The inception of Symplectic Geometry: the works of Lagrange and Poisson during the years 1808-1810

The concept of a symplectic structure first appeared in the works of Lagrange on the so-called "method of variation of the constants". These works are presented, together with those of Poisson, who first defined the composition law called today the "Poisson bracket". The method of variation of the constants is presented using today's mathematical concepts and notations.Comment: Presented at the meeting "Poisson 2008" in Lausanne, July 2008. Published in Letters in Mathematical Physics. 22 page

Sestrins as a therapeutic bridge between ROS and autophagy in cancer

The regulation of Reactive Oxygen Species (ROS) levels and the contribution therein from networks regulating cell metabolism, such as autophagy and the mTOR-dependent nutrient-sensing pathway, constitute major targets for selective therapeutic intervention against several types of tumors, due to their extensive rewiring in cancer cells as compared to healthy cells. Here, we discuss the sestrin family of proteins-homeostatic transducers of oxidative stress, and drivers of antioxidant and metabolic adaptation-as emerging targets for pharmacological intervention. These adaptive regulators lie at the intersection of those two priority nodes of interest in antitumor intervention-ROS control and the regulation of cell metabolism and autophagy-therefore, they hold the potential not only for the development of completely novel compounds, but also for leveraging on synergistic strategies with current options for tumor therapy and classification/stadiation to achieve personalized medicine

The Non-Trapping Degree of Scattering

We consider classical potential scattering. If no orbit is trapped at energy E, the Hamiltonian dynamics defines an integer-valued topological degree. This can be calculated explicitly and be used for symbolic dynamics of multi-obstacle scattering. If the potential is bounded, then in the non-trapping case the boundary of Hill's Region is empty or homeomorphic to a sphere. We consider classical potential scattering. If at energy E no orbit is trapped, the Hamiltonian dynamics defines an integer-valued topological degree deg(E) < 2. This is calculated explicitly for all potentials, and exactly the integers < 2 are shown to occur for suitable potentials. The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that the boundary of Hill's Region in configuration space is either empty or homeomorphic to a sphere. However, in many situations one can decompose a potential into a sum of non-trapping potentials with non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering.Comment: 25 pages, 1 figure Revised and enlarged version, containing more detailed proofs and remark

Interplay between ROS and Autophagy in Cancer and Aging: From Molecular Mechanisms to Novel Therapeutic Approaches

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Linear frictional forces cause orbits to neither circularize nor precess

For the undamped Kepler potential the lack of precession has historically been understood in terms of the Runge-Lenz symmetry. For the damped Kepler problem this result may be understood in terms of the generalization of Poisson structure to damped systems suggested recently by Tarasov[1]. In this generalized algebraic structure the orbit-averaged Runge-Lenz vector remains a constant in the linearly damped Kepler problem to leading order in the damping coeComment: 16 pages. 1 figure, Rewrite for resubmissio

Supersymmetry and discrete transformations of the Dirac operators in Taub-NUT geometry

It is shown that the N=4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N=4 supersymmetry.Comment: 14 pages, latex, no figure

Projective dynamics and classical gravitation

Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a "field of projective forces", we define a law of dynamics such that the position of the particle is a "ray" i.e. a half-line drawn from the origin of V. The impulsion is a bivector whose support is a 2-plane containing the ray. Throwing the particle with a given initial impulsion defines a projective trajectory. It is a curve in the space of rays S(V), together with an impulsion attached to each ray. In the simplest example where the force is identically zero, the curve is a straight line and the impulsion a constant bivector. A striking feature of projective dynamics appears: the trajectories are not parameterized. Among the projective force fields corresponding to a central force, the one defining the Kepler problem is simpler than those corresponding to other homogeneities. Here the thrown ray describes a quadratic cone whose section by a hyperplane corresponds to a Keplerian conic. An original point of view on the hidden symmetries of the Kepler problem emerges, and clarifies some remarks due to Halphen and Appell. We also get the unexpected conclusion that there exists a notion of divergence-free field of projective forces if and only if dim V=4. No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure

Strontium isotopic geochemistry of Pan-African/Brasiliano rocks, Chapada copper deposit, Goiás, Brazil

