203 research outputs found
Computations in formal symplectic geometry and characteristic classes of moduli spaces
We make explicit computations in the formal symplectic geometry of Kontsevich
and determine the Euler characteristics of the three cases, namely commutative,
Lie and associative ones, up to certain weights.From these, we obtain some
non-triviality results in each case. In particular, we determine the integral
Euler characteristics of the outer automorphism groups Out F_n of free groups
for all n <= 10 and prove the existence of plenty of rational cohomology
classes of odd degrees. We also clarify the relationship of the commutative
graph homology with finite type invariants of homology 3-spheres as well as the
leaf cohomology classes for transversely symplectic foliations. Furthermore we
prove the existence of several new non-trivalent graph homology classes of odd
degrees. Based on these computations, we propose a few conjectures and problems
on the graph homology and the characteristic classes of the moduli spaces of
graphs as well as curves.Comment: 33 pages, final version, to appear in Quantum Topolog
Relative information entropy and Weyl curvature of the inhomogeneous Universe
Penrose conjectured a connection between entropy and Weyl curvature of the
Universe. This is plausible, as the almost homogeneous and isotropic Universe
at the onset of structure formation has negligible Weyl curvature, which then
grows (relative to the Ricci curvature) due to the formation of large-scale
structure and thus reminds us of the second law of thermodynamics. We study two
scalar measures to quantify the deviations from a homogeneous and isotropic
space-time, the relative information entropy and a Weyl tensor invariant, and
show their relation to the averaging problem. We calculate these two quantities
up to second order in standard cosmological perturbation theory and find that
they are correlated and can be linked via the kinematic backreaction of a
spatially averaged universe model.Comment: 8 pages, matches the published version in Physical Review
Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra
We determine the abelianizations of the following three kinds of graded Lie
algebras in certain stable ranges: derivations of the free associative algebra,
derivations of the free Lie algebra and symplectic derivations of the free
associative algebra. In each case, we consider both the whole derivation Lie
algebra and its ideal consisting of derivations with positive degrees. As an
application of the last case, and by making use of a theorem of Kontsevich, we
obtain a new proof of the vanishing theorem of Harer concerning the top
rational cohomology group of the mapping class group with respect to its
virtual cohomological dimension.Comment: 30 pages, 18 figures. Title modified, final version, to appear in
Duke Math.
Information Entropy in Cosmology
The effective evolution of an inhomogeneous cosmological model may be
described in terms of spatially averaged variables. We point out that in this
context, quite naturally, a measure arises which is identical to a fluid model
of the `Kullback-Leibler Relative Information Entropy', expressing the
distinguishability of the local inhomogeneous mass density field from its
spatial average on arbitrary compact domains. We discuss the time-evolution of
`effective information' and explore some implications. We conjecture that the
information content of the Universe -- measured by Relative Information Entropy
of a cosmological model containing dust matter -- is increasing.Comment: LateX, PRLstyle, 4 pages; to appear in PR
土壌拡散のCO2放出量測定にもとづく火山体内部での熱水流体上昇を支配する要因の解明 : 浅間火山での事例研究
学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 中井 俊一, 東京大学教授 中田 節也, 東京大学教授 武尾 実, 東京大学准教授 上嶋 誠, 東京大学准教授 森 俊哉University of Tokyo(東京大学
Extending Lagrangian perturbation theory to a fluid with velocity dispersion
We formulate a perturbative approximation to gravitational instability, based
on Lagrangian hydrodynamics in Newtonian cosmology. We take account of
`pressure' effect of fluid, which is kinematically caused by velocity
dispersion, to aim hydrodynamical description beyond shell crossing. Master
equations in the Lagrangian description are derived and solved perturbatively
up to second order. Then, as an illustration, power spectra of density
fluctuations are computed in a one-dimensional model from the Lagrangian
approximations and Eulerian linear perturbation theory for comparison. We find
that the results by the Lagrangian approximations are different from those by
the Eulerian one in weakly non-linear regime at the scales smaller than the
Jeans length. We also show the validity of the perturbative Lagrangian
approximations by consulting difference between the first-order and the
second-order approximations.Comment: 14 pages, 4 figures, LaTeX 2.09 using mn.sty, accepted for
publication in MNRA
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