Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs

Several recently developed multisymplectic schemes for Hamiltonian PDEs have been shown to preserve associated local conservation laws and constraints very well in long time numerical simulations. Backward error analysis for PDEs, or the method of modified equations, is a useful technique for studying the qualitative behavior of a discretization and provides insight into the preservation properties of the scheme. In this paper we initiate a backward error analysis for PDE discretizations, in particular of multisymplectic box schemes for the nonlinear Schrodinger equation. We show that the associated modified differential equations are also multisymplectic and derive the modified conservation laws which are satisfied to higher order by the numerical solution. Higher order preservation of the modified local conservation laws is verified numerically.Comment: 12 pages, 6 figures, accepted Math. and Comp. Simul., May 200

Towards a spin foam model description of black hole entropy

We propose a way to describe the origin of black hole entropy in the spin foam models of quantum gravity. This stimulates a new way to study the relation of spin foam models and loop quantum gravity.Comment: 5 pages, 1 figur

Observables in 3-dimensional quantum gravity and topological invariants

In this paper we report some results on the expectation values of a set of observables introduced for 3-dimensional Riemannian quantum gravity with positive cosmological constant, that is, observables in the Turaev-Viro model. Instead of giving a formal description of the observables, we just formulate the paper by examples. This means that we just show how an idea works with particular cases and give a way to compute 'expectation values' in general by a topological procedure.Comment: 24 pages, 47 figure

Feynman diagams coupled to three-dimensional quantum gravity

A framework for quantum field theory coupled to three-dimensional quantum gravity is proposed. The coupling with quantum gravity regulates the Feynman diagrams. One recovers the usual Feynman amplitudes in the limit as the cosmological constant tends to zero.Comment: 7 pages. v2: minor corrections, added re

Phosphorus release kinetics in a soil amended with biosolids and vermicompost

Wastewater biosolids are large potential sources of macronutrients for agriculture, conservation and restoration of soils; there are, however, few studies on phosphorus (P) release in soils amended with biosolids. Biosolids and vermicomposted biosolids were tested in concentrations (5–30 g amendment kg-1 soil) equivalent to 18–100 Mg ha-1. Desorption of P was determined by successive extractions for 65 days. Soil P was low, and biosolid and vermicompost addition released 8 and 6 times more P, respectively, than soil alone. To describe the release of P, zero-, first- and second-order equations, simple Elovich and power functions and the parabolic diffusion lawwere compared based on their coefficient of determination (r2) and standard error (SE). In all treatments, the power function and especially the parabolic diffusion law were the best fit, with 0.898–0.996 r2 and 0.022–0.732 SE. The general behavior of the kinetic parameters mostly depended on the amendment doses. Eutrophication posited to start beyond 16 mg P kg-1 soil was more likely allayed by a maximum vermicompost dose of 50 Mg ha-1, higher than the 36 Mg ha-1 maximum biosolid dose. The higher vermicompost P addition and lower P release could favor gradual and longer-term P absorption by plants and may reduce leaching or runoff P losses

56Ni dredge-up in the type IIp Supernova 1995V

We present contemporary infrared and optical spectra of the plateau type II SN 1995V in NGC 1087 covering four epochs, approximately 22 to 84 days after shock breakout. The data show, for the first time, the infrared spectroscopic evolution during the plateau phase of a typical type II event. In the optical region P Cygni lines of the Balmer series and of metals lines were identified. The infrared (IR) spectra were largely dominated by the continuum, but P Cygni Paschen lines and Brackett gamma lines were also clearly seen. The other prominent IR features are confined to wavelengths blueward of 11000 \AA and include Sr II 10327, Fe II 10547, C I 10695 and He I 10830 \AA. We demonstrate the presence of He I 10830 \AA on days 69 and 85. The presence of this line at such late times implies re-ionisation. A likely re-ionising mechanism is gamma-ray deposition following the radioactive decay of 56Ni. We examine this mechanism by constructing a spectral model for the He I 10830 \AA line based on explosion model s15s7b2f of Weaver & Woosley (1993). We find that this does not generate the observed line owing to the confinement of the 56Ni to the central zones of the ejecta. In order to reproduce the He I line, it was necessary to introduce additional upward mixing of the 56Ni, with 10^{-5} of the total nickel mass reaching above the helium photosphere. In addition, we argue that the He I line-formation region is likely to have been in the form of pure helium clumps in the hydrogen envelope.Comment: Accepted for publication in MNRAS, 32 pages including 11 figures (uses psfig.sty - included

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

The cubic nonlinear Schrödinger (NLS) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The nonlinear spectrum of the associated Lax pair reveals topological properties of the NLS phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete. © 2006 Elsevier Inc. All rights reserved

Hidden Quantum Gravity in 3d Feynman diagrams

In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results.Comment: 35 pages, 4 figures, some comments adde

Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory

We study the no gravity limit G_{N}-> 0 of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the G_{N} expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagatorsComment: 46 pages, the wrong file was first submitte

The Hilbert space of Chern-Simons theory on the cylinder. A Loop Quantum Gravity approach

As a laboratory for loop quantum gravity, we consider the canonical quantization of the three-dimensional Chern-Simons theory on a noncompact space with the topology of a cylinder. Working within the loop quantization formalism, we define at the quantum level the constraints appearing in the canonical approach and completely solve them, thus constructing a gauge and diffeomorphism invariant physical Hilbert space for the theory. This space turns out to be infinite dimensional, but separable.Comment: Minor changes and some references added. Latex, 16 pages, 1 figur