8,948 research outputs found
On the diagonalization of the discrete Fourier transform
The discrete Fourier transform (DFT) is an important operator which acts on
the Hilbert space of complex valued functions on the ring Z/NZ. In the case
where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors
for the DFT. The transition matrix from the standard basis to the canonical
basis defines a novel transform which we call the discrete oscillator transform
(DOT for short). Finally, we describe a fast algorithm for computing the
discrete oscillator transform in certain cases.Comment: Accepted for publication in the journal "Applied and Computational
Harmonic Analysis": Appl. Comput. Harmon. Anal. (2009),
doi:10.1016/j.acha.2008.11.003. Key words: Discrete Fourier Transform, Weil
Representation, Canonical Eigenvectors, Oscillator Transform, Fast Oscillator
Transfor
Linear approach to the orbiting spacecraft thermal problem
We develop a linear method for solving the nonlinear differential equations
of a lumped-parameter thermal model of a spacecraft moving in a closed orbit.
Our method, based on perturbation theory, is compared with heuristic
linearizations of the same equations. The essential feature of the linear
approach is that it provides a decomposition in thermal modes, like the
decomposition of mechanical vibrations in normal modes. The stationary periodic
solution of the linear equations can be alternately expressed as an explicit
integral or as a Fourier series. We apply our method to a minimal thermal model
of a satellite with ten isothermal parts (nodes) and we compare the method with
direct numerical integration of the nonlinear equations. We briefly study the
computational complexity of our method for general thermal models of orbiting
spacecraft and conclude that it is certainly useful for reduced models and
conceptual design but it can also be more efficient than the direct integration
of the equations for large models. The results of the Fourier series
computations for the ten-node satellite model show that the periodic solution
at the second perturbative order is sufficiently accurate.Comment: 20 pages, 11 figures, accepted in Journal of Thermophysics and Heat
Transfe
Banach algebras of pseudodifferential operators and their almost diagonalization
We define new symbol classes for pseudodifferntial operators and investigate
their pseudodifferential calculus. The symbol classes are parametrized by
commutative convolution algebras. To every solid convolution algebra over a
lattice we associate a symbol class. Then every operator with such a symbol is
almost diagonal with respect to special wave packets (coherent states or Gabor
frames), and the rate of almost diagonalization is described precisely by the
underlying convolution algebra. Furthermore, the corresponding class of
pseudodifferential operators is a Banach algebra of bounded operators on . If a version of Wiener's lemma holds for the underlying convolution algebra,
then the algebra of pseudodifferential operators is closed under inversion. The
theory contains as a special case the fundamental results about Sj\"ostrand's
class and yields a new proof of a theorem of Beals about the H\"ormander class
of order 0.Comment: 28 page
Symmetry-Adapted Phonon Analysis of Nanotubes
The characteristics of phonons, i.e. linearized normal modes of vibration,
provide important insights into many aspects of crystals, e.g. stability and
thermodynamics. In this paper, we use the Objective Structures framework to
make concrete analogies between crystalline phonons and normal modes of
vibration in non-crystalline but highly symmetric nanostructures. Our strategy
is to use an intermediate linear transformation from real-space to an
intermediate space in which the Hessian matrix of second derivatives is
block-circulant. The block-circulant nature of the Hessian enables us to then
follow the procedure to obtain phonons in crystals: namely, we use the Discrete
Fourier Transform from this intermediate space to obtain a block-diagonal
matrix that is readily diagonalizable. We formulate this for general Objective
Structures and then apply it to study carbon nanotubes of various chiralities
that are subjected to axial elongation and torsional deformation. We compare
the phonon spectra computed in the Objective Framework with spectra computed
for armchair and zigzag nanotubes. We also demonstrate the approach by
computing the Density of States. In addition to the computational efficiency
afforded by Objective Structures in providing the transformations to
almost-diagonalize the Hessian, the framework provides an important conceptual
simplification to interpret the phonon curves.Comment: To appear in J. Mech. Phys. Solid
Measuring the galaxy power spectrum and scale-scale correlations with multiresolution-decomposed covariance -- I. method
We present a method of measuring galaxy power spectrum based on the
multiresolution analysis of the discrete wavelet transformation (DWT). Since
the DWT representation has strong capability of suppressing the off-diagonal
components of the covariance for selfsimilar clustering, the DWT covariance for
popular models of the cold dark matter cosmogony generally is diagonal, or
(scale)-diagonal in the scale range, in which the second scale-scale
correlations are weak. In this range, the DWT covariance gives a lossless
estimation of the power spectrum, which is equal to the corresponding Fourier
power spectrum banded with a logarithmical scaling. In the scale range, in
which the scale-scale correlation is significant, the accuracy of a power
spectrum detection depends on the scale-scale or band-band correlations. This
is, for a precision measurements of the power spectrum, a measurement of the
scale-scale or band-band correlations is needed. We show that the DWT
covariance can be employed to measuring both the band-power spectrum and second
order scale-scale correlation. We also present the DWT algorithm of the binning
and Poisson sampling with real observational data. We show that the alias
effect appeared in usual binning schemes can exactly be eliminated by the DWT
binning. Since Poisson process possesses diagonal covariance in the DWT
representation, the Poisson sampling and selection effects on the power
spectrum and second order scale-scale correlation detection are suppressed into
minimum. Moreover, the effect of the non-Gaussian features of the Poisson
sampling can be calculated in this frame.Comment: AAS Latex file, 44 pages, accepted for publication in Ap
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