Implementation of higher-order absorbing boundary conditions for the Einstein equations

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition B_L of order L=l yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions B_L with L<l, which include the widely used freezing-Psi_0 boundary condition that imposes the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in Class. Quantum Grav

Explicit solution of the linearized Einstein equations in TT gauge for all multipoles

We write out the explicit form of the metric for a linearized gravitational wave in the transverse-traceless gauge for any multipole, thus generalizing the well-known quadrupole solution of Teukolsky. The solution is derived using the generalized Regge-Wheeler-Zerilli formalism developed by Sarbach and Tiglio.Comment: 9 pages. Minor corrections, updated references. Final version to appear in Class. Quantum Gra

Regularity of the Einstein Equations at Future Null Infinity

When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal factor and thus appear to be ill-behaved at this (exterior) boundary. In this article however we show, through an enforcement of the Hamiltonian and momentum constraints to the needed order in a Taylor expansion, that these apparently singular terms are not only regular at the boundary but can in fact be explicitly evaluated there in terms of conformally regular geometric data. Though we employ a rather rigidly constrained and gauge fixed formulation of the field equations, we discuss the extent to which we expect our results to have a more 'universal' significance and, in particular, to be applicable, after minor modifications, to alternative formulations.Comment: 43 pages, no figures, AMS-TeX. Minor revisions, updated to agree with published versio

Analysis of fixed-point and floating-point arithmetic representations’ impact on synthesized area of a digital integrated circuit

Abstract. This thesis compared fixed-point and floating-point representations, using signal-to-quantization-noise-ratio (SQNR) and synthesized area as key comparison methods. Good-enough SQNR was set to 40 dB, and the goal was to choose area that was as small as possible, but still had sufficient dynamic range (DR), and also fulfilled the SQNR requirement. Quantization models for both representations were implemented with Matlab. For examination of the SQNR, an algorithm was chosen and aforementioned quantization models were added inside it. The chosen algorithm was memory-based 64-point FFT, implemented with radix-2 butterfly. The performance drop inside algorithm caused by arithmetic representation quantization was examined using SQNR. To be able to calculate the error value, a reference model was implemented, and that was done using FFT-function of Matlab. When SQNR-analysis had been executed, synthesis was run for arithmetic operation models, for area and power estimate calculation. From these results, a conclusion of impact on area of FXP and FLP on different FFT models was done and a superiority comparison was possible.Kiinteän pilkun luvun ja liukuvan pilkun luvun aritmeettisten esitystapojen vaikutusten analysointi digitaalisen mikropiirin syntetisoituun pinta-alaan. Tiivistelmä. Tässä työssä vertailtiin kiinteän pilkun lukuja ja liukuvan pilkun lukuja, käyttäen tärkeimpinä vertailuparametreina signaalikvantisointikohinasuhdetta (SQNR) sekä synteesistä saatavaa pinta-alaa. SQNR tavoitearvoksi asetettiin 40 dB ja tavoitteena oli valita mahdollisimman pieni pinta-ala, jolla vielä saavutettiin tarpeeksi suuri dynaaminen alue (DR) ja SQNR tavoite täyttyi. SQNR:n laskentaan tarvittiin molemmille aritmeettisille esitystavoille kvantisointimallit, jotka tehtiin Matlab-ohjelmalla. Lopulta kvantisointikohinan tarkempaan tarkasteluun valittiin algoritmi, jonka sisälle edellä mainitut kvantisointimallit asetettiin. Valittu algoritmi oli muistipohjainen 64-näytteinen FFT, joka on toteutettu radix-2 perhoslaskennalla. Algoritmin sisällä tapahtuvaa aritmeettisesta esitystavasta johtuvaa suorituskyvyn muutosta tutkittiin SQNR:n avulla. Jotta virhe voitiin laskea, myös referenssimalli täytyi implementoida, ja siihen käytettiin Matlabin valmista FFT-funktiota. Kun SQNR-analyysi oli suoritettu, ajettiin aritmeettisille operaatio malleille synteesit, joista voitiin laskea algoritmin vaatima pinta-ala. Näistä tuloksista voitiin yhteenvetää liukuvan pilkun ja kiinteän pilkun lukujen vaikutukset FFT mallien pinta-aloihin, ja siten tehdä paremmuusvertailua niiden välillä

An axisymmetric evolution code for the Einstein equations on hyperboloidal slices

