Galois action on knots I: Action of the absolute Galois group

Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties. Particularly an action of the absolute Galois group of the rational number field on profinite knots is rigorously established, which is attained by our extending the action of Drinfel0d’s Grothendieck–Teichmüller group on profinite braid groups into on profinite knots

Synthesis of nanoparticulate anatase and rutile crystallites at low temperatures in the Pluronic F127 microemulsion system

A low-temperature synthesis method for preparing nanosized TiO(2) crystallites has been developed based on a Pluronic F127 microemulsion system. Both anatase and rutile polymorphs can be prepared, and there exists a temperature window between 40 and 50 degrees C where the formation of rutile is favored over anatase. At 60 degrees C and above, anatase is kinetically favored and only very slowly transforms to rutile at 60 degrees C. The results differ from previous observations regarding formation kinetics and temperature range for rutile formation as well as in the microscopic aggregation of the formed nanoparticles. This development of a low-temperature synthesis of crystalline titania nanoparticles within the Pluronic block copolymer system is an important and enabling step toward devising a direct synthesis route for the formation of ordered mesoporous and crystalline titania

M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra

We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by grt_1, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) E_n operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for grt_1-elements implies the hexagon equation

Chandra observation of the central galaxies in A1060 cluster of galaxies

Chandra observation of the central region of the A1060 cluster of galaxies resolved X-ray emission from two giant elliptical galaxies, NGC 3311 and NGC 3309. The emission from these galaxies consists of two components, namely the hot interstellar medium (ISM) and the low-mass X-ray binaries (LMXBs). We found the spatial extent of the ISM component was much smaller than that of stars for both galaxies, while the ratios of X-ray to optical blue-band luminosities were rather low but within the general scatter for elliptical galaxies. After subtracting the LMXB component, the ISM is shown to be in pressure balance with the intracluster medium of A1060 at the outer boundary of the ISM. These results imply that the hot gas supplied from stellar mass loss is confined by the external pressure of the intracluster medium, with the thermal conduction likely to be suppressed. The cD galaxy NGC 3311 does not exhibit the extended potential structure which is commonly seen in bright elliptical galaxies, and we discuss the possible evolution history of the very isothermal cluster A1060.Comment: 12 pages, 7 figures, Latex2e(emulateapj5), accepted in Ap

Detection of an X-Ray Hot Region in the Virgo Cluster of Galaxies with ASCA

Based on mapping observations with ASCA, an unusual hot region with a spatial extent of 1 square degree was discovered between M87 and M49 at a center coordinate of R. A. = 12h 27m 36s and Dec. = $9^\circ18'$ (J2000). The X-ray emission from the region has a 2-10 keV flux of $1 \times 10^{-11}$ ergs s$^{-1}$ cm$^{-2}$ and a temperature of $kT \gtrsim 4$ keV, which is significantly higher than that in the surrounding medium of $\sim 2$ keV. The internal thermal energy in the hot region is estimated to be $V n k T \sim 10^{60}$ ergs with a gas density of $\sim 10^{-4}$ cm$^{-3}$. A power-law spectrum with a photon index $1.7-2.3$ is also allowed by the data. The hot region suggests there is an energy input due to a shock which is probably caused by the motion of the gas associated with M49, infalling toward the M87 cluster with a velocity $\gtrsim 1000$ km s$^{-1}$.Comment: 12 pages, 3 figures, accepted to ApJ