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Young's cellular automaton, recently applied to study the spatiotemporal evolution of binary patterns for favorable/hostile environments, has now been modified from a different point of view. In this model, each differentiated cell (DC) produces two diffusing morphogens: a short-range activator and a long-range inhibitor. Their combination creates the so-called local 'w' field. Undifferentiated cells (UCs) are passive. The question arises how to adapt it to modelling patterning processes in anisotropic substrates with a biaxial dependence of the morphogen diffusion rate. We use activation/inhibition regions with appropriate shape geometry defined by the so-called deformation parameter p. We complement this model by adding a physically reasonable transition zone with controlled local field slope. The patterning process uses the morphogenetic field W calculated separately for each cell, which is the sum of the 'w' values generated by all regional DCs surrounding the cell. It acts as a chemical signal determining the next state of the cell. We also introduce a threshold W* defining the required absolute chemical signal strength. The state of each cell can change depending on the rules for W and W*. This improves the stability of the model evolution and extends its applications. Using two pairs of two-point orthogonal correlation functions, we reveal their directional sensitivity to changes in biaxial anisotropy. Finally, we illustrate the general two-parameter dependence of the average final DC concentration on the geometry of the activation/inhibition regions and on the value of the long-range inhibitor. This facilitates the recognition of characteristic features of the evolution of DC concentration in our model., Comment: 17 pages, 15 figures

We study two uncoupled oscillators, horizontal and vertical, residing in rectilinear polygons (with only vertical and horizontal sides) and impacting elastically from their boundary. The main purpose of the article is to analyze the occurrence of resonance in such systems, depending on the shape of the analytical potentials that determine the oscillators. We define resonant energy levels; roughly speaking, these are levels for which the resonance phenomenon occurs more often than rarely. We focus on unimodal analytic potentials with the minimum at zero. The most important result of the work describes the size of the set of resonance levels in the form of the following trichotomy: it is mostly empty or is one-element or is large, i.e. non-empty and open. We also indicate which classes of potentials each of the three possibilities can occur in. From this point of view, the last case (strongly resonant) is the most interesting. Then, the potentials belong to a special class of potentials, denoted by $\mathcal{SP}$, which seems unknown in the literature. The presented results appear to be new, even in the simplest case, when the uncoupled oscillators are not trapped in any set., Comment: 37 pages, 1 figure. The paper is closely related to the recent preprint: arXiv:2012.07342

This paper presents a computer-aided cytology diagnosis system designed for animals, focusing on image quality assessment (IQA) using Convolutional Neural Networks (CNNs). The system's building blocks are tailored to seamlessly integrate IQA, ensuring reliable performance in disease classification. We extensively investigate the CNN's ability to handle various image variations and scenarios, analyzing the impact on detecting low-quality input data. Additionally, the network's capacity to differentiate valid cellular samples from those with artifacts is evaluated. Our study employs a ResNet18 network architecture and explores the effects of input sizes and cropping strategies on model performance. The research sheds light on the significance of CNN-based IQA in computer-aided cytology diagnosis for animals, enhancing the accuracy of disease classification.

This paper explores the innovative use of simulation environments to enhance data acquisition and diagnostics in veterinary medicine, focusing specifically on gait analysis in dogs. The study harnesses the power of Blender and the Blenderproc library to generate synthetic datasets that reflect diverse anatomical, environmental, and behavioral conditions. The generated data, represented in graph form and standardized for optimal analysis, is utilized to train machine learning algorithms for identifying normal and abnormal gaits. Two distinct datasets with varying degrees of camera angle granularity are created to further investigate the influence of camera perspective on model accuracy. Preliminary results suggest that this simulation-based approach holds promise for advancing veterinary diagnostics by enabling more precise data acquisition and more effective machine learning models. By integrating synthetic and real-world patient data, the study lays a robust foundation for improving overall effectiveness and efficiency in veterinary medicine.

