Algebraic entropy for algebraic maps

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations

Maximum Flag-Rank Distance Codes

In this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes

Backlund transformations and Hamiltonian flows

In this work we show that, under certain conditions, parametric Backlund transformations (BTs) for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non-autonomous Hamiltonian. The two systems share the same constants of motion. This observation lead to the identification of the Hamiltonian interpolating the iteration of the discrete map defined by the transformations, that indeed will be a linear combination of the integrals appearing in the spectral curve of the Lax matrix. An application to the Toda periodic lattice is given.Comment: 19 pages, 2 figures. to appear in J. Phys.

Integrable maps in 4D and modified Volterra lattices

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g$. As well as proving that each such discrete system is an integrable map in the Liouville sense, we also showed it to be an algebraic completely integrable system. In the discrete setting, the latter means that the generic level set of the invariants is an affine part of an abelian variety, in this case the Jacobian of the hyperelliptic curve, and each iteration of the map corresponds to a translation by a fixed vector on the Jacobian. In addition, we demonstrated that, by combining the discrete integrable dynamics with the flow of one of the commuting Hamiltonian vector fields, these maps provide genus $g$ algebro-geometric solutions of the infinite Volterra lattice, which justified naming them Volterra maps, denoted ${\cal V}_g$. The original motivation behind our work was the fact that, in the particular case $g=2$, we could recover an example of an integrable symplectic map in four dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified birational maps in 4D admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Hence, in this particular case, the map ${\cal V}_2$ yields genus two solutions of the Volterra lattice. The purpose of this note is to point out how two of the other 4D integrable maps obtained in the classification of Gubbiotti et al. correspond to genus two solutions of two different forms of the modified Volterra lattice, being related via a Miura-type transformation to the $g=2$ Volterra map ${\cal V}_2$. We dedicate this work to a dear friend and colleague, Decio Levi

One-year evolution of symptoms and health status of the copd multi-dimensional phenotypes: Results from the follow-up of the storico observational study

Aim: Describing the 1-year evolution of symptoms and health status in COPD patients enrolled in the STORICO study (observational study on characterization of 24-h symptoms in patients with COPD) classified in multidimensional phenotypes (m-phenotypes). Methods: In our previous study, we performed an exploratory factor analysis to identify clinical and pathophysiological variables having the greatest classificatory properties, followed by a cluster analysis to group patients into m-phenotypes (mild COPD (MC), mild emphysematous (ME), severe bronchitic (SB), severe emphysematous (SE), and severe mixed COPD (SMC)). COPD symptoms were recorded at baseline, 6-, and 12-month follow-up and their evolution was described as frequency of patients with always present, always absent, arising’, ‘no more present symptoms. QoL and quality of sleep were evaluated using the SGRQ and CASIS questionnaires, respectively. Results: We analyzed 379 subjects (144 MC, 71 ME, 96 SB, 14 SE, 54 SMC). M-phenotypes were stable over time in terms of presence of symptoms and health status with selected differences in evolution of symptoms in mild vs severe m-phenotypes. Indeed, 28.1% SB, 50.0% SE and 24.1% SMC vs 0.7% MC and 5.6% ME with night-time symptoms at baseline had no more symptoms at 6-month (p-value night-time symptom evolution MC vs SB, SE, SMC and ME vs SB, SE, SMC <0.0001). All m-phenotypes improved in quality of sleep, more markedly the severe than the mild ones (p-values CASIS score change between baseline and 6-or 12-month in MC, ME vs SB, SE, SMC <0.0001). QoL did not change during observation, irrespectively of m-phenotype. Conclusion: Over 1 year, severe m-phenotypes showed an improvement in night-time symptoms and quality of sleep, but not QoL. Being stable over time, m-phenotypes seem worthy of testing for classificatory and prognostic purposes

Clinical evolution and quality of life in clinically based copd chronic bronchitic and emphysematous phenotypes: Results from the 1-year follow-up of the storico italian observational study

Introduction: Understanding clinical evolution of chronic obstructive pulmonary disease (COPD) is crucial for improving disease management. Materials and Methods: STORICO (NCT03105999), an Italian, multicenter, non-inter-ventional, observational study conducted in 40 pulmonology centers, aimed to describe the 1-year clinical evolution and health status of clinicallbased phenotypes. Baseline and follow-up data of COPD subjects with a chronic bronchitis (CB) or emphysema (EM) phenotype were collected. The frequency of COPD symptoms during the 24 hours (gathered via the night-time, morning and day-time symptoms of COPD questionnaire) and the anxiety and depression levels (via the HADS Scale) were recorded at each visit. Results: A total of 261 CB and 159 EM patients were analyzed. CB patients with ≥1 night-time symptom seemed to be more frequent (51.7%, 41.8% and 41.4% at baseline, 6-month and 12-month follow-up, respectively) than EM (37.7%, 32.1% and 30.2% at study visits) even if no statistical differences were observed at time points between phenotypes (chi-square test p-values presence/absence of night-time symptoms in CB vs EM at study visits >0.0007). In the first 6 months, the frequency of patients with ≥1 night-time symptom decreased of 9.9% in CB and of 5.6% in EM. A clinically relevant decline of DLCO % predicted over 1 year in EM was observed, the mean (SD) being 61.5 (20.8) % at baseline and 59.1 (17.4) % at 12-month follow-up. EM had higher levels of anxiety and depression than CB (median (25th-75th percentile) HADS total score in CB: 7.0 (4.0–13.0) and 7.0 (3.0–12.0), in EM: 9.0 (3.0–14.0) and 9.5 (3.0–14.0) both at baseline and at 6-month follow-up, respectively), considering 1.17 as minimally clinical important difference (MCID) for the total score. Conclusion: EM patients, evaluated in a real-world setting, seem to suffer from a worse clinical condition and health status compared to CB patients, appearing to have “more treatable” traits