Let L be a σ-Dedekind complete Riesz space. In (8), H. Nakano uses an extension of the multiplication operator on a Riesz space into itself (analagous to the closed operator on a Hilbert space) to obtain a representation space for the Riesz space L. He calls such an operator a “dilatator operator on L.” More specifically, he shows that the set of all dilatator operators , when suitable operations are defined, is a Dedekind complete Riesz space which is isomorphic to the space of all functions defined and continuous on an open dense subset of some fixed totally disconnected Hausdorff space. The embedding of L in the function space is then obtained by showing that L is isomorphic to a Riesz subspace of . Moreover, when L is Dedekind complete, it is an ideal in , and the topological space is extremally disconnected.

In this paper we consider the following problem. Let (E, M, N, π) be a differentiable fibre bundle, where E is the total space, M the base space, N the fibre, and π: E → M the projection map. Then, given a Cr triangulation (ƒ, D) of M, can one obtain a Cr triangulation (F, K) of E such that the induced map ƒ–1πF: K → D is linear? R. Lashof and M. Rothenberg (3) have obtained this result for vector bundles.

Using methods quite different from theirs, we obtain a solution in the general case. The methods we use are the geometric methods developed by J. H. C. Whitehead. (7) in his triangulation of differentiable manifolds, as found in (5). In fact, our solution consists of generalizing his techniques in a fibre bundle setting.

A set of ordered pairs of integers {(ai, mi)} is said to cover the integers if each integer x satisfies the congruence x ≡ ai (mod mi) for some i. We may assume that the mi are positive. Trivially {(0, 1)} covers, as does {(0, m), (1, m), (2, m), … , (m — 1, m)}. In order to arrive at some non-trivial problems concerning covers, the following definition is given: A finite set of ordered pairs of integers with mi > 1 and mi ≠ mj if i ≠ j, is called a covering class of residues if every integer x satisfies the congruence x ≡ ai (mod mi) for some i.

A finite p-group G is said to be regular if for every pair of elements a, b in G,

1

where c is an element of the derived group of the subgroup generated by a and b. We shall say that G is nearly regular if (i) there is a central element z of order p such that G/⧼z⧽ is regular, and (ii) the subgroup generated by x and y is regular whenever x ∈ G and y ∈ G′, the derived group of G.

A. Zirakzadeh (1) has determined for n = 3 the minimal value of the perimeter length of a polygon A1 A2 … An, where A1, A2, … , An–1, and An divide the perimeter of a convex curve C, of perimeter length l, into n parts of equal length; further he has stated a conjecture concerning the general case. In the following a simpler proof for the case n = 3 is given; the minimum for even values of n, which confirms the conjecture of A. Zirakzadeh, is determined; and a fairly precise estimation for odd values of n, which refutes the conjecture of A. Zirakzadeh, is given. For n = 3 we have the following theorem.

Throughout this paper R and Rn will denote v-rings, that is, complete discrete rank-one valuation rings of characteristic zero, having a common residue field k of characteristic p. R is assumed unramified and Rn has ramification index n. Let π be a prime element in Rn. Then Rn = R[π], where π is a root of an Eisenstein polynomial ƒ = xn + pƒn-1 xn-1 + … + pƒ0 with coefficients in R and ƒ0 a unit. Thus Rn is inertially isomorphic to R[[x]]/ƒR[(x)], that is, the rings are isomorphic by a mapping which induces the identity mapping on the common residue field. R[[x]] represents the power series ring in the indeterminate x over R. In this paper we identify Rn with R[[x]]/ƒR[[x]], R with its natural embedding in Rn and π with x + ƒR[[x]].

In this section we fix some notations and give a definition of an area measure on a differentiate manifold, where throughout the paper the word differentiable implies differentiability of class C∞. Let M denote a differentiate manifold of dimension n and call a set of m linearly independent vectors {e1, … , em} at a point of M an m-frame of M. The set E′ of all such m-frames can be given the structure of a differentiable fibre bundle over M and we denote the projection of E' onto M by π′.

Let X1 X2, … , Xn be mutually independent random variables with a common continuous distribution function F(t). Let Fn(t) be the corresponding empirical distribution function, that is Fn(t) = (number of Xi ⩽ t, 1 ⩽ i ⩽ n)/n.

Consider the following sieve process. Let A(1) be the sequence of integers greater than 1. Let A(n+1) be obtained from A(n) = {a1(n), a2(n), …} by eliminating one element from each of the intervals Ik(n), where

We let an = an(n) and A = {an} be the sequence of integers that survive the sieve. M. C. Wunderlich (8) has found a necessary and sufficient condition for an ∼ n log n and, in a more recent paper, M. Wunderlich and W. E. Briggs (9) have studied a subclass of the sequences defined above for which an ∼ n log n.

