The concept of acyclic coloring was introduced by Grünbaum [5] and is a generalization of point arboricity.

A proper k-coloring of the vertices of a graph Gis said to be acyclic if G contains no two-colored cycle. The acyclic chromatic number of a graph G, denoted by a(G), is the minimum value of k for which G has an acyclic k-coloring. Let a(n) denote the maximum value of the acyclic chromatic number among all graphs of genus n. In [5], Grünbaum conjectured that a(0) = 5 and proved that a(0)≤9. The conjecture was proved by Borodin [3] after the upper bound was improved three times in [7], [1] and [6]. In [2], we proved that a(1)≤a(0) + 3. The purpose of this paper is to prove the following

Theorem. Any graph of genus n>0 can be acyclically colored with 4n + 4 colors.

It is not known for any n>0 whether a(n)>H(n), the Heawood number [8].

The integral equation of the title is

It was studied in [4], though h(x) was written as x-1g(x-1) there, and using a method involving orthogonal Watson transformations, it was shown there that if h ∈ L2(0, ∞), then the equation has a solution f ∈ L2(0, ∞), and that / is given by

In this paper, using the techniques of [3], we shall show that the equation can be solved for ℎ in the space ℒμ, p of [3] for 1 ≤ p < ∘, μ > 0, and that for these spaces, which include L2(0, ∘), f is given by the simpler formula

We shall further show that these results can be extended to the spaces ℒw, μ, p of [3]. This forms the content of our theorem below.

In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, ℋ may be a congruence, and so on.

The nature of the eigenvalues of a square quaternion matrix had been considered by Lee [1] and Brenner [2]. In this paper the author gives another elementary proof of the theorems on the eigenvalues of a square quaternion matrix by considering the equation Gy = μȳ, where G is an n x n complex matrix, y is a non-zero vector in Cn, μ is a complex number, and ȳ is the conjugate of y. The author wishes to thank Professor Y. C. Wong for his supervision during the preparation of this paper.

Let fr:Xr → BO(r) be a sequence of fibrations with maps gr:Xr → Xr+1 such that the usual diagram commutes. For such a situation R. Lashof defines the concept of an X-structure on manifolds (see [3]), and proves a Thom-isomorphism for the cobordism groups of such manifolds. Let n, m be positive integers which are fixed throughout this paper. If r is very big in comparison with n + m then Xr is a simply connected CW-complex and the map (gr)*:H*(Xr; Q)→ H*(Xr+l; Q) is an isomorphism up to dimension n. Let γ be the pull-back over Xr of the universal r-linear bundle (which is, of course, a bundle over BO(r)). If r is very big in comparison with n + m, then we put X = Xr, and we assume that γ is orientable and oriented. The elements of H*(X; Q) of dimension not greater than n, will be called rational universal X-characteristic classes. It is well-known that many of the usual classes of manifolds may be described in terms of X-structures, (e.g. SO, SU, Spin-manifolds etc.).

In this paper we continue the examination of the question of similarity of operators A and B begun in reference [3]. In that article, a similarity result was obtained based on a measure of closeness, or proximity, of the uniformly continuous semigroups etA and etB, t>0. The operators considered were elements of ℬ(ℋ), the algebra of bounded operators on a Hilbert space ℬWe now wish to relax this requirement and replace ℬ(ℋ) by a complex Banach algebra ℬ with unit I. In Section 2 we give a necessary condition for the similarity of A, B ∈ ℋ. We then give a condition sufficient to guarantee A and B are approximately similar (as defined in reference [5]). In Section 3 we restrict our attention to the case where ℋ = ℋ(ℋ). There we give a condition which guarantees A, B ∈ ℋ(ℋ) are intertwined by a Fredholm operator. This leads naturally into a discussion of proximity-similarity in the Calkin algebra si. This is the subject of Section 4. Following reference [7] we define a metric p on N(ℋ), the normal elements of ℋ We show (N(ℋ), p) is a complete metric space and that the unitary orbit of ℋ (N(ℋ) p)is the p-connected component of a in N (ℋ).

