On the Approximation Performance of Fictitious Play in Finite Games

We study the performance of Fictitious Play, when used as a heuristic for finding an approximate Nash equilibrium of a 2-player game. We exhibit a class of 2-player games having payoffs in the range [0,1] that show that Fictitious Play fails to find a solution having an additive approximation guarantee significantly better than 1/2. Our construction shows that for n times n games, in the worst case both players may perpetually have mixed strategies whose payoffs fall short of the best response by an additive quantity 1/2 - O(1/n^(1-delta)) for arbitrarily small delta. We also show an essentially matching upper bound of 1/2 - O(1/n)

Assessing Knowledge Structures in a Constructive Statistical Learning Environment

In this report, the method of free recall is put forward as a tool to evaluate a prototypical statistical learning environment. A number of students from the faculty of Health Sciences, Maastricht University, the Netherlands, were required to write down whatever they could remember of a statistics course in which they had participated. By means of examining the free recall protocols of the participants, insight can be obtained into the mental representations they had formed with respect to three statistical concepts. Quantitative as well as qualitative analyses of the free recall protocols showed that the effect of the constructive learning environment was not in line with the expectations. Despite small-group discussions on the statistical concepts, students appeared to have disappointingly low levels of conceptual understanding

The sub-energetic GRB 031203 as a cosmic analogue to GRB 980425

Over the six years since the discovery of the gamma-ray burst GRB 980425, associated with the nearby (distance, ~40 Mpc) supernova 1998bw, astronomers have fiercely debated the nature of this event. Relative to bursts located at cosmological distances, (redshift, z~1), GRB 980425 was under-luminous in gamma-rays by three orders of magnitude. Radio calorimetry showed the explosion was sub-energetic by a factor of 10. Here, we report observations of the radio and X-ray afterglow of the recent z=0.105 GRB 031203 and demonstrate that it too is sub-energetic. Our result, when taken together with the low gamma-ray luminosity, suggest that GRB 031203 is the first cosmic analogue to GRB 980425. We find no evidence that this event was a highly collimated explosion viewed off-axis. Like GRB 980425, GRB 031203 appears to be an intrinsically sub-energetic gamma-ray burst. Such sub-energetic events have faint afterglows. Intensive follow-up of faint bursts with smooth gamma-ray light curves (common to both GRBs 031203 and 980425) may enable us to reveal their expected large population.Comment: To Appear in Nature, August 5, 200

The bright optical afterglow of the nearby gamma-ray burst of 29 March 2003

Many past studies of cosmological gamma-ray bursts (GRBs) have been limited because of the large distance to typical GRBs, resulting in faint afterglows. There has long been a recognition that a nearby GRB would shed light on the origin of these mysterious cosmic explosions, as well as the physics of their fireballs. However, GRBs nearer than z=0.2 are extremely rare, with an estimated rate of localisation of one every decade. Here, we report the discovery of bright optical afterglow emission from GRB 030329. Our prompt dissemination and the brilliance of the afterglow resulted in extensive followup (more than 65 telescopes) from radio through X-ray bands, as well as measurement of the redshift, z=0.169. The gamma-ray and afterglow properties of GRB 030329 are similar to those of cosmological GRBs (after accounting for the small distance), making this the nearest known cosmological GRB. Observations have already securely identified the progenitor as a massive star that exploded as a supernova, and we anticipate futher revelations of the GRB phenomenon from studies of this source.Comment: 13 pages, 4 figures. Original tex

A Common Origin for Cosmic Explosions Inferred from Fireball Calorimetry

Past studies suggest that long-duration gamma-ray bursts (GRBs) have a standard energy of E_gamma ~ 10^51 erg in ultra-relativistic ejecta when corrected for asymmetry ("jets"). However, recently a group of sub-energetic bursts, including the peculiar GRB 980425 associated with SN 1998bw (E_gamma ~ 10^48 erg), has been identified. Here we report radio observations of GRB 030329, the nearest burst to date, which allow us to undertake calorimetry of the explosion. Our observations require a two-component explosion: a narrow (5 degrees) ultra-relativistic component responsible for the gamma-rays and early afterglow, and a wide, mildly relativistic component responsible for the radio and optical afterglow beyond 1.5 days. While the gamma-rays are energetically minor, the total energy release, dominated by the wide component, is similar to that of other GRBs. Given the firm link of GRB 030329 with SN 2003dh our result suggests a common origin for cosmic explosions in which, for reasons not understood, the energy in the highest velocity ejecta is highly variableComment: Accepted to Natur

Relativistic ejecta from XRF 060218 and the rate of cosmic explosions

Over the last decade, long-duration gamma-ray bursts (GRBs) including the subclass of X-ray flashes (XRFs) have been revealed to be a rare variety of Type Ibc supernova (SN). While all these events result from the death of massive stars, the electromagnetic luminosities of GRBs and XRFs exceed those of ordinary Type Ibc SNe by many orders of magnitude. The essential physical process that causes a dying star to produce a GRB or XRF, and not just an SN, remains the crucial open question. Here we present radio and X-ray observations of XRF 060218 (associated with SN 2006aj), the second nearest GRB identified to-date, which allow us to measure its total energy and place it in the larger context of cosmic explosions. We show that this event is 100 times less energetic but ten times more common than cosmological GRBs. Moreover, it is distinguished from ordinary Type Ibc SNe by the presence of 10^48 erg coupled to mildly-relativistic ejecta, along with a central engine (an accretion-fed, rapidly rotating compact source) which produces X-rays for weeks after the explosion. This suggests that the production of relativistic ejecta is the key physical distinction between GRBs/XRFs and ordinary SNe, while the nature of the central engine (black hole or magnetar) may distinguish typical bursts from low-luminosity, spherical events like XRF 060218.Comment: To appear in Nature on August 31 2006 (15 pages, 3 figures, 1 table, including Supplementary Information

