469 research outputs found
Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes
New algorithms for computing of asymptotic expansions for stationary
distributions of nonlinearly perturbed semi-Markov processes are presented. The
algorithms are based on special techniques of sequential phase space reduction,
which can be applied to processes with asymptotically coupled and uncoupled
finite phase spaces.Comment: 83 page
Computing generalized inverses using LU factorization of matrix product
An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and
the Moore-Penrose inverse of a given rational matrix A is established. Classes
A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R*
and T*(AT*)+, where R and T are rational matrices with appropriate dimensions
and corresponding rank. The proposed algorithm is based on these general
representations and the Cholesky factorization of symmetric positive matrices.
The algorithm is implemented in programming languages MATHEMATICA and DELPHI,
and illustrated via examples. Numerical results of the algorithm, corresponding
to the Moore-Penrose inverse, are compared with corresponding results obtained
by several known methods for computing the Moore-Penrose inverse
Rates of convergence for the approximation of dual shift-invariant systems in
A shift-invariant system is a collection of functions of the
form . Such systems play an important role in
time-frequency analysis and digital signal processing. A principal problem is
to find a dual system such that each
function can be written as . The
mathematical theory usually addresses this problem in infinite dimensions
(typically in or ), whereas numerical methods have to operate
with a finite-dimensional model. Exploiting the link between the frame operator
and Laurent operators with matrix-valued symbol, we apply the finite section
method to show that the dual functions obtained by solving a finite-dimensional
problem converge to the dual functions of the original infinite-dimensional
problem in . For compactly supported (FIR filter banks) we
prove an exponential rate of convergence and derive explicit expressions for
the involved constants. Further we investigate under which conditions one can
replace the discrete model of the finite section method by the periodic
discrete model, which is used in many numerical procedures. Again we provide
explicit estimates for the speed of convergence. Some remarks on tight frames
complete the paper
Numerical calculation of three-point branched covers of the projective line
We exhibit a numerical method to compute three-point branched covers of the
complex projective line. We develop algorithms for working explicitly with
Fuchsian triangle groups and their finite index subgroups, and we use these
algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate
Holographic Calculation for Large Interval R\'enyi Entropy at High Temperature
In this paper, we study the holographic R\'enyi entropy of a large interval
on a circle at high temperature for the two-dimensional conformal field theory
(CFT) dual to pure AdS gravity. In the field theory, the R\'enyi entropy is
encoded in the CFT partition function on -sheeted torus connected with each
other by a large branch cut. As proposed by Chen and Wu [Large interval limit
of R\'enyi entropy at high temperature, arXiv:1412.0763], the effective way to
read the entropy in the large interval limit is to insert a complete set of
state bases of the twist sector at the branch cut. Then the calculation
transforms into an expansion of four-point functions in the twist sector with
respect to . By using the operator product expansion of
the twist operators at the branch points, we read the first few terms of the
R\'enyi entropy, including the leading and next-to-leading contributions in the
large central charge limit. Moreover, we show that the leading contribution is
actually captured by the twist vacuum module. In this case by the Ward identity
the four-point functions can be derived from the correlation function of four
twist operators, which is related to double interval entanglement entropy.
Holographically, we apply the recipe in [T. Faulkner, The entanglement R\'enyi
entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221] and [T. Barrella
et al., Holographic entanglement beyond classical gravity, J. High Energy Phys.
09 (2013) 109] to compute the classical R\'enyi entropy and its one-loop
quantum correction, after imposing a new set of monodromy conditions. The
holographic classical result matches exactly with the leading contribution in
the field theory up to and , while the holographical
one-loop contribution is in exact agreement with next-to-leading results in
field theory up to and as well.Comment: minor corrections, match with the published versio
Low-complexity computation of plate eigenmodes with Vekua approximations and the Method of Particular Solutions
This paper extends the Method of Particular Solutions (MPS) to the
computation of eigenfrequencies and eigenmodes of plates. Specific
approximation schemes are developed, with plane waves (MPS-PW) or
Fourier-Bessel functions (MPS-FB). This framework also requires a suitable
formulation of the boundary conditions. Numerical tests, on two plates with
various boundary conditions, demonstrate that the proposed approach provides
competitive results with standard numerical schemes such as the Finite Element
Method, at reduced complexity, and with large flexibility in the implementation
choices
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
Decimated generalized Prony systems
We continue studying robustness of solving algebraic systems of Prony type
(also known as the exponential fitting systems), which appear prominently in
many areas of mathematics, in particular modern "sub-Nyquist" sampling
theories. We show that by considering these systems at arithmetic progressions
(or "decimating" them), one can achieve better performance in the presence of
noise. We also show that the corresponding lower bounds are closely related to
well-known estimates, obtained for similar problems but in different contexts
- …