A space S has property P-1 if S is nonempty. For n > — 1, S has property Pn if it is locally connected, has property Pn-1 and if whenever it is written as a union, S = A ∪ B where each of A and B is closed and has property Pn-1, then A ∩ B also has property Pn-1. The purpose of this paper is to establish that for locally compact spaces, each of the properties Pn is both cyclicly extensible and reducible.

A Pisot sequence is a sequence of integers defined recursively by the formula - . If 0 < a0 < a1 then an+1/an converges to a limit θ. We ask whether any rational p/q other than an integer can ever occur as such a limit. For p/q > q/2, the answer is no. However, if p/q < q/2 then the question is shown to be equivalent to a stopping time problem related to the notorious 3x + 1 problem and to a question of Mahler concerning the powers of 3/2. Although some interesting statistical properties of these stopping time problems can be established, we are forced to conclude that the question raised in the title of this paper is perhaps more intractable than it might appear.

The paper investigates the relationship between Perron - Stieltjes integrability and Lebesgue-Stieltjes integrability within the generalized Riemann approach. The main result states that with certain restrictions a Perron-Stieltjes integrable function is locally Lebesgue-Stieltjes integrable on an open dense set. This is then applied to show that a nonnegative Perron-Stieltjes integrable function is Lebesgue-Stieltjes integrable. Finally, measure theory is invoked to remove the restrictions in the main result.

The paper proves that the group of infinite bounded Nielsen transformations is generated by elementary simultaneous Nielsen transformations modulo the subgroup of those transformations which are equivalent to the identical transformation while acting in a free abelian group. This can be formulated somewhat differently: the group of bounded automorphisms of a free abelian group of countably infinite rank is generated by the elementary simultaneous automorphisms. This proves D. Solitar's conjecture for the abelian case.

In this article we give a sufficient condition for quasisimilar analytic Toeplitz operators to be unitarily equivalent. We also use a result of Deddens and Wong to give a sufficient condition for an operator intertwining two analytic Toeplitz operators to intertwine their inner parts too. Analytic Toeplitz operators with univalent symbols satisfying a suitable normalization that are quasisimilar are shown to have equal symbols.

Let P be a translation plane containing a net N which is replaceable by . Let P' denote the transposed plane. We note that N' is replaceable by ()'. This result shows how to relate the various constructions of the two translation planes of order 16 that admit PSL(2, 7).

For K/k a Zp-extension with Galois group Γ Iwasawa in [4], poses the question of determining the cohomology groups Hn (Γ, E) of the unit group E of K. In this article we compute the cohomology of the units (up to finite groups) for a certain class of Zp-extensions.

It is shown that there are no almost-quaternion substructures on the complex projective space Pn(ℂ).

The Poisson integral of a Denjoy-Perron integrable function defined on the boundary of an open disc is harmonic in this disc. Moreover, almost everywhere on the boundary, the nontangential limits of the integral coincide with the boundary condition. This extends the classical result for Lebesgue integrable boundary conditions. By means of conformai maps, a generalization to domains bounded by a sufficiently smooth Jordan curve is also obtained.

We show that the Fourier transforms of the admissible irreducible representations of the group GL2 over a nonarchimedian local field F are characterized by a functional equation (MF). We also prove that the functions satisfying (MF) and having at most one pole are exactly the Fourier transforms of the irreducible representations of the quaternion group H over F. The Jacquet-Langlands correspondence between irreducible representations of H and discrete series of GL2 then follows immediately from our criteria.

In this note, we construct a model with a normal measure U over a measurable cardinal κ so that the cardinal structures of V and Vκ/U are the same ≤2κ. We then show that it is possible to construct a model where this is not true.

This paper proves that if р < s, then 0 is the only function that multiplies a Bergman Lр space into a Bergman Ls space.

Many — conjecturally all — elliptic curves E/ have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X0(N) to E such that the pull-back of a holomorphic differential on E is a modular form (newform) f of weight 2 and level N. We describe an algorithm for computing the degree of φ as a branched covering, discuss the relationship of this degree to the "congruence primes" for f (the primes modulo which there are congruences between f and other newforms), and give estimates for the size of this degree as a function of N.

Let R be a periodic ring, N the set of nilpotents, and D the set of right zero divisors of R. Suppose that (i) N is commutative, and (ii) every x in R can be uniquely written in the form x = e + a, where e2 = e and a ∊ N. Then N is an ideal in R and R/N is a Boolean ring. If (i) is satisfied but (ii) is now assumed to hold merely for those elements x ∊ D, and if 1 ∊ R, then N is still an ideal in R and R/N is a subdirect sum of fields. It is further shown that if (i) is satisfied but (ii) is replaced by: "every right zero divisor is either nilpotent or idempotent," and if 1 ∊ R, then N is still an ideal in R and R/N is either a Boolean ring or a field.

The homogeneous Besov-Lipschitz spaces, usually defined by difference operators or Fourier transform, are studied in terms of mean oscillation, and several equivalent characterisations are given.

If n is a natural number, let A(n) denote the number of integers, k, such that 0 < k < n and n divides . Let ϕ(n) denote Euler's totient function. Necessary and sufficient conditions are given so that A(n) = ϕ(n) when n is square-free.

In this paper we show that the homotopy-theoretic fibre of the double suspension map E2:S2n-1 → Ω2S2n+1 is an H-space.

We modify the construction of the mod 2 Dyer-Lashof (co)-algebra to obtain a (co)-algebra W which is (also) unstable over the Steenrod algebra A*. W has canonical sub-coalgebras W[k] such that the hom-dual W[k:]* is isomorphic as an A-algebra to the ring of upper triangular invariants in ℤ/2ℤ [x1, . . . , xk].

Finiteness criteria are established for the negative spectra of higher order elliptic operators on Rn. The results are obtained by establishing isomorphism theorems for elliptic operators and applying the abstract finiteness criterion of Konno-Kuroda.