Numerical Integration of Nonlinear Wave Equations for General Relativity

A second-order numerical implementation is given for recently derived nonlinear wave equations for general relativity. The Gowdy T$^3$ cosmology is used as a test bed for studying the accuracy and convergence of simulations of one-dimensional nonlinear waves. The complete freedom in space-time slicing in the present formulation is exploited to compute in the Gowdy line-element. Second-order convergence is found by direct comparison of the results with either analytical solutions for polarized waves, or solutions obtained from Gowdy's reduced wave equations for the more general unpolarized waves. Some directions for extensions are discussed.Comment: 19 pages (LaTex), 3 figures (ps

Comparison of different objective functions for parameterization of simple respiration models

The eddy covariance measurements of carbon dioxide fluxes collected around the world offer a rich source for detailed data analysis. Simple, aggregated models are attractive tools for gap filling, budget calculation, and upscaling in space and time. Key in the application of these models is their parameterization and a robust estimate of the uncertainty and reliability of their predictions. In this study we compared the use of ordinary least squares (OLS) and weighted absolute deviations (WAD, which is the objective function yielding maximum likelihood parameter estimates with a double exponential error distribution) as objective functions within the annual parameterization of two respiration models: the Q10 model and the Lloyd and Taylor model. We introduce a new parameterization method based on two nonparametric tests in which model deviation (Wilcoxon test) and residual trend analyses (Spearman test) are combined. A data set of 9 years of flux measurements was used for this study. The analysis showed that the choice of the objective function is crucial, resulting in differences in the estimated annual respiration budget of up to 40%. The objective function should be tested thoroughly to determine whether it is appropriate for the application for which the model will be used. If simple models are used to estimate a respiration budget, a trend test is essential to achieve unbiased estimates over the year. The analyses also showed that the parameters of the Lloyd and Taylor model are highly correlated and difficult to determine precisely, thereby limiting the physiological interpretability of the parameter

Gravitational wave frequencies and energies in hypernovae

A torus develops a state of suspended accretion against a magnetic wall around a rapidly rotating black hole formed in core-collapse hypernovae. It hereby emits about 10% of the black hole spin-energy in gravitational radiation from a finite number of multipole mass moments. We quantify the relation between the frequency of quadrupole gravitational radiation and the energy output $E_w$ in torus winds by $f_{gw}\simeq 470{Hz}(E_w/4\times 10^{52}{erg})^{1/2}(7M_\odot/M)^{3/2}$, where $M$ denotes the mass of the black hole. We propose that $E_w$ irradiates the remnant stellar envelope from within. We identify $E_w$ with energies $\sim 10^{52}$ erg inferred from X-ray observations on matter injecta; and the poloidal curvature in the magnetic wall with the horizon opening angle in baryon poor outflows that power true GRB energies of $E_\gamma\simeq 3\times 10^{51}$ erg.Comment: To appear in AP

Calorimetry of gamma-ray bursts: echos in gravitational waves

Black holes surrounded by a disk or torus may drive the enigmatic cosmological gamma-ray bursts (GRBs). Equivalence in poloidal topology to pulsar magnetospheres shows a high incidence of the black hole-luminosity $L_H$ into the surrounding magnetized matter. We argue that this emission is re-radiated into gravitational waves at $L_{GW}\simeq L_H/3$ in frequencies of order 1kHz, winds and, potentially, MeV neutrinos. The total energy budget and input to the GRB from baryon poor jets are expected to be standard in this scenario, consistent with recent analysis of afterglow data. Collimation of these outflows by baryon rich disk or torus winds may account for the observed spread in opening angles up to about $35^o$. This model may be tested by future LIGO/VIRGO observations.Comment: To appear in ApJ

