# R. Shen - Hebrew University of Jerusalem

## Contact Details

NameR. Shen |
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AffiliationHebrew University of Jerusalem |
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CityJerusalem |
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Location |
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## Pubs By Year |
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## Pub CategoriesPhysics - Mesoscopic Systems and Quantum Hall Effect (16) Mathematics - Analysis of PDEs (9) High Energy Astrophysical Phenomena (9) Physics - Strongly Correlated Electrons (5) Physics - Superconductivity (4) Quantum Physics (3) Mathematics - Mathematical Physics (3) Mathematical Physics (3) Solar and Stellar Astrophysics (2) Mathematics - Quantum Algebra (2) Physics - Materials Science (2) Mathematics - Classical Analysis and ODEs (2) Mathematics - Group Theory (2) Computer Science - Computer Vision and Pattern Recognition (2) Mathematics - General Topology (1) Statistics - Machine Learning (1) Physics - Statistical Mechanics (1) Cosmology and Nongalactic Astrophysics (1) Physics - Disordered Systems and Neural Networks (1) Mathematics - Representation Theory (1) Statistics - Applications (1) Physics - Soft Condensed Matter (1) Physics - Chemical Physics (1) |

## Publications Authored By R. Shen

Sparse coding (SC) is an automatic feature extraction and selection technique that is widely used in unsupervised learning. However, conventional SC vectorizes the input images, which breaks apart the local proximity of pixels and destructs the elementary object structures of images. In this paper, we propose a novel two-dimensional sparse coding (2DSC) scheme that represents the input images as the tensor-linear combinations under a novel algebraic framework. Read More

Sparse coding (SC) is an unsupervised learning scheme that has received an increasing amount of interests in recent years. However, conventional SC vectorizes the input images, which destructs the intrinsic spatial structures of the images. In this paper, we propose a novel graph regularized tensor sparse coding (GTSC) for image representation. Read More

We obtain a generalization of the DeGiorgi Lemma to the infinitely degenerate regime and apply it to obtain sharp H\"older continuity of weak solutions to certain infinitely degenerate equations. This establishes H\"older continuity for the same weak solutions that were proved to be locally bounded in arXiv:1608.01630v2. Read More

The surface states of three-dimensional topological insulators posses the unique property of spin-momentum interlocking. This property gives rise to the interesting inverse Edelstein effect (IEE), in which an applied spin bias $\mu$ is converted to a measurable charge voltage difference $V$. We develop a semiclassical theory for the IEE of the surface states of $\text{Bi}_2\text{Se}_3$ thin films, which is applicable from the ballistic regime to diffusive regime. Read More

We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form equations, and the local boundedness turns out to be sharp in more than two dimensions, answering the `Moser gap' problem left open in arXiv:1506.09203v5. Finally we obtain a maximum principle for weak solutions under a very mild condition on the degeneracy function f (x), essentially that -ln f (x) is merely doubling on (0;1). Read More

**Affiliations:**

^{1}Hebrew University,

^{2}Tel Aviv University,

^{3}Hebrew University

Recent observation of some luminous transient sources with low color temperatures suggests that the emission is dominated by optically thick winds driven by super-Eddington accretion. We present a general analytical theory of the dynamics of radiation pressure-driven, optically thick winds. Unlike the classical adiabatic stellar wind solution whose dynamics are solely determined by the sonic radius, here the loss of the radiation pressure due to photon diffusion also plays an important role. Read More

The point contact tunnel junctions between a one-dimensional topological superconductor and single-channel quantum Hall (QH) liquids are investigated theoretically with bosonization technology and renormalization group methods. For the $\nu=1$ integer QH liquid, the universal low-energy tunneling transport is governed by the perfect Andreev reflection fixed point with quantized zero-bias conductance $G(0)=2e^{2}/h$, which can serve as a definitive fingerprint of the existence of a Majorana fermion. For the $\nu =1/m$ Laughlin fractional QH liquids, its transport is governed by the perfect normal reflection fixed point with vanishing zero-bias conductance and bias-dependent conductance $G(V) \sim V^{m-2}$. Read More

