A Mixed Discontinuous Galerkin Method for Incompressible Magnetohydrodynamics

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous P^3_{k}-P_{k-1} elements whereas the magnetic part of the equations is approximated by discontinuous P^3_{k}-P_{k+1} elements. We carry out a complete a-priori error analysis and prove that the energy norm error is convergent of order O(h^k) in the mesh size h. We also show that the method is able to correctly capture and resolve the strongest magnetic singularities in non-convex polyhedral domains. These results are verified in a series of numerical experiments.

Non-planar double-box, massive and massless pentabox Feynman integrals in the negative-dimensional approach

The negative-dimensional integration method is a technique which can be applied, with success, in usual covariant gauge calculations. We consider three two-loop diagrams: the scalar massless non-planar double-box with six propagators and the scalar pentabox in two cases, where six virtual particles have the same mass, and in the case all of them are massless. Our results are given in terms of hypergeometric functions of Mandelstam variables and also for arbitrary exponents of propagators and dimension D.

Further development of discrete computational techniques for calculation of restricted diffusion propagators in porous media

Magnetic resonance is a well-established tool for structural characterisation of porous media. Features of pore-space morphology can be inferred from NMR diffusion-diffraction plots or the time-dependence of the apparent diffusion coefficient. Diffusion NMR signal attenuation can be computed from the restricted diffusion propagator, which describes the distribution of diffusing particles for a given starting position and diffusion time. We present two techniques for efficient evaluation of restricted diffusion propagators for use in NMR porous-media characterisation. The first is the Lattice Path Count (LPC). Its physical essence is that the restricted diffusion propagator connecting points A and B in time t is proportional to the number of distinct length-t paths from A to B. By using a discrete lattice, the number of such paths can be counted exactly. The second technique is the Markov transition matrix (MTM). The matrix represents the probabilities of jumps between every pair of lattice nodes within a single timestep. The propagator for an arbitrary diffusion time can be calculated as the appropriate matrix power. For periodic geometries, the transition matrix needs to be defined only for a single unit cell. This makes MTM ideally suited for periodic systems. Both LPC and MTM are closely related to existing computational techniques: LPC, to combinatorial techniques; and MTM, to the Fokker-Planck master equation. The relationship between LPC, MTM and other computational techniques is briefly discussed in the paper. Both LPC and MTM perform favourably compared to Monte Carlo sampling, yielding highly accurate and almost noiseless restricted diffusion propagators. Initial tests indicate that their computational performance is comparable to that of finite element methods. Both LPC and MTM can be applied to complicated pore-space geometries with no analytic solution. We discuss the new methods in the context of diffusion propagator calculation in porous materials and model biological tissues.

Wheeler-Feynman absorber revisited: a useful technique to calculate decay rates and lifetimes in small scale optical systems

The Wheeler-Feynman (WF) absorber theory of radiation though no more of interest in explaining self interaction of an electron, can be very useful in today's research in small scale optical systems. The significance of the WF absorber is the use of time-symmetrical solution of Maxwell's equations as opposed to only the retarded solution. The radiative coupling of emitters to nano wires in the near field and change in their lifetimes due to small mode volume enclosures have been elucidated with the retarded solutions before. These solutions have also been shown to agree with quantum electrodynamics, thus allowing for classical electromagnetic approaches in such problems. It is here assumed that the radiative coupling of the emitter with a body is in proportion to its contribution to the classical force of radiative reaction as derived in the WF absorber theory. Representing such nano structures as a partial WF absorber acting on the emitter makes the computations considerably easier than conventional electromagnetic solutions for full boundary conditions.

Isolated singularities of polyharmonic operator in even dimension

We consider the equation Delta(2)u = g(x, u) >= 0 in the sense of distribution in Omega' = Omega\textbackslash {0} where u and -Delta u >= 0. Then it is known that u solves Delta(2)u = g(x, u) + alpha delta(0) - beta Delta delta(0), for some nonnegative constants alpha and beta. In this paper, we study the existence of singular solutions to Delta(2)u = a(x) f (u) + alpha delta(0) - beta Delta delta(0) in a domain Omega subset of R-4, a is a nonnegative measurable function in some Lebesgue space. If Delta(2)u = a(x) f (u) in Omega', then we find the growth of the nonlinearity f that determines alpha and beta to be 0. In case when alpha = beta = 0, we will establish regularity results when f (t) <= Ce-gamma t, for some C, gamma > 0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma). Later, we discuss its analogous generalization for the polyharmonic operator.

