Small oscillations and the Heisenberg Lie algebra

The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems for quadratic Hamiltonians of $\mathbb R^{2n}$ on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra $\mathfrak g$ that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that one on $\mathbb R^{2n}$. This system is a Lax pair equation whose solution can be computed with help of the Adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits in $\mathfrak g$ is related to the commutativity of a family of derivations of the 2n+1-dimensional Heisenberg Lie algebra $\mathfrak h_n$. Therefore the complete integrability is related to the existence of an n-dimensional abelian subalgebra of certain derivations in $\mathfrak h_n$. For instance, the motion of n-uncoupled harmonic oscillators near an equilibrium position can be described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of the AKS schem

Reproductive biology of the biofuel plant Jatropha curcas in its center of origin

In this work, we studied the main characteristics of flowering, reproductive system and diversity of pollinators for the biofuel plant Jatropha curcas (L.) in a site of tropical southeastern Mexico, within its center of origin. The plants were monoecious with inflorescences of unisexual flowers. The male flowers produced from 3062–5016 pollen grains (266–647 per anther). The plants produced fruits with both geitonogamy and xenogamy, although insect pollination significantly increased the number and quality of fruits. A high diversity of flower visiting insects (36 species) was found, of which nine were classified as efficient pollinators. The native stingless bees Scaptotrigona mexicana (Guérin-Meneville) and Trigona (Tetragonisca) angustula (Latreille) were the most frequent visitors and their presence coincided with the hours when the stigma was receptive. It is noteworthy that the female flowers open before the male flowers, favoring xenogamy, which may explain the high genetic variability reported in J. curcas for this region of the world

Oleosomas de semillas de <em>Jatropha curcas</em> L. como estimadores de diversidad en poblaciones del Sur de México

In this work, we used oleosome data to estimate the variability in eight <em>J. curcas</em> populations of Southern Mexico, using a multivariate approach. We found spherical oleosomes with diameters between 2.18 μm and 4.15 μm, which have two types of spatial arrangements: clustered and dispersed. A discriminant analysis showed that populations with more differentiation are Oaxaca and Veracruz, while Isthmus was the most diverse. The most informative variables for their contribution to the discriminant functions were oleosome size and number of oleosomes·μL of tissue⁻¹. A heritability study with seeds harvested in two consecutive years showed that the variables oleosomes·μL of tissue⁻¹ and oleosomes per cell have high broad sense heritabilities (h²_bs=94.3% and h²_bs=92.2%, respectively). A cluster analysis revealed high levels of dissimilarity between individuals. Correlation was found between the ratio oleosomes/tissue (volume/volume) and total oil content of the seed. The identification of genotypes with contrasting characteristics could provide the basis for future studies of the inheritance of characters in <em>J. curcas</em>.<br><br>Se estudiaron los caracteres de oleosomas de la planta para biocombustibles <em>Jatropha curcas</em> L. y se estimó la diversidad de ocho poblaciones del sur de México. Se encontraron oleosomas esféricos, con diámetros entre 2.18 μm y 4.15 μm, presentando dos tipos de arreglos espaciales: agrupados y dispersos. Un análisis discriminante mostró que las poblaciones más diferenciadas del resto son Oaxaca y Veracruz, mientras que Istmo fue la más diversa. Las variables más informativas por su contribución a las funciones discriminantes fueron tamaño de oleosoma y número de oleosomas·μL de tejido⁻¹. En un estudio de heredabilidad con semillas cosechadas en dos años consecutivos se encontró que las variables oleosomas·μL de tejido⁻¹ y oleosomas por célula tienen muy altas heredabilidades en sentido amplio (h²_bs=94.3% y h²_bs=92.2%, respectivamente). Un análisis de conglomerados reveló altos niveles de disimilitud entre individuos. Los análisis de asociación de variables encontraron correlación entre la relación porcentual oleosomas/tejido (volumen/volumen) y el contenido total de aceite de la semilla