Topological insulators (TIs) are a new quantum state of matter discovered recently, which are characterized by unconventional bulk topological invariants. Proposals for practical applications of the TIs are mostly based upon their metallic surface or edge states. Here, we report the theoretical discovery of a bulk quantum pumping effect in a two-dimensional TI electrically modulated in adiabatic cycles. Read More

In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $p \in [3,5)$. We prove that if initial data $(u_0, u_1)$ are radial so that $\|\nabla u_0\|_{L^2 ({\mathbb R}^3; d\mu)}, \|u_1\|_{L^2 ({\mathbb R}^3; d\mu)} \leq \infty$, where $d \mu = (|x|+1)^{1+2\varepsilon}$ with $\varepsilon > 0$, then the corresponding solution $u$ must exist for all time $t \in {\mathbb R}$ and scatter. The key ingredients of the proof include a transformation $\mathbf{T}$ so that $v = \mathbf{T} u$ solves the equation $v_{\tau \tau} - \Delta_y v = - \left(\frac{|y|}{\sinh |y|}\right)^{p-1} e^{-(p-3)\tau} |v|^{p-1}v$ with a finite energy, and a couple of global space-time integral estimates regarding a solution $v$ as above. Read More

In this paper, we classify the finite simple groups with an abelian Sylow subgroup. Read More

In this work we consider an energy subcritical semi-linear wave equation ($3 < p < 5$) \[ \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] with initial data $(u,u_t)|_{t=0} = (u_0,u_1)\in \dot{H}^{s_p} \times \dot{H}^{s_p-1}({\mathbb R}^3)$, where $s_p = 3/2 - 2/(p-1)$ and the function $\phi: {\mathbb R}^3 \rightarrow [-1,1]$ is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation $-\Delta W = \phi(x) |W|^{p-1} W$ has a nonzero radial solution $W \in C^2 ({\mathbb R}^3) \cap \dot{H}^{s_p} ({\mathbb R}^3)$, any radial solution $u$ with a finite uniform upper bound on the critical Sobolev norm $\|(u(\cdot,t), \partial_t u(\cdot,t))\|_{\dot{H}^{s_p}\times \dot{H}^{s_p}({\mathbb R}^3)}$ for all $t$ in the maximal lifespan must be a global solution in time and scatter. Read More

We develop subrepresentation inequalities for infinitely degenerate metrics, and obtain corresponding Poincare and Sobolev inequalities. We then derive conditions on the degenerate metric under which weak solutions to associated infinitely degenerate equations with rough coefficients are locally bounded, satisfy a maximum principle, or are continuous. As an application we obtain W-hypoellipticity of certain infinitely degenerate quasilinear equations with smooth coefficients having mild nonlinearities and degeneracies. Read More

In this work we consider a semi-linear energy critical wave equation in ${\mathbb R}^d$ ($3\leq d \leq 5$) \[ \partial_t^2 u - \Delta u = \pm \phi(x) |u|^{4/(d-2)} u, \qquad (x,t)\in {\mathbb R}^d \times {\mathbb R} \] with initial data $(u, \partial_t u)|_{t=0} = (u_0,u_1) \in \dot{H}^1 \times L^2 ({\mathbb R}^d)$. Here the function $\phi \in C({\mathbb R}^d; (0,1])$ converges to zero as $|x| \rightarrow \infty$. We follow the same compactness-rigidity argument as Kenig and Merle applied on the Cauchy problem of the equation \[ \partial_t^2 u - \Delta u = |u|^{4/(d-2)} u \] and obtain a similar result when $\phi$ satisfies some technical conditions. Read More

**Affiliations:**

^{1}Hebrew University of Jerusalem,

^{2}Hebrew University of Jerusalem,

^{3}Tel Aviv University,

^{4}Hebrew University of Jerusalem

**Category:**High Energy Astrophysical Phenomena

The nature of ultra-luminous X-ray sources (ULXs) has long been plagued by an ambiguity about whether the central compact objects are intermediate-mass (IMBH, >~ 10^3 M_sun) or stellar-mass (a few tens M_sun) black holes (BHs). The high luminosity (~ 10^39 erg/s) and super-soft spectrum (T ~ 0.1 keV) during the high state of the ULX source X-1 in the galaxy M101 suggest a large emission radius (>~ 10^9 cm), consistent with being an IMBH accreting at a sub-Eddington rate. Read More

