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

Oculofacial alterations in NBAS-SOPH like mutations: Case report

PURPOSE:: To describe the clinical features of a rare case of NBAS-SOPH-like mutations; to emphasize special aspects of the ocular and oro-facial regions. METHODS:: Case report. CASE DESCRIPTION:: We present a 5-year-old girl initially examined for her dysmorphic features, mental delay, strabismus, and high myopia. During the funduscopic examination, we observed optic atrophy with narrow thinned arterioles with the light brown reflex of the central retina. A genetic assessment revealed NBAS-SOPH like mutation. An assessment by a team of orthodontists defined typical characteristics. CONCLUSIONS:: NBAS mutations can also cause complex disease with a broad clinical spectrum ranging from isolated recurrent acute liver failure (RALF) to a multisystemic phenotype. Due to the heterogeneity of the expressions, a multispeciality approach to this situation is recommended

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

drag forces in statistically stationary and homogeneous submerged granular flows

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Traumatic anterior dislocation of a prosthetic knee, from trauma to delayed onset of vascular injury

Anterior dislocation of a prosthetic knee is a rare event. Only few episodes have been described in the past and have never been linked with neurovascular injury. This could lead orthopaedic surgeons to focus on the implant instability factors and underrate other complications. The authors report a case of a patient who experienced popliteal artery dissection 80 hours after traumatic anterior knee dislocation of a previously well-functioning total knee arthroplasty. Given that there is a lack of clinical cases that have established guidelines for management of this problem, the authors have focused on the importance of performing computed tomography angiography in the management of an anterior traumatic knee dislocation of a prosthetic knee

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

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

Biological Prosthesis (Hollow 3D-Printed Titanium Custom-Made Prosthesis and Bone Graft) for Humeral Reconstruction in Pediatric Oncologic Patients: Surgical Indications and Results

This study presents the mid-term outcomes of a novel “biological prosthesis” for pediatric humerus reconstruction after major bone tumor removal. This approach involves a hollow 3D-printed titanium custom-made prosthesis combined with bone grafting. The primary aim was to preserve and revitalize the unaffected autologous proximal or distal humeral stump. Between 2017 and 2021, we treated five pediatric patients (mean age 11.2 years; range 7–17) with humeral bone sarcomas. A one-stage surgical procedure involved tumor resection and implanting a hollow 3D-printed custom-made prosthesis. In two cases, we preserved the proximal humerus; in two, the distal part; and in one, both. Graft materials included homologous bone chips in three cases and free vascularized fibular grafts in two cases. All patients were clinically and radiographically assessed after a mean follow-up of 32.2 months (range of 14–68). No significant complications were observed, and no implant revisions were needed. Osseointegration was evident in all cases within eight months post-surgery; vascular support for the remaining autologous stump was demonstrated in all cases. Our hollow 3D-printed custom-made prosthesis and bone grafting offer the potential for partial or complete articular surface preservation. This approach encourages revascularization of the epiphysis, leading to satisfactory outcomes in humerus reconstruction within the pediatric population

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