2,066 research outputs found
The discrete sine transform and the spectrum of the finite q-ary tree.
We compute the spectrum of the finite q-ary tree using radon transforms and the discrete sine transform
Harmonic analysis of finite lamplighter random walks
Recently, several papers have been devoted to the analysis of lamplighter
random walks, in particular when the underlying graph is the infinite path
. In the present paper, we develop a spectral analysis for
lamplighter random walks on finite graphs. In the general case, we use the
-symmetry to reduce the spectral computations to a series of eigenvalue
problems on the underlying graph. In the case the graph has a transitive
isometry group , we also describe the spectral analysis in terms of the
representation theory of the wreath product . We apply our theory to
the lamplighter random walks on the complete graph and on the discrete circle.
These examples were already studied by Haggstrom and Jonasson by probabilistic
methods.Comment: 29 page
Multi-scale initial conditions for cosmological simulations
We discuss a new algorithm to generate multi-scale initial conditions with
multiple levels of refinements for cosmological "zoom-in" simulations. The
method uses an adaptive convolution of Gaussian white noise with a real space
transfer function kernel together with an adaptive multi-grid Poisson solver to
generate displacements and velocities following first (1LPT) or second order
Lagrangian perturbation theory (2LPT). The new algorithm achieves RMS relative
errors of order 10^(-4) for displacements and velocities in the refinement
region and thus improves in terms of errors by about two orders of magnitude
over previous approaches. In addition, errors are localized at coarse-fine
boundaries and do not suffer from Fourier-space induced interference ringing.
An optional hybrid multi-grid and Fast Fourier Transform (FFT) based scheme is
introduced which has identical Fourier space behaviour as traditional
approaches. Using a suite of re-simulations of a galaxy cluster halo our real
space based approach is found to reproduce correlation functions, density
profiles, key halo properties and subhalo abundances with per cent level
accuracy. Finally, we generalize our approach for two-component baryon and
dark-matter simulations and demonstrate that the power spectrum evolution is in
excellent agreement with linear perturbation theory. For initial baryon density
fields, it is suggested to use the local Lagrangian approximation in order to
generate a density field for mesh based codes that is consistent with
Lagrangian perturbation theory instead of the current practice of using the
Eulerian linearly scaled densities.Comment: 22 pages, 24 figures. MNRAS in press. Updated affiliation
Orthonormal and biorthonormal filter banks as convolvers, and convolutional coding gain
Convolution theorems for filter bank transformers are introduced. Both uniform and nonuniform decimation ratios are considered, and orthonormal as well as biorthonormal cases are addressed. All the theorems are such that the original convolution reduces to a sum of shorter, decoupled convolutions in the subbands. That is, there is no need to have cross convolution between subbands. For the orthonormal case, expressions for optimal bit allocation and the optimized coding gain are derived. The contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum. With one of the convolved sequences taken to be the unit pulse function,,e coding gain expressions reduce to those for traditional subband and transform coding. The filter-bank convolver has about the same computational complexity as a traditional convolver, if the analysis bank has small complexity compared to the convolution itself
Rolling tachyon in anti-de Sitter space-time
We study the decay of the unstable D-particle in three-dimensional anti-de
Sitter space-time using worldsheet boundary conformal field theory methods. We
test the open string completeness conjecture in a background for which the
phase space available is only field-theoretic. This could present a serious
challenge to the claim. We compute the emission of closed strings in the AdS(3)
x S^3 x T^4 background from the knowledge of the exact corresponding boundary
state we construct. We show that the energy stored in the brane is mainly
converted into very excited long strings. The energy stored in short strings
and in open string pair production is much smaller and finite for any value of
the string coupling. We find no "missing energy" problem. We compare our
results to those obtained for a decay in flat space-time and to a background in
the presence of a linear dilaton. Some remarks on holographic aspects of the
problem are made.Comment: JHEP style, 45 pages, one figure; v2: typos corrected, references
added, version to appear in JHE
Orbifold boundary states from Cardy's condition
Boundary states for D-branes at orbifold fixed points are constructed in
close analogy with Cardy's derivation of consistent boundary states in RCFT.
Comments are made on the interpretation of the various coefficients in the
explicit expressions, and the relation between fractional branes and wrapped
branes is investigated for orbifolds. The boundary states
are generalised to theories with discrete torsion and a new check is performed
on the relation between discrete torsion phases and projective representations.Comment: LaTeX2e, 50 pages, 5 figures. V3: final version to appear on JHEP
(part of a section moved to an appendix, titles of some references added, one
sentence in the introduction expanded
D-brane Decay in Two-Dimensional String Theory
We consider unstable D0-branes of two dimensional string theory, described by
the boundary state of Zamolodchikov and Zamolodchikov [hep-th/0101152]
multiplied by the Neumann boundary state for the time coordinate . In the
dual description in terms of the matrix model, this D0-brane is described
by a matrix eigenvalue on top of the upside down harmonic oscillator potential.
As suggested by McGreevy and Verlinde [hep-th/0304224], an eigenvalue rolling
down the potential describes D-brane decay. As the eigenvalue moves down the
potential to the asymptotic region it can be described as a free relativistic
fermion. Bosonizing this fermion we get a description of the state in terms of
a coherent state of the tachyon field in the asymptotic region, up to a
non-local linear field redefinition by an energy-dependent phase. This coherent
state agrees with the exponential of the closed string one-point function on a
disk with Sen's marginal boundary interaction for which describes D0-brane
decay.Comment: 19 pages, harvmac, minor change
Kondo lattice on the edge of a two-dimensional topological insulator
We revisit the problem of a single quantum impurity on the edge of a
two-dimensional time-reversal invariant topological insulator and show that the
zero temperature phase diagram contains a large local moment region for
antiferromagnetic Kondo coupling which was missed by previous poor man's
scaling treatments. The combination of an exact solution at the so-called
decoupling point and a renormalization group analysis \`a la
Anderson-Yuval-Hamann allows us to access the regime of strong
electron-electron interactions on the edge and strong Kondo coupling. We apply
similar methods to the problem of a regular one-dimensional array of quantum
impurities interacting with the edge liquid. When the edge electrons are at
half-filling with respect to the impurity lattice, the system remains gapless
unless the Luttinger parameter of the edge is less than 1/2, in which case
two-particle backscattering effects drive the system to a gapped phase with
long-range Ising antiferromagnetic order. This is in marked contrast with the
gapped disordered ground state of the ordinary half-filled one-dimensional
Kondo lattice.Comment: 18 pages, 3 figures; fixed typos, updated reference
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