19,916 research outputs found
New criteria to identify spectrum
In this paper we give some new criteria for identifying the components of a
probability measure, in its Lebesgue decomposition. This enables us to give new
criteria to identify spectral types of self-adjoint operators on Hilbert
spaces, especially those of interest.Comment: 10 page
Direct integrals and spectral averaging
A one parameter family of selfadjoint operators gives rise to a corresponding
direct integral. We show how to use the Putnam Kato theorem to obtain a new
method for the proof of a spectral averaging result
Persistence probabilities in centered, stationary, Gaussian processes in discrete time
Lower bounds for persistence probabilities of stationary Gaussian processes
in discrete time are obtained under various conditions on the spectral measure
of the process. Examples are given to show that the persistence probability can
decay faster than exponentially. It is shown that if the spectral measure is
not singular, then the exponent in the persistence probability cannot grow
faster than quadratically. An example that appears (from numerical evidence) to
achieve this lower bound is presented.Comment: 9 pages; To appear in a special volume of the Indian Journal of Pure
and Applied Mathematic
On the Evolution of Jet Energy and Opening Angle in Strongly Coupled Plasma
We calculate how the energy and the opening angle of jets in SYM
theory evolve as they propagate through the strongly coupled plasma of that
theory. We define the rate of energy loss and the jet opening
angle in a straightforward fashion directly in the gauge theory before
calculating both holographically, in the dual gravitational description. In
this way, we rederive the previously known result for without
the need to introduce a finite slab of plasma. We obtain a striking
relationship between the initial opening angle of the jet, which is to say the
opening angle that it would have had if it had found itself in vacuum instead
of in plasma, and the thermalization distance of the jet. Via this
relationship, we show that SYM jets with any initial energy that
have the same initial opening angle and the same trajectory through the plasma
experience the same fractional energy loss. We also provide an expansion that
describes how the opening angle of the SYM jets increases slowly
as they lose energy, over the fraction of their lifetime when their fractional
energy loss is not yet large. We close by looking ahead toward potential
qualitative lessons from our results for QCD jets produced in heavy collisions
and propagating through quark-gluon plasma.Comment: 40 pages, 9 figures, v2: minor clarifications adde
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