The Brown measure of a sum of two free random variables, one of which is triangular elliptic

The triangular elliptic operators are natural extensions of the elliptic deformation of circular operators. We obtain a Brown measure formula for the sum of a triangular elliptic operator $g_{\alpha\beta, \gamma}$ with an arbitrary random variable $x_0$, which is $*$-free from $g_{\alpha\beta, \gamma}$ with amalgamation over certain unital subalgebra. Let $c_t$ be a circular operator. We prove that the Brown measure of $x_0 + g_{\alpha\beta, \gamma}$ is the push-forward measure of the Brown measure of $x_0 + c_t$ by an explicitly defined map on $\mathbb{C}$ for some suitable $t$. We show that the Brown measure of $x_0+c_t$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{C}$ and its density is bounded by $1/\pi{t}$. This work generalizes earlier results on the addition with a circular operator, semicircular operator, or elliptic operator to a larger class of operators. We extend operator-valued subordination functions, due to Biane and Voiculescu, to certain unbounded operators. This allows us to extend our results to unbounded operators.Comment: 42 pages. Section 7 was rewritte

Spectrum of Laplacian matrices associated with large random elliptic matrices

A Laplacian matrix is a square matrix whose row sums are zero. We study the limiting eigenvalue distribution of a Laplacian matrix formed by taking a random elliptic matrix and subtracting the diagonal matrix containing its row sums. Under some mild assumptions, we show that the empirical spectral distribution of the Laplacian matrix converges to a deterministic probability distribution as the size of the matrix tends to infinity. The limiting measure can be interpreted as the Brown measure of the sum of an elliptic operator and a freely independent normal operator with a Gaussian distribution.Comment: 42 pages; minor corrections, added additional reference

Signal Processing for Nondifferentiable Data Defined on Cantor Sets: A Local Fractional Fourier Series Approach

From the signal processing point of view, the nondifferentiable data defined on the Cantor sets are investigated in this paper. The local fractional Fourier series is used to process the signals, which are the local fractional continuous functions. Our results can be observed as significant extensions of the previously known results for the Fourier series in the framework of the local fractional calculus. Some examples are given to illustrate the efficiency and implementation of the present method

The spectrum of local random Hamiltonians

The spectrum of a local random Hamiltonian can be represented generically by the so-called $\epsilon$-free convolution of its local terms' probability distributions. We establish an isomorphism between the set of $\epsilon$-noncrossing partitions and permutations to study its spectrum. Moreover, we derive some lower and upper bounds for the largest eigenvalue of the Hamiltonian.Comment: 22 page

Is the late near-infrared bump in short-hard GRB 130603B due to the Li-Paczynski kilonova?

Short-hard gamma-ray bursts (GRBs) are widely believed to be produced by the merger of two binary compact objects, specifically by two neutron stars or by a neutron star orbiting a black hole. According to the Li-Paczynski kilonova model, the merger would launch sub-relativistic ejecta and a near-infrared/optical transient would then occur, lasting up to days, which is powered by the radioactive decay of heavy elements synthesized in the ejecta. The detection of a late bump using the {\em Hubble Space Telescope} ({\em HST}) in the near-infrared afterglow light curve of the short-hard GRB 130603B is indeed consistent with such a model. However, as shown in this Letter, the limited {\em HST} near-infrared lightcurve behavior can also be interpreted as the synchrotron radiation of the external shock driven by a wide mildly relativistic outflow. In such a scenario, the radio emission is expected to peak with a flux of $\sim 100 \mu$Jy, which is detectable for current radio arrays. Hence, the radio afterglow data can provide complementary evidence on the nature of the bump in GRB 130603B. It is worth noting that good spectroscopy during the bump phase in short-hard bursts can test validity of either model above, analogous to spectroscopy of broad-lined Type Ic supernova in long-soft GRBs.Comment: 4 pages, 2 figures, published in ApJ Lette

The lightcurve of the macronova associated with the long-short burst GRB 060614

The {\it Swift}-detected GRB 060614 was a unique burst that straddles an imaginary divide between long- and short-duration gamma-ray bursts (GRBs), and its physical origin has been heavily debated over the years. Recently, a distinct very-soft F814W-band excess at $t\sim 13.6$ days after the burst was identified in a joint-analysis of VLT and HST optical afterglow data of GRB~060614, which has been interpreted as evidence for an accompanying Li-Paczynski macronova (also called a kilonova). Under the assumption that the afterglow data in the time interval of $1.7-3.0$ days after the burst are due to external forward shock emission, when this assumption is extrapolated to later times it is found that there is an excess of flux in several multi-band photometric observations. This component emerges at $\sim$4 days after the burst, and it may represent the first time that a multi-epoch/band lightcurve of a macronova has been obtained. The macronova associated with GRB 060614 peaked at $t\lesssim 4$ days after the burst, which is significantly earlier than that observed for a supernova associated with a long-duration GRB. Due to the limited data, no strong evidence for a temperature evolution is found. We derive a conservative estimate of the macronova rate of $\sim 16.3^{+16.3}_{-8.2}~{\rm Gpc^{-3}}{\rm yr^{-1}}$, implying a promising prospect for detecting the gravitational wave radiation from compact object mergers by upcoming Advanced LIGO/VIRGO/KAGRA detectors (i.e., the rate is ${\cal R}_{\rm GW} \sim 0.5^{+0.5}_{-0.25}(D/200~{\rm Mpc})^{3}~{\rm yr^{-1}}$).Comment: The version published in ApJL. Fig.1 has been updated, main conclusions are unchange

A possible Macronova in the late afterglow of the `long-short' burst GRB 060614

Long-duration ($>2$ s) $\gamma$-ray bursts that are believed to originate from the death of massive stars are expected to be accompanied by supernovae. GRB 060614, that lasted 102 s, lacks a supernova-like emission down to very stringent limits and its physical origin is still debated. Here we report the discovery of near-infrared bump that is significantly above the regular decaying afterglow. This red bump is inconsistent with even the weakest known supernova. However, it can arise from a Li-Paczy\'{n}ski macronova $-$ the radioactive decay of debris following a compact binary merger. If this interpretation is correct GRB 060614 arose from a compact binary merger rather than from the death of a massive star and it was a site of a significant production of heavy r-process elements. The significant ejected mass favors a black hole-neutron star merger but a double neutron star merger cannot be ruled out.Comment: Minor revision; The version published in Nature Communication