28,721 research outputs found
How unprovable is Rabin's decidability theorem?
We study the strength of set-theoretic axioms needed to prove Rabin's theorem
on the decidability of the MSO theory of the infinite binary tree. We first
show that the complementation theorem for tree automata, which forms the
technical core of typical proofs of Rabin's theorem, is equivalent over the
moderately strong second-order arithmetic theory to a
determinacy principle implied by the positional determinacy of all parity games
and implying the determinacy of all Gale-Stewart games given by boolean
combinations of sets. It follows that complementation for
tree automata is provable from - but not -comprehension.
We then use results due to MedSalem-Tanaka, M\"ollerfeld and
Heinatsch-M\"ollerfeld to prove that over -comprehension, the
complementation theorem for tree automata, decidability of the MSO theory of
the infinite binary tree, positional determinacy of parity games and
determinacy of Gale-Stewart games are all
equivalent. Moreover, these statements are equivalent to the
-reflection principle for -comprehension. It follows in
particular that Rabin's decidability theorem is not provable in
-comprehension.Comment: 21 page
Fringe spacing and phase of interfering matter waves
We experimentally investigate the outcoupling of atoms from Bose-Einstein
condensates using two radio-frequency (rf) fields in the presence of gravity.
We show that the fringe separation in the resulting interference pattern
derives entirely from the energy difference between the two rf fields and not
the gravitational potential difference. We subsequently demonstrate how the
phase and polarisation of the rf radiation directly control the phase of the
matter wave interference and provide a semi-classical interpretation of the
results.Comment: 4 pages, 3 figure
Magnetic effects in a holographic Fermi-like liquid
We explore the magnetic properties of the Fermi-like liquid represented by
the D3-D7' system. The system exhibits interesting magnetic properties such as
ferromagnetism and an anomalous Hall effect, which are due to the Chern-Simons
term in the effective gravitational action. We investigate the spectrum of
quasi-normal modes in the presence of a magnetic field and show that the
magnetic field mitigates the instability towards a striped phase. In addition,
we find a critical magnetic field above which the zero sound mode becomes
massive.Comment: 18 pages, 15 figure
Probing the classical field approximation - thermodynamics and decaying vortices
We review our version of the classical field approximation to the dynamics of
a finite temperature Bose gas. In the case of a periodic box potential, we
investigate the role of the high momentum cut-off, essential in the method. In
particular, we show that the cut-off going to infinity limit decribes the
particle number going to infinity with the scattering length going to zero. In
this weak interaction limit, the relative population of the condensate tends to
unity. We also show that the cross-over energy, at which the probability
distribution of the condensate occupation changes its character, grows with a
growing scattering length. In the more physical case of the condensate in the
harmonic trap we investigate the dissipative dynamics of a vortex. We compare
the decay time and the velocities of the vortex with the available analytic
estimates.Comment: 7 pages, 8 eps figures, submitted to J. Optics B for the proceedings
of the "Atom Optics and Interferometry" Lunteren 2002 worksho
Chiral primary one-point functions in the D3-D7 defect conformal field theory
JHEP is an open-access journal funded by SCOAP3 and licensed under CC BY 4.0archiveprefix: arXiv primaryclass: hep-th reportnumber: NORDITA-2012-81 slaccitation: %%CITATION = ARXIV:1210.7015;%%archiveprefix: arXiv primaryclass: hep-th reportnumber: NORDITA-2012-81 slaccitation: %%CITATION = ARXIV:1210.7015;%%C.F.K. and D.Y. were supported in part by FNU through grant number 272-08-0329.
G.W.S. is supported by NSERC of Canada and by the Villum foundation through their
Velux Visiting Professor program
Gravitational-wave astronomy: the high-frequency window
This contribution is divided in two parts. The first part provides a
text-book level introduction to gravitational radiation. The key concepts
required for a discussion of gravitational-wave physics are introduced. In
particular, the quadrupole formula is applied to the anticipated
``bread-and-butter'' source for detectors like LIGO, GEO600, EGO and TAMA300:
inspiralling compact binaries. The second part provides a brief review of high
frequency gravitational waves. In the frequency range above (say) 100Hz,
gravitational collapse, rotational instabilities and oscillations of the
remnant compact objects are potentially important sources of gravitational
waves. Significant and unique information concerning the various stages of
collapse, the evolution of protoneutron stars and the details of the
supranuclear equation of state of such objects can be drawn from careful study
of the gravitational-wave signal. As the amount of exciting physics one may be
able to study via the detections of gravitational waves from these sources is
truly inspiring, there is strong motivation for the development of future
generations of ground based detectors sensitive in the range from hundreds of
Hz to several kHz.Comment: 21 pages, 5 figures, Lectures presented at the 2nd Aegean Summer
School on the Early Universe, Syros, Greece, September 200
Collective excitations of a trapped Bose-condensed gas
By taking the hydrodynamic limit we derive, at , an explicit solution of
the linearized time dependent Gross-Pitaevskii equation for the order parameter
of a Bose gas confined in a harmonic trap and interacting with repulsive
forces. The dispersion law for the
elementary excitations is obtained, to be compared with the prediction
of the noninteracting harmonic oscillator model.
Here is the number of radial nodes and is the orbital angular
momentum. The effects of the kinetic energy pressure, neglected in the
hydrodynamic approximation, are estimated using a sum rule approach. Results
are also presented for deformed traps and attractive forces.Comment: uuencoded file including 12 pages REVTEX and 1 figur
O(4) texture with a cosmological constant
We investigate O(4) textures in a background with a positive cosmological
constant. We find static solutions which co-move with the expanding background.
There exists a solution in which the scalar field is regular at the horizon.
This solution has a noninteger winding number smaller than one. There also
exist solutions in which scalar-field derivatives are singular at the horizon.
Such solutions can complete one winding within the horizon. If the winding
number is larger than some critical value, static solutions including the
regular one are unstable under perturbations.Comment: 25 pages, revtex, 6 eps figure
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