Das Trägergestein der Chapada Kupferlagerstätte in Goiás ist in metamorpher Hoch-Amphibolitfazies ausgebildet. Gesamtgesteins- und individuelle Mineral Rb-Sr-Isochronen mit einem 87 Sr/ 86 Sr-Initialwert (I) von 0,70414±0.00005 ergeben für die Kristallisation der Vulkanite und Magmatite, die die Metamorphose vordatieren, ein Alter von 561±9 Mio Jahre. Die Datierung der Metamorphose während der panafrikanisch/brasilianischen Orogenese (700 bis 450 Mio Jahre) ergibt 532±1 Mio Jahre. Das Mineral-Isochronenalter eines Dioritkörpers porphyrischen Gefüges, der in die Schiefer intrudierte, beträgt 534±16 Mio Jahre (I=0,70415 ±0,00004). Das Ergebnis niedriger Initial-Werte dieser Isochronen ergibt für die Schiefer bei Chapada ein Ausgangs-Material im Bereich des oberen Mantels oder der tieferen Kruste; diese Ausgangswerte sind charakteristisch für magmatische Tätigkeit in Verbindung mit Inselbögen. Das unterkambrische Alter des Trägergesteins bei Chapada steht im Widerspruch zu dem bisher angenommenen mittelproterozoischen Alter und ist ein Hinweis auf eine mögliche Phase intensiverer pan-afrikanisch/brasilianischer magmatischer Aktivität in Zentral-Brasilien als ursprünglich angenommen wurde. Whole rock and mineral Rb-Sr isochrons from the upper amphibolite grade metamorphic host rocks of the Chapada copper deposit in Goiás, Brazil, yield an age of 561±9 Ma, with a 87 Sr/ 86 Sr initial (I) of 0.70414±0.00005, for the crystallization of pre-metamorphic volcanic and igneous rocks and 532 ±1 Ma for their metamorphism during the Pan-African/Brasiliano orogeny (700-450 Ma). A porphyritic diorite stock which intruded the host schists yields a mineral isochron age of 534±16 Ma (I=0.70415±0.00004). The low I values determined from these isochrons suggest that the host schists at Chapada were derived from upper mantle or lower crust material and are characteristic of magmatic products associated with island arcs. The Early Cambrian age of the host rocks at Chapada contradicts the previously assigned mid-Proterozoic age and suggests that Pan-African/Brasiliano magmatic activity may have been more extensive in central Brazil than previously thought. Les roches qui contiennent le gisement de cuivre de Chapada (Goias, Brésil) sont des schistes cristallins appartenent au facies dupérieur des amphibolites. Dans des volcanites et des roches orthomagmatiques appartenant à cet ensemble, des isochrones Rb-Sr sur roches totales et sur minéraux donnent un âge de 561±9 Ma avec un rapport initial de 0,70414±0,00005 pour la cristallisation pré-métamorphique et un âge de 532±1 Ma pour le métamorphisme, lié à l'orogenèse pan-africaine / brésilienne. Un pluton de diorite porphyrique, qui intrude les schistes cristallins donne, par isochrone sur minéraux, un âge de 534±16 Ma (rapport initial de 0,70415±0,00004). Les valeurs basses des rapports initiaux de ces diverses roches suggèrent que les schistes de Chapada sont dérivés de matériaux mantelliques ou crustaux profonds et présentent les caractères des produits magmatiques associés aux arcs insulaires. Leur âge éo-cambrien, en contradiction avec l'âge mésoprotérozoïque admis jusqu'ici, permet de penser que l'activité magmatique pan-africaine/brésilienne dans le centre du Brésil a été plus intense que ce qu'on croyait jusqu'ici. Вмещающие породы зал ежи меди Chapada в Goiбs представляют собой а мфиболитовую фацию в ысшей степени метаморфизм а. Цельная порода и отд ельные минералы дают для кри сталлизации вулкани тов и магматитов до метаморфизма нача льное значение соотн ошения стронция в 0,70414±0,00005, что раз решает принять их возраст в 561± 9 Мио лет, а возраст мет аморфизма в 532±1 Мио, т. е. отнести его к периоду панафриканского -бра зильского орогенеза / 700 до 450 Мио /. При определении в озраста минералов из диоритных тел порфиритовых тек стур, интрудированны х в сланец, получили дат ы в 534±16 Мио лет / 1 = 0,70415 = 0,00004 /. Такое низкое значе ние исходных величин соотношения изотопов разрешает п редполагать, что материал для Chapada при несен был из мантии, или глубинных регион ов коры. Эти исходные з начения характерны для магма тической деятельнос ти типа островных дуг. Во зраст вмещающих поро д считают средне-протерозойск им; это стоит в противо речии с новейшими данными, п о которым он может быт ь только нижнекембрий ским. Это также указыв ает на то, что фаза пан-африка нско-бразильской маг матической активности в централ ьной Бразилии протек ала, возможно, гораздо инт енсивнее, чем принима ли это до сих пор.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47849/1/531_2005_Article_BF01830182.pd