Measurability implies continuity for solutions of functional equations - even with few variables

Summary. We prove that - under certain conditions - measurable solutions $f$ of the functional equation $f(x)=h(x,y,f(g_{1}(x,y)),\ldots,f(g_{n}(x,y))),\quad(x,y)\in D \subset \mathbb{R}^{s} \times \mathbb{R}^{l}$ are continuous, even if $1\le l\le s$. As a tool we introduce new classes of functions which - roughly speaking - interpolate between continuous and Lebesgue measurable functions. Connection between these classes are also investigated.     

Exploring interactions in high-dimensional genomic data: an overview of Logic Regression, with applications

Logic Regression is an adaptive regression methodology mainly developed to explore high-order interactions in genomic data. Logic Regression is intended for situations where most of the covariates in the data to be analyzed are binary. The goal of Logic Regression is to find predictors that are Boolean (logical) combinations of the original predictors. In this article, we give an overview of the methodology and discuss some applications. We also describe the software for Logic Regression, which is available as an R and S-Plus package.     

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

For a commutative ring R with set of zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we show that Γ(T(R)) and Γ(R) are isomorphic as graphs, where T(R) is the total quotient ring of R, and that Γ(R) is uniquely complemented if and only if either T(R) is von Neumann regular or Γ(R) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring.     

A categorical characterization of varieties

A simple, direct proof of the following characterization of varieties of (finitary) algebras is presented: a cocomplete category is equivalent to a variety iff it has an algebraic generator, i.e., a regular generator which is exactly projective and finitely generated. This improves somewhat a recent restatement, due to Pedicchio and Wood, of the classical characterization theorem of Lawvere. A bijective correspondence between algebraic theories and algebraic generators is established.     

On essential extensions of reduced rings and domains

A ring is said to be a left essential extension of a reduced ring (domain) if it contains a left ideal which is a reduced ring (domain) and intersects nontrivially every nonzero twosided ideal of the ring. We prove that every ring which is a left essential extension of a reduced ring is a subdirect sum of rings which are essential extensions of domains, but the converse implication does not hold. We give some applications of this result and discuss several related questions.Received: 6 January 2003     

A metric approach to asymptotic analysis

We introduce a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. We relate this notion to a concept of “firm” asymptotic function. We use these notions to study boundedness properties which can be applied to continuity questions for some operations on sets and functions. Such questions arise in stability analysis of Hamilton-Jacobi equations. We present some other applications such as an extension of a theorem of Dieudonné and existence results in optimization and fixed point theory.     

Small generators of number fields

If K is a number field of degree n over Q with discriminant D _K and if α∈K generates K, i.e. K=Q(α), then the height of α satisfies with . The paper deals with the existence of small generators of number fields in this sense. We show: (1) For each $n$ there are infinitely many number fields K of degree $n$ with a generator α such that . (2) There is a constant d ² such that every imaginary quadratic number field has a generator α which satisfies .?(3) If K is a totally real number field of prime degree n then one can find an integral generator α with . Received: 10 January 1997 / Revised version: 13 January 1998     

Défaut d'exactitude des modéles¶de Néron toriques sur un corps local

Let R be a complete discrete valuation ring with mixed characteristic. Denote by K its field of fractions and by k its residue field. Let 0 →A ^K →B ^K →C ^K → 0 be an exact sequence of abelian varieties over K and consider the corresponding complex of Nérons models 0 →A→B→C→ 0, over R. We assume that the identity component B _k ⁰ of the special fibre B _k of B is a torus and we study the defect of exactmess at B in this last sequence.

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Re?u: 4 décembre 1997/ Version revisée: 15 décembre 1997