Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces

We model a problem motivated by road design as a feasibility problem. Projections onto the constraint sets are obtained, and projection methods for solving the feasibility problem are studied. We present results of numerical experiments which demonstrate the efficacy of projection methods even for challenging nonconvex problems

Jet trails and Mach cones: The interaction of microquasars with the ISM

A sub-set of microquasars exhibit high peculiar velocity with respect to the local standard of rest due to the kicks they receive when being born in supernovae. The interaction between the radio plasma released by microquasar jets from such high-velocity binaries with the ISM must lead to the production of trails and bow shocks similar to what is observed in narrow-angle tailed radio galaxies and pulsar wind nebulae. We present a set of numerical simulations of this interaction that illuminate the long term dynamical evolution and the observational properties of these microquasar bow shock nebulae and trails. We find that this interaction always produces a structure that consists of a bow shock, a trailing neck, and an expanding bubble. Using our simulations to model emission, we predict that the shock surrounding the bubble and the neck should be visible in H{\alpha} emission, the interior of the bubble should be visible in synchrotron radio emission, and only the bow shock is likely to be detectable in X-ray emission. We construct an analytic model for the evolution of the neck and bubble shape and compare this model with observations of X-ray binary SAX J1712.6-3739.Comment: 33 pages, 13 figures, 1 table; Accepted to Ap

Dating of ice cores from Vernagtferner (Austria) with fission products and lead-210

Fission product (90Sr_ 90y, I37CS, total beta) and 2tOPb_210pO activities were measured in core samples from the temperate vernagtferner (3150 m altitude, Oetztal Alps, Austria). The results show that the investigated fission products are transported with water resulting from melting processes, and are sorbed on dust or dirt horizons. These products are, therefore, not suited for dating temperate glaciers. 210Pb is also transported with water and displaced from its original deposition. However, despite large fluctuations, the specific activity of 210Pb decreases with depth, and can be used to estimate accumulation rates and the age of the ice. The average annual accumulation rate amounts to about 80 cm water equivalent, and the deepest sample (81 m i. e. "" 65 m w. e.) was deposited in the beginning of this century. These results agree with data obtained from other observations on this glacier and show that the 210Pb_method is suitable to date temperate glaciers, if the ice cores cover a time interval of about 100 years (i. e. "" 4 half-lives of 210Pb). The surface activity of 210Pb was found to be 5 ± I dpm per kg of ice in agreement with other locations in the Alps and with measurements of fresh snow

Multiple muons in MACRO

An analysis of the multiple muon events in the Monopole Astrophysics and Cosmic Ray Observatory detector was conducted to determine the cosmic ray composition. Particular emphasis is placed on the interesting primary cosmic ray energy region above 2000 TeV/nucleus. An extensive study of muon production in cosmic ray showers has been done. Results were used to parameterize the characteristics of muon penetration into the Earth to the location of a detector

The degenerate C. Neumann system I: symmetry reduction and convexity

The C. Neumann system describes a particle on the sphere S^n under the influence of a potential that is a quadratic form. We study the case that the quadratic form has l+1 distinct eigenvalues with multiplicity. Each group of m_\sigma equal eigenvalues gives rise to an O(m_\sigma)-symmetry in configuration space. The combined symmetry group G is a direct product of l+1 such factors, and its cotangent lift has an Ad^*-equivariant Momentum mapping. Regular reduction leads to the Rosochatius system on S^l, which has the same form as the Neumann system albeit for an additional effective potential. To understand how the reduced systems fit together we use singular reduction to construct an embedding of the reduced Poisson space T^*{S^n}/G into R^{3l+3}$. The global geometry is described, in particular the bundle structure that appears as a result of the superintegrability of the system. We show how the reduced Neumann system separates in elliptical-spherical co-ordinates. We derive the action variables and frequencies as complete hyperelliptic integrals of genus l. Finally we prove a convexity result for the image of the Casimir mapping restricted to the energy surface.Comment: 36 page

Hyperbolicity and Constrained Evolution in Linearized Gravity

Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution). Since computational solution of differential equations introduces almost inevitable errors, it is clearly "more correct" to introduce a scheme which actively maintains the constraints by solution ({\it constrained} evolution). This has shown promise in computational settings, but the analysis of the resulting mixed elliptic hyperbolic method has not been completely carried out. We present such an analysis for one method of constrained evolution, applied to a simple vacuum system, linearized gravitational waves. We begin with a study of the hyperbolicity of the unconstrained Einstein equations. (Because the study of hyperbolicity deals only with the highest derivative order in the equations, linearization loses no essential details.) We then give explicit analytical construction of the effect of initial data setting and constrained evolution for linearized gravitational waves. While this is clearly a toy model with regard to constrained evolution, certain interesting features are found which have relevance to the full nonlinear Einstein equations.Comment: 18 page