31,130 research outputs found
Determining the luminosity function of Swift long gamma-ray bursts with pseudo-redshifts
The determination of luminosity function (LF) of gamma-ray bursts (GRBs) is
of an important role for the cosmological applications of the GRBs, which is
however hindered seriously by some selection effects due to redshift
measurements. In order to avoid these selection effects, we suggest to
calculate pseudo-redshifts for Swift GRBs according to the empirical L-E_p
relationship. Here, such a relationship is determined by reconciling
the distributions of pseudo- and real redshifts of redshift-known GRBs. The
values of E_p taken from Butler's GRB catalog are estimated with Bayesian
statistics rather than observed. Using the GRB sample with pseudo-redshifts of
a relatively large number, we fit the redshift-resolved luminosity
distributions of the GRBs with a broken-power-law LF. The fitting results
suggest that the LF could evolve with redshift by a redshift-dependent break
luminosity, e.g., L_b=1.2\times10^{51}(1+z)^2\rm erg s^{-1}. The low- and
high-luminosity indices are constrained to 0.8 and 2.0, respectively. It is
found that the proportional coefficient between GRB event rate and star
formation rate should correspondingly decrease with increasing redshifts.Comment: 5 pages, 5 figures, accepted for publication in ApJ
Nonnegative Matrix Inequalities and their Application to Nonconvex Power Control Optimization
Maximizing the sum rates in a multiuser Gaussian channel by power control is a nonconvex NP-hard problem that finds engineering application in code division multiple access (CDMA) wireless communication network. In this paper, we extend and apply several fundamental nonnegative matrix inequalities initiated by Friedland and Karlin in a 1975 paper to solve this nonconvex power control optimization problem. Leveraging tools such as the Perron–Frobenius theorem in nonnegative matrix theory, we (1) show that this problem in the power domain can be reformulated as an equivalent convex maximization problem over a closed unbounded convex set in the logarithmic signal-to-interference-noise ratio domain, (2) propose two relaxation techniques that utilize the reformulation problem structure and convexification by Lagrange dual relaxation to compute progressively tight bounds, and (3) propose a global optimization algorithm with ϵ-suboptimality to compute the optimal power control allocation. A byproduct of our analysis is the application of Friedland–Karlin inequalities to inverse problems in nonnegative matrix theory
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