The primary objective of this research was to enhance the quality of semantic segmentation in cytology images by incorporating super-resolution (SR) architectures. An additional contribution was the development of a novel dataset aimed at improving imaging quality in the presence of inaccurate focus. Our experimental results demonstrate that the integration of SR techniques into the segmentation pipeline can lead to a significant improvement of up to 25% in the mean average precision (mAP) segmentation metric. These findings suggest that leveraging SR architectures holds great promise for advancing the state of the art in cytology image analysis.

Continuing the research initiated in \cite{Fr-Ki2}, we study the existence of solutions and their regularity for the cohomological equations $X u=f$ for locally Hamiltonian flows (determined by the vector field $X$) on a compact surface $M$ of genus $g\geq 1$. We move beyond the case studied so far by Forni in \cite{Fo1,Fo3}, when the flow is minimal over the entire surface and the function $f$ satisfies some Sobolev regularity conditions. We deal with the flow restricted to any its minimal component and any smooth function $f$ whenever the flow satisfies the Full Filtration Diophantine Condition (FFDC) (this is a full measure condition). The main goal of this article is to quantify optimal regularity of solutions. For this purpose we construct a family of invariant distributions $\mathfrak{F}_{\bar t}$, $\bar{t}\in\mathscr{TF}^*$ that play the roles of the Forni's invariant distributions introduced in \cite{Fo1,Fo3} by using the language of translation surfaces. The distributions $\mathfrak{F}_{\bar t}$ are global in nature (as emphasized in the title of the article), unlike the distributions $\mathfrak{d}^k_{\sigma,j}$, $(\sigma,k,j)\in\mathscr{TD}$ and $\mathfrak{C}^k_{\sigma,l}$, $(\sigma,k,l)\in\mathscr{TC}$ introduced in \cite{Fr-Ki2}, which are defined locally. All three families are used to determine the optimal regularity of the solutions for the cohomological equation, see Theorem 1.1 and 1.2. As a by-product, we also obtained, interesting in itself, a spectral result (Theorem 1.3) for the Kontsevich-Zorich cocycle acting on functional spaces arising naturally at the transition to the first-return map., Comment: The paper is closely related to the recent preprint: arXiv:2112.05939, arXiv:2112.13030 and arXiv:2305.16884

We study the cohomological equation $Xu=f$ for smooth locally Hamiltonian flows on compact surfaces. The main novelty of the proposed approach is that it is used to study the regularity of the solution $u$ when the flow has saddle loops, which has not been systematically studied before. Then we need to limit the flow to its minimum components. We show the existence and (optimal) regularity of solutions regarding the relations with the associated cohomological equations for interval exchange transformations (IETs). Our main theorems state that the regularity of solutions depends not only on the vanishing of the so-called Forni's distributions (cf.\ \cite{Fo1,Fo3}), but also on the vanishing of families of new invariant distributions (local obstructions) reflecting the behavior of $f$ around the saddles. Our main results provide some key ingredient for the complete solution to the regularity problem of solutions (in cohomological equations) for a.a.\ locally Hamiltonian flows (with or without saddle loops) to be shown in \cite{Fr-Ki3}. The main contribution of this article is to define the aforementioned new families of invariant distributions $\mathfrak{d}^k_{\sigma,j}$, $\mathfrak{C}^k_{\sigma,l}$ and analyze their effect on the regularity of $u$ and on the regularity of the associated cohomological equations for IETs. To prove this new phenomenon, we further develop local analysis of $f$ near degenerate singularities inspired by tools from \cite{Fr-Ki} and \cite{Fr-Ul2}. We develop new tools of handling functions whose higher derivatives have polynomial singularities over IETs., Comment: The paper is closely related to the recent preprint: arXiv:2112.05939, arXiv:2112.13030 and arXiv:2306.02340

