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

Flat families by strongly stable ideals and a generalization of Groebner bases

Let J be a strongly stable monomial ideal in S=K[x_1,...,x_n] and let Mf(J) be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K-vector basis of the quotient S/I. We show that an ideal I belongs to Mf(J) if and only if it is generated by a special set of polynomials, the J-marked basis of I, that in some sense generalizes the notion of reduced Groebner basis and its constructive capabilities. Indeed, although not every J-marked basis is a Groebner basis with respect to some term order, a sort of normal form modulo I (with the ideal I in Mf(J)) can be computed for every homogeneous polynomial, so that a J-marked basis can be characterized by a Buchberger-like criterion. Using J-marked bases, we prove that the family Mf(J) can be endowed, in a very natural way, with a structure of affine scheme that turns out to be homogeneous with respect to a non-standard grading and flat in the origin (the point corresponding to J), thanks to properties of J-marked bases analogous to those of Groebner bases about syzygies.Comment: This paper includes and extends the paper posed at arXiv:1005.0457. Revised version for publication. Added reference

Off-label long acting injectable antipsychotics in real-world clinical practice: a cross-sectional analysis of prescriptive patterns from the STAR Network DEPOT study

Introduction Information on the off-label use of Long-Acting Injectable (LAI) antipsychotics in the real world is lacking. In this study, we aimed to identify the sociodemographic and clinical features of patients treated with on- vs off-label LAIs and predictors of off-label First- or Second-Generation Antipsychotic (FGA vs. SGA) LAI choice in everyday clinical practice. Method In a naturalistic national cohort of 449 patients who initiated LAI treatment in the STAR Network Depot Study, two groups were identified based on off- or on-label prescriptions. A multivariate logistic regression analysis was used to test several clinically relevant variables and identify those associated with the choice of FGA vs SGA prescription in the off-label group. Results SGA LAIs were more commonly prescribed in everyday practice, without significant differences in their on- and off-label use. Approximately 1 in 4 patients received an off-label prescription. In the off-label group, the most frequent diagnoses were bipolar disorder (67.5%) or any personality disorder (23.7%). FGA vs SGA LAI choice was significantly associated with BPRS thought disorder (OR = 1.22, CI95% 1.04 to 1.43, p = 0.015) and hostility/suspiciousness (OR = 0.83, CI95% 0.71 to 0.97, p = 0.017) dimensions. The likelihood of receiving an SGA LAI grew steadily with the increase of the BPRS thought disturbance score. Conversely, a preference towards prescribing an FGA was observed with higher scores at the BPRS hostility/suspiciousness subscale. Conclusion Our study is the first to identify predictors of FGA vs SGA choice in patients treated with off-label LAI antipsychotics. Demographic characteristics, i.e. age, sex, and substance/alcohol use co-morbidities did not appear to influence the choice towards FGAs or SGAs. Despite a lack of evidence, clinicians tend to favour FGA over SGA LAIs in bipolar or personality disorder patients with relevant hostility. Further research is needed to evaluate treatment adherence and clinical effectiveness of these prescriptive patterns

The kinds of truth of geometry theorems

Proof by refutation of a geometry theorem that is not universally true produces a Gröbner basis whose elements, called side polynomials, may be used to give inequations that can be added to the hypotheses to give a valid theorem. We show that (in a certain sense) all possible subsidiary conditions are implied by those obtained from the basis; that what we call the kind of truth of the theorem may be derived from the basis; and that the side polynomials may be classified in a useful way. We analyse the relationship between side polynomials and kinds of truth, and we give a unified algorithmic treatment of side polynomials, with examples generated by an implementation