We study topological phase transitions in one dimensional (1-D) Rashba nanowire under a spatially varying Zeeman field when coupled to an $s$-wave superconductor substrate. We show that this system supports both Majorana bound states (MBS) and fractionally charged bound states (FBS) of Jackiw-Rebbi type. By disassembling Zeeman Hamiltonian into multiple helical components, we find that each helical component is relating to a corresponding topological region, characterized by the emergence of MBS. Read More

In this paper we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space ${\mathbb H}^n$ with $3 \leq n \leq 5$: \[ \partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = \zeta |u|^{4/(n-2)} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}. \] Here $\zeta = \pm 1$ and $\rho = (n-1)/2$ are constants. We introduce a family of Strichartz estimates compatible with initial data in the energy space $H^{0,1} \times L^2 ({\mathbb H}^n)$ and then establish a local theory with these initial data. Read More

**Category:**

A solution to the stability of capacitor-less low-dropout regulators with a 4pF Miller capacitor in Multi-level current amplifier is proposed. With the Miller compensation, a more than 50{\deg}phase margin is guaranteed in full load. An extra fast transient circuit is adopted to reduce stable time and peak voltage. Read More

In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space \[ \partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = - |u|^{p-1} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}; \] and introduce a Morawetz-type inequality \[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} |u|^{p+1} d\mu dt < C E, \] where $E$ is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in $H^{1/2,1/2} \times H^{1/2,-1/2}({\mathbb H}^n)$ if $2 \leq n \leq 6$ and $1

0. Read More

We propose a one-dimensional electron model with parameters modulated adiabatically in closed cycles, which can continuously pump spin to leads. By defining the spin-polarized Wannier functions, we show that the spin pump is protected by the spin Chern numbers, so that it is stable to perturbations violating the time-reversal symmetry and spin conservation. Our work demonstrates the possibility and principle to realize topological spin pumps independent of any symmetries, and also suggests a possible way to experimentally observe the bulk topological invariants. Read More

We propose an Andreev interferometer, based on a branched Y-junction, to detect the finite momentum pairing in Fulde-Ferrell (FF) superconductors. In this interferometer, the oscillation of subgap conductance is a unique function of phase difference between the two channels of the Y-junction, which is determined by the phase modulation of the order parameter in the FF superconductors. This interferometer has the potential not only to determine the magnitude but also the direction of the momentum of Cooper pairs in the FF superconductor. Read More

In this paper, we construct a new class of modules for the Schr\"{o}dinger algebra $\mS$, called quasi-Whittaker module. Different from \cite{[ZC]}, the quasi-Whittaker module is not induced by the Borel subalgebra of the Schr\"{o}dinger algebra related with the triangular decomposition, but its Heisenberg subalgebra $\mH$. We prove that, for a simple $\mS$-module $V$, $V$ is a quasi-Whittaker module if and only if $V$ is a locally finite $\mH$-module; Furthermore, we classify the simple quasi-Whittaker modules by the elements with the action similar to the center elements in $U(\mS)$ and their quasi-Whittaker vectors. Read More

We propose that a Floquet Weyl semimetal state can be induced in three-dimensional topological insulators, either nonmagnetic or magnetic, by the application of off-resonant light. The virtual photon processes play a critical role in renormalizing the Dirac mass and so resulting in a topological semimetal with vanishing gap at Weyl points. The present mechanism via off-resonant light is quite different from that via on-resonant light, the latter being recently suggested to give rise to a Floquet topological state in ordinary band insulators. Read More

We propose to implement quantum computing based on electronic spin qubits by controlling the propagation of the electron wave packets through the helical edge states of quantum spin Hall systems (QSHs). Specfically, two non-commutative single-qubit gates, which rotate a qubit around z and y axes, can be realized by utilizing gate voltages either on a single QSH edge channel or on a quantum point contact structure. The more challenging two-qubit controlled phase gate can be implemented through the on-demand capacitive Coulomb interaction between two adjacent edge channels from two parallel QSHs. Read More