We present the first stable dynamical numerical evolutions of the Einstein equations in terms of a conformally rescaled metric on hyperboloidal hypersurfaces extending to future null infinity. Axisymmetry is imposed in order to reduce the computational cost. The formulation is based on an earlier axisymmetric evolution scheme, adapted to time slices of constant mean curvature. Ideas from a previous study by Moncrief and the author are applied in order to regularize the formally singular evolution equations at future null infinity. Long-term stable and convergent evolutions of Schwarzschild spacetime are obtained, including a gravitational perturbation. The Bondi news function is evaluated at future null infinity.Comment: 21 pages, 4 figures. Minor additions, updated to agree with journal versio

Biomass production and feeding value of whole-crop cereal-legume-silages

In eastern Finland, several mixtures of spring wheat, spring barley, spring oats and/or rye with vetches and/or peas were evaluated in field experiments in 2005-2007 for their dry matter procuction, crude protein concentration and digestibility using three different harvesting times

Testing outer boundary treatments for the Einstein equations

Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a `reference' numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests use a first-order generalized harmonic formulation of the Einstein equations. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman-Penrose scalar Psi_0, and control gauge reflections.Comment: Modified to agree with version accepted for publication in Class. Quantum Gra

Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations

This paper is concerned with the initial-boundary value problem for the Einstein equations in a first-order generalized harmonic formulation. We impose boundary conditions that preserve the constraints and control the incoming gravitational radiation by prescribing data for the incoming fields of the Weyl tensor. High-frequency perturbations about any given spacetime (including a shift vector with subluminal normal component) are analyzed using the Fourier-Laplace technique. We show that the system is boundary-stable. In addition, we develop a criterion that can be used to detect weak instabilities with polynomial time dependence, and we show that our system does not suffer from such instabilities. A numerical robust stability test supports our claim that the initial-boundary value problem is most likely to be well-posed even if nonzero initial and source data are included.Comment: 27 pages, 4 figures; more numerical results and references added, several minor amendments; version accepted for publication in Class. Quantum Gra

Structure alignment based on coding of local geometric measures

BACKGROUND: A structure alignment method based on a local geometric property is presented and its performance is tested in pairwise and multiple structure alignments. In this approach, the writhing number, a quantity originating from integral formulas of Vassiliev knot invariants, is used as a local geometric measure. This measure is used in a sliding window to calculate the local writhe down the length of the protein chain. By encoding the distribution of writhing numbers across all the structures in the protein databank (PDB), protein geometries are represented in a 20-letter alphabet. This encoding transforms the structure alignment problem into a sequence alignment problem and allows the well-established algorithms of sequence alignment to be employed. Such geometric alignments offer distinct advantages over structural alignments in Cartesian coordinates as it better handles structural subtleties associated with slight twists and bends that distort one structure relative to another. RESULTS: The performance of programs for pairwise local alignment (TLOCAL) and multiple alignment (TCLUSTALW) are readily adapted from existing code for Smith-Waterman pairwise alignment and for multiple sequence alignment using CLUSTALW. The alignment algorithms employed a blocked scoring matrix (TBLOSUM) generated using the frequency of changes in the geometric alphabet of a block of protein structures. TLOCAL was tested on a set of 10 difficult proteins and found to give high quality alignments that compare favorably to those generated by existing pairwise alignment programs. A set of protein comparison involving hinged structures was also analyzed and TLOCAL was seen to compare favorably to other alignment methods. TCLUSTALW was tested on a family of protein kinases and reveal conserved regions similar to those previously identified by a hand alignment. CONCLUSION: These results show that the encoding of the writhing number as a geometric measure allow high quality structure alignments to be generated using standard algorithms of sequence alignment. This approach provides computationally efficient algorithms that allow fast database searching and multiple structure alignment. Because the geometric measure can employ different window sizes, the method allows the exploration of alignments on different, well-defined length scales

Seafloor deposit models, geochemistry, and petrology of the mafic-ultramafic hosted Big Lake VMS occurrence, Marathon, Ontario

The Big Lake volcanogenic massive sulphide (VMS) occurrence, located in the Schreiber-Hemlo belt of the Superior Province, was discovered in March 2006 near Marathon, Ontario. It is hosted in a mafic-ultramafic metavolcanic sequence lacking felsic volcanic or volcaniclastic rock, and consists of a thin, locally anastomosing sheet of veined pyrrhotite, chalcopyrite, and sphalerite currently defined over a plan area of approximately 0.5x0.5 km, along the base of a series of peridotite and pyroxenite cumulates termed the Big Lake Ultramafic Complex (BLUC)