A discontinuous Galerkin method for the Vlasov-Poisson system

A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.Comment: To appear in Journal for Computational Physics, 2011. 63 pages, 86 figure

On Krein-like theorems for noncanonical Hamiltonian systems with continuous spectra: application to Vlasov-Poisson

The notions of spectral stability and the spectrum for the Vlasov-Poisson system linearized about homogeneous equilibria, f_0(v), are reviewed. Structural stability is reviewed and applied to perturbations of the linearized Vlasov operator through perturbations of f_0. We prove that for each f_0 there is an arbitrarily small delta f_0' in W^{1,1}(R) such that f_0+delta f_0$is unstable. When$f_0$ is perturbed by an area preserving rearrangement, f_0 will always be stable if the continuous spectrum is only of positive signature, where the signature of the continuous spectrum is defined as in previous work. If there is a signature change, then there is a rearrangement of f_0 that is unstable and arbitrarily close to f_0 with f_0' in W^{1,1}. This result is analogous to Krein's theorem for the continuous spectrum. We prove that if a discrete mode embedded in the continuous spectrum is surrounded by the opposite signature there is an infinitesimal perturbation in C^n norm that makes f_0 unstable. If f_0 is stable we prove that the signature of every discrete mode is the opposite of the continuum surrounding it.Comment: Submitted to the journal Transport Theory and Statistical Physics. 36 pages, 12 figure

Galaxy Luminosity Functions from Deep Spectroscopic Samples of Rich Clusters

Using a new spectroscopic sample and methods accounting for spectroscopic sampling fractions that vary in magnitude and surface brightness, we present R-band galaxy luminosity functions (GLFs) for six nearby galaxy clusters with redshifts 4000 < cz < 20000 km/s and velocity dispersions 700 < sigma < 1250 km/s. In the case of the nearest cluster, Abell 1060, our sample extends to M_R=-14 (7 magnitudes below M*), making this the deepest spectroscopic determination of the cluster GLF to date. Our methods also yield composite GLFs for cluster and field galaxies to M_R=-17 (M*+4), including the GLFs of subsamples of star forming and quiescent galaxies. The composite GLFs are consistent with Schechter functions (M*_R=-21.14^{+0.17}_{-0.17}, alpha=-1.21^{+0.08}_{-0.07} for the clusters, M*_R=-21.15^{+0.16}_{-0.16}, alpha=-1.28^{+0.12}_{-0.11} for the field). All six cluster samples are individually consistent with the composite GLF down to their respective absolute magnitude limits, but the GLF of the quiescent population in clusters is not universal. There are also significant variations in the GLF of quiescent galaxies between the field and clusters that can be described as a steepening of the faint end slope. The overall GLF in clusters is consistent with that of field galaxies, except for the most luminous tip, which is enhanced in clusters versus the field. The star formation properties of giant galaxies are more strongly correlated with the environment than those of fainter galaxies.Comment: 53 pages, 8 figures, 1 ASCII table; accepted for publication in Ap

Mode signature and stability for a Hamiltonian model of electron temperature gradient turbulence

Stability properties and mode signature for equilibria of a model of electron temperature gradient (ETG) driven turbulence are investigated by Hamiltonian techniques. After deriving the infinite families of Casimir invariants, associated with the noncanonical Poisson bracket of the model, a sufficient condition for stability is obtained by means of the Energy-Casimir method. Mode signature is then investigated for linear motions about homogeneous equilibria. Depending on the sign of the equilibrium "translated" pressure gradient, stable equilibria can either be energy stable, i.e.\ possess definite linearized perturbation energy (Hamiltonian), or spectrally stable with the existence of negative energy modes (NEMs). The ETG instability is then shown to arise through a Kre\u{\i}n-type bifurcation, due to the merging of a positive and a negative energy mode, corresponding to two modified drift waves admitted by the system. The Hamiltonian of the linearized system is then explicitly transformed into normal form, which unambiguously defines mode signature. In particular, the fast mode turns out to always be a positive energy mode (PEM), whereas the energy of the slow mode can have either positive or negative sign

Decay of the free-theory vacuum of scalar field theory in de Sitter spacetime in the interaction picture

A free-theory vacuum state of an interacting field theory, e.g. quantum gravity, is unstable at tree level in general due to spontaneous emission of Fock-space particles in any spacetime with no global timelike Killing vectors, such as de Sitter spacetime, in the interaction picture. As an example, the rate of spontaneous emission of Fock-space particles is calculated in phi^4 theory in de Sitter spacetime. It is possible that this apparent spontaneous emission does not correspond to any physical processes because the states are not evolved by the true Hamiltonian in the interaction picture. Nevertheless, the constant spontaneous emission of Fock-space particles in the interaction picture clearly demonstrates that the in- and out-vacuum states are orthogonal to each other as emphasized by Polyakov and that the in-out perturbation theory, which presupposes some overlap between these two vacuum states, is inadequate. Other possible implications of apparent vacuum instability of this kind in the interaction picture are also discussed.Comment: title changed, 7 page

Geometric invariant theory of syzygies, with applications to moduli spaces

We define syzygy points of projective schemes, and introduce a program of studying their GIT stability. Then we describe two cases where we have managed to make some progress in this program, that of polarized K3 surfaces of odd genus, and of genus six canonical curves. Applications of our results include effectivity statements for divisor classes on the moduli space of odd genus K3 surfaces, and a new construction in the Hassett-Keel program for the moduli space of genus six curves.Comment: v1: 23 pages, submitted to the Proceedings of the Abel Symposium 2017, v2: final version, corrects a sign error and resulting divisor class calculations on the moduli space of K3 surfaces in Section 5, other minor changes, In: Christophersen J., Ranestad K. (eds) Geometry of Moduli. Abelsymposium 2017. Abel Symposia, vol 14. Springer, Cha

Cross-Protection against ALS-Inhibiting Herbicides in Non-Transgenic Switchgrass Selected for Imazapic Resistance

Switchgrass (Panicum virgatum L.) is a North American native warm-season perennial grass that is a cornerstone species in conservation systems and can also be utilized as a high-quality forage. ‘Tusca’ is a cultivar of lowland switchgrass selected from ‘Alamo’ (USDA NRCS, Knox City, TX) for resistance to the herbicide imazapic. This study was conducted to determine if selection in Tusca conferred cross-protection to other ALS-inhibiting herbicides. Five ALS-inhibiting herbicides, including imazapic (IPIC), imazamox (IMOX), imazapyr (IPYR), imazethapyr (ITHR), and metsulfuron methyl (MSUL), were tested on Tusca, Alamo, and wildtype johnsongrass (Sorghum halepense L. (Pers.)) at five rates (25, 50, 75, 100, and 125% of the recommended label rate) plus an untreated control, under laboratory (PRE) and greenhouse (POST) conditions. Alamo was used as a negative control and johnsongrass was used as a reference species to confirm efficacy of herbicide treatments. Six replications of 25 seed of both switchgrass cultivars and johnsongrass were screened for response to herbicide treatment at germination, as well as the 3-leaf stage. Mean germination percentage for untreated Tusca, Alamo, and johnsongrass were 71.5, 24.8, and 40.8, respectively. Compared to controls, mean germination percentage of Tusca remained \u3e 50% at all rates of ITHR and IMOX. Alamo exposed to ITHR had \u3c 25% germination and \u3c 50% for IMOX. While Tusca shows some improved resistance to IPIC at germination, greater resistance was found to ITHR, IPYR, and IMOX, whereas mean germination percentage of Alamo was significantly reduced by all treatments