**Affiliations:**

^{1}University of Toronto,

^{2}University of Toronto

**Category:**High Energy Astrophysical Phenomena

During a stellar tidal disruption event (TDE), an accretion disk forms as stellar debris returns to the disruption site and circularizes. Rather than being confined within the circularizing radius, the disk can spread to larger radii to conserve angular momentum. A spreading disk is a source of matter for re-accretion at rates which can exceed the later stellar fallback rate, although a disk wind can suppress its contribution to the central black hole accretion rate. Read More

The electron-hole conversion at the normal-metal superconductor interface in inversion-symmetric Weyl semimetals is investigated with an effective two-band model. We find that the specular Andreev reflection of Weyl fermions has two unusual features. The Andreev conductance for s-wave BCS pairing states is anisotropic, depending on the angle between the line connecting a pair of Weyl points and the normal of the junction, due to opposite chirality carried by the paired electrons. Read More

We propose an entanglement detector composed of two quantum spin Hall insulators and a side gate deposited on one of the edge channels. For an ac gate voltage, the differential noise contributed from the entangled electron pairs exhibits the nontrivial step structures, from which the spin entanglement concurrence can be easily obtained. The possible spin dephasing effects in the quantum spin Hall insulators are also included. Read More

High resolution microarrays and second-generation sequencing platforms are powerful tools to investigate genome-wide alterations in DNA copy number, methylation and gene expression associated with a disease. An integrated genomic profiling approach measures multiple omics data types simultaneously in the same set of biological samples. Such approach renders an integrated data resolution that would not be available with any single data type. Read More

We propose to realize Majorana fermions (MFs) on an edge of a two-dimensional topological insulator in the proximity with s-wave superconductors and in the presence of transverse exchange field h. It is shown that there appear a pair of MFs localized at two junctions and that a reverse in direction of h can lead to permutation of two MFs. With decreasing h, the MF states can either be fused or form one Dirac fermion on the {\pi}-junctions, exhibiting a topological phase transition. Read More

We analyze the reading and initialization of a topological qubit encoded by Majorana fermions in one-dimensional semiconducting nanowires, weakly coupled to a single level quantum dot (QD). It is shown that when the Majorana fermions are fused by tuning gate voltage, the topological qubit can be read out directly through the occupation of the QD in an energy window. The initialization of the qubit can also be realized via adjusting the gate voltage on the QD, with the total fermion parity conserved. Read More

The quantum spin Hall (QSH) effect is known to be unstable to perturbations violating time-reversal symmetry. We show that creating a narrow ferromagnetic (FM) region near the edge of a QSH sample can push one of the counterpropagating edge states to the inner boundary of the FM region, and leave the other at the outer boundary, without changing their spin polarizations and propagation directions. Since the two edge states are spatially separated into different "lanes", the QSH effect becomes robust against symmetry-breaking perturbations. Read More

Topological phase transitions in a three-dimensional (3D) topological
insulator (TI) with an exchange field of strength $g$ are studied by
calculating spin Chern numbers $C^\pm(k_z)$ with momentum $k_z$ as a parameter.
When $|g|$ exceeds a critical value $g_c$, a transition of the 3D TI into a
Weyl semimetal occurs, where two Weyl points appear as critical points
separating $k_z$ regions with different first Chern numbers. For $|g|

The Chern number is often used to distinguish between different topological phases of matter in two-dimensional electron systems. A fast and efficient coupling-matrix method is designed to calculate the Chern number in finite crystalline and disordered systems. To show its effectiveness, we apply the approach to the Haldane model and the lattice Hofstadter model, the quantized Chern numbers being correctly obtained. Read More

In this paper, Lie bialgebra structures on generalized Heisenberg-Virasoro algebra $\mathfrak{L}$ are considered. Also, $H^1({\mathfrak{L}} ,\mathfrak{L}\bigotimes\mathfrak{L})$ is given explicitly. Moreover, it is proved that all Lie bialgebra structure on centerless generalized Heisenberg-Virasoro algebra $\bar{\mathfrak{L}}$ are coboundary triangular. Read More

