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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Resolving GRB Light Curves
Robert L. Wolpert
Duke University
w/Mary E Broadbent (Duke) & Tom Loredo (Cornell)
2014 August 03 17:15{17:45
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
Robert L. Wolpert Resolving GRB Light Curves 2 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
Robert L. Wolpert Resolving GRB Light Curves 3 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Overview of L�evy Adaptive Regression Kernels
I Goal: inference on unknown function f
I Kernel regression approximates unknown function withweighted sum of functions
I Adaptive kernel regression infers the parameters of the kernelinstead of using �xed dictionary
f (x) �JX
j=1
ujK (x j sj ; �j)
where sj is the location of the kernel on the domain of f .I LARK models the number, weights, and locations of the
kernels as the largest jump discontinuities in a L�evy processI Main advantage: stochastic process gives coherent way to
incorporate threshold for approximation Skip ahead
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
L�evy (In�nitely Divisible) Processes
I For a one-dimensional stochastic process X (t) to be a L�evyProcess the increments X (t2)�X (t1) for t1 < t2 must satisfy:
Stationarity The distribution of an increment dependsonly on the length of the interval (t2 � t1).
Independence Disjoint increments are independent
I The L�evy-Khinchine theorem states that for anin�nitely-divisible random variable X , the natural log of thecharacteristic function
log E expfi!0Xg = i!0m�12!
0�!+
ZR
�e i!
0u � 1� i!0h(u)��(du)
I The �(du) corresponding to an in�nitely divisible process iscalled its L�evy measure
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IntroductionGamma-ray bursts
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Discussion
L�evy processes as a Poisson integral (W+Ickstadt, 1998)
I For a measure space (S ;F ; �(du)), the Poisson randommeasure H(du) is a random function from F to N such that
for disjoint Ai 2 F , H(Ai )ind� Po
��(Ai )
�.
I When �(du) is the L�evy measure for an in�nitely-divisiblerandom variable X , then
H[u] =
ZR
uH(du) =1Xj=1
ujd= X
for (random) support fujg of H.
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
L�evy Processes as Poisson Random Fields
I Previous slide: ID RV can be represented as the countablesum of the support of H � Po(�(du)).
I For stochastic process case, we can write
H�1f0<s�tg
�=
ZR�(0;t]
uH(du ds) =X
fuj j 0 < sj � tg d= X (t)
where fsjg are distributed uniformly on S and uj aredistributed proportionally to �(du)
I In simulation, only a �nite number of pairs f(sj ; uj)g aredrawn, creating an approximation to the true process;
J� � Po��(�;1)
�; fuj j uj > �g iid� �(du)1fu>�g=�(�;1)
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
L�evy Adaptive Regression Kernels
f (t) :=
ZK (t j s; �)1fs�tgH(du ds d�)
=Xsj�t
ujK (t j sj ; �j)
where (sj ; uj) represent the countable innovations resulting fromthe random integral representation of a L�evy process. For tractablesimulation, we use a truncation parameter � > 0, and simulate from
f�(t) :=X
ujK (t j sj ; �j)1fsj�tg1fuj>�g
We use functions of this form to do Bayesian inference onunknown functions, such as light curves.
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Example of a LARK model
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
LARK is coherent under aggregation
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Anatomy of a gamma-ray burst
Photo credit: NASA Goddard Space CenterRobert L. Wolpert Resolving GRB Light Curves 12 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Scienti�c Questions
I Number and shape of pulses
I Interaction between time and energy spectra
I Inverse problem from transformation of data by telescope
Robert L. Wolpert Resolving GRB Light Curves 13 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Pulse Number and Shape
I Figure: A variety of GRBphoton rate time series(known as light curves)
I Timescale: 0:5s to 100s
I Number of pulses: 1 to 5 ormore
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Inverse Problem Induced by Detector
I Figure: Photons are sortedinto 4 energy channels,based on the energydeposited
(not the incident energy)
I Channel 1 is lowest energy;Channel 4 is highest
I Energy deposited is less thanincident energy; scienti�cinterest is in incident space
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Interaction of Time and Energy
BATSE GRB Trigger 501
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
Robert L. Wolpert Resolving GRB Light Curves 17 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Modeling a Single Light Curve
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Model
I Observe counts (Y1; : : : ;YN) in intervals (t0; t1] : : : (tN�1; tN ].
