Piecewise Linear Models for the Quasiperiodic Transition to Chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic ``sine-circle'' map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction.Comment: 75 pages, plain TeX, 47 figures (available on request

Prognostic model to predict postoperative acute kidney injury in patients undergoing major gastrointestinal surgery based on a national prospective observational cohort study.

Background: Acute illness, existing co-morbidities and surgical stress response can all contribute to postoperative acute kidney injury (AKI) in patients undergoing major gastrointestinal surgery. The aim of this study was prospectively to develop a pragmatic prognostic model to stratify patients according to risk of developing AKI after major gastrointestinal surgery. Methods: This prospective multicentre cohort study included consecutive adults undergoing elective or emergency gastrointestinal resection, liver resection or stoma reversal in 2-week blocks over a continuous 3-month period. The primary outcome was the rate of AKI within 7 days of surgery. Bootstrap stability was used to select clinically plausible risk factors into the model. Internal model validation was carried out by bootstrap validation. Results: A total of 4544 patients were included across 173 centres in the UK and Ireland. The overall rate of AKI was 14·2 per cent (646 of 4544) and the 30-day mortality rate was 1·8 per cent (84 of 4544). Stage 1 AKI was significantly associated with 30-day mortality (unadjusted odds ratio 7·61, 95 per cent c.i. 4·49 to 12·90; P < 0·001), with increasing odds of death with each AKI stage. Six variables were selected for inclusion in the prognostic model: age, sex, ASA grade, preoperative estimated glomerular filtration rate, planned open surgery and preoperative use of either an angiotensin-converting enzyme inhibitor or an angiotensin receptor blocker. Internal validation demonstrated good model discrimination (c-statistic 0·65). Discussion: Following major gastrointestinal surgery, AKI occurred in one in seven patients. This preoperative prognostic model identified patients at high risk of postoperative AKI. Validation in an independent data set is required to ensure generalizability

Restriction fragment analysis of bacteriophage SPP1 in vitro transcription by host RNA polymerase

In vitro transcription of SPP1 DNA occurred on only one of the two strands, the same which is predominantly transcribed in SPP1-infected cells. Transcripts were distributed in several size classes. Analysis of elongation kinetics and of size distribution, coupled with hybridization to DNA restriction fragments, showed that some regions of the template have more initiation sites than others; some have none. Some regions were transcribed directly, some were transcribed from initiation sites located in other regions, and one was never transcribed. Several transcription initiation sites on SPP1 DNA are located on EcoRI fragment 1; four to five others are distributed among other fragments. Cutting the DNA with EcoRI did not introduce artifactual initiation sites. In vitro transcription units can be localized and oriented with respect to the EcoRI restriction map of SPP1 DNA.</jats:p

Unchained polygons and the N-body problem

To the memory of J. Moser, with admiration The simplest solutions of the N-body problem –symmetric relative equilibria – are shown to be organizing centers from which stem some recently studied classes of periodic solutions. We focus in particular on the relative equilibrium of the equal-mass regular N-gon, assumed horizontal, and study the families of Lyapunov quasi-periodic solutions bifurcating from them in the vertical direction. The proof of the local existence of such solutions relies on the fact that the restriction to the corresponding directions of the quadratic part of the energy is positive definite. We then discuss the possibility of continuing the families globally as action minimizers under symmetry constraints by using the fact that, in rotating frames where they become periodic, these solutions are highly symmetric. The paradigmatic examples are the “Eight ” families for an odd number of bodies and the “Hip-Hop ” families for an even number. The first ones generalize Marchal’s P12 family for 3 bodies, which starts with the equilateral triangle and ends with the Eight ([CM, Ma2, CFM, CF, S]); the second ones generalize the Hip-Hop family for 4 bodies, which starts from the square and ends with the Hip-Hop ([CV, CF, TV]). We argue that it is precisely for these two families that global minimization may be used. We also study the relation with the regular N-gon, of the so-called “chain ” choreographies (see [S]): here, only a local minimization property is true (except for N = 3) and moreover the parity plays a deciding role, in particular through the value of the angular momentum. This paper develops [CF]