Extensions of commutative rings in which trees of prime ideals contract to trees

Résumé  Une extensionA⊂B des anneaux (commutatifs) satisfait à la propriété si tout arbre dans Spec(B) couvre un arbre dans Spec(A). Il est possible qu'une extension entière d'un anneau Noethérien ne satisfait pas à . SiA⊂B soit unei-extension satisfaisante à soit “going-up” soit “going-down”, alorsA⊂B satisfait à . Cependant, une extension d'anneaux satisfaisante à “going-up”, “going-down”, et peut être nonunibranche dans hauteur >1. Un anneau intègreA a le spectre d'un arbre si et seulement siA⊂B satisfait àP pour tout anneau intègreB contenantA (resp., suranneau de BézoutB deA). De plus, si un anneau intègreA n'ait pas de spectre d'un arbre mais soit localement de dimension finie, (par exemple, tout anneau intègre Noethérien de dimension au moins 2), alors il existe un suranneau de BézoutB deA et un arbre saturé dans Spec(B) de sorte que card=4 et l'image de à l'égard de la flèche canonique Spec(B)→Spec(A) est un ensemble saturé tel que card =3 mais n'est pas d'arbre. On donne également des caractérisations associées des classes desi-domaines et des ai-domaines.      

A characterization of going-down-ring pairs of commutative rings

SoientA ⊆B des anneaux (commutaifs et unitaires). On dit que (A,B) est une paire d’anneaux de going-down siD est un anneau de going-down pour tout anneauD tel queA ⊆D ⊆B. On preuve que (A,B) est une paire d’anneaux de going-down si et seulement siA[b ₁,b ₂] est un anneau de going-down pour toutb ₁,b ₂ εB.     

The normal density of prime ideals in small regions

For a Gaussian prime (i) define ()=min|–| where runs through Gaussian primes satisfying ||>||. We prove that, subject to the Riemann Hypothesis for appropriateL-functions

Some algebraic invariants related to mixed product ideals

We compute some algebraic invariants (e.g. depth, Castelnuovo-Mumford regularity) for a special class of monomial ideals, namely the ideals of mixed products. As a consequence, we characterize the Cohen-Macaulay ideals of mixed products. Received: 25 October 2007     

On the set of <Emphasis Type="Italic">t</Emphasis>-linked overrings of an integral domain

Let R be an integral domain with quotient field L. An overring T of R is t-linked over R if I ⁻¹ = R implies that (T : IT)  =  T for each finitely generated ideal I of R. Let O _t (R) denotes the set of all t-linked overrings of R and O(R) the set of all overrings of R. The purpose of this paper is to study some finiteness conditions on the set O _t (R). Particularly, we prove that if O _t (R) is finite, then so is O(R) and O _t (R) = O(R), and if each chain of t-linked overrings of R is finite, then each chain of overrings of R is finite. This yields that the t-linked approach is more efficient than the Gilmer’s treatment (Proc Am Math Soc 131:2337–2346, 2002). We also examine the finiteness conditions in some Noetherian-like settings such as Mori domain, quasicoherent Mori domain, Krull domain, etc. We establish a connection between O _t (R) and the set of all strongly divisorial ideals of R and we conclude by a characterization of domains R that are t-linked under all their overrings. This work was funded by KFUPM under Project # FT/18-2005.     

Fragmented domains have infinite Krull dimension

Résumé  On dit qu'un anneau intègreR est fragmenté si pour tout élément non-inversibler deR, il existe un élément non-inversibles deR tel que r∈∩Rs ⁿ. On montre, pour un anneau intègreR qui n'est pas un corps, qu'il existe un idéal maximal deR qui contient une cha?ne strictement croissante d'idéaux premiers deR. Si, de plus,R n'ax qu'un nombre fini d'idéaux maximaux, alors on peut reformuler l'affirmation précédente pour tout idéal maximal deR. Il découle que toute anneau intègreR, qui n'est pas un corps et qui possède un idéal premierP tel queR+PR _p soit fragmenté, doit être de dimension infinie (au sens de Krull). On donne un exemple d'un tel anneauR qui n'est pas fragmenté.      

Vector-Valued capacities

It is well-known that ifX is a compact metric space and ifI is a capacity onX then every analytic subset ofX isI-capacitable [2], [1]. We introduce the notion of vector-valued capacity in the case when the codomain is a Banach lattice and we prove an analogous theorem for analytic subsets of a Polish space. Finally, we show that every vectorvalued outer measure is a capacity and, in connection with the so-called “marginal problem”, we give an example of a capacity taking values in a reflexive Banach lattice.     

