Rational periodic sequences for the Lyness recurrence

Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

This paper studies non-autonomous Lyness type recurrences of the form x_{n+2}=(a_n+x_n)/x_{n+1}, where a_n is a k-periodic sequence of positive numbers with prime period k. We show that for the cases k in {1,2,3,6} the behavior of the sequence x_n is simple(integrable) while for the remaining cases satisfying k not a multiple of 5 this behavior can be much more complicated(chaotic). The cases k multiple of 5 are studied separately.

Integrability and non-integrability of periodic non-autonomous Lyness recurrences (revised and enlarged version)

This paper studies non-autonomous Lyness type recurrences of the form xn+2 = (an+xn+1)=xn, where fang is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k 2 f1; 2; 3; 6g the behavior of the sequence fxng is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some di erent features.

On periodic solutions of 2-periodic Lyness difference equations

We study the existence of periodic solutions of the non--autonomous periodic Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/u_n, where {a_n} is a cycle with positive values a,b and with positive initial conditions. It is known that for a=b=1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a,b) different from (1,1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a is not equal to b, then any odd period, except 1, appears.

Efficacy and safety of a phospholipid emulsion (GR270773) in Gram-negative severe sepsis: results of a phase II multicenter, randomized, placebo-controlled, dose-finding clinical trial.

OBJECTIVE: To assess the survival benefit and safety profile of low-dose (850 mg/kg) and high-dose (1350 mg/kg) phospholipid emulsion vs. placebo administered as a continuous 3-day infusion in patients with confirmed or suspected Gram-negative severe sepsis. Preclinical and ex vivo studies show that lipoproteins bind and neutralize endotoxin, and experimental animal studies demonstrate protection from septic death when lipoproteins are administered. Endotoxin neutralization correlates with the amount of phospholipid in the lipoprotein particles. DESIGN: A three-arm, randomized, blinded, placebo-controlled trial. SETTING: Conducted at 235 centers worldwide between September 2004 and April 2006. PATIENTS: A total of 1379 patients participated in the study, 598 patients received low-dose phospholipid emulsion, and 599 patients received placebo. The high-dose phospholipid emulsion arm was stopped, on the recommendation of the Independent Data Monitoring Committee, due to an increase in life-threatening serious adverse events at the fourth interim analysis and included 182 patients. MEASUREMENTS AND MAIN RESULTS: A 28-day all-cause mortality and new-onset organ failure. There was no significant treatment benefit for low- or high-dose phospholipid emulsion vs. placebo for 28-day all-cause mortality, with rates of 25.8% (p = .329), 31.3% (p = .879), and 26.9%, respectively. The rate of new-onset organ failure was not statistically different among groups at 26.3%, 31.3%, 20.4% with low- and high-dose phospholipid emulsion, and placebo, respectively (one-sided p = .992, low vs. placebo; p = .999, high vs. placebo). Of the subjects treated, 45% had microbiologically confirmed Gram-negative infections. Maximal changes in mean hemoglobin levels were reached on day 10 (-1.04 g/dL) and day 5 (-1.36 g/dL) with low- and high-dose phospholipid emulsion, respectively, and on day 14 (-0.82 g/dL) with placebo. CONCLUSIONS: Treatment with phospholipid emulsion did not reduce 28-day all-cause mortality, or reduce the onset of new organ failure in patients with suspected or confirmed Gram-negative severe sepsis.

SWIVIT - Swiss video-intubation trial evaluating video-laryngoscopes in a simulated difficult airway scenario: study protocol for a multicenter prospective randomized controlled trial in Switzerland.

BACKGROUND: Video-laryngoscopes are marketed for intubation in difficult airway management. They provide a better view of the larynx and may facilitate tracheal intubation, but there is no adequately powered study comparing different types of video-laryngoscopes in a difficult airway scenario or in a simulated difficult airway situation. METHODS/DESIGN: The objective of this trial is to evaluate and to compare the clinical performance of three video-laryngoscopes with a guiding channel for intubation (Airtraq?, A. P. Advance?, King Vision?) and three video-laryngoscopes without an integrated tracheal tube guidance (C-MAC?, GlideScope?, McGrath?) in a simulated difficult airway situation in surgical patients. The working hypothesis is that each video-laryngoscope provides at least a 90% first intubation success rate (lower limit of the 95% confidence interval >0.9). It is a prospective, patient-blinded, multicenter, randomized controlled trial in 720 patients who are scheduled for elective surgery under general anesthesia, requiring tracheal intubation at one of the three participating hospitals. A difficult airway will be created using an extrication collar and taping the patients' head on the operating table to substantially reduce mouth opening and to minimize neck movement. Tracheal intubation will be performed with the help of one of the six devices according to randomization. Insertion success, time necessary for intubation, Cormack-Lehane grade and percentage of glottic opening (POGO) score at laryngoscopy, optimization maneuvers required to aid tracheal intubation, adverse events and technical problems will be recorded. Primary outcome is intubation success at first attempt. DISCUSSION: We will simulate the difficult airway and evaluate different video-laryngoscopes in this highly realistic and clinically challenging scenario, independently from manufacturers of the devices. Because of the sufficiently powered multicenter design this study will deliver important and cutting-edge results that will help clinicians decide which device to use for intubation of the expected and unexpected difficult airway. TRIAL REGISTRATION: NCT01692535.

