Channel model and lower capacity bound for the transmission based on nonlinear Fourier transform

The integrability of the nonlinear Schräodinger equation (NLSE) by the inverse scattering transform shown in a seminal work [1] gave an interesting opportunity to treat the corresponding nonlinear channel similar to a linear one by using the nonlinear Fourier transform. Integrability of the NLSE is in the background of the old idea of eigenvalue communications [2] that was resurrected in recent works [3{7]. In [6, 7] the new method for the coherent optical transmission employing the continuous nonlinear spectral data | nonlinear inverse synthesis was introduced. It assumes the modulation and detection of data using directly the continuous part of nonlinear spectrum associated with an integrable transmission channel (the NLSE in the case considered). Although such a transmission method is inherently free from nonlinear impairments, the noisy signal corruptions, arising due to the ampli¯er spontaneous emission, inevitably degrade the optical system performance. We study properties of the noise-corrupted channel model in the nonlinear spectral domain attributed to NLSE. We derive the general stochastic equations governing the signal evolution inside the nonlinear spectral domain and elucidate the properties of the emerging nonlinear spectral noise using well-established methods of perturbation theory based on inverse scattering transform [8]. It is shown that in the presence of small noise the communication channel in the nonlinear domain is the additive Gaussian channel with memory and signal-dependent correlation matrix. We demonstrate that the effective spectral noise acquires colouring", its autocorrelation function becomes slow decaying and non-diagonal as a function of \frequencies", and the noise loses its circular symmetry, becoming elliptically polarized. Then we derive a low bound for the spectral effiency for such a channel. Our main result is that by using the nonlinear spectral techniques one can significantly increase the achievable spectral effiency compared to the currently available methods [9]. REFERENCES 1. Zakharov, V. E. and A. B. Shabat, Sov. Phys. JETP, Vol. 34, 62{69, 1972. 2. Hasegawa, A. and T. Nyu, J. Lightwave Technol., Vol. 11, 395{399, 1993. 3. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4312{4328, 2014. 4. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4329{4345 2014. 5. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4346{4369, 2014. 6. Prilepsky, J. E., S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett., Vol. 113, 013901, 2014. 7. Le, S. T., J. E. Prilepsky, and S. K. Turitsyn, Opt. Express, Vol. 22, 26720{26741, 2014. 8. Kaup, D. J. and A. C. Newell, Proc. R. Soc. Lond. A, Vol. 361, 413{446, 1978. 9. Essiambre, R.-J., G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol., Vol. 28, 662{701, 2010.

Demonstration of nonlinear inverse synthesis transmission over transoceanic distances

Nonlinear Fourier transform (NFT) and eigenvalue communication with the use of nonlinear signal spectrum (both discrete and continuous), have been recently discussed as promising transmission methods to combat fiber nonlinearity impairments. In this paper, for the first time, we demonstrate the generation, detection and transmission performance over transoceanic distances of 10 Gbaud and nonlinear inverse synthesis (NIS) based signal (4 Gb/s line rate), in which the transmitted information is encoded directly onto the continuous part of the signal nonlinear spectrum. By applying effective digital signal processing techniques, a reach of 7344 km was achieved with a bit-error-rate (BER) (2.1×10-2) below the 20% FEC threshold. This represents an improvement by a factor of ~12 in data capacity x distance product compared with other previously demonstrated NFT-based systems, showing a significant advance in the active research area of NFT-based communication systems.

Nonlinear Fourier transform for optical communications

The nonlinear Fourier transform, also known as eigenvalue communications, is a transmission and signal processing technique that makes positive use of the nonlinear properties of fibre channels. I will discuss recent progress in this field.

Theoretical and experimental analysis of excessively tilted fiber gratings

We have theoretically and experimentally investigated the dual-peak feature of tilted fiber gratings with excessively tilted structure (named as Ex-TFGs). We have explained the dual-peak feature by solving eigenvalue equations for TM0m and TE0m of a circular waveguide, in which the TE (transverse electric) and TM (transverse magnetic) core modes are coupled into TE and TM cladding modes, respectively. Meanwhile, in the experiment, we have verified that one of the dual peaks at the shorter wavelength is due to the TM mode coupling whereas the other one at the longer wavelength arises from TE mode coupling when a linearly polarized light launched into the Ex-TFG. We have also investigated the peak separation of TE and TM cladding mode for different surrounding medium refractive indexes (SRI), revealed that the dual peaks separation is decreasing as increasing of SRI, which agrees very well with the theoretical analysis results.

