For Fredholm equations of the first kind with continuous kernels we investigate the uniform convergence of a general class of regularization methods. Applications are made to Tikhonov regularization and Landweber's iteration method.

Expectation values of one-particle and two-particle operators are evaluated in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics. Certain reductions, corresponding to the “quenching” of interactions by the Pauli exclusion principle, are carried out quite generally. More specific reductions, which lead to immediately useful expressions, are possible on the assumption of extreme Bose-Einstein condensation of the correlated pairs.

The fractional iteration of ex and solutions of the functional equation have frequently been discussed in literature. G. H. Hardy has shown (in [3], and in greater detail in [4]) that the asymptotic behaviour of the solutions of (1) cannot be expressed in terms of the logarithmico-exponential scale, although they are comparable with each member of the scale.1 Hence solutions of (1) provide a remarkably simple instance of functions whose manner of growth does not fit into the scale of L-functions but requires non-elementary orders of infinity for an accurate representation. This raises quite naturally the question whether there exists a most regularly growing solution of equation (1) which might serve as a prototype for this kind of growth. In a slightly more general context we may ask whether there exists a ‘best’ family of fractional iterates fσ(x), satisfying.

We show that the group F discovered by Richard Thompson in 1965 has a subexponential upper bound for its Dehn function. This disproves a conjecture by Gersten. We also prove that F has a regular terminating confluent presentation.

Let f be analytic in D = {z: |z| < 1} with f(0) = f′(0)−1=0. For γ > 0, the largest α (γ) and β(γ) are found such that . The results solve the inclusion problem for convex and starlike functions defined in a sector.

1. In [2,5,6,7] a.o. several interpretations of the inequality for all such that were given and the following was proved.

Pfaffian graphs are those which can be oriented so that the 1-factors have equal sign, as calculated according to the prescription of Kasteleyn. We consider various operations on graphs and examine their effect on the Pfaffian property. We show that the study of Pfaffian graphs may be reduced to the case of subcubic graphs (graphs in which no vertex has degree greater than 3) or bricks (3-connected bicritical graphs).

Let G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.

Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

We compare two influential ways of defining a generalized notion of space. The first, inspired by Gelfand duality, states that the category of ‘noncommutative spaces’ is the opposite of the category of C*-algebras. The second, loosely generalizing Stone duality, maintains that the category of ‘point-free spaces’ is the opposite of the category of frames (that is, complete lattices in which the meet distributes over arbitrary joins). Earlier work by the first three authors shows how a noncommutative C*-algebra gives rise to a commutative one internal to a certain sheaf topos. The latter, then, has a constructive Gelfand spectrum, also internal to the topos in question. After a brief review of this work, we compute the so-called external description of this internal spectrum, which in principle is a fibred point-free space in the familiar topos of sets and functions. However, we obtain the external spectrum as a fibred topological space in the usual sense. This leads to an explicit Gelfand transform, as well as to a topological reinterpretation of the Kochen–Specker theorem of quantum mechanics.

The Hurewicz theorem, Mayer-Vietoris sequence, and Whitehead's certain exact sequence are proved for simplicial Lie algebras. These results are applied, using crossed module techniques, to obtain information on the low dimensional homology of a Lie algebra, and information on aspherical presentations of Lie algebras.

The height (as in [3] and [4]) of a point in a rooted tree is the length of the path (that is, the number of lines in the path) from it to the root; the total height of a rooted tree is the sum of the heights of its points. The latter arises naturally in studies of random neural networks made by one of us (N.J.A.S.), where the enumeration of greatest interest is that of trees with all points distinctly labeled.

J. C. Kelly [2] introduced the concept of a bitopological space. Lane [3], Patty [4] and Pervin [5] have continued his work. Our purpose in this paper is to identify the projective objects in a suitable category of bitopological spaces after the manner of Gleason [1] and generalize his theorem that in the category of compact Hausdoriff topological spaces, the projective spaces are precisely the extremally disconnected ones.

We describe a generalization of the Hardy theorem on the motion group. We prove that for some weight functions νω growing very rapidly and a measurable function f, the finiteness of the Lp-norm of vf and the Lq-norm of ωf implies f=0 (almost everywhere).

We study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that l1, c0 and l∞ are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class ∧. We also present examples of weighted Orlicz spaces with the Schur property which are not L1-spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.

This note is concerned with arithmetic properties of power series with integral coefficients that are lacunary in the following sense. There are two infinite sequences of integers {rn} and {sn}, satisfying such that It is also assumed that f(z) has a positive radius of convergence, Rf say, where naturally . A power series with these properties will be called admissible.

Let K:= Q(α) be an algebraic number field which is given by specifying the minimal polynomial f(X) for α over Q. We describe a procedure for finding the subfields L of K by constructing pairs (w(X), g(X)) of polynomials over Q such that L= Q(w(α)) and g(X) is the minimal polynomial for w(α). The construction uses local information obtained from the Frobenius-Chebotarev theorem about the Galois group Gal(f), and computations over p-adic extensions of Q.

We establish a necessary condition (E) for a semigroup variety to be closed under the taking of epimorphisms and a necessary condition (S) for a variety to consist entirely of saturated semigroups. Condition (S) is shown to be sufficient for heterotypical varieties and a stronger condition (S′) is shown to be sufficient for homotypical varieties.

Some oscillation criteria for solutions of a general second order ordinary superlinear differential equation with alternating coefficients are given. The results generalize and complement some existing results in the literature.

We examine the concepts of nowhere simplicity in a wide class of abstract dependence systems. Initially we examine how many of the existing results valid for L(ω), the lattice of r.e. sets, have analogues valid for more general lattices. For example, we show that any r. e. subspace of V∞ can be decomposed into a pair of nowhere simple subspaces.

Let P be a finite, connected partially ordered set containing no crowns and let Q be a subset of P. Then the following conditions are equivalent: (1) Q is a retract of P; (2) Q is the set of fixed points of an order-preserving mapping of P to P; (3) Q is obtained from P by dismantling by irreducibles.