I Model: Ykind� Po(�k) where
�k =
Z tk
tk�1
f (t)dt
for some uncertain function f .
I We model f in LARK form: f (t) = B +PJ
j=1 AjK (t j Tj ; �j)
I The number of kernels, J, is unknown,as are f(Aj ;Tj ; �j)g for 1 � j � J
I We �x the baseline B and treat it as a known quantity
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Kernel Selection
I Fast rise, exponential decay (FRED) shapeI Norris+ (2005):
KN(t j T ; �1; �2) / exp
�� �1t � T
� t � T
�2
�1ft>Tg
I Considered broader \GiG" class but data were inconclusive:
KGiG (t j T ; p; �1; �2) / (t�T )p exp
�� �1t � T
� t � T
�2
�1ft>Tg
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Summary of Poisson Random Field Representation
I Model pulse start times Tj 's, and their maximum amplitudesAj 's as the jumps in a L�evy (e.g., Gamma or �-Stable) process
I The Poisson random �eld representation allows us to see theGamma process as the sum of countably many jumps
I whose times (Tj) are uniformly distributed on an intervalI whose heights (Aj) larger than any � > 0 are distributed
proportionally to
��(du) = �u�1 expf��ug1fu>�g du:I In�nitely many jumps lie below any threshold �, but we only
simulate the �nitely many jumps larger than �.I The number J� of jumps larger than � in a time interval of
length L has Po�LR1� �(du)
�distribution.
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Computation
I RJ-MCMC is a Metropolis-Hastings algorithm, but with atrans-dimensional proposal distribution, needed to doinference on our parameter space, the dimension of which israndom and variable.
I Our proposal distribution is a mixture of �ve di�erentproposals: birth, death, walk, split, and merge.
I Parallel thinning, a new variation on parallel tempering whichexploits the in�nite divisibility of LARK distributions to createinterpretable auxiliary chains, also aids in mode-�nding andmixing between modes
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 1
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 1
(a) (b)
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 1
(a) (b)
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 2
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 2
(a) (b)
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 2
(a) (b)
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 3
Robert L. Wolpert Resolving GRB Light Curves 31 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 3
(a) (b)
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 3
(a) (b)
Robert L. Wolpert Resolving GRB Light Curves 33 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 4
Robert L. Wolpert Resolving GRB Light Curves 34 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 4
(a) (b)
Robert L. Wolpert Resolving GRB Light Curves 35 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
GRB 501{ Channel 4
(a) (b)
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelSample of Results
Models in di�erent channels are incoherent
Number of pulses in each channel
(a): Channel 1 (b): Channel 2
(c): Channel 3 (d): Channel 4
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
Robert L. Wolpert Resolving GRB Light Curves 38 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
Robert L. Wolpert Resolving GRB Light Curves 39 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
Model
i : time indexk : channel indexj : pulse index
E(Yik) =�ik
�ik =
Z ti+1
ti
Bkdt
+ Rk
�Z ti+1
ti
JXj=1
Vj �(t j E ; �j)� �(E j �j)dt dE�
Rk is the response function for channel kBk is the empirically-determined baseline photon rate
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
Kernel Function { Time Spectrum for a Single Pulse
Time uence conditioned on energy:
(t j E ;T ; ; �) = �t � T + ln(E=100)
�=K
(x) = exp(��x)1fx>0g:
I This model has the pulse arrive instantly(time T , for photons of energy E = 100keV)
I �: time decay parameter
I : controls the fact that pulses arrive at di�erent times atdi�erent energies (typically, more energetic ones arrive earlier)
I K is a normalizing constant needed for the interpretation ofVj as pulse uence
Robert L. Wolpert Resolving GRB Light Curves 41 / 60
IntroductionGamma-ray bursts
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ModelGRB 501GRB 493
Kernel Function { Energy Spectrum for a Single Pulse
Comptonized Energy Spectrum:
�(E j �;Ec) =
�E
100
��
expf�E=Ecg1fE>0g
I Energy spectrum decays as a power law with index � < 0
I After cuto� Ec , spectrum decays exponentially
I Inference is challenging due to having only 4 channels ofDISCLA (Discriminator Large Area) data
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
How � and interact to create pulse shape
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IntroductionGamma-ray bursts
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Discussion
ModelGRB 501GRB 493
�-Stable process
I Volumes of pulses > � are modeled by the largest jumps of afully-skewed �-Stable process with � = 3=2
f (v) = 3�p8�
V�5=21fV>�g
I � = 3=2, re ects the observation that photon uence decaysas a power law with index �5=2
I � = 1 photon/cm2
I � = 0:1074 so that E(J�) = 3
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
Overdispersion
I To compensate for model misspeci�cation, an overdispersionparameter, r , is introduced. This parameter does not dependon the time interval or the energy channel.