This paper shows the machine learning system which performs instance segmentation of cytological images in veterinary medicine. Eleven cell types were used directly and indirectly in the experiments, including damaged and unrecognized categories. The deep learning models employed in the system achieve a high score of average precision and recall metrics, i.e. 0.94 and 0.8 respectively, for the selected three types of tumors. This variety of label types allowed us to draw a meaningful conclusion that there are relatively few mistakes for tumor cell types. Additionally, the model learned tumor cell features well enough to avoid misclassification mistakes of one tumor type into another. The experiments also revealed that the quality of the results improves with the dataset size (excluding the damaged cells). It is worth noting that all the experiments were done using a custom dedicated dataset provided by the cooperating vet doctors.

We consider smooth locally Hamiltonian flows on compact surfaces of genus $g\geq 2$ to prove their deviation of Birkhoff integrals for smooth observables. Our work generalizes results of Forni and Bufetov which prove the existence of deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals can display a power spectrum of behaviours with exactly $g$ positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich-Zorich, a renormalization cocycle over the Teichm\"uller flow on a stratum of the moduli space of translation surfaces. Our paper extends the study of the spectrum of deviations of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in deviation spectrum related to non-vanishing of the derivatives of observables at fixed points. The proof of this new phenomenon is based on tools developed in the recent work of the first author and Ulcigrai for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of IETs., Comment: The paper is closely related to the recent preprint arXiv:2112.05939

We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus $g\geq 1$ and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a \emph{power deviation spectrum} and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni (Annals 2002) and Bufetov (Annals 2014) and generalizing them to observables which are non-zero at fixed points. This in particular completes the proof of the original formulation of the Kontsevitch-Zorich conjecture. Our proof is based on building suitable \emph{correction operators} for cocycles with logarithmic singularities over a full measure set of interval exchange transformations (IETs), in the spirit of Marmi-Moussa-Yoccoz work on piecewise smooth cocycles over IETs. In the case of symmetric singularities, exploiting former work of the second author (Annals 2011), we prove a tightness result for a finite codimension class of observables. We then apply the latter result to prove the existence of ergodic infinite extensions for a full measure set of locally Hamiltonian flows with non-degenerate saddles in any genus $g\geq 2$.

The ergodic properties of two uncoupled oscillators, a horizontal and vertical one, residing in a class of non rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly larger than one. When at least one of the oscillators is un-harmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is unique ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components co-exist. We prove that for almost all partial energies in such segments the motion is unique ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.

We prove that neither a prime nor {an l-almost prime} number theorem hold in the class of regular Toeplitz subshifts. But, {when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler's totient function is assumed}, then the two theorems hold., Comment: 25 pages, no figures

The main goal of our research is to develop an effective method with a wide range of applications for the statistical reconstruction of heterogeneous microstructures with compact inclusions of any shape, such as highly irregular grains. The devised approach uses multi-scale extended entropic descriptors (ED) that quantify the degree of spatial non-uniformity of configurations of finite-sized objects. This technique is an innovative development of previously elaborated entropy methods for statistical reconstruction. Here, we discuss the two-dimensional case, but this method can be generalized into three dimensions. At the first stage, the developed procedure creates a set of black synthetic clusters that serve as surrogate inclusions. The clusters have the same individual areas and interfaces as their target counterparts, but random shapes. Then, from a given number of easy-to-generate synthetic cluster configurations, we choose the one with the lowest value of the cost function defined by us using extended ED. At the second stage, we make a significant change in the standard technique of simulated annealing (SA). Instead of swapping pixels of different phases, we randomly move each of the selected synthetic clusters. To demonstrate the accuracy of the method, we reconstruct and analyze two-phase microstructures with irregular inclusions of silica in rubber matrix as well as stones in cement paste. The results show that the two-stage reconstruction (TSR) method provides convincing realizations for these complex microstructures. The advantages of TSR include the ease of obtaining synthetic microstructures, very low computational costs, and satisfactory mapping in the statistical context of inclusion shapes. Finally, its simplicity should greatly facilitate independent applications., Comment: extended, published version with 7 figures, Supplementary Materials are at the end of the main manuscript

