Soit P un polynôme à coefficients entiers de degré 2 et Δ son discriminant. La quantité τP(n), définie par τP(n) := card {P(m) > 0: P(m)[mid ]n, m ∈ ℤ}, compte le nombre de diviseurs d de n qui s'écrivent sous la forme d = P(m). Lorsque P(X) = X(X + 1), le cardinal τX(X+1)(n) est égal au nombre de diviseurs consécutifs de n.

Let [afr ] be an ideal of a local ring (R, [mfr ]) and let N be a finitely generated R-module of dimension d: It is shown that Hd[afr ](N) ≃ Hd[mfr ](N)/[sum ]n∈ℕ〈[mfr ]〉 (0: Hd[mfr ](N)[afr ]n); where for an Artinian R-module X we put 〈[mfr ]〉X = ∩n∈ℕ[mfr ]nX. As a consequence several vanishing and connectedness results are deduced.

Let C be a germ of curve singularity embedded in (kn, 0). It is well known that the blowing-up of C centred on its closed ring, Bl(C), is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number of steps, we find only non-singular curves. This is the desingularization process. The main idea of this paper is to linearize the blowing-up of curve singularities Bl(C) → C. We perform this by studying the structure of [Oscr ]Bl(C)/[Oscr ]C as W-module, where W is a discrete valuation ring contained in [Oscr ]C. Since [Oscr ]Bl(C)/[Oscr ]C is a torsion W-module, its structure is determined by the invariant factors of [Oscr ]C in [Oscr ]Bl(C). The set of invariant factors is called in this paper as the set of micro-invariants of C (see Definition 1·2).

In the first section we relate the micro-invariants of C to the Hilbert function of C (Proposition 1·3), and we show how to compute them from the Hilbert function of some quotient of [Oscr ]C (see Proposition 1·4).

The main result of this paper is Theorem 3·3 where we give upper bounds of the micro-invariants in terms of the regularity, multiplicity and embedding dimension. As a corollary we improve and we recover some results of [6]. These bounds can be established as a consequence of the study of the Hilbert function of a filtration of ideals g = {g[r,i+1]}i [ges ] 0 of the tangent cone of [Oscr ]C (see Section 2). The main property of g is that the ideals g[r,i+1] have initial degree bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ]C.

Section 4 is devoted to computation the micro-invariants of branches; we show how to compute them from the semigroup of values of C and Bl(C) (Proposition 4·3). The case of monomial curve singularities is especially studied; we end Section 4 with some explicit computations.

In the last section we study some geometric properties of C that can be deduced from special values of the micro-invariants, and we specially study the relationship of the micro-invariants with the Hilbert function of [Oscr ]Bl(C). We end the paper studying the natural equisingularity criteria that can be defined from the micro-invariants and its relationship with some of the known equisingularity criteria.

Throughout the ground field is always supposed to be algebraically closed of characteristic zero. Let X be a smooth projective threefold of general type, denote by ϕm the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system [mid ]mKX[mid ]. Since, once given such a 3-fold X, ϕm is birational whenever m [Gt ] 0, quite an interesting thing to find is the optimal bound for such an m. This bound is important because it is not only crucial to the classification theory, but also strongly related to other problems. For example, it can be applied to determine the order of the birational automorphism group of X [21, remark in section 1]. To fix the terminology we say that ϕm is stably birational if ϕt is birational onto its image for all t [ges ] m. It is well known that the parallel problem in the surface case was solved by Bombieri [1] and others. In the 3-dimensional case, many authors have studied the problem, in quite different ways. Because, in this paper, we are interested in the results obtained by Hanamura [7], we do not plan to mention more references here. According to 3-dimensional MMP, X has a minimal model which is a normal projective 3-fold with only ℚ-factorial terminal singularities. Though X may have many minimal models, the singularity index (namely the canonical index) of any of its minimal models is uniquely determined by X. Denote by r the canonical index of minimal models of X. When r = 1 we know that ϕ6 is stably birational by virtue of [3, 6, 13 and 14]. When r [ges ] 2, Hanamura proved the following theorem.

We show that there is a well-defined cap-product structure on the Fintushel–Stern spectral sequence and the induced cap-product structure on the ℤ8-graded instanton Floer homology. The cap-product structure provides an essentially new property of the instanton Floer homology, from a topological point of view, which multiplies a finite-dimensional cohomlogy class by an infinite-dimensional homology class (Floer cycles) to get another infinite-dimensional homology class.

An invariant of a three-manifold with boundary is derived from topological quantum field theory. This invariant is then used as an obstruction to embedding one 3-manifold into another.

Fix a prime integer p. We show that a closed orientable 3-manifold M admits an action of Zp with the fixed point set S1 if and only if M can be obtained as the result of surgery on a p-periodic framed link L and Zp acts freely on the components of L. We prove a similar theorem for free Zp-actions. As an interesting application, we prove the following, rather unexpected result: for any M as above and for any odd prime p, H1(M, Zp) ≠ Zp. We also prove a similar criterion of 2-periodicity for rational homology 3-spheres.

Several authors [1, 5, 9] have investigated the algebraic and transcendental values of the Gaussian hypergeometric series

(formula here)

for rational parameters a, b, c and algebraic and rational values of z ∈ (0, 1). This led to several new identities such as

(formula here)

and

(formula here)

where Γ(x) denotes the gamma function. It was pointed out by the present authors [6] that these results, and others like it, could be derived simply by combining certain classical F transformation formulae with the singular values of the complete elliptic integral of the first kind K(k), where k denotes the modulus.

Here, we pursue the methods used in [6] to produce further examples of the type (1·2) and (1·3). Thus, we find the following results:

(formula here)

The result (1·6) is of particular interest because the argument and value of the F function are both rational.

This paper is concerned with the exponential sums

(formula here)

and with Heilbronn's type exponential sums

(formula here)

where (and below) p is prime, m, n, a and b are integers with m > n > 0 and p [nmid ] ab, and er(x) = exp(2πix/r) as usual.

In this paper we shall consider differential operators that admit a periodic structure (cf. [1, 2])

(formula here),

where

(formula here),

are Hölder continuous for some α > 0 and periodic, so that, say, without loss of generality, we have

(formula here).

We shall assume that Q is elliptic, i.e. that the matrix A = (aij) satisfies

(formula here),

for some

(formula here).

The principal result of the paper reduces the study of certain weakly compact sets in Banach spaces of measurable operators to that of the corresponding sets of generalized singular value functions. In particular, under natural conditions, it is shown that the orbit of a relatively weakly compact subset of the Köthe dual of a symmetric space of measurable operators affiliated with some semi-finite von Neumann algebra is again relatively weakly compact.