On the distribution of surface extrema in several one- and two-dimensional random landscapes

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one- and two-dimensional substrates focusing our analysis on the probability distribution function $P(M,L)$ of the number $M$ of maximal points (i.e., local ``peaks'') of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one--dimensional ballistic growth process in the steady state can be mapped onto ''rise-and-descent'' sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. \cite{ov}, that different characteristics of ``rise-and-descent'' patterns in random permutations can be interpreted in terms of a certain continuous--space Hammersley--type process. For one--dimensional system we compute $P(M,L)$ exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long--ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as $L \to \infty$.Comment: 25 pages, 12 figure

Narrow-escape times for diffusion in microdomains with a particle-surface affinity: Mean-field results

We analyze the mean time t_{app} that a randomly moving particle spends in a bounded domain (sphere) before it escapes through a small window in the domain's boundary. A particle is assumed to diffuse freely in the bulk until it approaches the surface of the domain where it becomes weakly adsorbed, and then wanders diffusively along the boundary for a random time until it desorbs back to the bulk, and etc. Using a mean-field approximation, we define t_{app} analytically as a function of the bulk and surface diffusion coefficients, the mean time it spends in the bulk between two consecutive arrivals to the surface and the mean time it wanders on the surface within a single round of the surface diffusion.Comment: 8 pages, 1 figure, submitted to JC

Random patterns generated by random permutations of natural numbers

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time $n$, whose moves to the right or to the left are induced by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site $X$ at time $n$, obtain the probability measure of different excursions and define the asymptotic distribution of the number of "U-turns" of the trajectories - permutation "peaks" and "through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers $1,2,3, >...,L$ on sites of a 1d or 2d square lattices containing $L$ sites. We calculate the distribution function of the number of local "peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss some surprising collective behavior emerging in this model.Comment: 16 pages, 5 figures; submitted to European Physical Journal, proceedings of the conference "Stochastic and Complex Systems: New Trends and Expectations" Santander, Spai

Current-mediated synchronization of a pair of beating non-identical flagella

The basic phenomenology of experimentally observed synchronization (i.e., a stochastic phase locking) of identical, beating flagella of a biflagellate alga is known to be captured well by a minimal model describing the dynamics of coupled, limit-cycle, noisy oscillators (known as the noisy Kuramoto model). As demonstrated experimentally, the amplitudes of the noise terms therein, which stem from fluctuations of the rotary motors, depend on the flagella length. Here we address the conceptually important question which kind of synchrony occurs if the two flagella have different lengths such that the noises acting on each of them have different amplitudes. On the basis of a minimal model, too, we show that a different kind of synchrony emerges, and here it is mediated by a current carrying, steady-state; it manifests itself via correlated "drifts" of phases. We quantify such a synchronization mechanism in terms of appropriate order parameters $Q$ and $Q_{\cal S}$ - for an ensemble of trajectories and for a single realization of noises of duration ${\cal S}$, respectively. Via numerical simulations we show that both approaches become identical for long observation times ${\cal S}$. This reveals an ergodic behavior and implies that a single-realization order parameter $Q_{\cal S}$ is suitable for experimental analysis for which ensemble averaging is not always possible.Comment: 10 pages, 2 figure

Spectroscopic properties of a two-dimensional time-dependent Cepheid model II. Determination of stellar parameters and abundances

Standard spectroscopic analyses of variable stars are based on hydrostatic one-dimensional model atmospheres. This quasi-static approach has theoretically not been validated. We aim at investigating the validity of the quasi-static approximation for Cepheid variables. We focus on the spectroscopic determination of the effective temperature $T_\mathrm{eff}$, surface gravity $\log \,g$, microturbulent velocity $\xi_\mathrm{t}$, and a generic metal abundance $\log\,A$ -- here taken as iron. We calculate a grid of 1D hydrostatic plane-parallel models covering the ranges in effective temperature and gravity encountered during the evolution of a two-dimensional time-dependent envelope model of a Cepheid computed with the radiation-hydrodynamics code CO5BOLD. We perform 1D spectral syntheses for artificial iron lines in local thermodynamic equilibrium varying the microturbulent velocity and abundance. We fit the resulting equivalent widths to corresponding values obtained from our dynamical model. For the four-parametric case, the stellar parameters are typically underestimated exhibiting a bias in the iron abundance of \approx-0.2\,\mbox{dex}. To avoid biases of this kind it is favourable to restrict the spectroscopic analysis to photometric phases $\phi_\mathrm{ph}\approx0.3\ldots 0.65$ using additional information to fix effective temperature and surface gravity. Hydrostatic 1D model atmospheres can provide unbiased estimates of stellar parameters and abundances of Cepheid variables for particular phases of their pulsations. We identified convective inhomogeneities as the main driver behind potential biases. For obtaining a complete view on the effects when determining stellar parameters with 1D models, multi-dimensional Cepheid atmosphere models are necessary for variables of longer period than investigated here.Comment: accepted for publication in Astronomy & Astrophysic

Spectroscopic properties of a two-dimensional time-dependent Cepheid model I. Description and validation of the model

Standard spectroscopic analyses of Cepheid variables are based on hydrostatic one-dimensional model atmospheres, with convection treated using various formulations of mixing-length theory. This paper aims to carry out an investigation of the validity of the quasi-static approximation in the context of pulsating stars. We check the adequacy of a two-dimensional time-dependent model of a Cepheid-like variable with focus on its spectroscopic properties. With the radiation-hydrodynamics code CO5BOLD, we construct a two-dimensional time-dependent envelope model of a Cepheid with $T_\mathrm{eff}= 5600$ K, $\log g=2.0$, solar metallicity, and a 2.8-day pulsation period. Subsequently, we perform extensive spectral syntheses of a set of artificial iron lines in local thermodynamic equilibrium. The set of lines allows us to systematically study effects of line strength, ionization stage, and excitation potential. We evaluate the microturbulent velocity, line asymmetry, projection factor, and Doppler shifts. The mean Doppler shift is non-zero and negative, -1 km/s, after averaging over several full periods and lines. This residual line-of-sight velocity (related to the "K-term") is primarily caused by horizontal inhomogeneities, and consequently we interpret it as the familiar convective blueshift ubiquitously present in non-pulsating late-type stars. Limited statistics prevent firm conclusions on the line asymmetries. Our two-dimensional model provides a reasonably accurate representation of the spectroscopic properties of a short-period Cepheid-like variable star. Some properties are primarily controlled by convective inhomogeneities rather than by the Cepheid-defining pulsations

Discrete singular integrals in a half-space

We consider Calderon -- Zygmund singular integral in the discrete half-space $h{\bf Z}^m_{+}$, where ${\bf Z}^m$ is entire lattice ($h>0$) in ${\bf R}^m$, and prove that the discrete singular integral operator is invertible in $L_2(h{\bf Z}^m_{+}$) iff such is its continual analogue. The key point for this consideration takes solvability theory of so-called periodic Riemann boundary problem, which is constructed by authors.Comment: 9 pages, 1 figur

Group Theoretical Quantum Tomography

The paper is devoted to the mathematical foundation of the quantum tomography using the theory of square-integrable representations of unimodular Lie groups.Comment: 13 pages, no figure, Latex2e. Submitted to J.Math.Phy