Microstructural evolution of primary crystallization in diffusion-controlled grain growth

A model has been developed for evaluating grain size distributions in primary crystallizations where the grain growth is diffusion controlled. The body of the model is grounded in a recently presented mean-field integration of the nucleation and growth kinetic equations, modified conveniently in order to take into account a radius-dependent growth rate, as occurs in diffusion-controlled growth. The classical diffusion theory is considered, and a modification of this is proposed to take into account interference of the diffusion profiles between neighbor grains. The potentiality of the mean-field model to give detailed information on the grain size distribution and transformed volume fraction for transformations driven by nucleation and either interface- or diffusion-controlled growth processes is demonstrated. The model is evaluated for the primary crystallization of an amorphous alloy, giving an excellent agreement with experimental data. Grain size distributions are computed, and their properties are discussed.

Randomized controlled study demonstrating failure of LPV/r monotherapy in HIV: the role of compartment and CD4-nadir.

BACKGROUND: Long-term side-effects and cost of HIV treatment motivate the development of simplified maintenance. Monotherapy with ritonavir-boosted lopinavir (LPV/r-MT) is the most widely studied strategy. However, efficacy of LPV/r-MT in compartments remains to be shown. METHODS: Randomized controlled open-label trial comparing LPV/r-MT with continued treatment for 48 weeks in treated patients with fully suppressed viral load. The primary endpoint was treatment failure in the central nervous system [cerebrospinal fluid (CSF)] and/or genital tract. Treatment failure in blood was defined as two consecutive HIV RNA levels more than 400 copies/ml. RESULTS: The trial was prematurely stopped when six patients on monotherapy (none in continued treatment-arm) demonstrated a viral failure in blood. At study termination, 60 patients were included, 29 randomized to monotherapy and 13 additional patients switched from continued treatment to monotherapy after 48 weeks. All failures occurred in patients with a nadir CD4 cell count below 200/microl and within the first 24 weeks of monotherapy. Among failing patients, all five patients with a lumbar puncture had an elevated HIV RNA load in CSF and four of six had neurological symptoms. Viral load was fully resuppressed in all failing patients after resumption of the original combination therapy. No drug resistant virus was found. The only predictor of failure was low nadir CD4 cell count (P < 0.02). CONCLUSION: Maintenance of HIV therapy with LPV/r alone should not be recommended as a standard strategy; particularly not in patients with a CD4 cell count nadir less than 200/microl. Further studies are warranted to elucidate the role of the central nervous system compartment in monotherapy-failure.

Effect of multidimensional lifestyle intervention on fitness and adiposity in predominantly migrant preschool children (Ballabeina): cluster randomised controlled trial.

OBJECTIVE: To test the effect of a multidimensional lifestyle intervention on aerobic fitness and adiposity in predominantly migrant preschool children. DESIGN: Cluster randomised controlled single blinded trial (Ballabeina study) over one school year; randomisation was performed after stratification for linguistic region. SETTING: 40 preschool classes in areas with a high migrant population in the German and French speaking regions of Switzerland. PARTICIPANTS: 652 of the 727 preschool children had informed consent and were present for baseline measures (mean age 5.1 years (SD 0.7), 72% migrants of multicultural origins). No children withdrew, but 26 moved away. INTERVENTION: The multidimensional culturally tailored lifestyle intervention included a physical activity programme, lessons on nutrition, media use (use of television and computers), and sleep and adaptation of the built environment of the preschool class. It lasted from August 2008 to June 2009. MAIN OUTCOME MEASURES: Primary outcomes were aerobic fitness (20 m shuttle run test) and body mass index (BMI). Secondary outcomes included motor agility, balance, percentage body fat, waist circumference, physical activity, eating habits, media use, sleep, psychological health, and cognitive abilities. RESULTS: Compared with controls, children in the intervention group had an increase in aerobic fitness at the end of the intervention (adjusted mean difference: 0.32 stages (95% confidence interval 0.07 to 0.57; P=0.01) but no difference in BMI (-0.07 kg/m(2), -0.19 to 0.06; P=0.31). Relative to controls, children in the intervention group had beneficial effects in motor agility (-0.54 s, -0.90 to -0.17; P=0.004), percentage body fat (-1.1%, -2.0 to -0.2; P=0.02), and waist circumference (-1.0 cm, -1.6 to -0.4; P=0.001). There were also significant benefits in the intervention group in reported physical activity, media use, and eating habits, but not in the remaining secondary outcomes. CONCLUSIONS: A multidimensional intervention increased aerobic fitness and reduced body fat but not BMI in predominantly migrant preschool children.

Administration of growth hormone to underweight patients with chronic obstructive pulmonary disease. A prospective, randomized, controlled study.

