Finite determinacy relative to closed¶and finitely generated ideals

In this paper we propose to develop a theory of finite determinacy and unfolding in the spirit of Thom–Mather theory for C ^∞ function germs in a finitely generated and closed ideal with three additionnal properties. It may be used as the beginning of a more systematic study of non isolated real singularities. Received: 26 January 2000 / Revised version: 30 June 2000     

On the maximal regular subsemigroups of ideals of order-preserving or order-reversing transformations

We characterize the maximal regular subsemigroups of the ideals of the semigroup of all order-preserving transformations as well as of the semigroup of all order-preserving or order-reversing transformations on a finite ordered set.     

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.     

On a class of closed prime ideals in H ∞

Gorkin and Mortini introduced the concept of k-hulls, k(x), of points x in M(H ^∞)????D, and studied the ideal structures of H ^∞ and H ^∞ +C. They posed a problem for which x∈ M(H ^∞)????D the set I(k(x)) is a closed prime ideal. In this article, we give a partial answer for sparse points x.     

On isomorphisms of blowing-ups of complete ideals¶of a rational surface singularity

Given a rational surface singularity (S,P), we relate S-isomorphism classes of normal blowing-ups of S and properties of semifactorization for complete ideals in . This relation is understood via the natural cell decomposition of the characteristic cone P˜(X/S), where X is a desingularization of S. Received: 1 April 1998 / Revised version: 2 July 1998     

On the spectrum of a discrete Laplacian on ℤ with finitely supported potential

We study spectral properties of the discrete Laplacian L?=??Δ?+?V on ? with finitely supported potential V. We give sufficient and necessary conditions for L to satisfy that the number of negative (resp. positive) eigenvalues is equal to one of the points x on which V(x) is negative (resp. positive). In addition, we prove that L has at least one discrete eigenvalue. If ∑ _x∈? V(x)?=?0, then L has both negative and positive discrete eigenvalues.     

On entire functions whose derivatives are¶integer-valued on geometric progressions

In this paper we study functions which together with all their derivatives take values in a number field ? on a sequence (u _n ) in ? satisfying certain conditions. As one corollary we get a result on the growth rate of entire transcendental functions whose derivatives assume integral values on the geometric progression u ⁿ ,n= 0,1,…. In a second theorem we prove that the lower bound for the growth rate stated in the corollary is best possible. Received: 16 May 2000     

On the tautological ring of ℳ g

We prove that any product of tautological classes of _g of degreed (in the Chow ring of _g with rational coefficients) vanishes ford>g–2 and is proportional to the class of the hyperelliptic locus in degreeg–2.Oblatum 28-I-1195 & 15-IV-1995In memory of Nicolaas H. Kuiper (1920–1994)     

On the jordan H-submodules of a simple superalgebra with superinvolution ∗

We prove that if A is a simple associative superalgebra with superinvolution ? and H = H(A, ?) is the hermitian superalgebra of symmetric elements then K(A, ?) is simple as a Jordan H-module. We also give a characterization of all Jordan H-submodules of A.     

On the existence of strongly normal ideals overP κ λ

For every uncountable regular cardinal and any cardinal,P denotes the set . Furthermore, < denotes=" the=" binary=" operation=" defined=">P byx<> iffxy¦x<>.By anideal over P we mean a proper, non-principal,-complete ideal overP extending the ideal dual to the filter generated by . For any idealI overP ,I ⁺ denotes the setP –I, andI ^* the filter dual toI.     

On the continuity of the minima for a family of constrained optimization problems ∗

We consider a family of problems Py dealing with the minimization of a given function on a constraint set, both depending on a parameter y. We study continuity properties, with respect to a parameter, of the value and of the solution set of the problems. Working with convex functions and convex constraint sets, we show how the well-posedness of the problem allows to avoid compactness hypotheses usually requested to get the same stability results.     

On ∗-representations of deformations of canonical anticommutation relations

We consider irreducible ∗-representations of deformations of canonical anticommutation relations (CAR) that belong to the class of ∗-algebras generated by generalized quons.     

On the number of divisors which are values of a polynomial

Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let