Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.     

The Brauer group of Sweedler's Hopf algebra

We calculate the Brauer group of the four dimensional Hopf algebra introduced by M. E. Sweedler. This Brauer group is defined with respect to a (quasi-) triangular structure on , given by an element . In this paper is a field . The additive group of is embedded in the Brauer group and it fits in the exact and split sequence of groups: where is the well-known Brauer-Wall group of . The techniques involved are close to the Clifford algebra theory for quaternion or generalized quaternion algebras.

     

Testing multivariate uniformity and its applications

Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in . Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in , we obtain some asymptotic properties of the new statistics. By Monte Carlo simulation, it is found that the finite-sample distributions of the new statistics are well approximated by the standard normal distribution, , or the chi-squared distribution, . A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.

     

The spectra of nonnegative integer matrices via formal power series

We characterize the possible nonzero spectra of primitive integer matrices (the integer case of Boyle and Handelman's Spectral Conjecture). Characterizations of nonzero spectra of nonnegative matrices over and follow from this result. For the proof of the main theorem we use polynomial matrices to reduce the problem of realizing a candidate spectrum to factoring the polynomial as a product where the 's are polynomials in satisfying some technical conditions and is a formal power series in . To obtain the factorization, we present a hierarchy of estimates on coefficients of power series of the form to ensure nonpositivity in nonzero degree terms.

     

Functional MRI of working memory and selective attention in vibrotactile frequency discrimination

Background  

Focal lesions of the frontal, parietal and temporal lobe may interfere with tactile working memory and attention. To characterise the neural correlates of intact vibrotactile working memory and attention, functional MRI was conducted in 12 healthy young adults. Participants performed a forced-choice vibrotactile frequency discrimination task, comparing a cue stimulus of fixed frequency to their right thumb with a probe stimulus of identical or higher frequency. To investigate working memory, the time interval between the 2 stimuli was pseudo-randomized (either 2 or 8 s). To investigate selective attention, a distractor stimulus was occasionally presented contralaterally, simultaneous to the probe.     

Intuitionistic notions of boundedness in ℕ

We consider notions of boundedness of subsets of the natural numbers ? that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and we formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of ? to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most of the results do not require countable choice. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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