Throughout this paper, A will denote a distributive lattice with 0 and 1; we shall write spec A for the prime spectrum of A (i.e. the set of prime ideals of A, with the Stone–Zariski topology), and max A, min A for the subspaces of spec A consisting of maximal and minimal prime ideals respectively. These two subspaces have rather different topological properties: max A is always compact, but not always Hausdorff (indeed, any compact T1-space can occur as max A for some A), and min A is always Hausdorff (in fact zero-dimensional), but not always compact. (For more information on max A and min A, see Simmons[3].)

In (5), we and Fred Cohen gave some quite general splitting theorems. These described how to decompose the suspension spectra of certain filtered spaces CX as wedges of the suspension spectra of their successive filtration quotients Dq X. The spaces CX were of the form Cr × Xr/(˜) for suitable sequences of spaces {Cr} and {Xr}, and the construction CX was intended to be a reworking in ‘proper generality’ of the constructions introduced in (9).

It is proved that $fg$ is Henstock–Kurzweil integrable on a compact interval ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i]$ in ${\mathbb R}^m$ for each Henstock–Kurzweil integrable function $f$ if and only if there exists a finite signed Borel measure $\nu$ on ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i)$ such that $g$ is equivalent to $\nu({\mathop{\prod}_{i=1}^{m}}[a_i, \,{\cdot}\,))$ on ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i]$.

In the following (Q, m, k) will denote a quasi-unmixed local ring of Krull dimension d with maximal ideal m and residue field k. We recall that Q is quasi-unmixed if dim Q̅/p = d for every minimal prime ideal p of the completion of Q.

The spectrum of the hydrogen energy operator

(Δ is the Laplacian and r is the distance from the origin) consists of the non-negative real axis and a sequence of negative eigenvalues of finite multiplicities converging to O. In the present study we are interested in finding sufficient conditions on a potential q(x) such that the spectrum of the operator

in En has a ‘hydrogen-like’ spectrum, i.e. a spectrum consisting of

(a) the non-negative real axis,

(b) at most a denumerable set of negative eigenvalues of finite multiplicities having zero as its only possible limit point.

In this note we shall show that the conditions given by G. Torres in (7) do not suffice to characterize the first Alexander polynomial of a link. We shall recall below Torres' results (for the 2-component case) and state the result of J. H. Bailey on which our argument relies, and then prove the following theorem.

Let C be a complex algebraic curve in the projective space ℙ2. The purpose of this paper is to calculate the fundamental group G of the complement of C in the case when C = X ∩ H1 ∩ … ∩ Hn−2, where

and Hi are generic hyperplanes (i = 1, … n − 2).

Our problem and the motive for attacking it are the same as in (1). Using a different method, we obtain a result admitting disconnected domains and higher dimensions without symmetry assumptions. For a detailed introduction we refer to (1).

In a recent paper Collins (l) has shown that Copson's method of solution of the problem of determining the electrostatic field due to an electrified disk (2) can be extended to determine the surface charge density induced on an electrified disk when it is situated inside an earthed coaxial infinitely long hollow cylinder. In Collins's solution the problem is reduced to that of solving a Fredholm integral equation of the second kind. The object of the present note is to show that Collins's solution can be obtained more simply by making use of a technique in the theory of dual integral equations developed recently by the author (3) and Copson(4).

In an earlier paper (2) the author and Radhakrishna have, under certain conditions, obtained the following result.

Let $\mathscr{S}$ be an irreducible topological Markov shift, and let μ be a shift-invariant Gibbs measure on $\mathscr{S}$. Let (Xn)n ≥ 1 be a sequence of i.i.d. random variables with common law μ. In this paper, we focus on the size of the covering of $\mathscr{S}$ by the balls B(Xn, n−s). This generalises the original Dvoretzky problem by considering random coverings of fractal sets by non-homogeneously distributed balls. We compute the almost sure dimension of lim supn →+∞B(Xn, n−s) for every s ≥ 0, which depends on s and the multifractal features of μ. Our results include the inhomogeneous covering of $\mathbb{T}^d$ and Sierpinski carpets.

A normed algebra A is a pre-B*-algebra if its norm satisfies ∥x*x∥= ∥x∥2 for all elements x ∈ A; if A is also complete in its norm, then A is a B*-algebra (see (l), page 180). In the study of certain locally convex algebras, the problem arose of expressing the condition that an algebra be a pre-B*-algebra in terms of its properties as a locally convex algebra, rather than in terms of the norm. A solution to this problem is presented in this note; the application to the theory of locally convex algebras will appear elsewhere.

Let K⊂C be a field finitely generated over Q, K(a)⊂C the algebraic closure of K and G(K)=Gal (K(a)/K) its Galois group. For each positive integer m we write K(μm) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime [lscr ] we write K([lscr ]) for the subfield of K(a) obtained by adjoining to K all [lscr ]-power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab)⊂K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K=Q then Q(ab)=Q(c) (the Kronecker-Weber theorem). We write χ[lscr ][ratio ]G(K)→Z*[lscr ] for the cyclotomic character defining the Galois action on all [lscr ]-power roots of unity. We write χ[lscr ]= χ[lscr ] mod [lscr ][ratio ]G(K) →Z*[lscr ]→(Z/[lscr ]Z)* for the cyclotomic character defining the Galois action on the [lscr ]th roots of unity. The character χ[lscr ] identifies Gal (K([lscr ])/K) with a subgroup of Z*[lscr ]=Gal (Q([lscr ])/Q). Let μ(Z[lscr ]) be the finite cyclic group μ(Z[lscr ]) of all roots of unity in Z*[lscr ]. Its order is equal to [lscr ]−1 if [lscr ] is odd and 2 if [lscr ]=2. Let Q([lscr ])′ be the subfield of μ(Z[lscr ])-invariants in Q([lscr ]). Clearly, Gal (Q([lscr ])/Q([lscr ])′)=μ(Z[lscr ]) and Gal (Q([lscr ])′/Q)= Z*[lscr ]/μ(Z[lscr ]) is isomorphic to Z[lscr ].

The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.

We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster's operad for weak ω-categories.

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $\Omega$ be a non-exact 2-form. Let $\lambda$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la\,\Omega$ is a coboundary if and only if the Gaussian curvature is constant and $\Omega$ is a constant multiple of the area form.

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.

This note is concerned with the relations between the invariants Ωi (i = 0,1,…, d – 1) of the canonical system on an algebraic Vd. Here Ω0 is the grade of the canonical system, and Ωi is the virtual arithmetic genus of the Vi of intersection of d – i canonical Vd–1's.

The purpose of this note is to improve Theorem 1·1 of (1) to the following result:

Theorem 1. Let H be the ordinary free product of a sequence of groups Gn, and let m be any integer > 0. If

where each gi є Gi and is not the identity, then h is not equal to any product of fewer than m commutators from H.

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.