Lie point symmetries and first integrals: the Kowalevsky top

We show how the Lie group analysis method can be used in order to obtain first integrals of any system of ordinary differential equations. The method of reduction/increase of order developed by Nucci (J. Math. Phys. 37, 1772-1775 (1996)) is essential. Noether's theorem is neither necessary nor considered. The most striking example we present is the relationship between Lie group analysis and the famous first integral of the Kowalevski top.Comment: 23 page

Ermakov's Superintegrable Toy and Nonlocal Symmetries

We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2,R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Analytic Behaviour of Competition among Three Species

We analyse the classical model of competition between three species studied by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the approaches of singularity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations between critical values arising from the approach of dynamical systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic

A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g., motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different Lagrangian formulations.Comment: 17 page

On the Integrability, B\"Acklund Transformation and Symmetry Aspects of a Generalized Fisher Type Nonlinear Reaction-Diffusion Equation

The dynamics of nonlinear reaction-diffusion systems is dominated by the onset of patterns and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized Fisher equation in both (1+1) and (2+1) dimensions. A Painlev\'e singularity structure analysis singles out a special case ($m=2$) as integrable. More interestingly, a B\"acklund transformation is shown to give rise to a linearizing transformation for the integrable case. A Lie symmetry analysis again separates out the same $m=2$ case as the integrable one and hence we report several physically interesting solutions via similarity reductions. Thus we give a group theoretical interpretation for the system under study. Explicit and numerical solutions for specific cases of nonintegrable systems are also given. In particular, the system is found to exhibit different types of travelling wave solutions and patterns, static structures and localized structures. Besides the Lie symmetry analysis, nonclassical and generalized conditional symmetry analysis are also carried out.Comment: 30 pages, 10 figures, to appear in Int. J. Bifur. Chaos (2004

Combined use of coenzyme Q10 and citicoline: A new possibility for patients with glaucoma

Glaucoma is the leading cause of irreversible blindness worldwide. Several risk factors have been involved in the pathogenesis of the disease. By now, the main treatable risk factor is elevated intraocular pressure. Nevertheless, some patients, whose intraocular pressure is considered in the target level, still experience a progression of the disease. Glaucoma is a form of multifactorial ocular neurodegeneration with complex etiology, pathogenesis, and pathology. New evidence strongly suggests brain involvement in all aspects of this disease. This hypothesis and the need to prevent glaucomatous progression led to a growing interest in the pharmacological research of new neuroprotective, non-IOP-lowering, agents. The aim of this paper is to report evidence of the usefulness of Coenzyme Q10 and Citicoline, eventually combined, in the prevention of glaucomatous neurodegeneration

Distrofia Miotônica Tipo 1: Frequência Dos Achados Oftalmológicos

The purpose of the study was to evaluate the frequency of ophthalmologic abnormalities in a cohort of myotonic dystrophy type 1 (DM1) patients and to correlate them with motor function. We reviewed the pathophysiology of cataract and low intraocular pressure (IOP). Method: Patients were included after clinical and laboratory diagnosis and after signed informed consent. They were evaluated by Motor Function Measure scale, Portuguese version (MFM-P) and ophthalmic protocol. Results: We evaluated 42 patients aged 17 to 64 years (mean 40.7 ± 12.5), 22 of which were men. IOP (n = 41) was reduced in all but one. We found cataract or positivity for surgery in 38 (90.48%) and ptosis in 23 (54.76%). These signs but not IOP were significantly correlated with severity of motor dysfunction. Abnormalities in ocular motility and stereopsis were observed. Conclusion: Cataract and ptosis are frequent in DM1 and associated to motor dysfunction. Reduced IOP is also common, but appears not to be related with motor impairment. © 2016, Associacao Arquivos de Neuro-Psiquiatria. All rights reserved.74318318

New strategies for neuroprotection in glaucoma,a disease that affects the central nervous system

Glaucoma is a disease where retinal ganglion cells (RGC) are specifically affected though a number of evidences endorse the hypothesis that glaucoma is a neuro-degenerative disorder of the central nervous system and suggest a possible connection between glaucomatous damage and cerebrovascular alterations. The mechanisms underlying RGC loss are not yet fully known but alterations of the autophagy machinery have been recently proposed as a potential contributing factor as for Alzheimer's disease. Here we review the current literature on new strategies for neuroprotection in glaucoma, focusing on pharmacologic strategies to minimize RGC damage

Effects on muscular activity after surgically assisted rapid palatal expansion: a prospective observational study

The study aims to investigate the modifications in the temporalis and the masseter activity in adult patients before and after SARPE (Surgically Assisted Rapid Palatal Expansion) by measuring electromyographic and electrokinesographic activity. 24 adult patients with unilateral posterior crossbite on the right side were selected from the Orthodontic Department of the University of Milan. Three electromyographic and electrokinesographic surface readings were taken respectively before surgery (T0) and 8 months after surgery (T1). The electromyographic data of both right and left masseter and anterior temporalis muscles were recorded during multiple tests: standardized maximum voluntary contraction (MVC)s, after transcutaneous electrical nerve stimulation (TENS) and at rest. T0 and T1 values were compared with paired Student’s t-test (p < 0.05). Results: Significant differences were found in the activity of right masseter (p = 0.03) and right temporalis (p = 0.02) during clench, in the evaluation of right masseter at rest (p = 0.03), also the muscular activity of masseters at rest after TENS from T0 to T1 (pr = 0.04, pl = 0.04). No significant differences were found in the activity of left masseter (p = 0.41) and left temporalis (p = 0.39) during clench and MVC, in the evaluation of left masseter at rest (p = 0.57) and in the activity during MVC of right masseter (p = 0.41), left masseter (p = 0.34), right temporalis (p = 0.51) and left temporalis (p = 0.77). Results showed that the activity of the masseter and temporalis muscles increased significantly after SARPE during rest and clenching on the side where the cross-bite was treated