A Dynamic Input-Output Model with Renewable Resources

The paper studies a generalisation of the dynamic Leontief input-output model. The standard dynamic Leontief model will be extended with the balance equation of renewable resources. The renewable stocks will increase regenerating and decrease exploiting primary natural resources. In this study the controllability of this extended model is examined by taking the consumption as the control parameter. Assuming balanced growth for both consumption and production, we investigate the exhaustion of renewable resources in dependence on the balanced growth rate and on the rate of natural regeneration. In doing so, classic results from control theory and on eigenvalue problems in linear algebra are applied.

A Dynamic Input-Output Model with Renewable Resources

The paper studies a generalisation of the dynamic Leontief input-output model. The standard dynamic Leontief model will be extended with the balance equation of renewable resources. The renewable stocks will increase regenerating and decrease exploiting primary natural resources. In this study the controllability of this extended model is examined by taking the consumption as the control parameter. Assuming balanced growth for both consumption and production, we investigate the exhaustion of renewable resources in dependence on the balanced growth rate and on the rate of natural regeneration. In doing so, classic results from control theory and on eigenvalue problems in linear algebra are applied.

Dynamic modeling and analysis of single-stage boost inverters under normal and abnormal conditions

Inverters play key roles in connecting sustainable energy (SE) sources to the local loads and the ac grid. Although there has been a rapid expansion in the use of renewable sources in recent years, fundamental research, on the design of inverters that are specialized for use in these systems, is still needed. Recent advances in power electronics have led to proposing new topologies and switching patterns for single-stage power conversion, which are appropriate for SE sources and energy storage devices. The current source inverter (CSI) topology, along with a newly proposed switching pattern, is capable of converting the low dc voltage to the line ac in only one stage. Simple implementation and high reliability, together with the potential advantages of higher efficiency and lower cost, turns the so-called, single-stage boost inverter (SSBI), into a viable competitor to the existing SE-based power conversion technologies.^ The dynamic model is one of the most essential requirements for performance analysis and control design of any engineering system. Thus, in order to have satisfactory operation, it is necessary to derive a dynamic model for the SSBI system. However, because of the switching behavior and nonlinear elements involved, analysis of the SSBI is a complicated task.^ This research applies the state-space averaging technique to the SSBI to develop the state-space-averaged model of the SSBI under stand-alone and grid-connected modes of operation. Then, a small-signal model is derived by means of the perturbation and linearization method. An experimental hardware set-up, including a laboratory-scaled prototype SSBI, is built and the validity of the obtained models is verified through simulation and experiments. Finally, an eigenvalue sensitivity analysis is performed to investigate the stability and dynamic behavior of the SSBI system over a typical range of operation. ^

Um estudo sobre polinômios matriciais

This research work aims to make a study of the algebraic theory of matrix monic polynomials, as well as the definitions, concepts and properties with respect to block eigenvalues, block eigenvectors and solvents of P(X). We investigte the main relations between the matrix polynomial and the Companion and Vandermonde matrices. We study the construction of matrix polynomials with certain solvents and the extention of the Power Method, to calculate block eigenvalues and solvents of P(X). Through the relationship between the dominant block eigenvalue of the Companion matrix and the dominant solvent of P(X) it is possible to obtain the convergence of the algorithm for the dominant solvent of the matrix polynomial. We illustrate with numerical examples for diferent cases of convergence.

Vacuum spacetimes with constant Weyl eigenvalues

Einstein spacetimes (that is vacuum spacetimes possibly with a non-zero cosmological constant A) with constant non-zero Weyl eigenvalues are considered. For type Petrov II & D this assumption allows one to prove that the non-repeated eigenvalue necessarily has the value 2A/3 and it turns out that the only possible spacetimes are some Kundt-waves considered by Lewandowski which are type II and a Robinson-Bertotti solution of type D. For Petrov type I the only solution turns out to be a homogeneous pure vacuum solution found long ago by Petrov using group theoretic methods. These results can be summarised by the statement that the only vacuum spacetimes with constant Weyl eigenvalues are either homogeneous or are Kundt spacetimes. This result is similar to that of Coley et al. who proved their result for general spacetimes under the assumption that all scalar invariants constructed from the curvature tensor and all its derivatives were constant.

