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## On the global synchronization in hub clusters of power grids

We study stability of a synchronous regime in hub clusters of the power networks, which are simulated by ensembles of phase oscillators. An approach allowing one to estimate the regions in the parameter space, which correspond to the global asymptotic stability of this regime, is presented. The method is illustrated by an example of a hub cluster consisting of one generator and two consumers.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Using video on the Internet has become a common practice, but the television-like ‘passive viewer’ approach misses the benefits of the interactive nature of the Internet. The technological limitations of television can be overridden by the Internet. Having multiple sources of input does not mean they should be merged into one editor-controlled flat output. Treating streams as objects, it is possible to make viewers editors for their screens whenever they want, or let them watch a pre-edited version. Active streams are distributed to viewers to gain control over the scene layout. Recorded scenes can be remastered whenever needed and represented in different views simultaneously. For lectures and conference recordings, inline slide browsing is also possible. This approach was successfully tested in the Viditory.net project for the broadcasting and recording of conferences with multi-camera shots and remote speakers. Despite the Adobe Flash platform becoming obsolete, it is possible to implement similar capabilities on modern platforms and by using modern technologies.

Although schizophrenia was previously associated with affected spatial neuronal synchronization, surprisingly little is known about the temporal dynamics of neuronal oscillations in this disease. However, given that the coordination of neuronal processes in time represents an essential aspect of practically all cognitive operations, it might be strongly affected in patients with schizophrenia. In the present study we aimed at quantifying long-range temporal correlations (LRTC) in patients (18 with schizophrenia; 3 with schizoaffective disorder) and 28 healthy control subjects matched for age and gender. Ongoing neuronal oscillations were recorded with multi-channel EEG at rest condition. LRTC in the range 5-50s were analyzed with Detrended Fluctuation Analysis. The amplitude of neuronal oscillations in alpha and beta frequency ranges did not differ between patients and control subjects. However, LRTC were strongly attenuated in patients with schizophrenia in both alpha and beta frequency ranges. Moreover, the cross-frequency correlation between LRTC belonging to alpha and beta oscillations was stronger for patients than healthy controls, indicating that similar neurophysiological processes affect neuronal dynamics in both frequency ranges. We believe that the attenuation of LRTC is most likely due to the increased variability in neuronal activity, which was previously hypothesized to underlie an excessive switching between the neuronal states in patients with schizophrenia. Attenuated LRTC might allow for more random associations between neuronal activations, which in turn might relate to the occurrence of thought disorders in schizophrenia.

OBJECTIVE:

We present a novel method for the extraction of neuronal components showing cross-frequency phase synchronization.

METHODS:

In general the method can be applied for the detection of phase interactions between components with frequencies f1 and f2, where f2 ≈ rf1 and r is some integer. We refer to the method as cross-frequency decomposition (CFD), which consists of the following steps: (a) extraction of f1-oscillations with the spatio-spectral decomposition algorithm (SSD); (b) frequency modification of the f1-oscillations obtained with SSD; and (c) finding f2-oscillations synchronous with f1-oscillations using least-squares estimation.

RESULTS:

Our simulations showed that CFD was capable of recovering interacting components even when the signal-to-noise ratio was as low as 0.01. An application of CFD to the real EEG data demonstrated that cross-frequency phase synchronization between alpha and beta oscillations can originate from the same or remote neuronal populations.

CONCLUSIONS:

CFD allows a compact representation of the sets of interacting components. The application of CFD to EEG data allows differentiating cross-frequency synchronization arising due to genuine neurophysiological interactions from interactions occurring due to quasi-sinusoidal waveform of neuronal oscillations.

SIGNIFICANCE:

CFD is a method capable of extracting cross-frequency coupled neuronal oscillations even in the presence of strong noise.

Copyright © 2011 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.

We consider a minimal action of a finitely generated semigroup by homeomorphisms of the circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random dynamics of circle homeomorphisms: Antonov’s theorem and its corollaries.

Generalized error-locating codes are discussed. An algorithm for calculation of the upper bound of the probability of erroneous decoding for known code parameters and the input error probability is given. Based on this algorithm, an algorithm for selection of the code parameters for a specified design and input and output error probabilities is constructed. The lower bound of the probability of erroneous decoding is given. Examples of the dependence of the probability of erroneous decoding on the input error probability are given and the behavior of the obtained curves is explained.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.