The classical configuration connected with the names of Miquel and Clifford associates with a set of n straight lines in a plane a point called their Miquel point when n is even, and a circle, their Clifford circle, when n is odd. When the number of lines n in a set is even, the omission of these, one at a time, gives n subsets whose associated Clifford circles are concurrent at the Miquel point of the original set; when n is odd, the omission of the lines one at a time gives n subsets whose associated Miquel points are concyclic on the Clifford circle of the original set. To start the chain, it is only necessary to define the Miquel point of two lines as their common point.

Let G be a monoid; that is to say, G is a set such that with each pair σ, τ of elements of G there is associated a further element of G called the ‘product’ of σ and τ and written as στ. In addition it is required that multiplication be associative and that G shall have a unit element. The so-called ‘Homology Theory’† associates with each left G-module A and each integer n (n ≥ 0) an additive Abelian group Hn (G, A), called the nth homology group of G with coefficients in A. It is natural to ask what can be said about G if all the homology groups of G after the pth vanish identically in A. In this paper we give a complete answer to this question in the case when G is an Abelian group. Before describing the main result, however, it will be convenient to define what we shall call the homology type of G. We write Hn(G, A) ≡ 0 if Hn(G, A) = 0 for all left G-modules A.

1. Introduction. A Markov Renewal Process (MRP) with m(<∞) states is one which records at each time t, the number of times a system visits each of the m states up to time t, if the system moves from state to state according to a Markov chain with transition probability matrix P0 = [pij] and if the time required for each successive move is a random variable whose distribution function (d.f.) depends on the two states between which the move is made. Thus, if the system moves from state i to state j, the holding time in the state i has Fij(x) as its d.f. (i, j = 1,2, …, m).

The following matrix problems are well known in quantum mechanics:

(a) The one-dimensional harmonic oscillator. Given

determine the eigenvalues hjj of H, and the matrix elements of X, P if H is diagonal. It is found (Wigner (4)) that

We consider a local move on a knot diagram, where we denote the local move by λ. If two knots K1 and K2 are transformed into each other by a finite sequence of λ-moves, the λ-distance between K1 and K2 is the minimum number of times of λ-moves needed to transform K1 into K2. A λ-distance satisfies the axioms of distance. A two dimensional lattice of knots by λ-moves is the two dimensional lattice graph which satisfies the following: the vertex set consists of oriented knots and for any two vertices K1 and K2, the distance on the graph from K1 to K2 coincides with the λ-distance between K1 and K2, where the distance on the graph means the number of edges of the shortest path which connects the two knots. Local moves called Cn-moves are closely related to Vassiliev invariants. In this paper, we show that for any given knot K, there is a two dimensional lattice of knots by C2m-moves with the vertex K.

1. In the paper [Kai] it was observed that if A is a Banach algebra (over R or C) then the dual space is not only an A-A-bimodule, but is also injective as a left (or right) A-module. Furthermore, if M is a left (or right) Banach module over the unital Banach algebra A, then there is a natural bilinear map, there denoted TrA, from M × M′ to A′, defined by

formula here

(or 〈TrA(m, m′), a〉 = 〈m′, ma〉). The map TrA can be extended to the projective tensor product M[otimesas]M′, which is also an A–A–bimodule.It is easy to see that the map TrA is a bimodule homomorphism, so that the image is an A–A–submodule of A′. This module was denoted EA (M) in [Kai] and is in general not closed as a subspace of M′. It does, however, have a natural norm (as a quotient space of M[otimesas]M′) and the unit ball can be used to define a new norm ∥a∥new = sup {|〈e, a〉 | e∈ the unit ball of EA(M)} on A, and it is easy to see that this new norm is simply the operator norm of a as an operator on M. The conclusion is that if a′∈A′ is not only continuous with respect to the norm ∥a∥L(M) (which is of course in general smaller thatn the given norm on A) but also with respect to the weak topology on A given by the set of all functionals of the form (0·1), then a′ has a representation of the form

formula here.

Throughout the paper, A will denote a unital Banach algebra with centre Z over the complex field ℂ. An element a of A is called quasi-central if, for some K ≥ 1,

The set of elements a satisfying (1) for a particular value of K will be denoted by Q(K) and Q = UK≥1Q(K) will be called the quasi-centre of A.

The purpose of this paper is to investigate the asymptotic behaviour as r → 1− of the integrals

and f is an analytic function on the unit disk Δ which has non-tangential limits at almost every point on ∂Δ. The paper is divided into three parts. In the first part we consider the case where λ ≠ 1/k, in the second the somewhat more delicate case when λ = 1/k and in the third part we concentrate on some problems related to the case λ = k = 1.