Benthic foraminiferal record from Santa Barbara basin

The oxygen minimum zone (OMZ) of the late Quaternary California margin experienced abrupt and dramatic changes in strength and depth in response to changes in intermediate water ventilation, ocean productivity, and climate at orbital through millennial time scales. Expansion and contraction of the OMZ is exhibited at high temporal resolution (107-126 year) by quantitative benthic foraminiferal assemblage changes in two piston cores forming a vertical profile in Santa Barbara Basin (569 m, basin floor; 481 m, near sill depth) to 34 and 24 ka, respectively. Variation in the OMZ is quantified by new benthic foraminiferal groupings and new dissolved oxygen index based on documented relations between species and water-mass oxygen concentrations. Foraminiferal-based paleoenvironmental assessments are integrated with principal component analysis, bioturbation, grain size, CaCO3, total organic carbon, and d13C to reconstruct basin oxygenation history. Fauna responded similarly between the two sites, although with somewhat different magnitude and taxonomic expression. During cool episodes (Younger Dryas and stadials), the water column was well oxygenated, most strongly near the end of the glacial episode (17-16 ka; Heinrich 1). In contrast, the OMZ was strong during warm episodes (Bølling/Allerød, interstadials, and Pre-Boreal). During the Bølling/Allerød, the OMZ shoaled to <360 m of contemporaneous sea level, its greatest vertical expansion of the last glacial cycle. Assemblages were then dominated by Bolivina tumida, reflecting high concentrations of dissolved methane in bottom waters. Short decadal intervals were so severely oxygen-depleted that no benthic foraminifera were present. The middle to late Holocene (6-0 ka) was less dysoxic than the early Holocene.

Using phase-space localized basis functions to obtain vibrational energies of molecules

For decades scientists have attempted to use ideas of classical mechanics to choose basis functions for calculating spectra. The hope is that a classically-motivated basis set will be small because it covers only the dynamically important part of phase space. One popular idea is to use phase space localized (PSL) basis functions. This thesis improves on previous efforts to use PSL functions and examines the usefulness of these improvements. Because the overlap matrix, in the matrix eigenvalue problem obtained by using PSL functions with the variational method, is not an identity, it is costly to use iterative methods to solve the matrix eigenvalue problem. We show that it is possible to circumvent the orthogonality (overlap) problem and use iterative eigensolvers. We also present an altered method of calculating the matrix elements that improves the performance of the PSL basis functions, and also a new method which more efficiently chooses which PSL functions to include. These improvements are applied to a variety of single well molecules. We conclude that for single minimum molecules, the PSL functions are inferior to other basis functions. However, the ideas developed here can be applied to other types of basis functions, and PSL functions may be useful for multi-well systems.

Dispersive estimates for the Schrödinger equation

In this document we explore the issue of $L^1\to L^\infty$ estimates for the solution operator of the linear Schr\"{o}dinger equation, \begin{align*} iu_t-\Delta u+Vu&=0 &u(x,0)=f(x)\in \mathcal S(\R^n). \end{align*} We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto the absolutely continuous spectrum of $H=-\Delta+V$ satisfies the following estimate $\|e^{itH} P_{ac}(H)\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}$ under certain conditions on the potential $V$. Specifically, we prove the dispersive estimate is satisfied with optimal assumptions on smoothness, that is $V\in C^{\frac{n-3}{2}}(\R^n)$ for $n=5,7$ assuming that zero is regular, $|V(x)|\lesssim \langle x\rangle^{-\beta}$ and $|\nabla^j V(x)|\lesssim \langle x\rangle^{-\alpha}$, $1\leq j\leq \frac{n-3}{2}$ for some $\beta>\frac{3n+5}{2}$ and $\alpha>3,8$ in dimensions five and seven respectively. We also show that for the five dimensional result one only needs that $|V(x)|\lesssim \langle x\rangle^{-4-}$ in addition to the assumptions on the derivative and regularity of the potential. This more than cuts in half the required decay rate in the first chapter. Finally we consider a problem involving the non-linear Schr\"{o}dinger equation. In particular, we consider the following equation that arises in fiber optic communication systems, \begin{align*} iu_t+d(t) u_{xx}+|u|^2 u=0. \end{align*} We can reduce this to a non-linear, non-local eigenvalue equation that describes the so-called dispersion management solitons. We prove that the dispersion management solitons decay exponentially in $x$ and in the Fourier transform of $x$.

Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities

In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised solutions. For a scalar model with Heaviside firing rate function we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability and the shapes of the dominant growing modes. Our predictions are shown to be in excellent agreement with direct numerical simulations. Moreover, beyond the instability our simulations demonstrate the emergence of multi-bump and labyrinthine patterns. With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem.

Efficient domination through eigenvalues

The paper begins with a new characterization of (k,τ)(k,τ)-regular sets. Then, using this result as well as the theory of star complements, we derive a simplex-like algorithm for determining whether or not a graph contains a (0,τ)(0,τ)-regular set. When τ=1τ=1, this algorithm can be applied to solve the efficient dominating set problem which is known to be NP-complete. If −1−1 is not an eigenvalue of the adjacency matrix of the graph, this particular algorithm runs in polynomial time. However, although it does not work in polynomial time in general, we report on its successful application to a vast set of randomly generated graphs.