The b-completion of a smooth manifold with linear connection was introduced by Schmidt (6). He pointed out that, in a spacetime, an inextensible timelike curve of finite proper length and bounded acceleration has an endpoint on the b-boundary. However, it is not clear in what sense boundedness was meant. Geroch(1) suggested that any acceptable method of completing a spacetime should have this property with boundedness interpreted through the Lorentz norm. This result for the b-completion was given by Rosso(4), based on his earlier paper (3), which, however, contains some arguments that we do not accept. We give another proof of the result and improve it by weakening the condition of boundedness of the acceleration.

Earlier study by one of the authors on the convergence of non-homogeneous stochastic chains is continued here. Several results more general than those considered earlier relating convergence of such chains to ergodicity of certain smaller chains are presented.

The radial index of a 1-form on a singular set is a generalization of the classical Poincaré–Hopf index. We consider different classes of closed singular semi-analytic sets in n that contain 0 in their singular locus and we relate the radial index of a 1-form at 0 on these sets to Poincaré–Hopf indices at 0 of vector fields defined on n.

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 this note, we construct a sequence of l.t. surfaces (Xn)n ∈ ℕ such that KXn is ample for all n and such that (K2Xn)n ∈ ℕ is a strictly increasing series with limit equal to 1. This answers (in the affirmative) a question by Alexeev, cf. [Al], 11·1. Here, an l.t. surface is a normal complex projective surface with at most quotient singularities (which is the same as ‘at most log terminal singularities’). A main result of [Al] implies that it is impossible to find a sequence (Xn)n ∈ ℕ of l.t. surfaces with KXn ample for all n such that K2Xn is strictly decreasing. Although our construction is not too difficult, the example is new and has several interesting implications, see Section 4.

Without further explanation, we use some fundamental tools concerning l.t. surfaces like Mumford's intersection theory or the notion of minimality; the reader should consult [Blb] and the references quoted there.

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

Consider a Markoff chain (by which is meant a stochastic process defined for discrete values of both variable and parameter, and whose probability dependence does not extend to more than a unit interval) with completely specified transition probability matrix P = (pij), where pij is the conditional probability that the (r + 1)th observation belongs to the state Ei, given that the rth observation belongs to the state Ei for r≥ I. We assume that there are a states E1, E2, …, Ea, where a is finite. We also assume an ergodic property for the stochastic process, that is, we consider non-periodic chains for which all the possible initial states remain permanently available. This defines what has been called the positively regular case. Also under the above assumption the moduli of all the latent roots λr of the transition probability matrix P = (pij) other than the first simple latent root λ1 = 1 are less than one (see Bartlett(1)).

Definition A. The serieswith partial sumUn (or the sequence {Un}) is said to be summable by logarithmic means to the sum U, if

This paper constructs completely integrable convex Hamiltonians on the cotangent bundle of certain k bundles over l. A central role is played by the Lax representation of a Bogoyavlenskij–Toda lattice. The classification of these systems, up to iso-energetic topological conjugacy, is related to the classification of abelian groups of Anosov toral automorphisms by their topological entropy function.

The theory of collective electron ferromagnetism given by Stoner applies to a system of electrons in a single energy band; for iron, cobalt and nickel this is identified with the band of states derived from atomic 3d functions. To bring in the generally assumed overlapping of the 3d band by the wide 4s band in these metals, the theory has been extended to take account of the transfer of electrons from 3d to 4s states with change of temperature. A previous calculation of this transfer effect must be regarded as inadequate, for the part played by the exchange energy in determining the distribution of electrons between the two sets of states was omitted. The general equations are derived in § 2 and used as a basis for discussion of the properties of nickel-copper alloys at absolute zero in § 3. In §§4 and 5 numerical results are presented which show the effect of the overlapping 4s band on the magnetic properties of a system such as nickel both above and below its Curie point. Comparison with the measured paramagnetic susceptibility of pure nickel above the Curie point strongly suggests that for this metal the overlapping 4s band has only a minor influence, although in principle the effect could be very large (cf. Fig. 4). This result is not unambiguous, however, because values thus inferred for the two unknown parameters lead to inaccurate predictions below the Curie point. First, the predicted values for the spontaneous magnetization are too small. Secondly, the theory demands that the nickel-copper alloys should only be ferromagnetic below a copper content of about 20 %, whereas experimentally the limit is known to be about 60 %. In conclusion, it is suggested that the implicit assumption of Stoner's theory that the exchange integrals between all pairs of 3d states are equal to one another is a more serious restriction on the theory than the consideration of only a single energy band.

