10,497 research outputs found
Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools
The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and
exactly solvable models, is a method for solving integrable partial
differential equations governing wave propagation in certain nonlinear media.
The NFT decorrelates signal degrees-of-freedom in such models, in much the same
way that the Fourier transform does for linear systems. In this three-part
series of papers, this observation is exploited for data transmission over
integrable channels such as optical fibers, where pulse propagation is governed
by the nonlinear Schr\"odinger equation. In this transmission scheme, which can
be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing
commonly used in linear channels, information is encoded in the nonlinear
frequencies and their spectral amplitudes. Unlike most other fiber-optic
transmission schemes, this technique deals with both dispersion and
nonlinearity directly and unconditionally without the need for dispersion or
nonlinearity compensation methods. This first paper explains the mathematical
tools that underlie the method.Comment: This version contains minor updates of IEEE Transactions on
Information Theory, vol. 60, no. 7, pp. 4312--4328, July 201
Spectral pollution and second order relative spectra for self-adjoint operators
We consider the phenomenon of spectral pollution arising in calculation of
spectra of self-adjoint operators by projection methods. We suggest a strategy
of dealing with spectral pollution by using the so-called second order relative
spectra. The effectiveness of the method is illustrated by a detailed analysis
of two model examples.Comment: 36 pages, 18 figures, AMS-LaTe
Efficient computation of the second-Born self-energy using tensor-contraction operations
In the nonequilibrium Green's function approach, the approximation of the
correlation self-energy at the second-Born level is of particular interest,
since it allows for a maximal speed-up in computational scaling when used
together with the Generalized Kadanoff-Baym Ansatz for the Green's function.
The present day numerical time-propagation algorithms for the Green's function
are able to tackle first principles simulations of atoms and molecules, but
they are limited to relatively small systems due to unfavourable scaling of
self-energy diagrams with respect to the basis size. We propose an efficient
computation of the self-energy diagrams by using tensor-contraction operations
to transform the internal summations into functions of external low-level
linear algebra libraries. We discuss the achieved computational speed-up in
transient electron dynamics in selected molecular systems.Comment: 9 pages, 4 figures, 1 tabl
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