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

From Lagrangian to Quantum Mechanics with Symmetries

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schr\"odinger equation is the key that takes classical mechanics into quantum mechanics. Some examples are presented.Comment: To appear in: Proceedings of Symmetries in Science XV, Journal of Physics: Conference Series, (2012

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

Using a mathematical model to evaluate the efficacy of TB control measures.

We evaluated the efficacy of recommended tuberculosis (TB) infection control measures by using a deterministic mathematical model for airborne contagion. We examined the percentage of purified protein derivative conversions under various exposure conditions, environmental controlstrategies, and respiratory protective devices. We conclude that environmental control cannot eliminate the risk for TB transmission during high-risk procedures; respiratory protective devices, and particularly high-efficiency particulate air masks, may provide nearly complete protection if used with air filtration or ultraviolet irradiation. Nevertheless, the efficiency of these control measures decreases as the infectivity of the source case increases. Therefore, administrative control measures (e.g., indentifying and isolating patients with infectious TB) are the most effective because they substantially reduce the rate of infection

Mork and Mindy, Canola Oil and Mustard Gas: The Dilemma of Scientific Illiteracy in Decisions about Food and Health

People often use food to represent and communicate their role in society, or political or ideological beliefs. Food consumption is thus laden with meaning beyond health or nutrition. Multiple audience studies examining perceptions about food/technologies/health have shown that scientific illiteracy and confusion are key to decision making about food

The reaction to nailing or cementing of the femur in rats. A microangiographic and fluorescence study.

Bone reaction to cement and to a cementless stem was studied in the rat femur with histological fluorescence and microangiographic techniques. Periosteal and endosteal apposition, and consequent remodelling, appeared as a reaction to reaming rather than caused by cement or a cementless stem. Every change in bone began with proliferation, progression and orientation of the vessels. Endosteal apposition was absent in cemented femurs because the entire medulla was occupied by the acrylic cement, but remodelling of the subendosteal cortex followed medullary revascularisation which was far advanced after 90 days. In cementless stems, endosteal apposition of primary woven bone and remodelling was the basis for bony ingrowth and anchorage through bony bridges. Our results suggest that the pattern of blood supply is relevant to the structural organisation of mature lamellar bone around the implant. Cemented stems have maximum anchorage and stability as soon as they are inserted, but this decreases with time as revascularisation occurs. Cementless stems can reach maximum integration later after insertion, and revascularisation is less critical because they usually do not fill the canal completely