Bounds for algorithms in differential algebra

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal. We also give an algorithm for converting a characteristic decomposition of a radical differential ideal from one ranking into another. This algorithm performs all differentiations in the beginning and then uses a purely algebraic decomposition algorithm.Comment: 40 page

Canonical Characteristic Sets of Characterizable Differential Ideals

We study the concept of canonical characteristic set of a characterizable differential ideal. We propose an efficient algorithm that transforms any characteristic set into the canonical one. We prove the basic properties of canonical characteristic sets. In particular, we show that in the ordinary case for any ranking the order of each element of the canonical characteristic set of a characterizable differential ideal is bounded by the order of the ideal. Finally, we propose a factorization-free algorithm for computing the canonical characteristic set of a characterizable differential ideal represented as a radical ideal by a set of generators. The algorithm is not restricted to the ordinary case and is applicable for an arbitrary ranking.Comment: 26 page

Interleukin-11 Drives Early Lung Inflammation during Mycobacterium tuberculosis Infection in Genetically Susceptible Mice

IL-11 is multifunctional cytokine whose physiological role in the lungs during pulmonary tuberculosis (TB) is poorly understood. Here, using in vivo administration of specific antibodies against IL-11, we demonstrate for the first time that blocking IL-11 diminishes histopathology and neutrophilic infiltration of the lung tissue in TB-infected genetically susceptible mice. Antibody treatment decreased the pulmonary levels of IL-11 and other key inflammatory cytokines not belonging to the Th1 axis, and down-regulated IL-11 mRNA expression. This suggests the existence of a positive feedback loop at the transcriptional level, which is further supported by up-regulation of IL-11 mRNA expression in the presence of rIL-11 in in vitro cultures of lung cells. These findings imply a pathogenic role for IL-11 during the early phase of Mycobacterium tuberculosis-triggered disease in a genetically susceptible host

On Computation of Kolchin Characteristic Sets: Ordinary and Partial Cases

Abstract In this paper we study the problem of computing a Kolchin characteristic set of a radical differential ideal. The central part of the article is the presentation of algorithms solving this problem in two principal cases: for ordinary differential polynomials and in the partial differential case. Our computations are mainly performed with respect to orderly rankings. We also discuss the usefulness of regular and characteristic decompositions of radical differential ideals. In the partial differential case we give an algorithm for computing characteristic sets in the special case of radical differential ideals satisfying the property of consistency. For this class of ideals we show how to deal with arbitrary differential rankings