A positive integer n is called square-full if p2|n for every prime factor p of n. Let Q(x) denote the number of square-full integers up to x. It was shown by Bateman and Grosswald [1] that

Bateman and Grosswald also remarked that any improvement in the exponent would imply a ‘quasi-Riemann Hypothesis’ of the type for . Thus (1) is essentially as sharp as one can hope for at present. From (1) it follows that, for the number of square-full integers in a short interval, we have

when and y = o (x½). (It seems more suggestive) to write the interval as (x, x + x½y]) than (x, x + y], since only intervals of length x½ or more can be of relevance here.)

The aim of the paper is to propose a general scheme for the consideration of non-commutative stochastic integrals. The role of a probability space is played by a couple (, φ0), where is a von Neumann algebra and φ0 is a faithful normal state on . Our processes live in the algebra of all measurable operators associated with the crossed product of by the modular automorphism group The algebra contains all the (Haagerup's) Lp spaces over . The measure topology of the algebra has the nice feature of inducing the Lp norm topology on the Lp spaces, which makes it particularly suitable for defining stochastic integrals. The commutative theory fits smoothly into the scheme, although there exists no canonical way of embedding the algebra of (commutative) random variables into . In fact, for any commutative stochastic process we have a family of different non-commutative stochastic processes corresponding to the process. This arbitrariness seems to be quite natural in the non-commutative context. An appropriate example can be found at the end of the paper (Section 6, C4).

For each positive integer n, let Fn be a free group of rank n with basis (in other words, free generating set) {x1, …, xn} . If θ is an automorphism of Fn then {x1θ, …, xnθ} is also a basis of Fn and every basis of Fn has this form.

For any variety of groups [Vfr ], let [Vfr ](Fn) denote the verbal subgroup of Fn corresponding to [Vfr ]. (See [10] for information on varieties and related concepts.) Let Gn = Fn/[Vfr ](Fn). Then Gn is a relatively free group of rank n in the variety [Vfr ]. By a basis of Gn we mean a subset S such that every map of S into Gn extends, uniquely, to an endomorphism of Gn. Write x¯i = xi[Vfr ](Fn) for i = 1, …, n. Then {x¯1, …, x¯n} is a basis of Gn. If λ is an automorphism of Gn then {x¯1λ, …, x¯nλ} is also a basis of Gn and every basis of Gn has this form.

Any automorphism θ of Fn induces an automorphism θ of Gn in which x¯iθ = (xiθ)[Vfr ](Fn) for i = 1, …, n. Thus every basis of Fn induces a basis of Gn. The converse however is not always true; in general, there are automorphisms of Gn which are not induced by automorphisms of Fn.

The methods of a previous paper (9) are improved and extended. It is assumed that the eigenfunctions and eigenvalues of an eigenvalue problem given by an elliptic differential equation are known subject to given boundary conditions on a finite boundary. It is shown how the corresponding quantities can be obtained for a similar problem in which the original differential equation, boundary and boundary conditions are simultaneously perturbed. The introduction of a surface displacement vector allows of a Taylor expansion of all quantities and a subsequent separation of orders. The problem of finding the perturbed eigenfunctions for each order then reduces to the solution of an inhomogeneous differential equation subject to known boundary conditions. These equations are solved by a variational method. An application of Green's theorem at each stage enables us to find the perturbed eigenvalues. The method is applied to a problem of which an exact solution is known and good agreement is obtained.

The author is greatly indebted to Dr H. Fröhlich for many interesting discussions and some valuable suggestions.

Consider a renewal process on the nonnegative real line with non-arithmetic distribution function F(x). Denote by V(x; t) the distribution function of the forward recurrence time from t, t ≤ 0. If t is chosen at random with distribution function Ф(t), the corresponding unconditional forward recurrence time has distribution function

There are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems.

We establish some general results on universal models in Topos Theory and show that the investigation of such models can shed light on problems of definability in Logic as well as on De Morgan's law and the law of excluded middle for Grothendieck toposes.

In the study of polynomials acting on Banach spaces, the weak topology is not such a good tool as in the case of linear operators, due to the bad behaviour of the polynomials with respect to the weak convergence. For example,

is a continuous polynomial taking a weakly null sequence into a sequence having no weakly Cauchy subsequences. In this paper we show that the situation is not so bad for unconditional series. Recall that is a weakly unconditionally Cauchy series (in short a w.u.C. series) in a Banach space E if for every f ε E* we have that and is an unconditionally converging series (in short an u.c. series) if every subseries is norm convergent.

A symplectic bundle over an algebraic curve has a natural invariant sLag determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound on sLag which is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced by sLag on moduli spaces of symplectic bundles, and get a full picture for the case of rank four.

In the paper, we give a lower bound on the genus of an amalgamation of two Heegaard splittings V1, ∪S1, W1 for M1 and V2, ∪S2, W2 for M2 along boundary components F1 and F2 under some conditions on the distances of the factor Heegaard splittings. A direct consequence is that under some circumstances, g(M) = g(M1) + g(M2) − g(F) holds, and another is that under some circumstances we have a better lower bound.

