It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10300 (see Brent[1]). Moreover it has been proved by Hagis[4]and Chein[2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in priniciple be obtained solely by calculation, in view of the result of Pomerance[6] who showed that if N is an odd perfect number with at most k prime factors, then

Pomerance's work was preceded by a theorem of Dickson[3]showing that there can be only a finite number of such N. Clearly however the above bound is vastly too large to be of any practical use. The principal object of the present paper is to sharpen the estimate (1). Indeed we shall handle odd ‘multiply perfect’ numbers in general, as did Kanold[5], who extended Dickson's work, and Pomerance. Our result is the following.

A locally inverse semigroup is a regular semigroup S with the property that eSe is inverse for each idempotent e of S. Motivated by natural examples such as inverse semigroups and completely simple semigroups, these semigroups have been the subject of deep structure-theoretic investigations. The class ℒ ℐ of locally inverse semigroups forms an existence variety (or e-variety): a class of regular semigroups closed under direct products, homomorphic images and regular subsemigroups. We consider the lattice ℒ(ℒℐ) of e-varieties of such semigroups. In particular we investigate the operations of taking meet and join with the e-variety CS of completely simple semigroups. An important consequence of our results is a determination of the join of CS with the e-variety of inverse semigroups – it comprises the E-solid locally inverse semigroups. It is shown, however, that not every e-variety of E-solid locally inverse semigroups is the join of completely simple and inverse e-varieties.

Throughout this paper we work in the smooth category and assume that all knots are oriented and consider two knots K1 and K2 to be equivalent if and only if there is an orientation preserving homeomorphism h:S3→S3 which carries K1 onto K2 so that their orientations match. We write K1 ≅ K2 if K1 and K2 are equivalent and – K denotes the knot obtained from K by inverting its orientation.

In this paper we develop a method which may be used to compute the Chern-Simons invariants of a large class of representations on a large class of manifolds. This class includes all representations on all Seifert fiber spaces, all graph manifolds, and some hyperbolic manifolds. I owe many thanks to Peter Scott, John Harer, Frank Raymond, Ron Fintushel, Paul Kirk and Eric Klassen, without whose help and support this paper could not have been written.

In 1988, E. Witten proposed an array of invariants (quantum G-invariants) for a 3-manifold associated with a compact simple Lie group G based on the quantum field theory [22]. N. Reshetikhin and V. Turaev [20] gave a mathematical approach to these invariants for G = SU(2) by using representations of the quantum group Uq(sl(2, ℂ)). See also [2, 11, 12, 14, 15, 16] for other approaches to these invariants: [16] may be the simplest way (for knot theorists) to understand the existence of these invariants.

A Banach space E is said to have the Banach-Saks property (BS) if every bounded sequence (xn) in E has a subsequence (x′n) with norm convergent Cesaro means; that is, there is x in E such that

If this occurs for every weakly convergent sequence in E it is said that E has the Weak Banach-Saks property (WBS) (also called Banach-Saks-Rosenthal property).

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].

Two of the most important topologies on the space ℋ(E) of all holomorphic functions f:E → ℂ on a complex locally convex space E are the compact-open topology τ0 and the Nachbin-ported topology τε. We recall that a seminorm p on ℋ(E) is said to be τω-continuous if there is a compact K such that for every open set V with K contained in V there is a constant c > 0 satisfying

Clearly, the following natural question arises: when do the topologies τ0 and τω coincide? In the setting of Fréchet spaces equality of τ0 and τω forces E to be a Montel space; Mujica [21] proved τ0 = τω for Fréchet-Schwartz spaces and Ansemil-Ponte [1] showed that for Fréchet-Montel spaces this happens if and only if the space of all continuous n-homogeneous polynomials with the compact-open topology, (Pn(E), τ0), is barrelled. By duality, it turns out that the question τ0 = τω is intimately related to ‘Grothendieck's problème des topologies’ which asks whether or not for two Fréchet spaces E1 and E2 each bounded set B of the (completed) projective tensor product is contained in the closed absolutely convex hull of the set B1 ⊗ B2, where Bk is a bounded subset of Ek for k = 1, 2. If this is the case, then the pair (E1, E2) is said to have property (BB). Observe that every compact set B in can always be lifted by compact subsets Bk of Ek (see e.g. [20], 15·6·3). Hence, for Fréchet-Montel spaces E1 and E2, property (BB) of (El,E2) means that is Fréchet-Montel and vice versa. Taskinen[24] found the first counterexample to Grothendieck's problem. In [25] he constructed a Fréchet-Montel space E0 for which (E0,E0) does not have property (BB), and Ansemil-Taskinen [2] showed that τ0 ≠ τω on ℋ(E0).

We obtain an expression for the dimension of a mixing repeller of a non-conformal mapping analogous to the well-known Bowen-Ruelle formula for conformal repellers. The dimension is given in terms of a generalized pressure defined in the context of the thermodynamic formalism. In the course of the paper we develop a subadditive version of the thermodynamic formalism that is suited to our needs and also obtain a ‘bounded distortion’ principle applicable to the non-conformal situation.

A parametric bootstrap estimate (PB) may be more accurate than its non-parametric version (NB) if the parametric model upon which it is based is, at least approximately, correct. Construction of an optimal estimator based on both PB and NB is pursued with the aim of minimizing the mean squared error. Our approach is to pick an empirical estimate of the optimal tuning parameter ε∈[0, 1] which minimizes the mean square error of εNB+(1−ε) PB. The resulting hybrid estimator is shown to be more reliable than either PB or NB uniformly over a rich class of distributions. Theoretical asymptotic results show that the asymptotic error of this hybrid estimator is quite close in distribution to the smaller of the errors of PB and NB. All these errors typically have the same convergence rate of order . A particular example is also presented to illustrate the fact that this hybrid estimate can indeed be strictly better than either of the pure bootstrap estimates in terms of minimizing mean squared error. Two simulation studies were conducted to verify the theoretical results and demonstrate the good practical performance of the hybrid method.

We improve known Tauberian results concerning the power series method of summability Jp based on the sequence {pn} by removing the condition that pn be asymptotically logarithmico-exponential. We also prove an entirely new Tauberian result for rapidly decreasing pn.

Isomorphic measures on compact groups

BY S. Grekas

(Volume112 (1992), 349–360)

There is a gap in the proof of Theorem 5·8 of that paper. More precisely, it is not clear that the image of a completion regular measure under the map f of Lemma 5·7 is also completion regular, as f does not seem to preserve Borel measurability. Consequently, the proof of Theorems 5·A, 5·B and 5·C is not correct. However, in Remarks 5·13 there is a proof of Theorem 5·A using an alternative method. Thus, atleast Theorem 5-A is correct. The author believes that an analogous method can similarly be applied to Theorems 5·B and 5·C as well.

Invariance principles for logarithmic averages

By Miklos Csorgo and Lajos Horvath

(Volume112 (1992), 195–205)

In (1·4) and (1·5) of the paper above nt should be replaced by nt. Also, in the inequalities of (2·5) and (2·6) the expression should be replaced by

We are indebted and grateful to Professor Endre Csáki for pointing out these mistakes to us.