λ = (λ1, …, λk) ∈ Rk and satisfy (1.1) then we call λ an eigenvalue, x an eigenvector and (λ, x) an eigenpair. While our main thrust is towTards the general case of several parameters λn, the method ultimately involves reduction to a sequence of one parameter problems. Our chief contributions are (i) to generalise the conditions under which this reduction is possible, and (ii) to develop methods for the one parameter problem particularly suited to the multiparameter application. For example, we give rather general results on the magnitude and direction of the movement of non-linear eigenvalues under perturbation.

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l by

where y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfying

with inner product

where . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

Let ψ0, ψ1, … denote the Walsh-Paley functions and let ∔ denote the group operation which Fine [5] defined on the interval [0, 1). Thus, if k ≧ 0 is an integer and if u, t are points in the interval [0, 1) then

(where αk = 0 or 1 represents the kth coefficient of the binary expansion of t), and

A real-valued function ƒ, is said to be dyadically differentiable at a point x ∈ [0, 1) if ƒ is defined at x and at x ∔ 2–n–1, n = 0, 1, …;, and if the sequence

(1)

converges as N → ∞.