Spectral statistics of random geometric graphs

We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the spectrum via the nearest neighbour and next nearest neighbour spacing distribution and long range correlations via the spectral rigidity Delta_3 statistic. These correlations in the level spacings give information about localisation of eigenvectors, level of community structure and the level of randomness within the networks. We find a parameter dependent transition between Poisson and Gaussian orthogonal ensemble statistics. That is the spectral statistics of spatial random geometric graphs fits the universality of random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert and Watts-Strogatz random graph.Comment: 19 pages, 6 figures. Substantially updated from previous versio

On the rate of quantum ergodicity in Euclidean billiards

For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdi\`ere and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we first give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions like bouncing ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean, and find good agreement with a Gaussian distribution.Comment: 40 pages, LaTeX2e. This version does not contain any figures. A version with all figures can be obtained from http://www.physik.uni-ulm.de/theo/qc/ (File: http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp97-8.ps.gz) In case of any problems contact Arnd B\"acker (e-mail: [email protected]) or Roman Schubert (e-mail: [email protected]

Differential role of MAX2 and strigolactones in pathogen, ozone, and stomatal responses

Strigolactones are a group of phytohormones that control developmental processes including shoot branching and various plant-environment interactions in plants. We previously showed that the strigolactone perception mutant more axillary branches 2 (max2) has increased susceptibility to plant pathogenic bacteria. Here we show that both strigolactone biosynthesis (max3 and max4) and perception mutants (max2 and dwarf14) are significantly more sensitive to Pseudomonas syringae DC3000. Moreover, in response to P. syringae infection, high levels of SA accumulated in max2 and this mutant was ozone sensitive. Further analysis of gene expression revealed no major role for strigolactone in regulation of defense gene expression. In contrast, guard cell function was clearly impaired in max2 and depending on the assay used, also in max3, max4, and d14 mutants. We analyzed stomatal responses to stimuli that cause stomatal closure. While the response to abscisic acid (ABA) was not impaired in any of the mutants, the response to darkness and high CO2 was impaired in max2 and d14-1 mutants, and to CO2 also in strigolactone synthesis (max3, max4) mutants. To position the role of MAX2 in the guard cell signaling network, max2 was crossed with mutants defective in ABA biosynthesis or signaling. This revealed that MAX2 acts in a signaling pathway that functions in parallel to the guard cell ABA signaling pathway. We propose that the impaired defense responses of max2 are related to higher stomatal conductance that allows increased entry of bacteria or air pollutants like ozone. Furthermore, as MAX2 appears to act in a specific branch of guard cell signaling (related to CO2 signaling), this protein could be one of the components that allow guard cells to distinguish between different environmental conditions.Peer reviewe

Classical and quantum ergodicity on orbifolds

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.Comment: 14 page

The acquisition of Sign Language: The impact of phonetic complexity on phonology

Research into the effect of phonetic complexity on phonological acquisition has a long history in spoken languages. This paper considers the effect of phonetics on phonological development in a signed language. We report on an experiment in which nonword-repetition methodology was adapted so as to examine in a systematic way how phonetic complexity in two phonological parameters of signed languages — handshape and movement — affects the perception and articulation of signs. Ninety-one Deaf children aged 3–11 acquiring British Sign Language (BSL) and 46 hearing nonsigners aged 6–11 repeated a set of 40 nonsense signs. For Deaf children, repetition accuracy improved with age, correlated with wider BSL abilities, and was lowest for signs that were phonetically complex. Repetition accuracy was correlated with fine motor skills for the youngest children. Despite their lower repetition accuracy, the hearing group were similarly affected by phonetic complexity, suggesting that common visual and motoric factors are at play when processing linguistic information in the visuo-gestural modality

Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)

Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure

Political hashtag publics and counter-visuality: a case study of #fertilityday in Italy

In 2016 the Italian health ministry launched the ‘Fertility Day’ campaign, aimed at tackling Italy’s low birth rate. Under the accusation of delivering sexist and racist messages, the campaign became a trending topic on Twitter, and a protest was launched to be held during Fertility Day. By applying a combination of digital methods and visual content analysis to the #fertilityday Twitter stream, this paper contributes to existing research on the deliberative strength of political hashtag publics, with a particular focus on their power structures, communication patterns and visual content use. Findings on gatekeeping dynamics downsize optimistic views on the democratising potential of Twitter’s socio-technical infrastructure as they point to the emergence of online satirical media and ‘tweetstars’ – along with mainstream news media– as main producers of spreadable content, with ordinary users only surfacing when traditional media elites and new satirical actors lack or lose interest in the debate. Results confirm that political hashtag publics follow acute event communication patterns, with users highly engaged in retweeting and referencing external material and visual content playing a key role in these gatewatching practices. The transient counter-visuality – or critical stance – of tweets with user-manipulated images, however, also suggests that the deliberative potential of these publics is not easily sustainable over time

On the Lebesgue measure of Li-Yorke pairs for interval maps

We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure