Analysis of the Systematic Errors in the Positions of BATSE Catalog Bursts

We analyze the systematic errors in the positions of bursts in the BATSE 1B, 2B and 3B catalogs, using a likelihood approach. We use the BATSE data in conjunction with 196 single IPN arcs. We assume circular Gaussian errors, and that the total error is the sum in quadrature of the systematic error $\sigma_{\rm sys}$ and statistical error $\sigma_{\rm stat}$, as prescribed by the BATSE catalog. We find that the 3B burst positions are inconsistent with the value $\sigma_{\rm sys} = 1.6^\circ$ stated in the BATSE 3B catalog.Comment: A five page LateX file that uses the Revtex conference proceedings macro aipbook.sty, and includes five postscript figures using psfig. To Be published in the Proceedings of the Third Hunstville Symposium on Gamma-Ray Bursts, eds. C. Kouveliotou, M.S. Briggs and G.J. Fishman (New York:AIP

Likelihood Methods and Classical Burster Repetition

We develop a likelihood methodology which can be used to search for evidence of burst repetition in the BATSE catalog, and to study the properties of the repetition signal. We use a simplified model of burst repetition in which a number $N_{\rm r}$ of sources which repeat a fixed number of times $N_{\rm rep}$ are superposed upon a number $N_{\rm nr}$ of non-repeating sources. The instrument exposure is explicitly taken into account. By computing the likelihood for the data, we construct a probability distribution in parameter space that may be used to infer the probability that a repetition signal is present, and to estimate the values of the repetition parameters. The likelihood function contains contributions from all the bursts, irrespective of the size of their positional errors --- the more uncertain a burst's position is, the less constraining is its contribution. Thus this approach makes maximal use of the data, and avoids the ambiguities of sample selection associated with data cuts on error circle size. We present the results of tests of the technique using synthetic data sets.Comment: 5 pages, Revtex (aipbook.sty included), 2 PostScript figures included using psfig. To appear in the Proceedings of the 1995 La Jolla Workshop "High Velocity Neutron Stars and Gamma-Ray Bursts," eds. R. Rothschild and R. Lingenfelter, AIP, New Yor

Determining the Gamma-Ray Burst Rate as a Function of Redshift

We exploit the 14 gamma-ray bursts (GRBs) with known redshifts $z$ and the 7 GRBs for which there are constraints on $z$ to determine the GRB rate $R_{\rm GRB}(z)$, using a method based on Bayesian inference. We find that, despite the qualitative differences between the observed GRB rate and estimates of the SFR in the universe, current data are consistent with $R_{\rm GRB}(z)$ being proportional to the SFR.Comment: 3 pages, 3 figures, to appear in AIP proc. "Gamma-Ray Burst and Afterglow Astronomy 2001" Woods Hole, Massachusett

Determining the Gamma-Ray Burst Rate as a Function of Redshift

We exploit the 14 gamma-ray bursts (GRBs) with known redshifts z and the 7 GRBs for which there are constraints on z to determine the GRB rate R_{GRB}(z), using a method based on Bayesian inference. We find that, despite the qualitative differences between the observed GRB rate and estimates of the SFR in the universe, current data are consistent with R_{GRB}(z) being proportional to the SFR.Comment: To appear in Procs. of Gamma-Ray Bursts in the Afterglow Era: 2nd Workshop, 3 pages, 3 figures, LaTe

Determining the GRB (Redshift, Luminosity)-Distribution Using Burst Variability

We use the possible Cepheid-like luminosity estimator for the long-duration gamma-ray bursts (GRBs) developed by Reichart et al. (2000) to estimate the intrinsic luminosity, and thus the redshift, of 907 long-duration GRBs from the BATSE 4B catalog. We describe a method based on Bayesian inference which allows us to infer the intrinsic GRB burst rate as a function of redshift for bursts with estimated intrinsic luminosities and redshifts. We apply this method to the above sample of long-duration GRBs, and present some preliminary results

OS-net: Orbitally Stable Neural Networks

We introduce OS-net (Orbitally Stable neural NETworks), a new family of neural network architectures specifically designed for periodic dynamical data. OS-net is a special case of Neural Ordinary Differential Equations (NODEs) and takes full advantage of the adjoint method based backpropagation method. Utilizing ODE theory, we derive conditions on the network weights to ensure stability of the resulting dynamics. We demonstrate the efficacy of our approach by applying OS-net to discover the dynamics underlying the R\"{o}ssler and Sprott's systems, two dynamical systems known for their period doubling attractors and chaotic behavior