In [2], chapter S, and also in the ATLAS ([1], p. 131), we give two presentations for 3 · Suz: 2, the normalizer of a certain 3-element in Conway's largest sporadic simple group Co1. These presentations extend naturally to presentations for groups having Co1 × 2 as image, leading to conjectured presentations for Co1 × 2 and Co1 (see [1], p. 183, although the form of one of the presentations there is slightly different from what we give here).

Let

be a non-degenerate quadratic form with integral coefficients. Further, let Q(x) be a zero form, i.e. let there exist x ≠ 0 in ℤn such that Q(x) = 0. Then we know from Cassels[2], (Davenport[6] and ‘a slightly more general result’ from Birch and Davenport [1]) that there exists a ‘small’ solution x in ℤn of the equation Q(x) = 0; more precisely, if

then there exists x ≠ 0 in ℤn such that Q(x) = 0 and further

(Here, and throughout this section, k will denote a number, not necessarily the same at each occurrence, which depends only on n.) An analogue of this estimate for ‘integral’ quadratic forms over algebraic number fields was proved in [8], with the exponent (n − 1)/2 remaining intact.

In this paper we complete the enumeration, begun in [5], of the maximal subgroups of the Fischer group Fi22 and its automorphism group Fi22:2. It turns out that the lists of maximal subgroups given in [5] and [2] are complete. (Note however that in ([5], theorem (A) (iii)), ) should read .) More precisely, we prove

Main Theorem (i) The simple group Fi22 has exactly 14 conjugacy classes of maximal subgroups, as follows:

(ii) The group Aut (Fi22) ≅ Fi22: 2 has exactly 13 conjugacy classes of maximal subgroups, as follows:

A group G is called a triangle group if it can be presented in the form

It is well-known that G is isomorphic to a subgroup of PSL2(ℂ), that a is of order l, b is of order m and ab is of order n. If

then G contains the fundamental group of a positive genus orientable surface as a subgroup of finite index whenever s(G) ≤ 1; in particular G is infinite. Furthermore, if s(G) < 1, the genus of the surface is greater than 1 and consequently G contains a free group of rank 2.

The purpose of this paper is to classify the finite simple Moufang loops. A Moufang loop M is a loop which satisfies the identity

note that the equivalent identities ((xy)z)y = x(y(zy)), x(y(xz)) = ((xy)x)z also hold, by [2], p. 115. The Moufang loop M is simple if it has no non-trivial proper homomorphic images, or equivalently, if it has no non-trivial proper normal subloops. For basic definitions and properties of Moufang loops, see [2] – in particular, the Jordan–Hölder theorem holds for finite Moufang loops ([2], p. 67). Of course if the finite simple loop M is associative, then M is a simple group, and hence is determined by the classification of finite simple groups. In [9], Paige defines, for each finite field GF(q), a finite simple Moufang loop M(q) which is not associative – M(q) is essentially the set of units in the eight-dimensional split Cayley algebra over GF(q), modulo the centre (we shall describe M(q) in much more detail in §2).

Let (R, M) be a commutative Noetherian local ring having an identity, and assume the residue field R/M is infinite. If I is an ideal in R, recall that an ideal J contained in I is called a reduction of I if JIn = In + 1 for some non-negative integer n. A reduction of J of I is called a minimal reduction of I if it does not properly contain another reduction of I. Reductions (and minimal reductions) were introduced and studied by Northcott and Rees[8]. If J is a reduction of I we define the reduction number of I with respect to J, denoted rJ(I), to be the smallest non-negative integer n such that JIn = In + 1 (note that rJ(I) = 0 if and only if J = I). The reduction number of I (sometimes referred to as the reduction exponent) is defined as r(I) = min{rj(I)|J ⊆ I is a minimal reduction of I}.

This paper is largely inspired by the work and results of Iitaka in [6]. The study undertaken is of pairs X & C, up to birational equivalence, where X is a projective, non-singular surface and C is a reduced curve on X. In [6] Iitaka gives a classification of &-minimal pairs and introduces the important idea of the #-minimal model. The classification of &-minimal pairs given here is a slight improvement on that of Iitaka on account of the use of Mori theory in place of more classical methods. Before the classification in Section 3 we have included a section on the uniqueness of the &-minimal model which does not occur in the work of Iitaka.

