How do random Fibonacci sequences grow?

We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<p\le 1$). Our main result is that the exponential growth of $F_n$ for $0<p\le 1$ (linear case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by $$\int_0^\infty \log x d\nu_\alpha (x),$$ where $\alpha$ is an explicit function of $p$ depending on the case we consider, and $\nu_\alpha$ is an explicit probability distribution on \RR_+ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of $p$, since we prove that it is equal to zero for $0<p\le1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative

Phase separation of the Potts model in que square lattice

When the two dimensional q-color Potts model in the square lattice is quenched at zero temperature with Glauber dynamics, the energy decreases in time following an Allen-Cahn power law, and the system converges to a phase with energy higher than the ground state energy after an arbitrary large time when q>4. At low but finite temperature, it cesses to obey the power-law regime and orders after a very long time, which increases with q, and before which it performs a domain growth process which tends to be slower as q increases. We briefly present and comment numerical results on the ordering at nonzero temperature.Comment: 3 pages, 1 figure, proceedings of the "International Workshop on Complex sytems", June 2006 in Santander (Spain

Generalized quasiperiodic Rauzy tilings

We present a geometrical description of new canonical $d$-dimensional codimension one quasiperiodic tilings based on generalized Fibonacci sequences. These tilings are made up of rhombi in 2d and rhombohedra in 3d as the usual Penrose and icosahedral tilings. Thanks to a natural indexing of the sites according to their local environment, we easily write down, for any approximant, the sites coordinates, the connectivity matrix and we compute the structure factor.Comment: 11 pages, 3 EPS figures, final version with minor change

Effects of hyaluronic acid injections on pain and functioning in patients affected by tendinopathies: a narrative review

BACKGROUND: Tendinopathies are overuse tendon injuries showing load-dependant pain, stiffness, weakness of movement in the affected area, and impairment in the movements. The scientific interest on the role of Hyaluronic Acid (HA) for the management of tendinopathies has been increased due to its anti-inflammatory and lubricative properties. OBJECTIVE: To collect evidence regarding the effectiveness and safety of HA injections in reducing pain in patients affected by tendinopathies. METHODS: A scientific literature search was conducted using the PubMed, Medline and PEDro electronic databases. The databases were searched since their inception until July 2021. The search was limited to English language articles. Different combinations of the terms and MeSH terms 'tendinopathy', 'tendinosis', 'tendinitis', 'hyaluronic acid', 'hyaluronate', 'infiltration', 'hyaluronic injections', 'viscosupplementation' connected with various boolean operators were used for other electronic databases. RESULTS: One hundred and one records were identified from the selected databases plus three additional papers identified by the authors through other sources. After removing duplicated papers and title/abstract screening, 19 studies were included in our review (eight papers on shoulder, three on elbow, four on hand, one on knee, and three on ankle). CONCLUSION: The results showed that none of the studies report severe adverse effects and most of them support the use of HA injections in tendinopathies, with a special attention to pain reduction and functional assessment. Further studies are warranted to better investigate effects and methods of administration of HA in tendinopathies

Oxygen–ozone therapy in the rehabilitation field: state of the art on mechanisms of action, safety and effectiveness in patients with musculoskeletal disorders

In recent years, the interest in oxygen–ozone (O2O3) therapy application has considerably increased in the field of rehabilitation. Despite its widespread use in common clinical practice, the biochemical effects of O2O3 are still far from being understood, although its chemical properties seem to play a pivotal role in exerting its positive effects on different pathological conditions. Indeed, the effectiveness of O2O3 therapy might be partly due to the moderate oxidative stress produced by O3 interactions with biological components. O2O3 therapy is widely used as an adjuvant therapeutic option in several pathological conditions characterized by chronic inflammatory processes and immune over‐activation, and most musculoskeletal disorders share these pathophysiological processes. The present comprehensive review depicts the state‐of‐the‐art on the mechanisms of action, safety and effectiveness of O2O3 therapy in the complex scenario of the management of musculoskeletal disorders. Taken together, our findings suggest that O2O3 therapy seems to reduce pain and improve functioning in patients affected by low back pain and knee osteoarthritis, as reported by several studies in the literature. However, to date, further studies are warranted to clearly investigate the therapeutic effects of this promising therapy on other musculoskeletal disorders in the field of rehabilitation

Effect of virtual reality rehabilitation on functional outcomes for return-to-work patients with Parkinson's disease: an umbrella review of systematic reviews