Patients with chronic obstructive pulmonary disease (COPD) often develop weight loss, which is associated with increased mortality. Recombinant human growth hormone (rhGH) treatment has been proposed to improve nitrogen balance and to increase muscle strength in these patients. The aim of this study was to assess the effects of rhGH administration on the nutritional status, resting metabolism, muscle strength, exercise tolerance, dyspnea, and subjective well-being of underweight patients with stable COPD. Sixteen patients attending a pulmonary rehabilitation program (age: 66 +/- 9 yr; weight: 77 +/- 7% of ideal body weight; FEV1: 39 +/- 13% of predicted) were randomly treated daily with either 0.15 IU/kg rhGH or placebo during 3 wk in a double-blind fashion. Measurements were made at the beginning (DO) and at the end (D21) of treatment and 2 mo later (D81). Body weight was similar in the two groups during the study, but lean body mass was significantly higher in the rhGH group at D21 (p < 0.01) and D81 (p < 0.05). The increase in lean body mass was 2.3 +/- 1.6 kg in the rhGH group and 1.1 +/- 0.9 kg in the control group at D21 and 1.9 +/- 1.6 kg in the rhGH group and 0.7 +/- 2.1 kg in the control group at D81. At D21, the resting energy expenditure was increased in the rhGH group (107.8% of DO, p < 0.001 compared with the control group). At D21 and D81, the changes in maximal respiratory pressures, handgrip strength, maximal exercise capacity, and subjective well-being were similar in the two groups. At D21, the 6-min walking distance decreased in the rhGH group (-13 +/- 31%) and increased in the control group (+10 +/- 14%; p < 0.01). We conclude that the daily administration of 0.15 IU/kg rhGH during 3 wk increases lean body mass but does not improve muscle strength or exercise tolerance in underweight patients with COPD.

Colorectal polypectomy during insertion and withdrawal or only during withdrawal? A randomized controlled trial.

BACKGROUND AND STUDY AIMS: Removal of colorectal polyps is routinely performed during withdrawal of the endoscope. However, polyps detected during insertion of the colonoscope may be missed at withdrawal. We aimed to evaluate whether polypectomy during both insertion and withdrawal increases polyp detection and removal rates compared with polypectomy at withdrawal only, and to assess the duration of both approaches. PATIENTS AND METHODS: Patients were included into the study when the first polyp was detected, and randomized into two groups; in group A, polyps ≤ 10 mm in diameter were removed during insertion and withdrawal of the colonoscope, while in group B, these polyps were removed at withdrawal only. Main outcome measures were duration of colonoscopy, number of polyps detected during insertion but not recovered during withdrawal, technical ease, patient discomfort, and complications. RESULTS: 150 patients were randomized to group A and 151 to group B. Mean (± standard deviation [SD]) duration of colonoscopy did not differ between the groups (30.8 ± 15.6 min [A] vs. 28.5 ± 13.8 min [B], P = 0.176). In group A 387 polyps (mean 2.58 per colonoscopy) were detected and removed compared with 389 polyps detected (mean 2.58 per colonoscopy) in group B of which 376 were removed (13 polyps were missed, mean size [SD] 3.2 [1.3] mm; 7.3 % of patients). Patient tolerance was similar in the two groups. CONCLUSIONS: Removal of polyps ≤ 10 mm during withdrawal only is associated with a considerable polyp miss rate. We therefore recommend that these polyps are removed during both insertion and withdrawal.

Effectiveness of psychotropic medications in the maintenance phase of bipolar disorder:a meta-analysis of randomized controlled trials

The purpose of this meta-analysis was to examine the efficacy of maintenance treatments for bipolar disorder. Placebo-controlled or active comparator bipolar maintenance clinical trials of ≥6 months' duration with at least 15 patients/treatment group were identified using Medline, EMBASE, clinicaltrials.gov, and Cochrane databases (1993 to July 2010). The main outcome measure was relative risk for relapse for patients in remission. Twenty trials (5,364 patients) were identified. Overall, lithium and quetiapine were the most studied agents (eight and five trials, respectively). The majority of studies included patients who had previously responded to treatment for an acute episode. All interventions, with the exception of perphenazine+mood stabilizer, showed a relative risk for manic/mixed or depressive relapse below 1.0, although there was variation in the statistical significance of the findings vs. placebo. No monotherapy was associated with a significantly reduced risk for both manic/mixed and depressed relapse. Of the combination treatments, only quetiapine+lithium/divalproex, was associated with a significantly reduced risk vs. comparator (placebo+lithium/valproate) for relapse at both the manic/mixed and depressed poles of bipolar illness. Limitations for the analysis include differences in study durations and definitions of relapse. In conclusion, available maintenance therapies show considerable variation in efficacy. The efficacy of lithium and divalproex has been confirmed, but newer therapies, such as a number of atypical antipsychotics were also shown to be effective in bipolar disorder. Efficacy of all maintenance interventions needs to be balanced against the safety and tolerability profiles of individual agents.