Lie Algebras Represented as a Sum of Two Subalgebras

Let L be a Lie algebra represented as a sum of two subalgebras A and B. We prove that if L belongs to a subclass of the class of locally finite Lie algebras over a field of characteristic &#8800; 2 and both A and B are locally nilpotent, then L is locally soluble. We also prove that if L is a serially finite Lie algebra over a field of characteristic zero, then any common serial subalgebra of A and B is serial in L.</p

SERIALLY COALESCENT CLASSES OF LIE ALGEBRAS

We introduce the concept of serially coalescent classes of Lie algebras corresponding to those of coalescent classes and ascendantly coalescent classes. We show that the class of finite-dimensional and nilpotent, the class of finite-dimensional and the class of finite-dimensional and soluble Lie algebras, are serially coalescent classes for locally finite Lie algebras over any field of characteristic zero. We also introduce the concept of locally serially coalescent classes of Lie algebras and find some locally serially coalescent classes for locally finite Lie algebras.</p

Spectral Comparison of Weak Short Bursts to the Persistent X-rays from the Magnetar 1E 1547.0-5408 in its 2009 Outburst

In January 2009, the 2.1-sec anomalous X-ray pulsar 1E 1547.0-5408 evoked intense burst activity. A follow-up Suzaku observation on January 28 recorded enhanced persistent emission both in soft and hard X-rays (Enoto et al. 2010b). Through re-analysis of the same Suzaku data, 18 short bursts were identified in the X-ray events recorded by the Hard X-ray Detector (HXD) and the X-ray Imaging Spectrometer (XIS). Their spectral peaks appear in the HXD-PIN band, and their 10-70 keV X-ray fluences range from ~2e-9 erg cm-2 to 1e-7 erg cm-2. Thus, the 18 events define a significantly weaker burst sample than was ever obtained, ~1e-8-1e-4 erg cm-2. In the ~0.8 to ~300 keV band, the spectra of the three brightest bursts can be represented successfully by a two-blackbody model, or a few alternative ones. A spectrum constructed by stacking 13 weaker short bursts with fluences in the range (0.2-2)e-8 erg s-1 is less curved, and its ratio to the persistent emission spectrum becomes constant at ~170 above ~8 keV. As a result, the two-blackbody model was able to reproduce the stacked weaker-burst spectrum only after adding a power-law model, of which the photon index is fixed at 1.54 as measured is the persistent spectrum. These results imply a possibility that the spectrum composition employing an optically-thick component and a hard power-law component can describe wide-band spectra of both the persistent and weak-burst emissions, despite a difference of their fluxes by two orders of magnitude. Based on the spectral similarity, a possible connection between the unresolved short bursts and the persistent emission is discussed.Comment: 21 pages, 18 figures and 3 tables. Accepted for publication in Monthly Notices of the Royal Astronomical Society Main Journa

Probing the Structure of Gamma-Ray Burst Jets with Steep Decay Phase of their Early X-ray Afterglows

We show that the jet structure of gamma-ray bursts (GRBs) can be investigated with the tail emission of the prompt GRB. The tail emission which we consider is identified as a steep-decay component of the early X-ray afterglow observed by the X-ray Telescope onboard Swift. Using a Monte Carlo method, we derive, for the first time, the distribution of the decay index of the GRB tail emission for various jet models. The new definitions of the zero of time and the time interval of a fitting region are proposed. These definitions for fitting the light curve lead us an unique definition of the decay index, which is useful to investigate the structure of the GRB jet. We find that if the GRB jet has a core-envelope structure, the predicted distribution of the decay index of the tail has a wide scatter and has multiple peaks, which cannot be seen for the case of the uniform and the Gaussian jet. Therefore, the decay index distribution tells us the information on the jet structure. Especially, if we observe events whose decay index is less than about 2, both the uniform and the Gaussian jet models will be disfavored according to our simulation study.Comment: 21 pages, 10 figures, the paper with full resolution images is http://theo.phys.sci.hiroshima-u.ac.jp/~takami/research/achievements/papers/003_full.pd