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Regularity at infinity of real mappings and a Morse-Sard theorem
We prove a new Morse-Sard type theorem for the asymptotic critical values of
semi-algebraic mappings and a new fibration theorem at infinity for
mappings. We show the equivalence of three different types of regularity
conditions which have been used in the literature in order to control the
asymptotic behaviour of mappings. The central role of our picture is played by
the -regularity and its bridge toward the -regularity which implies
topological triviality at infinity
Formation of laser-induced periodic surface structures on niobium by femtosecond laser irradiation
Hamiltonian thermodynamics of three-dimensional dilatonic black holes
The action for a class of three-dimensional dilaton-gravity theories with a
cosmological constant can be recast in a Brans-Dicke type action, with its free
parameter. These theories have static spherically symmetric black
holes. Those with well formulated asymptotics are studied through a Hamiltonian
formalism, and their thermodynamical properties are found out. The theories
studied are general relativity (), a dimensionally reduced
cylindrical four-dimensional general relativity theory (), and a
theory representing a class of theories (). The Hamiltonian
formalism is setup in three dimensions through foliations on the right region
of the Carter-Penrose diagram, with the bifurcation 1-sphere as the left
boundary, and anti-de Sitter infinity as the right boundary. The metric
functions on the foliated hypersurfaces are the canonical coordinates. The
Hamiltonian action is written, the Hamiltonian being a sum of constraints. One
finds a new action which yields an unconstrained theory with one pair of
canonical coordinates , being the mass parameter and its
conjugate momenta The resulting Hamiltonian is a sum of boundary terms only. A
quantization of the theory is performed. The Schr\"odinger evolution operator
is constructed, the trace is taken, and the partition function of the canonical
ensemble is obtained. The black hole entropies differ, in general, from the
usual quarter of the horizon area due to the dilaton.Comment: 34 pages, 3 figures, references added, minor changes in the revised
versio
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