Disjunctive linear operators and partial multiplications in Riesz spaces

In dit proefschrift worden lineaire operatoren op Riesz ruimten bestudeerd, in het bijzonder disjunctieve lineaire operatoren. Enkele voorbeelden van niet orde begrensde disjunctieve lineaire operatoren van een Riesz ruimte op zichzelf worden gegeven, terwijl de orde begrensde disjunctieve lineaire operatoren in verband worden gebracht met een partieel gedefinieerde vermenigvuldigingsoperatie in de Riesz ruimte.Hoofdstuk I geeft een inleiding in de theorie van Riesz ruimten. In hoofdstuk II worden enkele typen van convergentie bestudeerd in verband met lineaire operatoren. Disjunctieve lineaire operatoren worden in hoofdstuk III bestudeerd. In het bijzonder wordt aandacht geschonken aan disjunctieve lineaire functionalen en orde begrensde disjunctieve lineaire operatoren van een Archimedische Riesz ruimte met een sterke orde eenheid op zichzelf. In hoofdstuk IV worden Riesz ruimten bestudeerd die voorzien zijn van een vrij star gedefinieerde vermenigvuldigingsstructuur, de zogenaamde f-algebra's. Enige overeenkomsten en verschillen tussen Riesz homomorfismen en ring homomorfismen worden aangegeven en een variant van de stelling van Ellis-Phelps wordt afgeleid. Met behulp van de in hoofdstuk V ontwikkelde theorie van inverteren en worteltrekken in regulator complete Φ-algebra's wordt een generalisatie verkregen van bovengenoemde stelling. In hoofdstuk VI worden partiële vermenigvuldigingsoperaties op Archimedische Riesz ruimten met een zwakke orde eenheid axiomatisch ingevoerd. Op iedere Archimedische Riesz ruimte met sterke orde eenheid wordt tenslotte een intrinsieke partidle vermenigvuldiging aangegeven en in verband gebracht met de ruimte van alle orde begrensde disjunctieve lineaire operatoren van de Riesz ruimte op zichzelf.</p

Modeling pathological brain rhythms: constructing a neural mass model from single cell dynamics

Neural mass models (NMM) describe neural activity on a macroscopic scale, which can be compared to the electroencephalogram (EEG). This allows a better understanding of the processes responsible for various EEG patterns, including pathological rhythms as diffuse slowing or burst-suppression [1]. Using available models which contain explicit expressions for the synaptic response and number of synapses [2], pathological conditions that modulate synaptic function, such as anesthetics [3] and hypoxia, can be included easily. However, it is less obvious how to incorporate conditions which alter the excitability of neurons, such as hyperkalemia or channel blockers. Here, we present a method for constructing a neural mass model by using the relation between synaptic input of a single cell model and its firing rate. This allows an easy implementation for pathological conditions. We describe the average firing rate of a single population of neurons receiving one type of synaptic input, but this can readily be extended to multiple populations. A set of differential equations describes, traditionally, the average synaptic conductance [2]. Assuming Poisson statistics for the input, we can derive another equation, which describes the time evolution of the standard deviation of the synaptic conductance across the population. The average and standard deviation of the conductance then determine the distribution and the corresponding average of the firing rates in the population. As initial verification, the constructed mean field model is numerically compared to a network of single cells. From the single cell model we determine the dependence of the firing rate on (constant) synaptic conductance numerically. Furthermore, we show that, for fluctuating inputs, the firing rate is well approximated by the instantaneous synaptic conductance. 120 Hodgkin-Huxley type cells were connected all-to-all with inhibitory synapses: a simple configuration which results in intrinsic oscillations. Each cell receives inhibitory external input as well, consisting of Poisson trains. We find a close agreement between the constructed neural mass model and the network simulation (Figure 1). Figure 1. Comparison of step response of the derived NMM and a detailed network model The proposed method can easily be extended to model heterogeneous populations, multiple types of synapses, spatial structures, propagation delays, and bursting dynamics [4]. Any pathophysiology can readily be incorporated by adapting the single cell model. This allows for testing hypotheses on processes underlying abnormal EEGs

Intra-Arterial Infusion with Methotrexate in the Rat

The superiority of intra-arterial infusion with methotrexate (MTX) over its systemic use in the treatment of head and neck tumours is still being questioned. A model in the rat, suitable for intra-arterial administration of MTX could be constructed. In this model 3 schedules have been investigated: (1) 7 days continuous intra-arterial infusion with MTX; (2) the same schedule combined with leucovorin (CF) 6-hourly intraperitoneally (i.p.) after Sullivan et al. (1959); (3) intermittent administration of MTX 2 × 24 h intra-arterial infusion on Day 1 and 4, while on Day 2, 3, 5, 6 and 7 the catheter is kept open by the continuous intra-arterial infusion of saline. For all the three schedules intra-arterial MTX proved to be superior to its systemic use

Differential Forms and Wave Equations for General Relativity

Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and York (AACY) have cast the 3+1 evolution equations of general relativity in gauge-covariant and causal ``first-order symmetric hyperbolic form,'' thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's 2-form, which in the ``time-gauge'' is built linearly from the ``extrinsic curvature 1-form.'' The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt-Deser-Misner gravitational momentum.Comment: REVTeX, 26 pages, no figures, 1 macr