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.

Leaf plasticity in successive selection cycles of 'Saracura' maize in response to periodic soil flooding

The objective of this work was to assess the effect of successive selection cycles on leaf plasticity of 'Saracura' maize BRS-4154 under periodical flooding in field conditions. Soil flooding started at the six-leaf stage with the application of a 20-cm depth water layer three times a week. At flowering, samples of leaves were collected and fixed. Paradermic and transverse sections were observed under photonic microscope. Several changes were observed throughout the selection cycles, such as modifications in the number and size of the stomata, higher amount of vascular bundles and the resulting decrease of the distance between them, smaller diameter of the metaxylem, decrease of cuticle and epidermis thickness, decrease of number and size of bulliform cells, increase of phloem thickness, smaller sclerenchyma area. Therefore, the successive selection cycles of 'Saracura' maize resulted in changes in the leaf anatomy, which might be favorable to the plant's tolerance to the intermittent flooding of the soil.

J2 effect and elliptic inclined periodic orbits in the collision three-body problem

The existence of a new class of inclined periodic orbits of the collision restricted three-body problem is shown. The symmetric periodic solutions found are perturbations of elliptic kepler orbits and they exist only for special values of the inclination and are related to the motion of a satellite around an oblate planet

Report on the Status of Findings and Recommendations from the City of Cambridge’s Periodic Examination Report dated September 5, 2014 for the period June 1, 2015 through November 30, 2015

Report on the Status of Findings and Recommendations from the City of Cambridge’s Periodic Examination Report dated September 5, 2014 for the period June 1, 2015 through November 30, 2015

Dissemination of periodic mammography and patterns of use, by birth cohort, in Catalonia (Spain)

Background: In Catalonia (Spain) breast cancer mortality has declined since the beginning of the 1990s. The dissemination of early detection by mammography and the introduction of adjuvant treatments are among the possible causes of this decrease, and both were almost coincident in time. Thus, understanding how these procedures were incorporated into use in the general population and in women diagnosed with breast cancer is very important for assessing their contribution to the reduction in breast cancer mortality. In this work we have modeled the dissemination of periodic mammography and described repeat mammography behavior in Catalonia from 1975 to 2006. Methods: Cross-sectional data from three Catalan Health Surveys for the calendar years 1994, 2002 and 2006 was used. The dissemination of mammography by birth cohort was modeled using a mixed effects model and repeat mammography behavior was described by age and survey year. Results: For women born from 1938 to 1952, mammography clearly had a period effect, meaning that they started to have periodic mammograms at the same calendar years but at different ages. The age at which approximately 50% of the women were receiving periodic mammograms went from 57.8 years of age for women born in 1938–1942 to 37.3 years of age for women born in 1963–1967. Women in all age groups experienced an increase in periodic mammography use over time, although women in the 50–69 age group have experienced the highest increase. Currently, the target population of the Catalan Breast Cancer Screening Program, 50–69 years of age, is the group that self-reports the highest utilization of periodic mammograms, followed by the 40–49 age group. A higher proportion of women of all age groups have annual mammograms rather than biennial or irregular ones. Conclusion: Mammography in Catalonia became more widely implemented during the 1990s. We estimated when cohorts initiated periodic mammograms and how frequently women are receiving them. These two pieces of information will be entered into a cost-effectiveness model of early detection in Catalonia.

Evidence-based provisional clinical classification criteria for autoinflammatory periodic fevers.

The objective of this work was to develop and validate a set of clinical criteria for the classification of patients affected by periodic fevers. Patients with inherited periodic fevers (familial Mediterranean fever (FMF); mevalonate kinase deficiency (MKD); tumour necrosis factor receptor-associated periodic fever syndrome (TRAPS); cryopyrin-associated periodic syndromes (CAPS)) enrolled in the Eurofever Registry up until March 2013 were evaluated. Patients with periodic fever, aphthosis, pharyngitis and adenitis (PFAPA) syndrome were used as negative controls. For each genetic disease, patients were considered to be 'gold standard' on the basis of the presence of a confirmatory genetic analysis. Clinical criteria were formulated on the basis of univariate and multivariate analysis in an initial group of patients (training set) and validated in an independent set of patients (validation set). A total of 1215 consecutive patients with periodic fevers were identified, and 518 gold standard patients (291 FMF, 74 MKD, 86 TRAPS, 67 CAPS) and 199 patients with PFAPA as disease controls were evaluated. The univariate and multivariate analyses identified a number of clinical variables that correlated independently with each disease, and four provisional classification scores were created. Cut-off values of the classification scores were chosen using receiver operating characteristic curve analysis as those giving the highest sensitivity and specificity. The classification scores were then tested in an independent set of patients (validation set) with an area under the curve of 0.98 for FMF, 0.95 for TRAPS, 0.96 for MKD, and 0.99 for CAPS. In conclusion, evidence-based provisional clinical criteria with high sensitivity and specificity for the clinical classification of patients with inherited periodic fevers have been developed.

