Ordered optimal solutions and parametric minimum cut problems

AbstractIn this paper, we present an algebraic sufficient condition for the existence of a selection of optimal solutions in a parametric optimization problem that are totally ordered, but not necessarily monotone. Based on this result, we present necessary and sufficient conditions that ensure the existence of totally ordered selections of minimum cuts for some classes of parametric maximum flow problems. These classes subsume the class studied by Arai et al. [Discrete Appl. Math. 41 (1993) 69–74] as a special case

Tight Bounds for The Price of Fairness

A central decision maker (CDM), who seeks an efficient allocation of scarce resources among a finite number of players, often has to incorporate fairness criteria to avoid unfair outcomes. Indeed, the Price of Fairness (POF), a term coined in Bertsimas et al. (2011), refers to the efficiency loss due to the incorporation of fairness criteria into the allocation method. Quantifying the POF would help the CDM strike an appropriate balance between efficiency and fairness. In this paper we improve upon existing results in the literature, by providing tight bounds for the POF for the proportional fairness criterion for any $n$, when the maximum achievable utilities of the players are equal or are not equal. Further, while Bertsimas et al. (2011) have already derived a tight bound for the max-min fairness criterion for the case that all players have equal maximum achievable utilities, we also provide a tight bound in scenarios where these utilities are not equal. Finally, we investigate the sensitivity of our bounds and Bertsimas et al. (2011) bounds for the POF to the variability of the maximum achievable utilities

GRB Energetics in the Swift Era

We examine the rest frame energetics of 76 gamma-ray bursts (GRBs) with known redshift that were detected by the Swift spacecraft and monitored by the satellite's X-ray Telescope (XRT). Using the bolometric fluence values estimated in Butler et al. 2007b and the last XRT observation for each event, we set a lower limit the their collimation corrected energy Eg and find that a 68% of our sample are at high enough redshift and/or low enough fluence to accommodate a jet break occurring beyond the last XRT observation and still be consistent with the pre-Swift Eg distribution for long GRBs. We find that relatively few of the X-ray light curves for the remaining events show evidence for late-time decay slopes that are consistent with that expected from post jet break emission. The breaks in the X-ray light curves that do exist tend to be shallower and occur earlier than the breaks previously observed in optical light curves, yielding a Eg distribution that is far lower than the pre-Swift distribution. If these early X-ray breaks are not due to jet effects, then a small but significant fraction of our sample have lower limits to their collimation corrected energy that place them well above the pre-Swift Eg distribution. Either scenario would necessitate a much wider post-Swift Eg distribution for long cosmological GRBs compared to the narrow standard energy deduced from pre-Swift observations. We note that almost all of the pre-Swift Eg estimates come from jet breaks detected in the optical whereas our sample is limited entirely to X-ray wavelengths, furthering the suggestion that the assumed achromaticity of jet breaks may not extend to high energies.Comment: 30 pages, 10 figures, Accepted to Ap

Pulse Width Evolution of Late Time X-rays Flares in GRBs: Evidence For Internal Shocks

We study the duration and variability of late time X-ray flares following gamma-ray bursts (GRBs) observed by the narrow field X-ray telescope (XRT) aboard the {\it Swift} spacecraft. These flares are thought to be indicative of late time activity by the central engine that powers the GRB and produced by means similar to those which produce the prompt emission. We use a non-parametric procedure to study the overall temporal properties of the flares and a structure function analysis to look for an evolution of the fundamental variability time-scale between the prompt and late time emission. We find a strong correlation in 28 individual x-ray flares in 18 separate GRBs between the flare duration and their time of peak flux since the GRB trigger. We also find a qualitative trend of decreasing variability as a function of time since trigger, with a characteristic minimum variability timescale $\Delta t/t=0.1$ for most flares. We interpret these results as evidence of internal shocks at collision radii that are larger than those that produced the prompt emission. Contemporaneous detections of high energy emission by GLAST could be a crucial test in determining if indeed these X-ray flares originate as internal shocks behind the afterglow, as any X-ray emission originating from behind the afterglow is expected to undergo inverse Compton scattering as it passes through the external shock.Comment: 26 pages, 5 figures, 1 table. Submitted to ApJ. This work expands upon and formalizes our previous report at the October 2006 AAS HEAD Meeting of the discovery of pulse width evolutio

GRB 090426: The Environment of a Rest-Frame 0.35-second Gamma-Ray Burst at Redshift z=2.609

We present the discovery of an absorption-line redshift of z = 2.609 for GRB 090426, establishing the first firm lower limit to a redshift for a gamma-ray burst with an observed duration of <2 s. With a rest-frame burst duration of T_90z = 0.35 s and a detailed examination of the peak energy of the event, we suggest that this is likely (at >90% confidence) a member of the short/hard phenomenological class of GRBs. From analysis of the optical-afterglow spectrum we find that the burst originated along a very low HI column density sightline, with N_HI < 3.2 x 10^19 cm^-2. Our GRB 090426 afterglow spectrum also appears to have weaker low-ionisation absorption (Si II, C II) than ~95% of previous afterglow spectra. Finally, we also report the discovery of a blue, very luminous, star-forming putative host galaxy (~2 L*) at a small angular offset from the location of the optical afterglow. We consider the implications of this unique GRB in the context of burst duration classification and our understanding of GRB progenitor scenarios.Comment: Submitted to MNRA

