We are concerned in this paper with category-theoretic aspects of homotopy theory. Originally, category theory developed as a simplifying language in the context of algebraic topology and yet one primary example: the category Π of spaces and homotopy classes of maps admits only limited use of the language owing to the very sparse occurrence of limits. Of course, full use has been made of them nevertheless: limits and colimits exist in the case of products and coproducts, and in almost no other case; yet, from this we obtain the theory of Samelson products, Whitehead products, and Hopf invariants which can all be expressed in Π see [8]. In addition, there are hosts of adjoint functors and yet the outcome is disappointing because the language applies only to special cases rather than to the situation as a whole.

Since Tutte initiated the systematic enumeration of planar maps [11], most of the literature on the subject has dealt with rooted maps (i.e., with maps whose automophism group has been trivialized by distinguishing a doubly-oriented edge). In particular, Tutte proved [11] that the number B′(n) of rooted planar 2-connected (i.e., non-separable) maps with n ≧ 1 edges is expressed by the formula

Recently one of the authors developed a general technique for enumerating unrooted planar maps considered up to orientationpreserving isomorphisms (see [6] and [8]). This technique, which is based on combinatorial map theory, Burnside’s lemma [3, p. 181] and the concept of a quotient map (see Section 1.4), was used to find, with little algebraic manipulation, simple counting formulae for the numbers of non-isomorphic planar maps of several types [7].

For nonlinear equations of the form

I

there has been considerable interest in determining the asymptotic forms of nonoscillatory solutions. We assume r(t) is continuous and positive on [0, ∞), and f(t, x) is continuous on [0, ∞) × R, and f(t, x) ≥ 0 for x ≠ 0. For n = 2, equation (I) was studied by Kusano and Naito [3], who found necessary and sufficient conditions for the existence of minimal and maximal nonoscillatory solutions. The former are the bounded solutions, while the later are those asymptotic to the function

1.1

Their method consisted of writing (I) in the form of an integral operator and applying the Schauder fixed point theorem. For arbitrary n, but for r(t) = 1, Kreith [2] found necessary and sufficient conditions for the existence of maximal solutions.

If k is a f.r. (= formally real) field which is partially ordered with positive cone, P, XP denotes the space of total orders T of k with P ⊂ T. Suppose you have a subset A ⊂ XP and an element T ∈ XP, T ∉ A. Then the main question investigated in this paper is the following: How can T be separated from A by using elements of k? To be more specific, this is split up into two different questions.

Question 1. Suppose A is closed. Then there is an n ∈ N and elements a1, …, an ∈ k such that the basic open set H = H(a1, …, an) is a neighborhood of T and has empty intersection with A. Now, if T is given, what is the least n ∊ N (if it exists) such that T has a neighborhood basis consisting of basic open sets of the form H(a1, …, an)?

The theory of Fourier transforms of tempered distributions as developed by Laurent Schwartz [17] is quite simple and elegant and has wide variety of applications, but there does not exist a corresponding neat and simple theory for the Hilbert transform of generalized functions (distributions) having wide applications. One of the objectives of this paper is to develop such a theory for the Hilbert transform of generalized functions and indicate its applicability to a variety of problems.

In problems of physics sometimes we need to find harmonic functions u(x, y) in the region y > 0 whose limit as y → 0+ does not exist in pointwise sense but does exist in the distributional sense. The theory of Hilbert transform of generalized functions that we are going to develop will provide an answer to the existence and uniqueness of this problem.

A preference order, or linear preorder, on a set X is a binary relation which is transitive, reflexive and total. This preorder partitions the set X into equivalence classes of the form . The natural relation induced by on the set of equivalence classes is a linear order. A well-founded preference order, or prewellordering, will similarly induce a well-ordering. A representation or Paretian utility function of a preference order is an order-preserving map f from X into the R of real numbers (provided with the standard ordering). Mathematicians and economists have studied the problem of obtaining continuous or measurable representations of suitably defined preference orders [4, 7]. Parametrized versions of this problem have also been studied [1, 7, 8]. Given a continuum of preference orders which vary in some reasonable sense with a parameter t, one would like to obtain a continuum of representations which similarly vary with t.

Let R be any division ring and let

1

be a polynomial, in the indeterminate X, with coefficients in R. Note that the powers of X are always to the right of the coefficients. We denote the set of all such polynomials by R[X].

B. Beck [3] proved the following theorem for the generalized quaternion division algebra; i.e., any division ring of dimension 4 over its center:

THEOREM 1. If f(X) is of degree n then f(X) has either infinitely many or at most n zeros in R.

Under a reasonable definition of multiplicity Beck also proved:

THEOREM 2. Let (c1, c2, …, cn) be a set of pairwise non-conjugate elements of R, and (m1, …, mN) positive integers such that Σmi = n = deg f(x).

In [4], H. Kamowitz considered the condition, to be satisfied by a bounded operator N on a Hilbert space , that

for all operators X on . Kamowitz discovered that such an N must be normal and its spectrum must lie on a line or a circle; that is, N must be of the form αJ + β, where α and β are complex numbers and J is either Hermitian or unitary. G. Weiss [5] showed that the Hilbert-Schmidt norm behaves differently: N need only be normal in order that

for all finite-rank operators X, and in fact this condition is equivalent to normality. Actually, the result in [5] removes the restriction that X be finite-rank, that is, if N is normal and X is any bounded operator, then

We define (as in [7]) integer sequences one for each positive integer k ≧ 2, by

1.1

where 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 that

In 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.

We examine questions related to approximating functions by sums of the form

We focus on approximations to functions given by the integral transformation

where γ is a positive measure. Approximations to this class of functions (Laplace transforms in the variable — lnx) are particularly well behaved (see Theorem 1). Questions concerning existence, uniqueness and characterization of such approximations have been thoroughly examined in the equivalent setting of exponential sum approximations (see [3], [4], [6] and [9]). Less well studied is the order of convergence of the approximation. This is the problem we address. Part of the motivation for using sums of the form (1), which we shall call Gaussian sums, stems from the observation that all analytic functions with Taylor series expansion having positive coefficients are of the form (2).

In 1979 Edgar asked for a characterization of those completely regular Hausdorff topological spaces X which have the property that any Boolean σ-homomorphism from the Baire σ-field of X into the measure algebra of an arbitrary complete probability space can be realized by a measurable point-mapping. Those spaces X will be called homomorphism-compact or, for short, H-compact hereafter. It is wellknown that compact spaces are H-compact (cf. [4], p. 637, Proposition 3.4). We will show that the same is true for strongly measure compact spaces. On the other hand H-compact spaces are easily seen to be real-compact. Since the notions of measure-compactness and liftingcompactness (cf. [3]) also lie between strong measure-compactness and real-compactness it is natural to investigate the relations among these notions. Here the results are mainly negative (cf. Sections 4 and 6). Concerning the structural properties of H-compactness not very much can be said so far (cf. Section 7): it is, for instance, unknown whether the product of two H-compact spaces is again H-compact.