199,428 research outputs found
Recurrence of cocycles and stationary random walks
We survey distributional properties of -valued cocycles of
finite measure preserving ergodic transformations (or, equivalently, of
stationary random walks in ) which determine recurrence or
transience.Comment: Published at http://dx.doi.org/10.1214/074921706000000112 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Basic zeta functions and some applications in physics
It is the aim of these lectures to introduce some basic zeta functions and
their uses in the areas of the Casimir effect and Bose-Einstein condensation. A
brief introduction into these areas is given in the respective sections. We
will consider exclusively spectral zeta functions, that is zeta functions
arising from the eigenvalue spectrum of suitable differential operators. There
is a set of technical tools that are at the very heart of understanding
analytical properties of essentially every spectral zeta function. Those tools
are introduced using the well-studied examples of the Hurwitz, Epstein and
Barnes zeta function. It is explained how these different examples of zeta
functions can all be thought of as being generated by the same mechanism,
namely they all result from eigenvalues of suitable (partial) differential
operators. It is this relation with partial differential operators that
provides the motivation for analyzing the zeta functions considered in these
lectures. Motivations come for example from the questions "Can one hear the
shape of a drum?" and "What does the Casimir effect know about a boundary?".
Finally "What does a Bose gas know about its container?"Comment: To appear in "A Window into Zeta and Modular Physics", Mathematical
Sciences Research Institute Publications, Vol. 57, 2010, Cambridge University
Pres
Stability and instability of Ricci solitons
We consider the volume-normalized Ricci flow close to compact shrinking Ricci
solitons. We show that if a compact Ricci soliton is a local maximum of
Perelman's shrinker entropy, any normalized Ricci flow starting close to it
exists for all time and converges towards a Ricci soliton. If is not a
local maximum of the shrinker entropy, we show that there exists a nontrivial
normalized Ricci flow emerging from it. These theorems are analogues of results
in the Ricci-flat and in the Einstein case.Comment: 23 pages, published versio
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