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non degenerate isomorphic saddle has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval exchange transformations with a hyperelliptic permutation (of any number of exchanged intervals), under a roof with symmetric logarithmic singularities. The result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay. A key ingredient in the proof, which is of independent interest, is a result on translation surfaces well approximated by single cylinders. We show that for almost every translation surface in any connected component of any stratum there exists a full measure set of directions which can be well approximated by a single cylinder of area arbitrarily close to one. The result, in the special case of the stratum $\mathcal{H}(1,1)$, yields rigidity sets needed for the singularity result., Comment: 32 pages, 7 figures

In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent. In this paper we focus on the problem of recurrence for elements of smooth curves in the moduli space. We give an effective criterion for the recurrence of almost every element of a smooth curve. The criterion relies on results developed by Minsky-Weiss in \cite{Mi-We}. Next we apply the criterion to the billiard flow on planar tables confined by arcs of confocal conics. The phase space of such billiard flow splits into invariant subsets determined by caustics. We prove that for almost every caustic the billiard flow restricted to the corresponding invariant set is uniquely ergodic. This answers affirmatively to a question raised by Zorich., Comment: 6 figures

We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for $\mathbb{Z}^d$-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents., Comment: 44 pages, 30 figures

We report a multiscale approach of broad applicability to stochastic reconstruction of multiphase materials, including porous ones. The approach devised uses an optimization method, such as the simulated annealing (SA) and the so-called entropic descriptors (EDs). For a binary pattern, they quantify spatial inhomogeneity or statistical complexity at discrete length-scales. The EDs extract dissimilar structural information to that given by two-point correlation functions (CFs). Within the SA, we use an appropriate cost function consisting of EDs or comprised of EDs and CFs. It was found that the stochastic reconstruction is computationally efficient when we begin with a preliminary synthetic configuration having in part desirable features. Another option is low-cost approximate reconstructing of the entire multiphase medium beyond the SA technique. The information included in the target ED-curve was utilized for this purpose. For a given volume fraction the low-cost trial microstructures can be generated in two ways. In the first one, applied to ceramics and carbonate samples, the interpenetrating spheres generate a number of trial configurations. In the second one, with phase-EDs, here used to the sandstone sample, the overlapping superspheres do it. Both methods use a radius determined from the EDs-linked two-exponent power-law. However, the supersphere deformation parameter allows controlling of the spatial inhomogeneity of prototypical microstructures. At last, even for a hypothetical ED-curve (under reasonable assumptions), the specific microstructure can be found, if it is realizable for a given volume fraction. In general, the EDs-based methods offer a compromise between computational efficiency and the accuracy of reconstructions., Comment: 34 pages, 17 figures, rewritten and clarified with modified references.arXiv admin note: text overlap with arXiv:1508.03857

A wide variety of real random composites can be studied by means of prototypes of multiphase microstructures with a controllable spatial inhomogeneity. To create them, we propose a versatile model of randomly overlapping super-spheres of a given radius and deformed in their shape by the parameter p. With the help of the so-called decomposable entropic measure, a clear dependence of the phase inhomogeneity degree on the values of the parameter p is found. Thus, a leading trend in changes of the phase inhomogeneity can be forecast. It makes searching for possible structure/property relations easier. For the chosen values of p, examples of two and three-phase prototypical microstructures show how the phase inhomogeneity degree evolves at different length scales. The approach can also be applied to preparing the optimal starting configurations in reconstructing real materials., Comment: 7 pages, 4 figures, published version

In this paper we prove that translation structures for which the corresponding vertical translation flows is weakly mixing and disjoint with its inverse, form a $G_\delta$-dense set in every non-hyperelliptic connected component of the moduli space $\mathcal M$. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is isomorphic to its inverse. To prove the main result, we study limits of the off-diagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the moduli space into the space of measure-preserving flows to obtain the $G_\delta$-condition.

