The determination of the apsidal angles and Bertrand's theorem

We derive an expression for the determination of the apsidal angles that holds good for arbitrary central potentials. Then we discuss under what conditions the apsidal angles remain independent of the mechanical energy and angular momentum in the central force problem. As a consequence, an alternative and non-perturbative proof of Bertrand's theorem is obtained.Comment: Latex file, one figure; submitted for publicatio

Thermodynamics of Abelian Gauge Fields in Real Hyperbolic Spaces

We work with $N-$dimensional compact real hyperbolic space $X_{\Gamma}$ with universal covering $M$ and fundamental group $\Gamma$. Therefore, $M$ is the symmetric space $G/K$, where $G=SO_1(N,1)$ and $K=SO(N)$ is a maximal compact subgroup of $G$. We regard $\Gamma$ as a discrete subgroup of $G$ acting isometrically on $M$, and we take $X_{\Gamma}$ to be the quotient space by that action: $X_{\Gamma}=\Gamma\backslash M = \Gamma\backslash G/K$. The natural Riemannian structure on $M$ (therefore on $X$) induced by the Killing form of $G$ gives rise to a connection $p-$form Laplacian ${\frak L}_p$ on the quotient vector bundle (associated with an irreducible representation of K). We study gauge theories based on abelian $p-$forms on the real compact hyperbolic manifold $X_{\Gamma}$. The spectral zeta function related to the operator ${\frak L}_p$, considering only the co-exact part of the $p-$forms and corresponding to the physical degrees of freedom, can be represented by the inverse Mellin transform of the heat kernel. The explicit thermodynamic fuctions related to skew-symmetric tensor fields are obtained by using the zeta-function regularization and the trace tensor kernel formula (which includes the identity and hyperbolic orbital integrals). Thermodynamic quantities in the high and low temperature expansions are calculated and new entropy/energy ratios established.Comment: Six pages, Revtex4 style, no figures; small typo correcte

Bacterial lipopolysaccharide induces apoptosis in the trout ovary

BACKGROUND: In mammals it is well known that infections can lead to alterations in reproductive function. As part of the innate immune response, a number of cytokines and other immune factors is produced during bacterial infection or after treatment with lipopolysaccharide (LPS) and acts on the reproductive system. In fish, LPS can also induce an innate immune response but little is known about the activation of the immune system by LPS on reproduction in fish. Therefore, we conducted studies to examine the in vivo and in vitro effects of lipopolysaccharide (LPS) on the reproductive function of sexually mature female trout. METHODS: In saline- and LPS -injected brook trout, we measured the concentration of plasma steroids as well as the in vitro steroidogenic response (testosterone and 17alpha-hydroxyprogesterone) of ovarian follicles to luteinizing hormone (LH), the ability of 17alpha,20beta-dihydroxy-4-pregnen-3-one to induce germinal vesicle breakdown (GVBD) in vitro, and that of epinephrine to stimulate follicular contraction in vitro. We also examined the direct effects of LPS in vitro on steroid production, GVBD and contraction in brook trout ovarian follicles. The incidence of apoptosis was evaluated by TUNEL analysis. Furthermore, we examined the gene expression pattern in the ovary of saline- and LPS-injected rainbow trout by microarray analysis. RESULTS: LPS treatment in vivo did not affect plasma testosterone concentration or the basal in vitro production of steroids, although a small but significant potentiation of the effects of LH on testosterone production in vitro was observed in ovarian follicles from LPS-treated fish. In addition, LPS increased the plasma concentration of cortisol. LPS treatment in vitro did not affect the basal or LH-stimulated steroid production in brook trout ovarian follicles. In addition, we did not observe any effects of LPS in vivo or in vitro on GVBD or follicular contraction. Therefore, LPS did not appear to impair ovarian steroid production, oocyte final maturation or follicular contraction under the present experimental conditions. Interestingly, LPS administration in vivo induced apoptosis in follicular cells, an observation that correlated with changes in the expression of genes involved in apoptosis, as evidenced by microarray analysis. CONCLUSION: These results indicate that female trout are particularly resistant to an acute administration of LPS in terms of ovarian hormone responsiveness. However, LPS caused a marked increase in apoptosis in follicular cells, suggesting that the trout ovary could be sensitive to the pro-apoptotic effects of LPS-induced inflammatory cytokines