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.Comment: 27 pages; 1 figur

On the set of periods of the 2-periodic Lyness’ Equation

PreprintWe study the periodic solutions of the non–autonomous periodic Lyness’ recurrence un+2 = (an +un+1)=un, where fangn is a cycle with positive values a,b and with positive initial conditions. Among other methodological issues we give an outline of the proof of the following results: (1) If (a;b) 6= (1;1), then there exists a value p0(a;b) such that for any p > p0(a;b) there exist continua of initial conditions giving rise to 2p–periodic sequences. (2) The set of minimal periods arising when (a;b) 2 (0;¥) 2 and positive initial conditions are considered, contains all the even numbers except 4, 6, 8, 12 and 20. If a 6= b, then it does not appear any odd period, except 1.Preprin

Integrable birational maps on the plane: blending dynamics and algebraic geometry

Contingut del Pòster presentat al congrés New Trends in Dynamical SystemsPeer ReviewedPreprin

On the set of periods of the 2-periodic Lyness' equation

Publicació amb motiu de la International Conference on Difference Equations and Applications (July 22-27, 2012, Barcelona, Spain) amb el títol Difference Equations, Discrete Dynamical Systems and ApplicationsWe study the periodic solutions of the non-autonomous periodic Lyness' recurrence u = (a + u )/u, where {a} is a cycle with positive values a,b and with positive initial conditions. Among other methodological issues we give an outline of the proof of the following results: (1) If (a, b) ≠ (1, 1), then there exists a value p(a, b) such that for any p > p(a, b) there exist continua of initial conditions giving rise to 2p-periodic sequences. (2) The set of minimal periods arising when (a, b) ∈ (0,∞) and positive initial conditions are considered, contains all the even numbers except 4, 6, 8, 12 and 20. If a ≠ b, then it does not appear any odd period, except 1

Éclairer les mathématiques par les exemples ?

Nous exposons tout d’abord le lien étroit qu’entretiennent mathématiques et exemples d’un point de vue des fondements mêmes de l’activité mathématique. Puis, nous présentons rapidement des tentatives didactiques pour développer un enseignement basé sur l’usage raisonné des exemples. Cette première approche s’appuie sur six entretiens conduits avec des enseignants-chercheurs avec lesquels nous abordons les différents types d’usages des exemples dans l’enseignement supérieur français. Ceci nous conduit à une conclusion quelque peu paradoxale. En effet, si les exemples sont reconnus comme un élément essentiel nécessaire à la réalisation d’un travail mathématique efficace, y compris dans l’enseignement, leur usage reste minoré dans les pratiques des enseignants de mathématiques du supérieur, notamment dans le travail en responsabilité des étudiants. Nous interprétons cela en termes de problématisation, perçue comme essentielle pour comprendre les enjeux des mathématiques, et d’adidacticité, pour que les connaissances émergent du travail en responsabilité des étudiants.The strong link between mathematics and examples is analysed as a very component of mathematical activity. Then some didactical tentative proposals are presented for developing mathematical training based on a reasoned use of examples. Interviews with researcher-teachers enable to identify different types of using examples at a universitary level in the French context. In fact examples are considerd as a crucial element that is necessary for conducing efficient mathematical work, including for students; nevertheless, their use remains marginal in mathematical teachers’ practives: their teaching privileges grand theoretical constructions and their emblematic theorems

Characterising lower-body musculoskeletal morphology and whole-body composition of elite female and male Australian Football players

Background: Physical demands and injury rates differ between elite female and male Australian Football (AF) players. To improve understanding of contributing physical factors to these differences, the purpose of this study was to investigate lower-body morphology and whole-body composition of elite footballers competing in the Australian Football League (AFL) and Australian Football League Women’s (AFLW). Methods: Lower-body morphology and whole-body composition of 23 AFL players and 23 AFLW players were assessed using peripheral Quantitative Computed Tomography and Dual-energy X-ray Absorptiometry at the beginning of pre-season. Differences between cohorts, with sub-analyses of kicking vs. support limbs, and experienced vs. inexperienced player status were assessed using two-sample independent t-tests. Magnitude of differences were assessed using Cohen’s d effect sizes. Results: AFL players had greater absolute (p \u3c 0.001; ES = 3.28) and relative (p \u3c 0.001; ES = 2.29) whole body lean soft-tissue mass, with less absolute (p = 0.004; ES = 0.91) and relative (p \u3c 0.001; ES = 2.29) fat mass than AFLW players. For AFLW players, no significant differences existed between kicking and support limbs with few differences observed between experienced and inexperienced players. Conclusions: Greater emphasis on physical development in AFLW players may be required to enable increases in muscle mass and skeletal robustness, to ensure they can tolerate the loads of elite competition

Pre-season body composition has minimal influence on in-season match availability, and match performance in female Australian Football League (AFLW) players

This study examined the relationship between pre-season body composition, in-season match performance, and match availability in female players competing in the Australian Football League Women\u27s (AFLW) competition. With the outlawing of body composition assessments as part of pre-draft player evaluations in the AFLW, this study seeks to examine whether this is justified. Twenty-two (n = 22) players had body composition assessed with dual-energy x-ray absorptiometry at the beginning of the 2021 AFLW pre-season (whole-body and regional fat mass and lean soft-tissue mass [LSTM]). In-season match availability and match performance data (Coaches Score [CS], Champion Data Player Rank, average disposals, disposal and kicking efficiency) were collected throughout the 2021 competition. Pearson correlations were performed to assess if associations existed between body composition and in-season match performance and availability. A median split was performed to divide players into higher and lower performing groups for match performance variables. Two-sample independent t-tests were then used to assess differences between groups. No body composition characteristics could differentiate between in-season match availability groups (100 % availability vs. \u3c 100 % availability) or higher and lower performing groups for all match performance variables. Total leg LSTM asymmetry shared a moderate negative association with CS. Body composition may not be important for determining in-season match availability and performance in female AFLW players. Thus, the repercussions following the removal of pre-draft body composition assessments across the league may not be as significant as is currently perceived. Other physiological, biomechanical, or performance qualities are more variable and may mask the effect of body composition in these players. AFLW practitioners should prioritize the development of other important attributes, such as aerobic fitness, muscular strength and power, and technical skill