A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity

In two previous papers [J. Differential Equations, 228 (2006), pp. 530 579; Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261 1300] we have developed fast algorithms for the computations of invariant tori in quasi‐periodic systems and developed theorems that assess their accuracy. In this paper, we study the results of implementing these algorithms and study their performance in actual implementations. More importantly, we note that, due to the speed of the algorithms and the theoretical developments about their reliability, we can compute with confidence invariant objects close to the breakdown of their hyperbolicity properties. This allows us to identify a mechanism of loss of hyperbolicity and measure some of its quantitative regularities. We find that some systems lose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. We find empirically that, close to the breakdown, the distances between the invariant bundles and the Lyapunov multipliers which are natural measures of hyperbolicity depend on the parameters, with power laws with universal exponents. We also observe that, even if the rigorous justifications in [J. Differential Equations, 228 (2006), pp. 530-579] are developed only for hyperbolic tori, the algorithms work also for elliptic tori in Hamiltonian systems. We can continue these tori and also compute some bifurcations at resonance which may lead to the existence of hyperbolic tori with nonorientable bundles. We compute manifolds tangent to nonorientable bundles.

Analysis of the genetic basis of periodic fever with aphthous stomatitis, pharyngitis, and cervical adenitis (PFAPA) syndrome.

PFAPA syndrome is the most common autoinflammatory syndrome in children from Western countries. In spite of its strong familial clustering, its genetic basis and inheritance pattern are still unknown. We performed a comprehensive genetic study on 68 individuals from 14 families. Linkage analysis suggested a susceptibility locus on chromosome 8, but direct molecular sequencing did not support this initial statistical finding. Exome sequencing revealed the absence of any gene that was mutated in all patients. Exhaustive screening of genes involved in other autoinflammatory syndromes or encoding components of the human inflammasome showed no DNA variants that could be linked to PFAPA molecular pathology. Among these, the previously-reported missense mutation V198M in the NLRP3 gene was clearly shown not to co-segregate with PFAPA. Our results on this relatively large cohort indicate that PFAPA syndrome is unlikely to be a monogenic condition. Moreover, none of the several genes known to be involved in inflammation or in autoinflammatory disorders seem to be relevant, alone, to its etiology, suggesting that PFAPA results from oligogenic or complex inheritance of variants in multiple disease genes and/or non-genetic factors.

Phenotypic and genotypic characteristics of cryopyrin-associated periodic syndrome: a series of 136 patients from the Eurofever Registry.

OBJECTIVE: To evaluate genetic, demographic and clinical features in patients with cryopyrin-associated periodic syndrome (CAPS) from the Eurofever Registry, with a focus on genotype-phenotype correlations and predictive disease severity markers. METHODS: A web-based registry retrospectively collected data on patients with CAPS. Experts in the disease independently validated all cases. Patients carrying NLRP3 variants and germline-mutation-negative patients were included. RESULTS: 136 patients were analysed. The median age at disease onset was 9 months, and the median duration of follow-up was 15 years. Skin rash, musculoskeletal involvement and fever were the most prevalent features. Neurological involvement (including severe complications) was noted in 40% and 12% of the patients, respectively, with ophthalmological involvement in 71%, and neurosensory hearing loss in 42%. 133 patients carried a heterozygous, germline mutation, and 3 patients were mutation-negative (despite complete NLRP3 gene screening). Thirty-one different NLRP3 mutations were recorded; 7 accounted for 78% of the patients, whereas 24 rare variants were found in 27 cases. The latter were significantly associated with early disease onset, neurological complications (including severe complications) and severe musculoskeletal involvement. The T348M variant was associated with early disease onset, chronic course and hearing loss. Neurological involvement was less strongly associated with V198M, E311 K and A439 V alleles. Early onset was predictive of severe neurological complications and hearing loss. CONCLUSIONS: Patients carrying rare NLRP3 variants are at risk of severe CAPS; onset before the age of 6 months is associated with more severe neurological involvement and hearing loss. These findings may have an impact on treatment decisions.