In a recent paper by the authors, Lie bialgebra structures on generalized Heisenberg- Virasoro algebra L are considered. In this paper, the explicit formula of the quantization on generalized Heisenberg-Virasoro algebra is presented. Read More

Evaporation is a fundamental physical phenomenon, of which many challenging questions remain unanswered. Enhanced evaporation of liquids in some occasions is of enormous practical significance. Here we report the enhanced evaporation of the nearly permanently stable silicone oil by dispersing with nanopariticles including CaTiO3, anatase and rutile TiO2. Read More

In this paper, we consider the wave equation in 3-dimensional space with an energy-subcritical nonlinearity, either in the focusing or defocusing case. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined in time and scatters. The proof depends on the compactness/rigidity argument, decay estimates for radial, "compact" solutions, gain of regularity arguments and the "channel of energy" method. Read More

We design an ingenious spintronic quantum eraser to quantitatively probe the two-electron entanglement. It is shown that the concurrence of two spin-entangled electrons is directly given by the Aharonov-Bohm oscillation amplitude of the Fano factor, a measurable current-current correlation, making it rather promising to experimentally quantify the two-electron entanglement. The singlet and triplet entangled states are distinguished by the opposite signs in the Fano factor. Read More

GRB 120422A is a low-luminosity Gamma-ray burst (GRB) associated with a bright supernova, which distinguishes itself by its relatively short T90 ~ 5 s and an energetic X-ray tail. We analyze the Swift BAT and XRT data and discuss the physical implications. We show that the early steep decline in the X-ray light curve can be interpreted as the curvature tail of a late emission episode around 58-86 s, with a curved instantaneous spectrum at the end of the emission episode. Read More

In this paper the author considers the global existence and well-posedness of the non-linear wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in 3-dimensional space, assuming that the initial data is in the space $(\dot{H}^s \cap \dot{H}^{s_p}) \times (\dot{H}^{s-1} \cap \dot{H}^{s_p-1})$, with $11/3

Read More

The disorder-driven metal-insulator transition in the quantum spin Hall systems is studied by scaling analysis of the Thouless conductance $g$. Below a critical disorder strength, the conductance is independent of the sample size $M$, an indication of critically delocalized electron states. The calculated beta function $\beta=d\ln g/d\ln M$ indicates that the metal-insulator transition is Kosterlitz-Thouless (KT) type, which is characterized by bounding and unbounding of vortex-antivortex pairs of the local currents. Read More

The paper deals with the defocusing case of the energy subcritical non-linear wave equation in $R^3$. We assume the initial data is in the space $\dot{H}^s \times \dot{H}^{s-1}$ and radial. If $s=1$, this is the energy space and the scattering results are known. Read More

We study the relationship between bipartite entanglement, subsystem particle number and topology in a half-filled free fermion system. It is proposed that the spin-projected particle numbers can distinguish the quantum spin Hall state from other states, and can be used to establish a new topological index for the system. Furthermore, we apply the new topological invariant to a disordered system and show that a topological phase transition occurs when the disorder strength is increased beyond a critical value. Read More

**Authors:**W. Zheng, R. F. Shen, T. Sakamoto, A. P. Beardmore, M. De Pasquale, X. F. Wu, J. Gorosabel, Y. Urata, S. Sugita, B. Zhang, A. Pozanenko, M. Nissinen, D. K. Sahu, M. Im, T. N. Ukwatta, M. Andreev, E. Klunko, A. Volnova, C. W. Akerlof, P. Anto, S. D. Barthelmy, A. Breeveld, U. Carsenty, S. Castillo-Carri'on, A. J. Castro-Tirado, M. M. Chester, C. J. Chuang, R. Cunniffe, A. De Ugarte Postigo, R. Duffard, H. Flewelling, N. Gehrels, T. Guver, S. Guziy, V. P. Hentunen, K. Y. Huang, M. Jelínek, T. S. Koch, P. Kub'anek, P. Kuin, T. A. McKay, S. Mottola, S. R. Oates, P. O'Brien, M. J. Page, S. B. Pandey, C. Perez del Pulgar, W. Rujopakarn, E. Rykoff, T. Salmi, R. S'anchez-Ramírez, B. E. Schaefer, A. Sergeev, E. Sonbas, A. Sota, J. C. Tello, K. Yamaoka, S. A. Yost, F. Yuan