There are several ways to approach the eigenfunction expansion problem for ordinary differential operators via the spectral theorem for self-ad joint linear operators in Hilbert space. One can examine the resolvent, which requires a detailed study of the Green's function (4, 5, 7), or one can use the spectral theorem for unbounded operators (2, 3, 9). Since the eigenf unction expansion theorem also requires some multiplicity theory, unless one is prepared to use a rather powerful form of the spectral theorem for unbounded operators, as in (2, 9), the proof requires a good deal of work in addition to the spectral theorem.

This paper presents two results. They are:

Theorem 1. Let G be a doubly transitive permutation group of degree nq + 1 where a is a prime and n < g. If G is neither alternating nor symmetric, then G has Sylow q-subgroup of order only q.

Result 2. There is no unsolvable transitive permutation group of degree p = 29, 53, 149, 173, 269, 293, or 317 properly contained in the alternating group of degree p.

Result 2 was demonstrated by a computation on the Illiac II computer at the University of Illinois.

Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on a linear associative algebra , with identity, over the real field R or the complex field C. In this paper we shall consider a semi-simple algebra = ⊕ … ⊕ over R or C with simple components . Let G be the group of all automorphisms or anti-automorphisms of which leave the ground field elementwise invariant, and let H be the subgroup of G such that Ω = (i = 1, 2, … , t) for each Ω in H.

The purpose of this paper is to investigate conditions under which the Hilbert transform defines a bounded linear operator from a given function space into itself. The spaces with which we deal have the property of rearrangement-invariance which is defined in §1. This class of spaces includes the Lebesgue, Orlicz, and Lorentz spaces.

We are concerned here with the existence of fixed or common fixed points of commuting monotone self-mappings of a partially ordered set into itself. Let X be a partially ordered set. A self-mapping ƒ of X into itself is called an isotone mapping if x ⩾ y implies ƒ(x) ⩾ ƒ(y). Similarly, a self-mapping ƒ of X into itself is called an antitone mapping if x ⩾ y implies ƒ(x) ⩽ ƒ(y). An element X0 ∈ X is called well-ordered complete if every well-ordered subset with x0 as its first element has a supremum. An element x0 ∈ X is called chain-complete if every non-empty chain C ⊆ X such that x ⩾ x0 for all x ∈ C, has a supremum. X is called a well-ordered-complete semi-lattice if every non-empty well-ordered subset has a supremum. X is called a complete semi-lattice if every non-empty subset of X has a supremum.

This paper is concerned with generalizations of the classical Hardy spaces (8, p. 39) and the question of boundary values for functions of these various spaces. The general setting is the “big disk” Δ discussed by Arens and Singer in (1, 2) and by Hoffman in (7). Analytic functions are defined in (1). Classes of such functions corresponding to the Hardy Hp spaces are considered and shown to possess boundary values in (2). Contrary to the classical case, such functions do not form a Banach space; hence they are not the functional analytic analogue of the classical spaces. In (3) quasi-analytic functions are defined while in (4) Hardy spaces of such functions are considered and are shown to have boundary values and to form a Banach space.

For various classes Q of metric spaces, there are several well-known results characterizing the local n-connectivity of a metric space in terms of n-ANR(Q)'s. Specifically, we have in mind the results of Kuratowski (13, p. 265) and Kodama (10, p. 79). The main purpose of this paper will be to obtain similar results along these lines for non-metric classes Q. In the last part of the paper we specify Q to be the class of totally normal spaces and characterize the local n-connectivity of an n-dimensional separable metric space in terms of ANR(Q)'s.

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that

1.1

Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.

A cover is directed (is well-ordered, totally ordered) if it is directed (is well-ordered, totally ordered, respectively) by set inclusion. Specifically is directed if, given U, V in , there exists W in such that U ⋃ V ⊂ W.

Directed covers have been used implicitly in numerous theorems dealing with paracompactness and related properties. In particular, characterizations of paracompactness in terms of product spaces seem to require the use of directed covers (3; 11, §2; or 12). However, the utility of directed covers has not been fully exploited.

Let p be a large prime and let ƒ(x) be a polynomial of fixed degree d ⩾ 4 with integral coefficients, say,

1.1

Recently Mordell (8) has considered the problem of estimating the least positive residue of ƒ(x) (mod p), that is, the unique integer l (0 ⩽ l ⩽ p — 1) such that the congruence

1.2

is soluble for r = l but not for r = 0, 1, … , l — 1.