Fermi Large Area Telescope Constraints on the Gamma-ray Opacity of the Universe

The Extragalactic Background Light (EBL) includes photons with wavelengths from ultraviolet to infrared, which are effective at attenuating gamma rays with energy above ~10 GeV during propagation from sources at cosmological distances. This results in a redshift- and energy-dependent attenuation of the gamma-ray flux of extragalactic sources such as blazars and Gamma-Ray Bursts (GRBs). The Large Area Telescope onboard Fermi detects a sample of gamma-ray blazars with redshift up to z~3, and GRBs with redshift up to z~4.3. Using photons above 10 GeV collected by Fermi over more than one year of observations for these sources, we investigate the effect of gamma-ray flux attenuation by the EBL. We place upper limits on the gamma-ray opacity of the Universe at various energies and redshifts, and compare this with predictions from well-known EBL models. We find that an EBL intensity in the optical-ultraviolet wavelengths as great as predicted by the "baseline" model of Stecker et al. (2006) can be ruled out with high confidence.Comment: 42 pages, 12 figures, accepted version (24 Aug.2010) for publication in ApJ; Contact authors: A. Bouvier, A. Chen, S. Raino, S. Razzaque, A. Reimer, L.C. Reye

Evidence for dust destruction from the early-time colour change of GRB 120119A

We present broad-band observations and analysis of Swift gamma-ray burst (GRB) 120119A. Our early-time afterglow detections began under 15 s after the burst in the host frame (redshift z = 1.73), and they yield constraints on the burst energetics and local environment. Late-time afterglow observations of the burst show evidence for a moderate column of dust (AV ≈ 1.1 mag) similar to, but statistically distinct from, dust seen along Small Magellanic Cloud sightlines. Deep late-time observations reveal a dusty, rapidly star-forming host galaxy. Most notably, our early-time observations exhibit a significant red-to-blue colour change in the first ∼200 s after the trigger at levels heretofore unseen in GRB afterglows. This colour change, which is coincident with the final phases of the prompt emission, is a hallmark prediction of the photodestruction of dust in GRB afterglows. We test whether dust-destruction signatures are significantly distinct from other sources of colour change, namely a change in the intrinsic spectral index β. We find that a time-varying power-law spectrum alone cannot adequately describe the observed colour change, and allowing for dust destruction (via a time-varying AV) significantly improves the fit. While not definitively ruling out other possibilities, this event provides the best support yet for the direct detection of dust destruction in the local environment of a GRB

Activation of Cytotoxic and Regulatory Functions of NK Cells by Sindbis Viral Vectors

Oncolytic viruses (OVs) represent a relatively novel anti-cancer modality. Like other new cancer treatments, effective OV therapy will likely require combination with conventional treatments. In order to design combinatorial treatments that work well together, a greater scrutiny of the mechanisms behind the individual treatments is needed. Sindbis virus (SV) based vectors have previously been shown to target and kill tumors in xenograft, syngeneic, and spontaneous mouse models. However, the effect of SV treatment on the immune system has not yet been studied. Here we used a variety of methods, including FACS analysis, cytotoxicity assays, cell depletion, imaging of tumor growth, cytokine blockade, and survival experiments, to study how SV therapy affects Natural Killer (NK) cell function in SCID mice bearing human ovarian carcinoma tumors. Surprisingly, we found that SV anti-cancer efficacy is largely NK cell-dependent. Furthermore, the enhanced therapeutic effect previously observed from Sin/IL12 vectors, which carry the gene for interleukin 12, is also NK cell dependent, but works through a separate IFNγ-dependent mechanism, which also induces the activation of peritoneal macrophages. These results demonstrate the multimodular nature of SV therapy, and open up new possibilities for potential synergistic or additive combinatorial therapies with other treatments

Regulation of Glucose Homeostasis by KSR1 and MARK2

Protein scaffolds control the intensity and duration of signaling and dictate the specificity of signaling through MAP kinase pathways. KSR1 is a molecular scaffold of the Raf/MEK/ERK MAP kinase cascade that regulates the intensity and duration of ERK activation. Relative to wild-type mice, ksr1-/- mice are modestly glucose intolerant, but show a normal response to exogenous insulin. However, ksr1-/- mice also demonstrate a three-fold increase in serum insulin levels in response to a glucose challenge, suggesting a role for KSR1 in insulin secretion. The kinase MARK2 is closely related to C-TAK1, a known regulator of KSR1. Mice lacking MARK2 have an increased rate of glucose disposal in response to exogenous insulin, increased glucose tolerance, and are resistant to diet-induced obesity. mark2-/-ksr1-/- (DKO) mice were compared to wild type, mark2-/-, and ksr1-/- mice for their ability to regulate glucose homeostasis. Here we show that disruption of KSR1 in mark2-/- mice reverses the increased sensitivity to exogenous insulin resulting from MARK2 deletion. DKO mice respond to exogenous insulin similarly to wild type and ksr1-/- mice. These data suggest a model whereby MARK2 negatively regulates insulin sensitivity in peripheral tissue through inhibition of KSR1. Consistent with this model, we found that MARK2 binds and phosphorylates KSR1 on Ser392. Phosphorylation of Ser392 is a critical regulator of KSR1 stability, subcellular location, and ERK activation. These data reveal an unexpected role for the molecular scaffold KSR1 in insulin-regulated glucose metabolism