We study the nonlinear boundary value problem (BVP) consisting of the equation −(p(t)y′)′ + q(t)y = w(t)f(y) on [a, b] and a general separated boundary condition (BC). By comparing it with a linear Sturm--Liouville problem (SLP) we obtain conditions for the existence and nonexistence of nodal solutions of this problem. More specifically, let λn, n = 0, 1, 2, . . ., be the nth eigenvalue of the corresponding linear SLP. Then the BVP has a pair of solutions with exactly n zeros in (a, b) if λn is in the interior of the range of f(y)/y; and does not have any solution with exactly n zeros in (a, b) if λn is outside this range. These conditions become necessary and sufficient when f(y)/y is monotone on (−∞, 0) and on (0, ∞). We also discuss the changes of the number of different types of nodal solutions as the equation or the BC changes. Our results are obtained without assuming the global existence and uniqueness of solutions of the corresponding initial value problems.

Let G be an affine, connected, semisimple, simply connected algebraic group over an algebraically closed field k of non-zero characteristic p and, for a positive integer n, let Gn be the nth infinitesimal subgroup (the kernel of the nth power of the Frobenius morphism). The purpose of this note is to point out that, for certain rational G-modules V, the natural homomorphism

is the zero map, for m ≽ 0. For such modules V, taking m = 0, we see from the 5-term exact sequence of Lyndon-Hochschild-Serre that the inflation-restriction sequence

is exact and thus the dimension of H1(G, V), when finite, may be calculated by adding the dimensions of the outer terms. The practical value of this result seems to lie in the special case n = 1 and V = L(λ) ⊗ L(µ), the tensor product of simple rational G-modules. In this case H1(G/G1) VG1 is H1(G, L(λ′) ⊗ L(¯′)) for λ′, ¯′ ‘smaller’ than λ µ and the above exact sequence gives an inductive procedure for computing Ext1G between simple rational (G-modules once the (G-module structure of Ext1G1 (equi valently Ext1u1 where% is the restricted enveloping algebra of the Lie algebra of G), for simple G1-modules, is known. Using the rather complicated structure of Weyl modules Cline, in (4), calculated Ext1G between simple modules for G = SL2(k) (this process was reversed in (5)). The present note has been used by S. EIB. Yehia, (17), in finding the corresponding result for SL3(k). We believe these Ext groups to be of increasing importance in view of their connection with Lusztig's conjecture, (14), on the characters of the simple rational G-modules and the related conjecture, 7–2 of (l), of Andersen.

Let X1, X2, X3,… be a sequence of independent, identically distributed, absolutely continuous random variables whose first moment is μ1. Let Sk = X1 + X2 + … + Xk, and let fk(x) be the frequency function of Sk, defined as the k-fold convolution of f1(x). When f1(x) has been defined for all x, fk(x) is uniquely defined for all k, x. Write

A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is the non-additive notion corresponding to that for a ring R stating that every finitely generated submodule of every finitely presented right R-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of a model companion of the class of right S-acts (right R-modules) and hence that the class of existentially closed right S-acts (right R-modules) is axiomatisable.

Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.

Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.

The main objective of the paper is to show that FLA(Ω) is right coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of right S-acts and of left S-acts.

This note is a continuation of our attempts (see [3]) to give a satisfactory representation-theoretic justification of the following formula of D. E. Littlewood:

where sI is the Schur symmetric function corresponding to a partition I, |I| is the weight of I, r(I) is the rank of I, and the summation ranges over all self-conjugate partitions (i.e. partitions I such that I = I where I is the partition conjugate to I).

Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao’s conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.

In (12), Takeuchi characterized those Fuchsian groups which are arithmetic in terms of the traces of the matrices involved in the group. Further, he used these criteria (13) to show that there are only finitely many arithmetic Fuchsian triangle groups and indeed to determine the signatures of all such groups. We note that, if two Fuchsian triangle groups have the same signature, then they are conjugate in PGL(2, ) and hence either all Fuchsian triangle groups of a given signature are arithmetic or none of them are. This will not, in general, be the case for non-triangle Fuchsian groups and in this paper, we examine Fuchsian groups with signature of one of the forms (1; n; 0), n ≽ 2 or (0; 2,2, 2, n; 0), n ≽ 3 and odd in which case the space of conjugacy classes has dimension two. Our principal results state that for each n, there are only finitely many conjugacy classes of arithmetic Fuchsian groups of given signature (1; n; 0) or (0; 2, 2, 2, n; 0) and for large enough n there are no arithmetic Fuchsian groups of that given signature (Theorems 3 and 4). Together with the results of (13) and known results for non-cocompact Fuchsian groups, these results show that there are only finitely many conjugacy classes of two-generator arithmetic Fuchsian groups (see §7).

Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.

In previous communications* the author has discussed the recombination of gaseous ions, subsequently extending the theory to include large ions† and also to apply to the coagulation of colloids and of smokes‡. The predictions of the original theory were in excellent agreement with experiment, but discrepancies appeared when it was generalized. The author is now indebted to Dr F. J. W. Whipple for the discovery of a numerical error in the first paper, the correction of which leads on the whole to a much more satisfactory agreement with experiment.