506 research outputs found
Regular vs. classical M\"obius transformations of the quaternionic unit ball
The regular fractional transformations of the extended quaternionic space
have been recently introduced as variants of the classical linear fractional
transformations. These variants have the advantage of being included in the
class of slice regular functions, introduced by Gentili and Struppa in 2006, so
that they can be studied with the useful tools available in this theory. We
first consider their general properties, then focus on the regular M\"obius
transformations of the quaternionic unit ball B, comparing the latter with
their classical analogs. In particular we study the relation between the
regular M\"obius transformations and the Poincar\'e metric of B, which is
preserved by the classical M\"obius transformations. Furthermore, we announce a
result that is a quaternionic analog of the Schwarz-Pick lemma.Comment: 14 page
Incommensurability and edge states in the one-dimensional S=1 bilinear-biquadratic model
Commensurate-incommensurate change on the one-dimensional S=1
bilinear-biquadratic model () is examined. The gapped
Haldane phase has two subphases (the commensurate Haldane subphase and the
incommensurate Haldane subphase) and the commensurate-incommensurate change
point (the Affleck-Kennedy-Lieb-Tasaki point, ). There have been
two different analytical predictions about the static structure factor in the
neighborhood of this point. By using the S{\o}rensen-Affleck prescription,
these static structure factors are related to the Green functions, and also to
the energy gap behaviors. Numerical calculations support one of the
predictions. Accordingly, the commensurate-incommensurate change is recognized
as a motion of a pair of poles in the complex plane.Comment: 29 pages, 15 figure
String Interactions from Matrix String Theory
The Matrix String Theory, i.e. the two dimensional U(N) SYM with N=(8,8)
supersymmetry, has classical BPS solutions that interpolate between an initial
and a final string configuration via a bordered Riemann surface. The Matrix
String Theory amplitudes around such a classical BPS background, in the strong
Yang--Mills coupling, are therefore candidates to be interpreted in a stringy
way as the transition amplitude between given initial and final string
configurations. In this paper we calculate these amplitudes and show that the
leading contribution is proportional to the factor g_s^{-\chi}, where \chi is
the Euler characteristic of the interpolating Riemann surface and g_s is the
string coupling. This is the factor one expects from perturbative string
interaction theory.Comment: 15 pages, 2 eps figures, JHEP Latex class, misprints correcte
Inversion of perturbation series
We investigate the inversion of perturbation series and its resummation, and
prove that it is related to a recently developed parametric perturbation
theory. Results for some illustrative examples show that in some cases series
reversion may improve the accuracy of the results
A reliable Pade analytical continuation method based on a high accuracy symbolic computation algorithm
We critique a Pade analytic continuation method whereby a rational polynomial
function is fit to a set of input points by means of a single matrix inversion.
This procedure is accomplished to an extremely high accuracy using a novel
symbolic computation algorithm. As an example of this method in action we apply
it to the problem of determining the spectral function of a one-particle
thermal Green's function known only at a finite number of Matsubara frequencies
with two example self energies drawn from the T-matrix theory of the Hubbard
model. We present a systematic analysis of the effects of error in the input
points on the analytic continuation, and this leads us to propose a procedure
to test quantitatively the reliability of the resulting continuation, thus
eliminating the black magic label frequently attached to this procedure.Comment: 11 pages, 8 eps figs, revtex format; revised version includes
reference to anonymous ftp site containing example codes (MapleVr5.1
worksheets) displaying the implementation of the algorithm, including the
padematinv.m library packag
Regular Moebius transformations of the space of quaternions
Let H be the real algebra of quaternions. The notion of regular function of a
quaternionic variable recently presented by G. Gentili and D. C. Struppa
developed into a quite rich theory. Several properties of regular quaternionic
functions are analogous to those of holomorphic functions of one complex
variable, although the diversity of the quaternionic setting introduces new
phenomena. This paper studies regular quaternionic transformations. We first
find a quaternionic analog to the Casorati-Weierstrass theorem and prove that
all regular injective functions from H to itself are affine. In particular, the
group Aut(H) of biregular functions on H coincides with the group of regular
affine transformations. Inspired by the classical quaternionic linear
fractional transformations, we define the regular fractional transformations.
We then show that each regular injective function from the Alexandroff
compactification of H to itself is a regular fractional transformation.
Finally, we study regular Moebius transformations, which map the unit ball B
onto itself. All regular bijections from B to itself prove to be regular
Moebius transformations.Comment: 12 page
Fluctuation force exerted by a planar self-avoiding polymer
Using results from Schramm Loewner evolution (SLE), we give the expression of
the fluctuation-induced force exerted by a polymer on a small impenetrable
disk, in various 2-dimensional domain geometries. We generalize to two polymers
and examine whether the fluctuation force can trap the object into a stable
equilibrium. We compute the force exerted on objects at the domain boundary,
and the force mediated by the polymer between such objects. The results can
straightforwardly be extended to any SLE interface, including Ising,
percolation, and loop-erased random walks. Some are relevant for extremal value
statistics.Comment: 7 pages, 22 figure
Quasisymmetric graphs and Zygmund functions
A quasisymmetric graph is a curve whose projection onto a line is a
quasisymmetric map. We show that this class of curves is related to solutions
of the reduced Beltrami equation and to a generalization of the Zygmund class
. This relation makes it possible to use the tools of harmonic
analysis to construct nontrivial examples of quasisymmetric graphs and of
quasiconformal maps.Comment: 21 pages, no figure
The Universal Phase Space of AdS3 Gravity
We describe what can be called the "universal" phase space of AdS3 gravity,
in which the moduli spaces of globally hyperbolic AdS spacetimes with compact
spatial sections, as well as the moduli spaces of multi-black-hole spacetimes
are realized as submanifolds. The universal phase space is parametrized by two
copies of the Universal Teichm\"uller space T(1) and is obtained from the
correspondence between maximal surfaces in AdS3 and quasisymmetric
homeomorphisms of the unit circle. We also relate our parametrization to the
Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the
holographic (Fefferman-Graham) description. In particular, we obtain a relation
between the generators of quasiconformal deformations in each T(1) sector and
the chiral Brown-Henneaux vector fields. We also relate the charges arising in
the holographic description (such as the mass and angular momentum of an AdS3
spacetime) to the periods of the quadratic differentials arising via the Bers
embedding of T(1)xT(1). Our construction also yields a symplectic map from
T*T(1) to T(1)xT(1) generalizing the well-known Mess map in the compact spatial
surface setting.Comment: 41 pages, 2 figures, revised version accepted for publication in
Commun.Math.Phy
Monte Carlo study of the hull distribution for the q=1 Brauer model
We study a special case of the Brauer model in which every path of the model
has weight q=1. The model has been studied before as a solvable lattice model
and can be viewed as a Lorentz lattice gas. The paths of the model are also
called self-avoiding trails. We consider the model in a triangle with boundary
conditions such that one of the trails must cross the triangle from a corner to
the opposite side. Motivated by similarities between this model, SLE(6) and
critical percolation, we investigate the distribution of the hull generated by
this trail (the set of points on or surrounded by the trail) up to the hitting
time of the side of the triangle opposite the starting point. Our Monte Carlo
results are consistent with the hypothesis that for system size tending to
infinity, the hull distribution is the same as that of a Brownian motion with
perpendicular reflection on the boundary.Comment: 21 pages, 9 figure
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