Let G be a group given in terms of generators and denning relations. The order problem is said to be solvable for (the given presentation of) the group G if, given any element W of G (as a word in the given generators of G), we can determine the order of W in G. The power problem is solvable for G if, given any pair X, Y of elements of G, we can determine whether or not X belongs to the cyclic subgroup {Y} of G generated by Y. It is easy to see that if either of these problems is solvable for G, then the word problem is also solvable for G.

An algebra L = L = (A; V, Λ, *, +, 0, 1) of type (2, 2, 1, 1, 0, 0) is called a distributive double p-algebra whenever its reduct (A, V, Λ, 0, 1) is a distributive (0, 1)-lattice that, for any a ∈ A, contains a greatest element a* such that a Λ a* = 0 and a least element a+ for which a v a+ = 1.

The purpose of this paper is to present a proof of the following theorem of Maclagan-Wedderburn.*

Every finite skew-field† is a field.

The proof depends on group theory and on the properties of Galois fields. As an introduction, §§1–4 are devoted to a systematic and self-contained account of the theory of Galois fields.

In the paper [5], Rema used the well-known fact that in a Boolean algebra the binary operation d: B × B → B defined by is a “metric“ operation to show that, if D is any dual ideal of ^, then the sets Up = {(x, y): d(x, y) <p}, where p ∈ D, form a base for a uniformity of }, the resulting topological space <B; T[D]> being called an auto-topologized Boolean algebra. Recently, Kent and Atherton [1, 4] exhibited a family of topologies on an arbitrary lattice ℒ defined in terms of ideals and dual ideals. More specifically, if I and D are respectively an ideal and a dual ideal of ℒ, then the T[I:D] topology on ℒ is the topology defined by taking the sets of the form a*⋂b+, where , as sub-base for the open sets. It is these topologies that are studied in this paper.

Let Rbe an associative ring, Jan infinite index set, and Rj the ring of all J × J row-finite matrices over R. The Jacobson radical of Rwill be denoted by Γ(R).

Suppose that T and S are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and S in the sense that SA = AT, then a classical result due to Rosenblum implies that the spectra σ(T) and σ(S) must overlap, see [12]. Actually, Davis and Rosenthal [5]have shown that the surjectivity spectrum σsu(T) will meet the approximate point spectrum σap(S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see [9], [12], [13].

We identify the extreme points of the unit sphere of the Lorentz space Lw,1 This yields a characterization of the surjective isometries of Lw,1(0,1). Our main result is that every element in the unit sphere of Lw,1 is the barycenter of a unique Borel probability measure supported on the extreme points of the unit sphere of Lw,1.

About fifteen years ago I. M. Isaacs and S. D. Smith [9] gave several character-theoretic characterizations of finite p-solvable groups G with p-length one, where p is a prime number. They proved that for a finite group G with a Sylow p-subgroup P, the following four conditions (a)–(d) are equivalent.

In § 2 the following two formulae will be established.

If, when p≧q + l, R(ar + k)>0, r = 1, 2,…, p and | ampz |<π,

If p≧q, the result holds if the integral is convergent.

In [8], [9] R. Rautmann has given a systematic development of error estimates for the spectral Galerkin approximations of the solution of the Navier–Stokes equations (spectral in the sense that one chooses as basis functions the eigenfunctions of the Stokes operator).

Error estimates are presented locally in time, valid on a finite interval determined by certain norms of the data. If one assumes the solution to be approximated is uniformly regular fort ≥ 0, the method gives an error estimate which grows exponentially with time. Without further assumptions this is the best estimate that one can expect. However, as pointed out in [2] by J. G. Heywood, if one assumes, additionally, that the solution to be approximated is stable, then one obtains an error estimate which is uniform in time.