As in [4], a specification is a list (r, s, t1 h0, hmc (l), …, c(ho)) such that

(i) each of r, s, t1 h0, h∞ is a non-negative integer,

(ii) for each i, c(i) is a positive integer,

(iii) if h∞ = 0 then ho = ∞,

(iv) if h∞ = 1 and t1 + h0 is finite then t1 is even,

(v) r + s + t1 + h0 + h∞ = ∞.

Let M be a compact connected boundaryless surface and f: M → ℝ3 a smooth immersion transverse to a straight line L. Thus there is an even number p of points xεM such that f(x)εL. Under further transversality assumptions on f (see §3) there is a finite number q of points x of M such that the plane containing f(x) and L touches f(M) at f(x). These assumptions are mild in the sense that they hold for any f in an open dense subset of the space of smooth immersions under consideration. Suppose that the Gaussian curvature of f(M) is positive at q+ of these points and negative at q−, with q = q++ q−. Then

where e(M) denotes the Euler number of M.

II est connu [4] que si A = U() est l'algèbre enveloppante d'une algèbre de Lie nilpotente de dimension finie sur un corps F de caractéristique 0, tout idéal (complètement) premier P a pourlocalisé R = Ap un anneau régulier au sens de [5]; c'est-à-dire que le radical de Jacobson de R est engendrè par une suite centralisante régulière de longueur n = K-dim R, soit (z1…, zn). Dans le cas très particulier où P est l'idéal d'augmentation de U() il suffit de prendre pour (z1…, zn) l'image dans U()p d'une base de sur F adaptée à la suite centrale ascendante de .

Recall that a Noetherian ring R is a Hilbert ring if the Jacobson radical of every factor ring of R is nilpotent. As one of the main results of [13], J. E. Roseblade proved that if J is a commutative Hilbert ring and G is a polycyclic-by-finite group then JG is a Hilbert ring. The main theorem of this paper is a generalisation of this result in the case where all the field images of J are absolute fields—we shall say that J is absolutely Hilbert. The result is stated in terms of the (Gabriel–Rentschler–) Krull dimension; the definition and basic properties of this may be found in [5]. Let M be a finitely generated right module over the ring R. We write AnnR(M) (or just Ann(M)) for the ideal {r ∈ R: Mr = 0}, the annihilator of M in R. If M is also a left module, its left annihilator will be denoted l-AnnR(M). If R is a group ring JG, put

Let G be a finite group and p a prime number. About five years ago I. M. Isaacs and S. D. Smith [5] gave several character-theoretic characterizations of finite p-solvable groups with p-length 1. Indeed, they proved that if P is a Sylow p-subgroup of G then the next four conditions (l)–(4) are equivalent:

(1) G is p-solvable of p-length 1.

(2) Every irreducible complex representation in the principal p-block of G restricts irreducibly to NG(P).

(3) Every irreducible complex representation of degree prime to p in the principal p-block of G restricts irreducibly to NG(P).

(4) Every irreducible modular representation in the principal p-block of G restricts irreducibly to NG(P).

In Schinzel [1] the following interesting question is asked: does there exist an absolute constant K such that every trinomial in ℚ[x] has a factor irreducible in ℚ[x] which has at most K terms? The only known result appears to be that of Mrs. H. Smyczek, given in the above paper, that if K exists, then K ≥ 6. We here extend this bound to K ≥ 8 by exhibiting a trinomial in ℤ[x] which splits into the product of two irreducible factors, each having 8 terms.

In [1, Corollary 5], Figiel gives an elegant demonstration that the modulus ofconvexity δ in real Banach space X is nondecreasing, where

It is deduced from this that in fact δ(ɛ)/ɛ is nondecreasing [Proposition 3]. During the course of the proof [Lemma 4] it is stated that if v ∊ Sx is a local maximum on Sx of φ ∈Sx*, then v is a global maximum (φ(v) = 1). This is false; it could be that v is a global minimum. It is easy to construct such an example in R2 endowed with the maximum norm. What is true is that v is a global maximum of |φ|.

The structure of semigroups whose subsemigroups form a chain under inclusion was determined by Tamura [9]. If we consider the analogous problem for inverse semigroups it is immediate that (since idempotents are singleton inverse subsemigroups) any inverse semigroup whose inverse subsemigroups form a chain is a group. We will therefore, continuing the approach of [5, 6], consider inverse semigroups whose full inverse subsemigroups form a chain: we call these inverse ▽-semigroups.

Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

1. For functions f ∈ LLoc[0, ∞) the Riemann-Liouville operator of fractional integration I∞ is defined by

and its adjoint operator, the Weyl operator Kα, is defined by

for functions f ∈ LLoc[0, ∞) having a suitable behaviour at infinity.

We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such that

This extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].

In the earlier article [7], I began the study of rational period functions for the modular group Γ(l) = SL(2, Z) (regarded as a group of linear fractional transformations) acting on the Riemann sphere. These are rational functions q(z) which occur in functional equations of the form

where k∈Z and F is a function meromorphic in the upper half-plane ℋ, restricted in growth at the parabolic cusp ∞. The growth restriction may be phrased in terms of the Fourier expansion of F(z) at ∞:

with some μ∈Z. If (1.1) and (1.2) hold, then we call F a modular integral of weight 2k and q(z) the period of F.

When Ramanujan died in 1920 he left behind three notebooks containing statements of a few thousand theorems, mostly without proofs. The second notebook is an enlarged edition of the first, and the third is short and fragmentary. Thus our primary attention may be directed toward the second notebook. In the decade following Ramanujan's death, G. N. Watson and B. M. Wilson agreed to perform the enormous task of editing the notebooks. Unfortunately, this task was never completed, possibly, in part, due to the premature death of Wilson in 1935. In 1957, a photostat edition [19] of the notebooks was published, but no editing whatsoever was undertaken.

In this note it is shown that if S is a free inverse semigroup of rank at least two and if e, f are idempotents of S such that e > f then S can be embedded in the partial semigroup eSe/fSf. The proof makes use of Scheiblich's construction for free inverse semigroups [7, 8] and of Reilly's characterisation of a set of free generators in an inverse semigroup [4, 5].