13,910 research outputs found
Fixed point property for a CAT(0) space which admits a proper cocompact group action
We prove that if a geodesically complete space admits a
proper cocompact isometric action of a group, then the Izeki-Nayatani invariant
of is less than . Let be a finite connected graph, be
the linear spectral gap of , and be the nonlinear spectral
gap of with respect to such a space . Then, the result
implies that the ratio is bounded from below by a
positive constant which is independent of the graph . It follows that any
isometric action of a random group of the graph model on such has a global
fixed point. In particular, any isometric action of a random group of the graph
model on a Bruhat-Tits building associated to a semi-simple algebraic group has
a global fixed point
Uniform estimates of nonlinear spectral gaps
By generalizing the path method, we show that nonlinear spectral gaps of a
finite connected graph are uniformly bounded from below by a positive constant
which is independent of the target metric space. We apply our result to an
-ball in the -regular tree, and observe that the asymptotic
behavior of nonlinear spectral gaps of as does not
depend on the target metric space, which is in contrast to the case of a
sequence of expanders. We also apply our result to the -dimensional Hamming
cube and obtain an estimate of its nonlinear spectral gap with respect to
an arbitrary metric space, which is asymptotically sharp as .Comment: to appear in Graphs and Combinatoric
q-Deformed Bi-Local Fields II
We study a way of -deformation of the bi-local system, the two particle
system bounded by a relativistic harmonic oscillator type of potential, from
both points of view of mass spectra and the behavior of scattering amplitudes.
In our formulation, the deformation is done so that , the square of center
of mass momentum, enters into the deformation parameters of relative
coordinates. As a result, the wave equation of the bi-local system becomes
nonlinear with respect to ; then, the propagator of the bi-local system
suffers significant change so as to get a convergent self energy to the second
order. The study is also made on the covariant -deformation in four
dimensional spacetime.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Generation of Dicke States with Phonon-Mediated Multi-level Stimulated Raman Adiabatic Passage
We generate half-excited symmetric Dicke states of two and four ions. We use
multi-level stimulated Raman adiabatic passage (STIRAP) whose intermediate
states are phonon Fock states. This process corresponds to the spin squeezing
operation and half-excited Dicke states are generated during multi-level
STIRAP. This method does not require local access for each ion or the
preparation of phonon Fock states. Furthermore, it is robust since it is an
adiabatic process. We evaluate the Dicke state using a witness operator and
determine the upper and lower bounds of the fidelity without using full quantum
tomography.Comment: 5pages, 4 figure
- …