**Category:**High Energy Astrophysical Phenomena

We present a comprehensive analysis of a bright, long duration (T90 ~ 257 s) GRB 110205A at redshift z= 2.22. The optical prompt emission was detected by Swift/UVOT, ROTSE-IIIb and BOOTES telescopes when the GRB was still radiating in the gamma-ray band. Read More

We mainly discuss the cardinal invariants and generalized metric properties on paratopological groups or rectifiable spaces, and show that: (1) If $A$ and $B$ are $\omega$-narrow subsets of a paratopological group $G$, then $AB$ is $\omega$-narrow in $G$, which give an affirmative answer for \cite[Open problem 5.1.9]{A2008}; (2) Every bisequential or weakly first-countable rectifiable space is metrizable; (3) The properties of Fr$\acute{e}$chet-Urysohn and strongly Fr$\acute{e}$chet-Urysohn are coincide in rectifiable spaces; (4) Every rectifiable space $G$ contains a (closed) copy of $S_{\omega}$ if and only if $G$ has a (closed) copy of $S_{2}$; (5) If a rectifiable space $G$ has a $\sigma$-point-discrete closed $k$-network, then $G$ contains no closed copy of $S_{\omega_{1}}$; (6) If a rectifiable space $G$ is pointwise canonically weakly pseudocompact, then $G$ is a Moscow space. Read More

**Affiliations:**

^{1}University of Toronto,

^{2}University of Toronto

**Category:**High Energy Astrophysical Phenomena

The plateaus observed in about one half of the early X-ray afterglows are the most puzzling feature in gamma-ray bursts (GRBs) detected by Swift. By analyzing the temporal and spectral indices of a large X-ray plateau sample, we find that 55% can be explained by external, forward shock synchrotron emission produced by a relativistic ejecta coasting in a \rho ~ r^{-2}, wind-like medium; no energy injection into the shock is needed. After the ejecta collects enough medium and transitions to the adiabatic, decelerating blastwave phase, it produces the post-plateau decay. Read More

Observational evidence suggests a link between long duration gamma ray bursts (LGRBs) and Type Ic supernovae. Here, we propose a potential mechanism for Type Ic supernovae in LGRB progenitors powered solely by accretion energy. We present spherically-symmetric hydrodynamic simulations of the long-term accretion of a rotating gamma-ray burst progenitor star, a "collapsar," onto the central compact object, which we take to be a black hole. Read More

Gamma-ray burst X-ray flares are believed to mark the late time activity of the central engine. We compute the temporal evolution of the average flare luminosity $< L >$ in the common rest frame energy band of 44 GRBs taken from the large \emph{Swift} 5-years data base. Our work highlights the importance of a proper consideration of the threshold of detection of flares against the contemporaneous continuous X-ray emission. Read More

Since learning is typically very slow in Boltzmann machines, there is a need to restrict connections within hidden layers. However, the resulting states of hidden units exhibit statistical dependencies. Based on this observation, we propose using $l_1/l_2$ regularization upon the activation possibilities of hidden units in restricted Boltzmann machines to capture the loacal dependencies among hidden units. Read More

The association of long-duration gamma-ray bursts (LGRBs) with Type Ic supernovae presents a challenge to supernova explosion models. In the collapsar model for LGRBs, gamma rays are produced in an ultrarelativistic jet launching from the magnetosphere of the black hole that forms in the aftermath of the collapse of a rotating progenitor star. The jet is collimated along the star's rotation axis, but the concomitant luminous supernova should be relatively--though certainly not entirely--spherical, and should synthesize a substantial mass of 56Ni. Read More

GRB 090417B was an unusually long burst with a T_90 duration of at least 2130 s and a multi-peaked light curve at energies of 15-150 keV. It was optically dark and has been associated with a bright star-forming galaxy at a redshift of 0.345 that is broadly similar to the Milky Way. Read More