Let $ G $ be a connected, simply connected nilpotent Lie group and $ \Gamma < G $ a lattice. We prove that each ergodic diffeomorphism $ \phi(x\Gamma)=uA(x)\Gamma $ on the nilmanifold $ G/\Gamma $, where $ u\in G $ and $ A:G\to G $ is a unipotent automorphism satisfying $ A(\Gamma)=\Gamma $, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow: (i) Sarnak's conjecture on M\"obius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the M\"obius orthogonality holds on so called typical short interval: $ \frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n

We propose an approximate reconstruction of random heterogeneous microstructures using the two-exponent power-law (TEPL). This rule originates from the entropic descriptor (ED) that is a multi-scale measure of spatial inhomogeneity for a given microstructure. A digitized target sample is a cube of linear size L in voxels. Then, a number of trial configurations can be generated by a model of overlapping spheres of a fixed radius, which are randomly distributed on a regular lattice. The TEPL describes the averaged maximum of the ED as a function of the phase concentration and the radius. Thus, it can be used to determine the radius. The suggested approach is tested on surrogate samples of ceramic and carbonate. In each of the cases, fifty low-cost trials provided a few good enough candidates to a selection of the optimal reconstruction. When a better accuracy is planned, the final reconstructions can serve as the starting configurations. Then, the resulting reconstructions should be competitive indeed to those starting with random configurations., Comment: In Figure 3, the repeated by mistake Fig. 2b is replaced by the correct Fig. 3b, published version

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices $ASL_2(\mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathcal{H}(1,1)$ of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragovic and Radnovic, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Fraczek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained., Comment: To appear in Journal of Modern Dynamics

The simple entropic method to statistical reconstructing of heterogeneous three-dimensional media from a single two-dimensional image is briefly reported. We apply the entropic descriptor quantifying spatial inhomogeneity that depends on length scale. The multiscale reconstructing procedure is boosted when we start from the synthetic 3D configuration. It is randomly generated with the overlapping spheres of a radius depending on the structure considered. In our scenario, the simulated annealing terminates when all assigned temperature loops are completed. Thus, the quality of the results for different samples can be fairly compared. The reliable 3D reconstructions of porous sandstone, ceramics and carbonate samples are obtained. They suggest the entropic method is a promising approach, which offers a kind of compromise between the computational efficiency and the accuracy of the statistical reconstructions. Furthermore, this approach is versatile enough to be extended for any multiphase medium., Comment: 10 pages, 6 figures, clarified version

In this paper, we consider generalized wind-tree models and $\Z^d$-covers over compact translation surfaces. Under suitable hypothesis, we prove recurrence of the linear flow in a generic direction and non-ergodicity of Lebesgue measure.

Recently, the effective medium approach using 2x2 basic cluster of model lattice sites to predict the conductivity of interacting droplets has been presented by Hattori et al. To make a step aside from pure applications, we have studied earlier a multi-scale percolation, employing any kxk basic cluster for non-interacting particles. Here, with interactions included, we examine in what way they alter the percolation threshold for any cluster case. We found that at a fixed length scale k the interaction reduces the range of shifts of the percolation threshold. To determine the critical concentrations, the simplified model is used. It diminishes the number of local conductivities into two main ones. In the presence of a dominance of the repulsive interaction over the thermal energy, the exact percolation thresholds at scales k=2 and 3 can be obtained from analytical formulas. Furthermore, by a simple reasoning, we obtain the limiting threshold formula for odd k. When k>>1, the odd-even difference becomes negligible. Hence, the 0.75 is the highest possible value of the threshold., Comment: 13 pages with 6 figures