I Instead of modeling Yij as Poisson,
Yij � NB
�n = r�ij ; p =
r
1 + r
�
I E(Yij) = �ijI V(Yij) = �ij
1+rr
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
Robert L. Wolpert Resolving GRB Light Curves 46 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
GRB 501
Robert L. Wolpert Resolving GRB Light Curves 47 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
GRB 501
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
GRB 501 { Results
Figure: 95% Credible Interval for Mean, GRB 501
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
GRB 501 { Results
Figure: 95% posterior predictive intervals for GRB 501
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IntroductionGamma-ray bursts
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Discussion
ModelGRB 501GRB 493
GRB 501 { Results
(a) (b)
Figure: Number of Pulses, GRB 501
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IntroductionGamma-ray bursts
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Discussion
ModelGRB 501GRB 493
GRB 501 { Overdispersion
Figure: Variance in ation
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IntroductionGamma-ray bursts
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Discussion
ModelGRB 501GRB 493
GRB 501 { Parameter Estimates
Parameter Pulse 1 Pulse 2 Pulse 3T (s) (�0:542;�0:344) (1:58; 2:44) (�0:602;�0:287)V ( /s) (10:2; 19:3) (1:02; 6:46) (1:00; 1:45) (s) (0:5923; 0:875) (0:0119; 1:370) (�0:665;�0:154)� (s�1) (0:344; 0:511) (0:415; 1:19) (1:01; 1:59)� (�0:229; 0:0) (�2:09; �0:001) (�1:42; �0:053)Ec (keV) (53:1; 71:5) (28:3; 386:0 (79:3; 362:0)
Table: Highest posterior density intervals (95%) for the parameters of thethree-pulse models, GRB 501.
Robert L. Wolpert Resolving GRB Light Curves 53 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
Robert L. Wolpert Resolving GRB Light Curves 54 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
GRB 493 { Doesn't always work as desired!
Figure: 95% Credible Interval for Mean, GRB 493
Robert L. Wolpert Resolving GRB Light Curves 55 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
ModelGRB 501GRB 493
GRB 493 { Doesn't always work as desired!
(a) (b)
Figure: Number of Pulses, GRB 493
Robert L. Wolpert Resolving GRB Light Curves 56 / 60
IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Outline
Introduction
Gamma-ray bursts
LARK for Time SeriesModelSample of Results
LARK for Joint Time and EnergyModelGRB 501GRB 493
Discussion
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IntroductionGamma-ray bursts
LARK for Time SeriesLARK for Joint Time and Energy
Discussion
Discussion
I Our model successfully models the number and shapes ofpulses for some GRBs, but not others, due to strict modelingof the rises of pulses as \instantaneous" in incident space.
I This modeling approach does not depend on �xed-length timeor energy bins
I Our approach successfully incorporates knowledge of thephoton detector and corrects for systematic bias in photonenergy assignment.
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IntroductionGamma-ray bursts
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Discussion
Future work
I Flexible model for better rise-time modeling
I Improve sampling for better e�ective sample sizes
I Incorporate 16-channel data (MER) instead of 4 (DISCLA) forimproved recovery of spectral parameters
I Share information among bursts to infer populationparameters
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IntroductionGamma-ray bursts
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Discussion
Thanks for your attention!
Thanks too to NSF (DMS & PHY) and NASA AISR Program,and to Jon Hakkila (College of Charleston, Dept Physics & Astro).
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