Let G be a locally compact group and let L1(G) be its group algebra. This paper demonstrates that the order structure of the second dual algebra L1(G)** identifies the topological group G.

Motivated by a problem from number theory about the relationship between Fermat curves and modular curves, a new class of loops is introduced, the Catalan loops. In the number-theoretic context, these loops turn out to be abelian precisely when the Fermat curves and modular curves coincide. General Catalan loops arise on certain transversals to diagonal subgroups in special linear groups over rings with a topologically nilpotent element. The transversals consist of products of certain affine shears. In a Catalan loop, the multiplication and right division are given by rational functions. The left division is algebraic, corresponding to a quadratic irrationality. The left division embodies generating functions for the Catalan numbers. Structurally, Catalan loops are shown to be residually nilpotent.

The purpose of this note is to present a few facts about the Julia set of a rational function that are well known to the experts in the subject of complex dynamics but whose documented exposition in the literature seems to need a little clarification. For example, under conditions set out in Theorem 1, the Julia set of a rational function can be expressed as the limit of a sequence of finite sets. In particular, for certain choices of a point α, the Julia set is the limit as n increases of the inverse image sets R−n(α). This formulation is widely exploited in the backwards iteration algorithm to produce computer illustrations of Julia sets (see for example Section 5·4 of [14]).

We prove the existence of a piecewise analytic expanding map associated to certain Kleinian groups without parabolics acting in the 3-dimensional hyperbolic space. These groups have a fundamental domain ℛ with the property that the geodesic planes containing each face are part of the tesselation. We use this map together with the methods of thermodynamic formalism to give another proof that the Selberg zeta function for such groups has a meromorphic extension to ℂ.

A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance inherent in general foliations of spacetimes. Moreover, in contrast to the situation of the more special constant mean curvature foliations, which play an important role in global analysis of spacetimes, this theorem overcomes the existence problem arising from topological restrictions for surfaces of constant mean curvature.

A class of groups that is closed with respect to subnormal subgroups and normal products is called a Fitting class. Given a finite soluble group G, one may ask for the Fitting class (G) generated by G, that is the intersection of all Fitting classes containing G. For simple or nilpotent groups G it is easy to compute (G), but in other cases the determination of (G) seems to be surprisingly difficult, and there is no general method of solving this problem. In recent years there has been a lot of work in this area, see for instance Bryce and Cossey[l], [2], Hawkes[6] (or [5], IX. 9. Var. II), Heineken[7] and McCann[10].

Introduction. In the classical Waring's problem, the following asymptotic formula was established by Vinogradov (see Vinogradov (2), Chapter VII).

If R is any n-dimensional convex region, symmetric about the origin, and V(R), Δ(R) denote the content and critical determinant of R respectively, then

Einleitung. Man verdankt Richard Dedekind (siehe etwa(1)) das erste Beispiel einer Umkehrungsformel einer endlichen Reihe; dabei spielt, wie man weiβ, die sogenannte Möbius'sche Funktion (der Zahlentheorie) eine groβe Rolle. Auch die sogenannten Aufzählungsmethoden der Gruppentheorie, usw., die mit der soeben erwähnten Dedekindschen Umkehrungsformel eng verbunden sind, stützen sich, wie hauptsächlich von Herren Louis Weisner(2) und Philip Hall (3) gezeigt wurde, auf eine passende Verallgemeinerung der gewöhnlichen Möbius'schen Funktion, die für alle teilweise geordneten Mengen einen Sinn hat, deren Quotienten endlich sind. Siehe auch Ward (4).

Matijaseviê [7] showed in 1970 that the problem of deciding whether an arbitrary Diophantine equation has an integer solution is algorithmically unsolvable. However, in 1972, Siegel [10] provided an algorithm for all equations of degree two.