The Adams operations {ψk} considered as self-maps of BU have the property that . Id2n on H2n(BU; ℚ). Sullivan in his M.I.T. notes [12] constructed ψk-type self-maps of BU(n) in the case in which k is prime to n!. Later work by Friedlander[5] and Wilkerson[14] gave constructions for all other compact connected Lie groups G, subject to the condition that k be prime to the order of the Weyl group W(G).

The fundamental group G of a μ-component link has the properties:

(1) G is finitely presented, with deficiency 1;

(2) G/G′ is free abelian on μ distinguished generators, say {t1 …, tμ}.

Let ψ: G → G/G′ → 〈t〉 be the composition of the canonical projection G → G/G′ and the epimorphism defined by . Then the Z〈t〉-module M = Ker ψ/(Ker ψ)′ (the so-called module of the link) has a square (say n × n) relation matrix N. We write [N] for the Z〈t〉-equivalence class of N (Fox [3], p. 199) and N′̅ for the conjugate transpose of N.

A link or knot in S3 is universal if it serves as common branching set for all closed, oriented 3-manifolds. A knot is simple if its exterior space is simple, i.e. any incompressible torus or annulus is parallel to the boundary. No iterated torus knot or link is universal, but we know of many knots and links that are universal. Thurston gave the first examples of universal links [8], and subsequently we proved that all 2-bridge knots and links that can be universal (no torus knots or links) are in fact universal [3]. Some other universal knots are described in [1] and [2], together with a general procedure for constructing such knots. For a general reference to knots see [9].

Let K be a knot in S3, and let K(a/b) denote the closed oriented 3-manifold obtained by a/b-Dehn surgery on K. (We parametrize the Dehn surgeries on K by ℚ ∩ {1/0} as in [9]; in particular, K(1/0) = S3.) If K is a (p, q)-cable knot, then K(pq) is always reducible (see Section 3), and it is conjectured by González-Acuña and Short in [5] that these are the only examples where Dehn surgery on a knot in S3 yields a reducible manifold. One of the results in this direction proved in [5] is that if π1 (K(a/b)) is a non-trivial free product then |b| ≤ 5. We show that this may be sharpened to the assertion in the title.

Let Mn be a smooth closed n-dimensional manifold and let DIFF (Mn) be the group of diffeomorphisms of Mn. Two diffeomorphisms f0, f1 ∈ DIFF(Mn) are said to be concordant (pseudo-isotopic) if there is a diffeomorphism F ∈ DIFF (Mn × I), where I = [0, 1], such that F(x, 0) = f0(x) and F(x, 1) = f1(x) for all x ∈ Mn.

In a previous paper [6] the Bergman projection in D = A ∪ B, the union of domains A, B ⊂ CN, was described by alternating projections procedure. It was shown that when N = 1, and A, B are two halfplanes, the procedure can be carried out by explicit analytic calculations. In the present paper we show that the same is true for A = {|Re z| < π} and B = {Re z > 0}. In this case the calculations are more involved, and an essential use is made of L2-theorems of Paley–Wiener type due to T. Genchev, M. Dzrbasjan and W. Martirosjan. Of central importance is an operator W which maps f(z) onto . We show that for every g ∈ L2H(B) the sequence Wng (n = 1, 2, …) converges to zero locally uniformly in B.

We give a quick proof of a result of A. Nagel and E. M. Stein concerning the boundedness of certain generalizations of the Hardy-Littlewood maximal operator.

1. We use Eα to denote the Euler transformation obtained as the special case of the Hausdorff transformation (H, μn) in which μn = αn (see [5], §§64, 72; in the notation of Hardy's book [1], our Eα is (E, q) with q = (1 − α)/α). Eα is regular if and only if 0 < α < 1, and in this range Eα increases in strength as α decreases since EαEβ = Eαβ. Also, E1 = I, the identity transformation.

A basic problem of the theory of Markov processes is to find a semigroup S of operators on the bounded Borel measurable functions on ℝd, such that S is given by a transition function, and

for all f belonging to the domain of a given second-order elliptic differential operator D. A process is constructed from the transition function associated with S, and then one hopes to realize the process as one with continuous paths [2].