Black Holes, Cosmological Solutions, Future Singularities, and Their Thermodynamical Properties in Modified Gravity Theorie

54 p.

Propagators and ridge jumps in a back-arc basin, the West Philippine Basin

We explore the tectono-magmatic processes in the western West Philippine Basin, Philippine Sea Plate, using bathymetric data acquired in 2003 and 2004. The northwestern part of the basin formed through a series of northwestward propagating rifts. We identify at least five sequences of propagating rifts, probably triggered by mantle flow away from the mantle thermal anomaly that is responsible for the origin of the Benham and Urdenata plateaus. Gravitational forces caused by along-axis topographic gradient and a similar to 30 degrees ridge reorientation appear to also be driving the rift propagations. The along-axis mantle flow appears to be reduced and deflected along the Luzon-Okinawa fracture zone, because the spreading system remained stable west of this major fault zone. North-east of the Benham plateau, a left-lateral fracture zone has turned into a NE-SW-trending spreading axis. As a result, a microplate developed at the triple junction.

On pairs of regular foliations in R(3) and singularities of map-germs

We study germs of pairs of codimension one regular foliations in R(3) . We show that the discriminant of the pair determines the topological type of the pair. We also consider various classifications of the singularities of the discriminant.

The light-cone gauge without prescriptions

Feynman integrals in the physical light-cone gauge are more difficult to solve than their covariant counterparts. The difficulty is associated with the presence of unphysical singularities due to the inherent residual gauge freedom in the intermediate boson propagators constrained within this gauge choice. In order to circumvent these non-physical singularities, the headlong approach has always been to call for mathematical devices - prescriptions - some successful and others not. A more elegant approach is to consider the propagator from its physical point of view, that is, an object obeying basic principles such as causality. Once this fact is realized and carefully taken into account, the crutch of prescriptions can be avoided altogether. An alternative, third approach, which for practical computations could dispense with prescriptions as well as avoiding the necessity of careful stepwise consideration of causality, would be of great advantage. and this third option is realizable within the context of negative dimensions, or as it has been coined, the negative dimensional integration method (NDIM).

Massless and massive one-loop three-point functions in negative dimensional approach

In this article we present the complete massless and massive one-loop triangle diagram results using the negative dimensional integration method (NDIM). We consider the following cases: massless internal fields; one massive, two massive with the same mass m and three equal masses for the virtual particles. Our results are given in terms of hypergeometric and hypergeometric-type functions of the external momenta (and masses for the massive cases) where the propagators in the Feynman integrals are raised to arbitrary exponents and the dimension of the space-time is D. Our approach reproduces the known results; it produces other solutions as yet unknown in the literature as well. These new solutions occur naturally in the context of NDIM revealing a promising technique to solve Feynman integrals in quantum field theories.

Reply to Comment on 'validity of Feynman's prescription of disregarding the Pauli principle in intermediate states'

In a recent paper, we raised a question on the validity of Feynman's prescription of disregarding the Pauli principle in intermediate states of perturbation theory. In the preceding Comment, Cavalcanti correctly pointed out that Feynman's prescription is consistent with the exact solution of the model that we used. This means that the Pauli principle does not necessarily apply to intermediate states. We discuss implications of this puzzling aspect.

Validity of Feynman's prescription of disregarding the Pauli principle in intermediate states

Regarding the Pauli principle in quantum field theory and in many-body quantum mechanics, Feynman advocated that Pauli's exclusion principle can be completely ignored in intermediate states of perturbation theory. He observed that all virtual processes (of the same order) that violate the Pauli principle cancel out. Feynman accordingly introduced a prescription, which is to disregard the Pauli principle in all intermediate processes. This ingenious trick is of crucial importance in the Feynman diagram technique. We show, however, an example in which Feynman's prescription fails. This casts doubts on the general validity of Feynman's prescription. [S1050-2947(99)04604-1].