Lifting chains of prime ideals to paravaluation rings

Résumé  SoitR ⊂T une extension des anneaux commutatifs et soit {P _α :α ∈I} une cha?ne croissante des idéaux premiers deR (I étant un ensemble totalement ordonné, peut-être infini). Alors il existe un anneau de paravaluationV deT et une cha?ne {Q _α} des idéaux premiers deV de sorte queR ⊂V etQ _α ∩R =P _α pour toutα ∈I. Tout d’abord, on établit le cas spécial dans lequelT est un corps; dans ce cas, on trouve en effet un tel anneau de valuationV deT. Ensuite, l’assertion ci-dessus pour le cas général découle comme conséquence. Dans le cas général, on peut aussi remplacer le mot “paravaluation” avec le mot “valuation” siR est un anneau de Marot etT est son anneau total de fractions.      

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

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     

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 left conjugacy closed loops in which the left multiplication group is normal

A loopQ is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. This papers investigates those LCC loops where the group generated by left translations is normal in the group generated by both left and right translations.     

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     

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.     

Extensions of positive projections

The main purpose of this paper is the following Theorem: letE be a Dedekind complete Riesz space (vector lattice), letA be a vector subspace of it which majorizesE and letx ₀ be an element ofE−A. IfT ₀ is a positive projection onA, then there existsy ₀∈E such thatT ₀ can be extended to a positive projection onA+ℝx ₀+ℝy ₀.     

On the structure of inner ideals of nest algebras

IfA is a nest algebra andA _s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A _s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A _s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A _s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A _s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.     

The fundamental theorem of geometric calculus via a generalized riemann integral

Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative ∇ _V F exists and is Lebesgue integrable. We also give sufficient conditions that ∇ _V F exists.     

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.     

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

Let

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.     

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.     

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.      

On two-point boundary value problems for systems of higher-order ordinary differential equations with singularities

The sufficient conditions of solvability and unique solvability of the two-point boundary value problems of Vallèe-Poussin and Cauchy-Niccoletti have been found for a system of ordinary differential equations of the form

A Classification of Homogeneous Operators in the Cowen-Douglas Class

A complete list of homogeneous operators in the Cowen-Douglas class is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc. This research was supported in part by a DST - NSF S&T Cooperation Programme and a PSC-CUNY grant.