Log canonical thresholds of binomial ideals

We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming. Dedicated to Professor Toshiyuki Katsura on the occasion of his sixtieth birthday     

Blow-up rings and rational singularities

Let A be a normal local ring which is essentially finite type over a field of characteristic zero. Let I⊂A be an ideal such that the Rees algebra R _A (I) is Cohen–Macaulay and normal. In this paper we address the question: “When does R _A (I) have rational singularities?” In particular, we study the connection between rational singularities of R _A (I) and the adjoint ideals of the powers I ⁿ (n∋ℕ). Received: 25 May 1998 / Revised version: 20 August 1998     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

An Improved Cauchy Formula for Hypermonogenic Functions

A new Cauchy-type formula for hypermonogenic functions is derived. Hypermonogenic functions, introduced in [6], are a generalization of holomorphic functions to several dimensions. The power function x^m is hypermonogenic. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš     

On the volume of a tilting module

We consider a finite dimensional k-algebraA and associate to each tilting module a cone in the Grothendieck groupK ₀ of finitely generated A-modules. We prove that the set of cones associated to tilting modules of projective dimension at most one defines a, not necessarily finite, fan Σ(A). IfA is of finite global dimension, the fan Σ(A) is smooth. Moreover, using the cone of a tilting module, we can associate a volume to each tilting module. Using the fan and the volume, we obtain new proofs for several classical results; we obtain certain convergent sums naturally associated to the algebraA and obtain criteria for the completeness of a list of tilting modules. Finally, we consider several examples. Dedicated to O. Riemenschneider on the occasion of his 65th birthday     

Calcolo degli autovalori di matrici tridiagonali non normali

This article is about the accurate computation of the eigenvalues of nonnormal tridiagonal matrices. Results of a comparison we made between several methods and algorithms are reported, and an algorithm derived from the method presented in [15], [16] is shown to be the only one able to give accurate outputs in case of ill-conditioned eigenvalues.

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Conferenza tenuta da L. Pasquini il 29 maggio 1995     

Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators

Let

Explicit formula for computing matrix exponential: An analytical approach

This paper is devoted to a practical formula for computing e ^tA, where A is anr×r matrix. Our main result is based on the combinatorial aspect of generalized Fibonacci sequences. Examples and applications are given. Date: June 9, 2003.     

On the derivability of Lipschitz functions

We show that given any Borel measure onR, every Lipschitz function is μ-a.e. differentiable with respect to μ.     

On a Polynomial Basis Generated from the Generalized Kolosov–Muskhelishvili Formulae

In this paper we are working with a generalized Kolosov-Muskhelishvili formula. This representation formula expresses the elastic displacements of a solid body in terms of two monogenic functions. Known polynomial systems of monogenic functions are used to construct a basis of polynomial solutions to the Lamé system from linear elasticity. Received: October, 2007. Accepted: February, 2008.     

Configuration Models For Moduli Spaces of Riemann surfaces with boundary

In this article we consider Riemann surfacesF of genus g ≥ 0 with n ≥ 1 incoming and m ≥ 1 outgoing boundary circles, where on each incoming circle a point is marked. For the moduli space m_g*(m, n) of all suchF of genusg ≥ 0 a configuration space model Rad_h(m, n) is described: it consists of configurations of h = 2g-2+m+n pairs of radial slits distributed over n annuli; certain combinatorial conditions must be satisfied to guarantee the genusg and exactly m outgoing circles. Our main result is a homeomorphism between Rad_h(m, n) and M_g*(m,n). The space Rad_h(m, n) is a non-compact manifold, and the complement of a subcomplex in a finite cell complex. This can be used for homological calculations. Furthermore, the family of spaces Rad_h(m, n ) form an operad, acting on various spaces connected to conformai field theories.     

Jordan decompositions in alternative loop rings

It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop. Research of the second author is supported by CSIR.     

Dynamics on Teichmüller spaces and self-covering of Riemann surfaces

A non-injective holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmüller space. We investigate the dynamics of such self-embeddings by applying our structure theorem of self-covering of Riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the Teichmüller space. We also extend our observation to quasiregular self-covers of Riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self-cover.