We say that a real vector bundle ξ over a finite C.W. complex X is stably trivial of type (n, k) or, simply, of type (n, k) if ξ ⊕ kε ≅ nε, where ε denotes a trivial line bundle. The following theorem is an immediate corollary (see (12)) of a theorem of T. Y. Lam ((9), theorem 2).

Let G be a locally compact topological group with left Haar measure, m; let M(G) denote the bounded regular Borel measures on G and let Lp(G) denote the equivalence classes of pth power integrable functions on G with respect to the left Haar measure.

Let a cyclic group of odd prime order p act on a ℤ(p)-Poincaré duality space X. We prove a relation between the Witt classes associated to the [ ]p-cohomology rings of the fixed point set of this action and of X. This is applied to show a similar result for actions of finite p-groups on ℤ(p)-homology manifolds.

Recent workers [1, 3] have proved density theorems about the rational points on K3 surfaces of the form

$$ V:\;X_0^4+cX_1^4=X_2^4+cX_3^4 $$

In this paper we consider Dehn surgery and essential annuli whose two boundary components are in distinct components of the boundary of a 3-manifold.

Let Nl be an orientable 3-manifold with boundary, Kl a knot in Nl, and N2 the 3-manifold obtained by performing γ-Dehn surgery Kl. In detail, let Vl be a regular neighbourhood Kl, X = Nl − int Vl the exterior of Kl, T the toral component ∂Vl of ∂X, and γ a slope on T. Then we obtain the 3-manifold N2 by attaching a solid torus V2 to X so that γ bounds a disc in V2. Let K2 be the core of V2. Let π be the slope of a meridian loop of Kl, and Δ the distance between the slopes π and γ, i.e. the minimal number of intersection points of the two slopes on T. Suppose for i = 1 and 2 that Ni contains a proper annulus Ai such that the two components of ∂Ai are essential loops on distinct incompressible components of ∂Ni. Then note that Ai is essential, i.e. incompressible and ∂-incompressible in Ni.

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q.

We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second ℓ-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle–Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero.

Let D be the unit disc in the complex plane ℂ with centre 0 and let ∂D be its boundary. By Möb (D) we denote the group of all Möbius transformations which leave D invariant. A Fuchsian group G acting on D is a discrete subgroup of Möb (D). The limit set of G is in ∂D. We decompose ∂D into the following three disjoint sets:

0. Introduction. The concept of a nilpotent action on a nilpotent group was studied in (2), and was crucial to the definition of a nilpotent map (or nilpotent fibration) in (4). These ideas had, of course, already been treated systematically in (1)†. The concept found a new, and related, application to homotopy theory in (3), where the following situation was studied. Let X be a nilpotent space and let G be a group which ‘acts’on X. Here it is not strictly speaking necessary to specify the nature of the action, provided that there is an induced action of G on the homotopy and homology groups of X. Thus G might be a group of base-point preserving homeomorphisms of X, or a group of (based) homotopy classes of self-homotopy-equivalences of X. We then discussed in (3) questions related to the nilpotency of the induced actions of G on the homotopy and homology groups of X and also relativized the theory to a discussion of fibrations on which the group G acts; to obtain interesting results related to nilpotency, it turned out to be at least necessary to suppose the fibration F → E → B quasinilpotent (7), meaning that all spaces are connected and π1B acts nilpotently on the homology groups of F.

The following note was suggested by an interesting paper written by F. P. White, where many references are given. It refers to a theorem given by W. F. Meyer, by whom the proof is indicated as possible by generalization of an intricate analytical proof given by him for a simple case. His result is that if on the rational curve of order r, in space [r], say the curve cr [r], there be an involution ∞k, of sets of m points, expressed, suppose, by an equation

then the spaces [r − 1], formed from r points of any one of the polyhedra of m points, are an aggregate ∞k of primes of this space [r], which is of class (m − k, m − r), the notation (p, q) meaning the binomial coefficient p! / q!(p − q)!. By Meyer, the conditions k < r ≤ m are supposed to be satisfied. But there is a theorem for r < k ≤ m − 1, relating to selected [r − 1], formed from r points of any one of the polyhedra of m points. The general theorem may be formulated thus: In a space [r], the equation of any prime may be expressed by λu + μv +... + ρw = 0, where u = 0, v = 0, …w = 0 are any r + 1 given independent primes, and λ, μ, …ρ are coefficients which may be described as prime coordinates of the [r].

Suppose we have a sample of N independent random variables X1, …, XN where Xi has the distribution F(X|θ, øi). θ is a k-dimensional ‘structural’ parameter (θ(1), …, θ(k)), and the øi are scalar or vector ‘incidental’ parameters in some given space. The Xi may be scalar or vector random variables which are either discrete in which case we write f(X|θ, øi) for the probability associated with a given point, or else continuous random variables with a probability density f(X|θ, øi). In either case we sup pose the support of the probability distribution to be fixed. We aim to estimate the true value of θ by maximum-likelihood methods.