Homotopy Classification of Generalized Phrases in Turaev's Theory of Words

In 2005 V. Turaev introduced the theory of topology of words and phrases. Turaev defined an equivalence relation on generalized words and phrases which is called homotopy. This is suggested by the Reidemeister moves in the knot theory. Then Turaev gave the homotopy classification of generalized words with less than or equal to five letters. In this paper we give the classification of generalized phrases up to homotopy with less than or equal to three letters. To do this we construct a new homotopy invariant for nanophrases over any $\alpha$.Comment: 12 page

Homotopy classification of nanophrases with less than or equal to four letters

In this paper we give the stable classification of ordered, pointed, oriented multi-component curves on surfaces with minimal crossing number less than or equal to 2 such that any equivalent curve has no simply closed curves in its components. To do this, we use the theory of words and phrases which was introduced by V. Turaev. Indeed we give the homotopy classification of nanophrases with less than or equal to 4 letters. It is an extension of the classification of nanophrases of length 2 with less than or equal to 4 letters which was given by the author in a previous paper. This is a corrected version of Hokkaido University Preprint Series in Mathematics #921. I corrected the subsection 5.3 and added proofs of propositions.Comment: 15 pages, 2 figures. This is a corrected version of Hokkaido University Preprint Series in Mathematics #92

An Interpretation of the Evidence for TeV Emission from Gamma-Ray Burst 970417a

The Milagrito collaboration recently reported evidence for emission of very high energy gamma-rays in the TeV range from one of the BATSE GRBs, GRB 970417a. Here I discuss possible interpretations of this result. Taking into account the intergalactic absorption of TeV gamma-rays by the cosmic infrared background, I found that the detection rate (one per 54 GRBs observed by the Milagrito) and energy fluence can be consistently explained with the redshift of this GRB at z \sim 0.7 and the isotropic total energy in the TeV range, E_{TeV, iso} >~ 10^{54} erg. This energy scale is not unreasonably large, but interestingly similar to the maximum total GRB energy observed to date, in the sub-MeV range for GRB 990123. On the other hand, the energy emitted in the ordinary sub-MeV range becomes E_{MeV, iso} \sim 10^{51} erg for the GRB 970417a, which is much smaller than the total energy in the TeV range by a factor of about 10^3. I show that the proton-synchrotron model of GRBs provides a possible explanation for these observational results. I also discuss some observational signatures expected in the future experiments from this model.Comment: 6 pages, 2 figures, Accepted by ApJ Letter

Mutual information of excited states and relative entropy of two disjoint subsystems in CFT

In this paper, we first study mutual information of excited states in the small subsystem size limit in generic conformal field theory. We then discuss relative entropy of two disjoint subsystems in the same limit.Comment: 18 page

Cosmological Fast Radio Bursts from Binary Neutron Star Mergers

Fast radio bursts (FRBs) at cosmological distances have recently been discovered, whose duration is about milliseconds. We argue that the observed short duration is difficult to explain by giant flares of soft gamma-ray repeaters, though their event rate and energetics are consistent with FRBs. Here we discuss binary neutron star (NS-NS) mergers as a possible origin of FRBs. The FRB rate is within the plausible range of NS-NS merger rate and its cosmological evolution, while a large fraction of NS-NS mergers must produce observable FRBs. A likely radiation mechanism is coherent radio emission like radio pulsars, by magnetic braking when magnetic fields of neutron stars are synchronized to binary rotation at the time of coalescence. Magnetic fields of the standard strength (~ 10^{12-13} G) can explain the observed FRB fluxes, if the conversion efficiency from magnetic braking energy loss to radio emission is similar to that of isolated radio pulsars. Corresponding gamma-ray emission is difficult to detect by current or past gamma-ray burst satellites. Since FRBs tell us the exact time of mergers, a correlated search would significantly improve the effective sensitivity of gravitational wave detectors.Comment: 4 pages, no figure. Matches the published version in PASJ. References added. This is an open access paper at the PASJ website http://pasj.asj.or.jp/v65/n5/65L012/65L012.pd