The paper presents StegBlocks, which defines a new concept for performing undetectable hidden communication. StegBlocks is a general approach for constructing methods of network steganography. In StegBlocks, one has to determine objects with defined properties which will be used to transfer hidden messages. The objects are dependent on a specific network protocol (or application) used as a carrier for a given network steganography method. Moreover, the paper presents the approach to perfect undetectability of network steganography, which was developed based on the rules of undetectability for general steganography. The approach to undetectability of network steganography was used to show the possibility of developing perfectly undetectable network steganography methods using the StegBlocks concept., Comment: 6 pages, 1 figure, Accepted to Fourth International Workshop on Cyber Crime (IWCC 2015), co-located with 10th International Conference on Availability, Reliability and Security (ARES 2015), Toulouse, France, 24-28 August 2015

The kinetic Monte Carlo method is applied to examine effects of hydrostatic pressure on the benzimidazolium azelate (BenAze) proton conductivity. Following the experimental indications the recently proposed model has been modified to simulate the transport phenomena under moderate pressure, resulting in a very good agreement between numerical and experimental results. We demonstrate that the pressure-induced changes in the proton conductivity can be attributed to solely two parameters: the length of the hydrogen bond and the amplitude of lattice vibrations while other processes play a more minor role. It may provide an insight into tailoring new materials with improved proton-conducting properties. Furthermore, in high-pressure regime we anticipate the crossover from the increasing to decreasing temperature dependence of the proton conductivity resulting from the changes in the hydrogen-bond activation barrier with increased pressure., Comment: 12 pages, 3 figures

Macroscopic properties of heterogeneous media are frequently modelled by regular lattice models, which are based on a relatively small basic cluster of lattice sites. Here, we extend one of such models to any cluster's size kxk. We also explore its modified form. The focus is on the percolation behaviour of the effective conductivity of random two- and three-phase systems. We consider only the influence of geometrical features of local configurations at different length scales k. At scales accessible numerically, we find that an increase in the size of the basic cluster leads to characteristic displacements of the percolation threshold. We argue that the behaviour is typical of materials, whose conductivity is dominated by a few linear, percolation-like, conducting paths. Such a system can be effectively treated as one-dimensional medium. We also develop a simplified model that permits of an analysis at any scale. It is worth mentioning that the latter approach keeps the same thresholds predicted by the former one. We also briefly discuss a three-phase system, where the double-thresholds paths appear on model surfaces., Comment: some parts are rewritten, references extended, 12 figures

The kinetic Monte Carlo method is used to model the dynamic properties of proton diffusion in anhydrous proton conductors. The results have been discussed with reference to a two-step process called the Grotthuss mechanism. There is a widespread belief that this mechanism is responsible for fast proton mobility. We showed in detail that the relative frequency of reorientation and diffusion processes is crucial for the conductivity. Moreover, the current dependence on proton concentration has been analyzed. In order to test our microscopic model the proton transport in polymer electrolyte membranes based on benzimidazole C7H6N2 molecules is studied., Comment: 19 pages, 8 figures, revtex

In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in \cite{Fr-Ku-Le}. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations $T:[0,1)\to[0,1)$ (IETs) and under piecewise absolutely continuous roof functions $f:[0,1)\to\mathbb{R}_+$. We show that for almost every IET $T$ if $f$ is absolutely continuous over exchanged intervals and has non-zero sum of jumps then the special flow $T^f$ is not isomorphic to its inverse. The same conclusion is valid for a typical piecewise constant roof function.

We show that vertical light rays in almost every periodic array of Eaton lenses do not leave certain strips of bounded width. The light rays are traced by leaves of a non-orientable foliation on a singular plane. We study the flow defined by the induced foliation on the orientation cover of the singular plane. The behavior of that flow and ultimately our claim for the light rays is based on an analysis of the Teichm\"uller flow and the Kontsevich-Zorich cocycle on the moduli space of two branched, two sheeted torus covers in genus two., Comment: To appear in Nonlinearity