8,886 research outputs found
A Note on the Diagonalization of the Discrete Fourier Transform
Following the approach developed by S. Gurevich and R. Hadani, an analytical
formula of the canonical basis of the DFT is given for the case where
is a prime number and (mod 4).Comment: 12 pages, accepted by Applied and Computational Harmonic Analysi
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
Efficient Quantum Transforms
Quantum mechanics requires the operation of quantum computers to be unitary,
and thus makes it important to have general techniques for developing fast
quantum algorithms for computing unitary transforms. A quantum routine for
computing a generalized Kronecker product is given. Applications include
re-development of the networks for computing the Walsh-Hadamard and the quantum
Fourier transform. New networks for two wavelet transforms are given. Quantum
computation of Fourier transforms for non-Abelian groups is defined. A slightly
relaxed definition is shown to simplify the analysis and the networks that
computes the transforms. Efficient networks for computing such transforms for a
class of metacyclic groups are introduced. A novel network for computing a
Fourier transform for a group used in quantum error-correction is also given.Comment: 30 pages, LaTeX2e, 7 figures include
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
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
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