Broadband dielectric spectroscopy on benzophenone: alpha relaxation, beta relaxation, and mode coupling theory

We have performed a detailed dielectric investigation of the relaxational dynamics of glass-forming benzophenone. Our measurements cover a broad frequency range of 0.1 Hz to 120 GHz and temperatures from far below the glass temperature well up into the region of the small-viscosity liquid. With respect to the alpha relaxation this material can be characterized as a typical molecular glass former with rather high fragility. A good agreement of the alpha relaxation behavior with the predictions of the mode coupling theory of the glass transition is stated. In addition, at temperatures below and in the vicinity of Tg we detect a well-pronounced beta relaxation of Johari-Goldstein type, which with increasing temperature develops into an excess wing. We compare our results to literature data from optical Kerr effect and depolarized light scattering experiments, where an excess-wing like feature was observed in the 1 - 100 GHz region. We address the question if the Cole-Cole peak, which was invoked to describe the optical Kerr effect data within the framework of the mode coupling theory, has any relation to the canonical beta relaxation detected by dielectric spectroscopy.Comment: 11 pages, 7 figures; revised version with new Fig. 5 and some smaller changes according to referees' demand

A non-autonomous flow system with Plykin type attractor

A non-autonomous flow system is introduced with an attractor of Plykin type that may serve as a base for elaboration of real systems and devices demonstrating the structurally stable chaotic dynamics. The starting point is a map on a two-dimensional sphere, consisting of four stages of continuous geometrically evident transformations. The computations indicate that in a certain parameter range the map has a uniformly hyperbolic attractor. It may be represented on a plane by means of a stereographic projection. Accounting structural stability, a modification of the model is undertaken to obtain a set of two non-autonomous differential equations of the first order with smooth coefficients. As follows from computations, it has the Plykin type attractor in the Poincar\'{e} cross-section.Comment: 9 pages, 4 figure

Desmoplastic myxoid tumor of pineal region, SMARCB1-mutant, in young adult

We present a young adult woman who developed a myxoid tumor of the pineal region having a SMARCB1 mutation, which was phenotypically similar to the recently described desmoplastic myxoid, SMARCB1-mutant tumor of the pineal region (DMT-SMARCB1). The 24-year-old woman presented with headaches, nausea, and emesis. Neuroimaging identified a hypodense lesion in CT scans that was T1-hypointense, hyperintense in both T2-weighted and FLAIR MRI scans, and displayed gadolinium enhancement. The resected tumor had an abundant, Alcian-blue positive myxoid matrix with interspersed, non-neoplastic neuropil-glial-vascular elements. It immunoreacted with CD34 and individual cells for EMA. Immunohistochemistry revealed loss of nuclear INI1 expression by the myxoid component but its retention in the vascular elements. Molecular analyses identified a SMARCB1 deletion and DNA methylation studies showed that this tumor grouped together with the recently described DMT-SMARCB1. A cerebrospinal fluid cytologic preparation had several cells morphologically similar to those in routine and electron microscopy. We briefly discuss the correlation of the pathology with the radiology and how this tumor compares with other SMARCB1-mutant tumors of the nervous system

The boundary of chaos for interval mappings

A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set

On the complexity of some birational transformations

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis, and the third method is more numerical, using integer arithmetics. Each method has its own domain of application, but they give corroborating results, and lead us to a conjecture on the complexity of a class of maps constructed from matrix inversions

Structure determination of the (√3×√3)R30° boron phase on the Si(111) surface using photoelectron diffraction

A quantitative structural analysis of the system Si(111)(√3×√3)R30°−B has been performed using photoelectron diffraction in the scanned energy mode. We confirm that the substitutional S5 adsorption site is occupied and show that the interatomic separations to the three nearest-neighbor Si atoms are 1.98(±0.04)Å, 2.14(±0.13)Å, and 2.21(±0.12)Å. These correspond to the silicon atom immediately below the boron atom, the adatom immediately above, and the three atoms to which it is coordinated symmetrically in the first layer

Parabolic maps with spin: Generic spectral statistics with non-mixing classical limit

We investigate quantised maps of the torus whose classical analogues are ergodic but not mixing. Their quantum spectral statistics shows non-generic behaviour, i.e.it does not follow random matrix theory (RMT). By coupling the map to a spin 1/2, which corresponds to changing the quantisation without altering the classical limit of the dynamics on the torus, we numerically observe a transition to RMT statistics. The results are interpreted in terms of semiclassical trace formulae for the maps with and without spin respectively. We thus have constructed quantum systems with non-mixing classical limit which show generic (i.e. RMT) spectral statistics. We also discuss the analogous situation for an almost integrable map, where we compare to Semi-Poissonian statistics.Comment: 29 pages, 20 figure

Extreme value laws in dynamical systems under physical observables

Extreme value theory for chaotic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for statistical prediction of large value occurrences. In this method, one performs inference for the parameters of the Generalised Extreme Value (GEV) distribution, using maxima over blocks of regularly sampled observations along an orbit of the system. The observables studied so far in the theory are expressed as functions of the distance with respect to a point, which is assumed to be a density point of the system's invariant measure. However, this is not the structure of the observables typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. In this paper we consider extreme value limit laws for observables which are not functions of the distance from a density point of the dynamical system. In such cases, the limit laws are no longer determined by the functional form of the observable and the dimension of the invariant measure: they also depend on the specific geometry of the underlying attractor and of the observable's level sets. We present a collection of analytical and numerical results, starting with a toral hyperbolic automorphism as a simple template to illustrate the main ideas. We then formulate our main results for a uniformly hyperbolic system, the solenoid map. We also discuss non-uniformly hyperbolic examples of maps (H\'enon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models). Our purpose is to outline the main ideas and to highlight several serious problems found in the numerical estimation of the limit laws

Ruelle-Perron-Frobenius spectrum for Anosov maps

We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem.Comment: 58 pages, LaTe

Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. [2006] have found analytical results.Comment: 34 pages, 7 figures; Journal of Statistical Physics 201