Estimating the expected value of an observable appearing in a non-equilibrium stochastic process usually involves sampling. If the observables variance is high, many samples are required. In contrast, we show that performing the same task without sam
pling, using tensor network compression, efficiently captures high variances in systems of various geometries and dimensions. We provide examples for which matching the accuracy of our efficient method would require a sample size scaling exponentially with system size. In particular, the high variance observable $mathrm{e}^{-beta W}$, motivated by Jarzynskis equality, with $W$ the work done quenching from equilibrium at inverse temperature $beta$, is exactly and efficiently captured by tensor networks.
The mechanism that causes the prompt-emission episode of gamma-ray bursts (GRBs) is still widely debated despite there being thousands of prompt detections. The favoured internal shock model relates this emission to synchrotron radiation. However, it
does not always explain the spectral indices of the shape of the spectrum, often fit with empirical functions. Multi-wavelength observations are therefore required to help investigate the possible underlying mechanisms that causes the prompt emission. We present GRB 121217A, for which we were able to observe its near-infrared (NIR) emission during a secondary prompt-emission episode with the Gamma-Ray Burst Optical Near-infrared Detector (GROND) in combination with the Swift and Fermi satellites, covering an energy range of 0.001 keV to 100 keV. We determine a photometric redshift of z=3.1+/-0.1 with a line-of-sight extinction of A_V~0 mag, utilising the optical/NIR SED. From the afterglow, we determine a bulk Lorentz factor of Gamma~250 and an emission radius of R<10^18 cm. The prompt-emission broadband spectral energy distribution is well fit with a broken power law with b1=-0.3+/-0.1, b2=0.6+/-0.1 that has a break at E=6.6+/-0.9 keV, which can be interpreted as the maximum injection frequency. Self-absorption by the electron population below energies of E_a<6 keV suggest a magnetic field strength of B~10^5 G. However, all the best fit models underpredict the flux observed in the NIR wavelengths, which also only rebrightens by a factor of ~2 during the second prompt emission episode, in stark contrast to the X-ray emission, which rebrightens by a factor of ~100, suggesting an afterglow component is dominating the emission. We present GRB 121217A one of the few GRBs for which there are multi-wavelength observations of the prompt-emission period and show that it can be understood with a synchrotron radiation model.
In the past few years the number of well-sampled optical to NIR light curves of long Gamma-Ray Bursts (GRBs) has greatly increased particularly due to simultaneous multi-band imagers such as GROND. Combining these densely sampled ground-based data se
ts with the Swift UVOT and XRT space observations unveils a much more complex afterglow evolution than what was predicted by the most commonly invoked theoretical models. GRB 100814A represents a remarkable example of these interesting well-sampled events, showing a prominent late-time rebrightening in the optical to NIR bands and a complex spectral evolution. This represents a unique laboratory to test the different afterglow emission models. Here we study the nature of the complex afterglow emission of GRB 100814A in the framework of different theoretical models. Moreover, we compare the late-time chromatic rebrightening with those observed in other well-sampled long GRBs. We analysed the optical and NIR observations obtained with the seven-channel Gamma-Ray burst Optical and Near-infrared Detector at the 2.2 m MPG/ESO telescope together with the X-ray and UV data detected by the instruments onboard the Swift observatory. The broad-band afterglow evolution, achieved by constructing multi-instrument light curves and spectral energy distributions, will be discussed in the framework of different theoretical models. We find that the standard models that describe the broad-band afterglow emission within the external shock scenario fail to describe the complex evolution of GRB 100814A, and therefore more complex scenarios must be invoked. [abridged]
To answer questions on the start and duration of the epoch of reionisation, periods of galaxy mergers and properties of other cosmological encounters, the cosmic star formation history (CSFH), is of fundamental importance. Using the association of lo
ng gamma-ray bursts (LGRBs) with the death of massive stars and their ultra-luminous nature, the CSFH can be probed to higher redshifts than current conventional methods. Unfortunately, no consensus has been reached on the manner in which the LGRB rate (LGRBR) traces the CSFH, leaving many of the questions mentioned mostly unexplored by this method. Observations by the GRB NIR detector (GROND) over the past 4 years have, for the first time, acquired highly complete LGRB samples. Driven by these completeness levels and new evidence of LGRBs also occurring in more massive and metal rich galaxies than previously thought, the possible biases of the LGRBR-CSFH connection are investigated over a large range of galaxy properties. The CSFH is modelled using empirical fits to the galaxy mass function and galaxy star formation rates. Biasing the CSFH by metallicity cuts, mass range boundaries, and other unknown redshift dependencies, a LGRBR is generated and compared to the highly complete GROND sample. It is found that there is no strong preference for a metallicity cut or fixed galaxy mass boundaries and that there are no unknown redshift effects, in contrast to previous work which suggest values of Z/Z_sun~0.1-0.3. From the best-fit models, we predict that ~1.2% of the LGRB burst sample exists above z=6. The linear relationship between the LGRBR and the CSFH suggested by our results implies that redshift biases present in previous LGRB samples significantly affect the inferred dependencies of LGRBs on their host galaxy properties. Such biases can lead to, e.g., an interpretation of metallicity limitations and evolving LGRB luminosity functions.
Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state
, continuous time Markov chains, we develop a theory of nonlinear expectations in the spirit of [Dynamically consistent nonlinear evaluations and expectations (2005) Shandong Univ.]. We prove basic properties of these expectations and show their applications to dynamic risk measures on such spaces. In particular, we prove comparison theorems for scalar and vector valued solutions to BSDEs, and discuss arbitrage and risk measures in the scalar case.
By analogy with the theory of Backward Stochastic Differential Equations, we define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. This paper considers these processes as constructions in their ow
n right, not as approximations to the continuous case. We establish the existence and uniqueness of solutions under weaker assumptions than are needed in the continuous time setting, and also establish a comparison theorem for these solutions. The conditions of this theorem are shown to approximate those required in the continuous time setting. We also explore the relationship between the driver $F$ and the set of solutions; in particular, we determine under what conditions the driver is uniquely determined by the solution. Applications to the theory of nonlinear expectations are explored, including a representation result.
This paper studies the set of $ntimes n$ matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and furthermore is iso
morphic to the set of arbitrary $(n-1)times (n-1)$ matrices. The Moore-Penrose pseudoinverse corresponds with the true inverse, (when it exists), in this lower dimension and an explicit representation of this pseudoinverse in terms of the lower dimensional space is given. This analysis is then extended to non-square matrices with all row or all column sums equal to zero.
We consider backward stochastic differential equations (BSDEs) related to finite state, continuous time Markov chains. We show that appropriate solutions exist for arbitrary terminal conditions, and are unique up to sets of measure zero. We do not re
quire the generating functions to be monotonic, instead using only an appropriate Lipschitz continuity condition.