Radio Emission from Ultra-Cool Dwarfs

The 2001 discovery of radio emission from ultra-cool dwarfs (UCDs), the very low-mass stars and brown dwarfs with spectral types of ~M7 and later, revealed that these objects can generate and dissipate powerful magnetic fields. Radio observations provide unparalleled insight into UCD magnetism: detections extend to brown dwarfs with temperatures <1000 K, where no other observational probes are effective. The data reveal that UCDs can generate strong (kG) fields, sometimes with a stable dipolar structure; that they can produce and retain nonthermal plasmas with electron acceleration extending to MeV energies; and that they can drive auroral current systems resulting in significant atmospheric energy deposition and powerful, coherent radio bursts. Still to be understood are the underlying dynamo processes, the precise means by which particles are accelerated around these objects, the observed diversity of magnetic phenomenologies, and how all of these factors change as the mass of the central object approaches that of Jupiter. The answers to these questions are doubly important because UCDs are both potential exoplanet hosts, as in the TRAPPIST-1 system, and analogues of extrasolar giant planets themselves.Comment: 19 pages; submitted chapter to the Handbook of Exoplanets, eds. Hans J. Deeg and Juan Antonio Belmonte (Springer-Verlag

Evaluation of a fiberoptic-based system for measurement of optical properties in highly attenuating turbid media

BACKGROUND: Accurate measurements of the optical properties of biological tissue in the ultraviolet A and short visible wavelengths are needed to achieve a quantitative understanding of novel optical diagnostic devices. Currently, there is minimal information on optical property measurement approaches that are appropriate for in vivo measurements in highly absorbing and scattering tissues. We describe a novel fiberoptic-based reflectance system for measurement of optical properties in highly attenuating turbid media and provide an extensive in vitro evaluation of its accuracy. The influence of collecting reflectance at the illumination fiber on estimation accuracy is also investigated. METHODS: A neural network algorithm and reflectance distributions from Monte Carlo simulations were used to generate predictive models based on the two geometries. Absolute measurements of diffuse reflectance were enabled through calibration of the reflectance system. Spatially-resolved reflectance distributions were measured in tissue phantoms at 405 nm for absorption coefficients (μ(a)) from 1 to 25 cm(-1 )and reduced scattering coefficients ([Formula: see text]) from 5 to 25 cm(-1). These data and predictive models were used to estimate the optical properties of tissue-simulating phantoms. RESULTS: By comparing predicted and known optical properties, the average errors for μ(a )and [Formula: see text] were found to be 3.0% and 4.6%, respectively, for a linear probe approach. When bifurcated probe data was included and samples with μ(a )values less than 5 cm(-1 )were excluded, predictive errors for μ(a )and [Formula: see text] were further reduced to 1.8% and 3.5%. CONCLUSION: Improvements in system design have led to significant reductions in optical property estimation error. While the incorporation of a bifurcated illumination fiber shows promise for improving the accuracy of [Formula: see text] estimates, further study of this approach is needed to elucidate the source of discrepancies between measurements and simulation results at low μ(a )values

The optical afterglow of the short gamma-ray burst GRB 050709

It has long been known that there are two classes of gamma-ray bursts (GRBs), mainly distinguished by their durations. The breakthrough in our understanding of long-duration GRBs (those lasting more than ~2 s), which ultimately linked them with energetic Type Ic supernovae, came from the discovery of their long-lived X-ray and optical afterglows, when precise and rapid localizations of the sources could finally be obtained. X-ray localizations have recently become available for short (duration <2 s) GRBs, which have evaded optical detection for more than 30 years. Here we report the first discovery of transient optical emission (R-band magnitude ~23) associated with a short burst; GRB 050709. The optical afterglow was localized with subarcsecond accuracy, and lies in the outskirts of a blue dwarf galaxy. The optical and X-ray afterglow properties 34 h after the GRB are reminiscent of the afterglows of long GRBs, which are attributable to synchrotron emission from ultrarelativistic ejecta. We did not, however, detect a supernova, as found in most nearby long GRB afterglows, which suggests a different origin for the short GRBs.Comment: 11 pages, 3 figures, press material at http://www.astro.ku.dk/dark

Visual parameter optimisation for biomedical image processing

Background: Biomedical image processing methods require users to optimise input parameters to ensure high quality output. This presents two challenges. First, it is difficult to optimise multiple input parameters for multiple input images. Second, it is difficult to achieve an understanding of underlying algorithms, in particular, relationships between input and output. Results: We present a visualisation method that transforms users’ ability to understand algorithm behaviour by integrating input and output, and by supporting exploration of their relationships. We discuss its application to a colour deconvolution technique for stained histology images and show how it enabled a domain expert to identify suitable parameter values for the deconvolution of two types of images, and metrics to quantify deconvolution performance. It also enabled a breakthrough in understanding by invalidating an underlying assumption about the algorithm. Conclusions: The visualisation method presented here provides analysis capability for multiple inputs and outputs in biomedical image processing that is not supported by previous analysis software. The analysis supported by our method is not feasible with conventional trial-and-error approaches