BACKGROUND: Parkinson's disease (PD) is a neurodegenerative disease characterized by loss of substantia nigra neurons with deficiency of dopamine. The main symptoms are tremor, rigidity and bradykinesia. Rehabilitation has an important role in the treatment of this condition and virtual reality (VR) is one of the most recent tools. OBJECTIVE: The purpose of this umbrella review is to evaluate the effectiveness of VR systems on gait control for return to work in patients with PD. METHODS: The electronic search, for reviews and meta-analysis studies that investigated the effectiveness of VR on gait control in PD patients, was performed through December 2021 using the following databases: PubMed, Scopus, PEDro, and Google Scholar. Mesh terms used were: Job integration/reintegration OR return-to-work AND Parkinson's disease AND virtual reality OR exergame. No limit on the year of publication of the article was used. CONCLUSIONS: A total of 14 articles were included in our analysis. The included evidence shows a stride length improvement in patients treated with VR compared to conventional active treatments. No difference was found in walking speed. Also, the included articles show an improvement on various measures of balance, motor function and severity of PD motor symptoms. In addition, the literature shows an improvement in the quality of life and neuropsychiatric symptoms in patients undergoing VR rehabilitation training. RESULTS: he results of our study suggest that VR rehabilitation improves gait performance, particularly stride length, thus being able to provide an improvement in the quality of life and a more effective return to work training in patients with PD

Generalized Inverse Participation Numbers in Metallic-Mean Quasiperiodic Systems

From the quantum mechanical point of view, the electronic characteristics of quasicrystals are determined by the nature of their eigenstates. A practicable way to obtain information about the properties of these wave functions is studying the scaling behavior of the generalized inverse participation numbers $Z_q \sim N^{-D_q(q-1)}$ with the system size $N$. In particular, we investigate $d$-dimensional quasiperiodic models based on different metallic-mean quasiperiodic sequences. We obtain the eigenstates of the one-dimensional metallic-mean chains by numerical calculations for a tight-binding model. Higher dimensional solutions of the associated generalized labyrinth tiling are then constructed by a product approach from the one-dimensional solutions. Numerical results suggest that the relation $D_q^{d\mathrm{d}} = d D_q^\mathrm{1d}$ holds for these models. Using the product structure of the labyrinth tiling we prove that this relation is always satisfied for the silver-mean model and that the scaling exponents approach this relation for large system sizes also for the other metallic-mean systems.Comment: 7 pages, 3 figure

Non-surgical and rehabilitative interventions in patients with frozen shoulder: umbrella review of systematic reviews

Background: Frozen shoulder (FS) is a painful condition characterized by progressive loss of shoulder function with passive and active range of motion reduction. To date, there is still no consensus regarding its rehabilitative treatment for pain management. Purpose: The aim of this umbrella review of systematic reviews was to analyze the literature, investigating the effects of non-surgical and rehabilitative interventions in patients suffering from FS. Patients and Methods: A review of the scientific literature was carried out from 2010 until April 2020 using the following search databases: PubMed, Medline, PEDro, Scopus and Cochrane Library of Systematic Reviews. A combination of terms was used for the search: frozen shoulder OR adhesive capsulitis AND systematic review OR meta-analysis AND rehabilitation NOT surgery NOT surgical intervention. We included systematic reviews that specifically dealt with adults with FS, treated with non-surgical approaches. All the systematic reviews and meta-analyses included in the study that met the inclusion criteria were assessed using the Assessment of Multiple Systematic Reviews as a quality assessment tool. Results: Out of 49 studies, only 14 systematic reviews respected the eligibility criteria and were included in this study. Their results showed an important heterogeneity of the studies and all of them agree on the lack of high-quality scientific work to prove unequivocally which rehabilitative treatment is better than the other. Due to this lack of gold standard criteria, there may be also a heterogeneity in the diagnosis of the reviews analyzed. Conclusion: Non-surgical and rehabilitative interventions are undoubtedly effective in treating FS, but there is no evidence that one approach is more effective than the other regarding the methods reported. Future high-quality RCTs are needed to standardize the treatment modalities of each physiotherapy intervention to provide strong recommendations in favor

Design S-N curves for old Portuguese and French riveted bridges connection based on statistical analyses

Maintenance of ancient road and railway metallic bridges has become a major concern for governmental agencies in the past few decades. Indeed, since the construction of these structures, between the end of the 19th century and the beginning of the 20th century, traffic conditions have evolved, both in weight and frequency. In the purpose to assess the remaining life of old metallic bridges, some critical structural details have been identified and associated to S-N curves in order to be used in damage estimation (using Palmgren-Miner’s rule for cumulative damage, for example). These constructional details are described by design rules of several European and North American standards, such as the Eurocode 3, BS 5400 and AASHTO standards. The particularity of ancient bridges is that hot riveted assemblies, commonly used for their construction, are not represented in most construction standards. Further experiences on the matter by numerous research teams have suggested detail category C71 from the Eurocode 3 as appropriate. In this paper, experimental data from double shear assemblies manufactures from three different metallic ancient bridges is used to identify, through a statistical analysis, the S-N curves that best fit this constructional detail. Portuguese and French puddled iron bridges were considered.The authors of this paper thank the National Society of French Railways and the SciTech - Science and Technology for Competitive and Sustainable Industries, R&D project NORTE-01-0145-FEDER-000022 cofinanced by Programa Operacional Regional do Norte ("NORTE2020"), through Fundo Europeu de Desenvolvimento Regional (FEDER) for their collaboration and support during this research works. The authors also acknowledge the Portuguese Science Foundation (FCT) for the financial support through the post-doctoral grant SFRH/BPD/107825/2015