The purpose of this article is to derive a set of ‘easily verifiable’ sufficient conditions for the asymptotic stability of the trivial solution of a class of linear neutral integro-delay-differential equations of the form

where ẋ denotes the right derivative, and then generalize the result to a vector system. Asymptotic stability of the trivial solution of the form (1·1) and several of its variants has been considered by many authors (see the discussion at the end). Also, there exists a well developed fundamental theory for neutral delay-differential equations (e.g. existence, uniqueness, continuous dependence etc. of solutions; see for instance the survey article by Akhmerov[1] and the references in it); however there exists no ‘easily verifiable’ sufficient condition for the asymptotic stability of the trivial solution of (1·1). By the term ‘easily verifiable’ we mean a verification which is as easy as in the case of Routh-Hurwitz criterion or the diagonal dominance condition of a matrix. A number of results which are valid for linear autonomous ordinary and delay-differential equations cannot be generalized (or extended) to neutral equations. It has been shown by Gromova and Zverkin [8] that a linear neutral differential equation can have unbounded solutions even though the related characteristic equation has only purely imaginary roots (see also Gromova[7], Datko[3], Snow [16], Brumley [2]); such behaviour is not possible in the case of ordinary or (non-neutral) delay-differential equations. It is known (theorem 6·1 of Henry [10]) that if the characteristic equation associated with a only with negative real parts and if all the roots are uniformly bounded away from the imaginary axis, then asymptotic stability of the trivial solution can be asserted.

Unlike their counterparts for the Laplace's equation, the known results on heat capacity and parabolic measure are rather incomplete. We make a few comments on these two areas.

The aim of this paper is to give some results on the determination of Hamiltonian structure for a set of evolutionary equations, either differential or non-local. There are several reasons why the recognition of Hamiltonian structure is important. Firstly, as in the case of Lagrangian systems, conservation laws are often connected with symmetry groups of the equations. This point is addressed in §4. Secondly, variational principles for steady solutions can often be found by restricting attention to the steady equations. Connected to the second point, methods have been developed by Arnol'd[2] and more recently by Marsden, Weinstein etc. (see, e.g. [24]) to study the nonlinear stability of steady solutions, on lines similar to the proof of nonlinear stability for the solitary-wave solution of the Korteweg–de Vries equation by Benjamin [4] and Bona[9].

‘On the global attractivity ina generalized delay-logistic differential equation’

In the proof of Theorem 3·1, the function space and its topology are incorrect, The correct function space and its topology with the related modifications are given below, which will replace the contents of the last line of page 188 and the first eight lines of page 189 of the above article.

Let B denote the real linear space of all continuous functions defined on [0, ∞)topologized by a separating family of seminorms Pn, where

Since p1 ≤ p2 ≤ p3 ≤ …, the sets

form a convex local base for the topology of B. The topology generated by {Vn} is also compatible with that of the metric d, where

Thus B becomes a metrizable, Hausdorif, locally convex linear topological space. The strong version of the Tychonoff fixed-point theorem of S. A. Morris and E. S. Noussair (Mat. Čs. 25 (1975), 165–172) states ‘If T is a continuous mapping of a convex subset A of a Hausdorff locally convex linear topological space B into a compact subset of A then T has a fixed point’. In the following, the compactness of the image T(A) ⊂ A will follow from an application of the generalized version of Arzela's theorem found in J. R. Munkres (Topology, A first course, Prentice-Hall, 1975, p. 292). Note that the set A is not closed in B but is convex; we will show below that the image set T(A) is contained in A and the members of T(A) are pointwise bounded by (3·16) and pointwise equicontinuous, from which the sequential compactness of T(A) in the topology of compact convergence will follow by the generalized Arzela's theorem; since the real line is metrizable, such a sequential compactness implies sequential compactness in the compact-open topology of T(A); but the set T(A) is itself metrizable by d with a topology compatible with the compact-open topology. Thus the sequential compactness of T(A) in the compact-open topology implies the compactness of the set T(A) in the compact-open topology. Thus it is sufficient to show the pointwise boundedness and equicontinuity of elements of T(A).