Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci

In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to {prove the existence of obstructions arising from} conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries

Oxygen and Sodium Abundances in M13 (NGC 6205) Giants: Linking Globular Cluster Formation Scenarios, Deep Mixing, and Post-RGB Evolution

We present O, Na, and Fe abundances, as well as radial velocities, for 113 red giant branch (RGB) and asymptotic giant branch (AGB) stars in the globular cluster M13. The abundances and velocities are based on spectra obtained with the WIYN-Hydra spectrograph, and the observations range in luminosity from the horizontal branch (HB) to RGB-tip. The results are examined in the context of recent globular cluster formation scenarios. We find that M13 exhibits many key characteristics that suggest its formation and chemical enrichment are well-described by current models. Some of these observations include: the central concentration of O-poor stars, the notable decrease in [O/Fe] (but small increase in [Na/Fe]) with increasing luminosity that affects primarily the "extreme" population, the small fraction of stars with halo-like composition, and the paucity of O-poor AGB stars. In agreement with recent work, we conclude that the most O-poor M13 giants are likely He-enriched and that most (all?) O-poor RGB stars evolve to become extreme HB and AGB-manqu\'e stars. In contrast, the "primordial" and "intermediate" population stars appear to experience standard HB and AGB evolution.Comment: Accepted for publication in The Astrophysical Journal Letters. 18 pages; 3 figures; 1 tabl

Hadronic decays of the (pseudo-)scalar charmonium states ηc\eta_c and χc0\chi_{c0} in the extended Linear Sigma Model

We study the phenomenology of the ground-state (pseudo-)scalar charmonia $\eta_c$ and $\chi_{c0}$ in the framework of a $U(4)_r \times U(4)_l$ symmetric linear sigma model with (pseudo-)scalar and (axial-) vector mesons. Based on previous results for the spectrum of charmonia and the spectrum and (OZI-dominant) strong decays of open charmed mesons, we extend the study of this model to OZI-suppressed charmonia decays. This includes decays into 'ordinary' mesons but also particularly interesting channels with scalar-isoscalar resonances $f_0(1370),\, f_0(1500),\, f_0(1710)$ that may include sizeable contributions from a scalar glueball. We study the variation of the corresponding decay widths assuming different mixings between glueball and quark-antiquark states. We also compute the decay width of the pseudoscalar $\eta_c$ into a pseudoscalar glueball. In general, our results for decay widths are in reasonable agreement with experimental data where available. Order of magnitude predictions for as yet unmeasured states and channels are potentially interesting for BESIII, Belle II, LHCb as well as the future PANDA experiment at the FAIR facility.Comment: 20 pages, 3 figures, 6 tabe

Darboux Integrability of Trapezoidal H4H^{4} and H6H^{6} Families of Lattice Equations II: General Solutions

In this paper we construct the general solutions of two families of quad-equations, namely the trapezoidal $H^{4}$ equations and the $H^{6}$ equations. These solutions are obtained exploiting the properties of the first integrals in the Darboux sense, which were derived in [Gubbiotti G., Yamilov R.I., J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages, arXiv:1608.03506]. These first integrals are used to reduce the problem to the solution of some linear or linearizable non-autonomous ordinary difference equations which can be formally solved

On the rationality of the moduli space of L\"uroth quartics

We prove that the moduli space M_L of L"uroth quartics in P^2, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL_3(CC) is rational, as is the related moduli space of Bateman seven-tuples of points in P^2.Comment: 7 page

Surface plasma resonance in small rare gas clusters by mixing IR and VUV laser pulses

The ionization dynamics of a Xenon cluster with 40 atoms is analyzed under a pum p probe scenario of laser pulses where an infrared laser pulse of 50 fs length follows with a well defined time delay a VUV pulse of the same length and peak intensity. The mechanism of resonant energy absorption due to the coinc idence of the IR laser frequency with the frequency of collective motion of quasi free electrons in the cluster is mapped out by varying the time delay between the pulses

Comparison of four different algorithms for the calculation of radioimmunoassay standard curves

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