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We study a difference Riccati equation $\Phi(x) + \rho(x)/\Phi(x-\omega) = v(x)$ with $1-$periodic continuos coefficients. Using continued fraction theory we investigate a problem of existence of continuos solutions for this equation. It is shown that convergence of a continued fraction representing a solution of the Riccati equation can be expressed in terms of hyperbolicity of a cocycle naturally associated to this continued fraction. We apply the critical set method to establish the uniform hyperbolicity of the cocycle and to obtain sufficient conditions for the convergence of a continued fraction giving a representation for a solution of the Riccati equation.

We study a slow-fast system with two slow and one fast variables. We assume that the slow manifold of the system possesses a fold and there is an equilibrium of the system in a small neighbourhood of the fold. We derive a normal form for the system in a neighbourhood of the pair "equilibrium-fold" and study the dynamics of the normal form. In particular, as the ratio of two time scales tends to zero we obtain an asymptotic formula for the Poincar\'e map and calculate the parameter values for the first period-doubling bifurcation. The theory is applied to a generalization of the FitzHugh-Nagumo system., Comment: 30 pages, 2 figures

In complex colloidal systems, interparticle interactions strongly affect the dynamics of the constituting particles. A study of the dynamical response also provides invaluable information on the character of those interactions. Here we demonstrate how tuning the electrostatic interactions by an ionic surfactant in dispersions of magnetic nanoplatelets leads to developing new dynamic modes in magnetic response spectra. The collective modes can be induced or suppressed by either varying the concentration ratio of the magnetic nanoplatelets (MP) to the surfactant or increasing the MP concentration reflecting the nanoscale characteristics of this fluid magnet.

We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ being a $C^{1}$-map has the form $A(x) = R(\varphi(x)) Z(\lambda(x))$, where $R(\varphi)$ is a rotation in $\mathbb{R}^{2}$ over the angle $\varphi$ and $Z(\lambda)= diag\{\lambda, \lambda^{-1}\}$ is a diagonal matrix. Assuming that $\lambda(x) \ge \lambda_{0} > 1$ with a sufficiently large constant $\lambda_{0}$ and the function $\varphi$ be such that $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some additional requirements on the derivative of the function $\varphi$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes hyperbolic in contrary to the case when secondary collisions can be partially eliminated.

This paper introduces a run-and-tumble model with self-reinforcing directionality and rests. We derive a single governing hyperbolic partial differential equation for the probability density of random walk position, from which we obtain the second moment in the long time limit. We find the criteria for the transition between superdiffusion and diffusion caused by the addition of a rest state. The emergence of superdiffusion depends on both the parameter representing the strength of self-reinforcement and the ratio between mean running and resting times. The mean running time must be at least $2/3$ of the mean resting time for superdiffusion to be possible. Monte Carlo simulations validate this theoretical result. This work demonstrates the possibility of extending the telegrapher's (or Cattaneo) equation by adding self-reinforcing directionality so that superdiffusion occurs even when rests are introduced.

Nowadays, loss-based TCP congestion controls in general and CUBIC specifically became the de facto standard for the Internet. BBR congestion control challenges the loss-based approach by modeling the network based on estimated bandwidth and round-trip time. At Dropbox, we've been using BBRv1 since 2017 and are accustomed to its pros and cons. BBRv2 introduces a set of improvements to network modeling (explicit loss targets and inflight limits) and fairness (differential probing and headroom for new flows.) In this paper, we go over experimental data gathered on the Dropbox Edge Network. We compare BBRv2 to BBRv1 and CUBIC showing that BBRv2 is a definite improvement over both of them. We also show that BBRv2 experimental results match its theoretical design principles., Comment: 7 pages, 15 figures, netdev 0x14 conference

We study a linear cocycle over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed the cocycle is generated by a $C^{1}$-map $A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which depends on a small parameter $\varepsilon\ll 1$ and has the form of the Poincar\'e map corresponding to a singularly perturbed Schr\"odinger equation. Under assumption the eigenvalues of $A_{\varepsilon}(x)$ to be of the form $\exp(\pm \lambda(x)/\varepsilon)$, where $\lambda(x)$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter $\varepsilon$. We show that in the limit $\varepsilon\to 0$ the cocycle "typically" exhibits ED only if it is exponentially close to a constant cocycle. In contrary, if the cocycle is not close to a constant one it does not posesses ED, whereas the Lyapunov exponent is "typically" large.

In this work, we study the effect of dipole-dipole interparticle interactions on the static thermodynamic and magnetic properties of an ensemble of immobilized monodisperse superparamagnetic nanoparticles. We assume that magnetic nanoparticles are embedded in the nodes of a regular cubic lattice, so that the particle translational degrees of freedom are turned off. The relaxation of the magnetic moments of the nanoparticles occurs by the Neel mechanism. The easy axes are aligned (i) parallel or (ii) perpendicular to the direction of an external field. These models are investigated using theory and computer simulation, taking microscopic discrete structure explicitly into account. The analytical expressions of the Helmholtz free energy, the static magnetization, and the initial magnetic susceptibility are derived for both configurations (i) and (ii) as functions of the height of the magnetic crystallographic anisotropy energy barrier, measured by parameter $\sigma$, and the intensity of the dipole-dipole interparticle interactions measured by $\lambda_e$. A good agreement between the theory and the results of MC simulations in the region of low and moderate values of $\lambda_e$ and $\sigma$ is obtained. For high values of $\lambda_e$ and $\sigma$, the structuring of magnetic moments in regularly orientated structures was found from MC simulations for configuration (i)., Comment: 16 pages, 13 figures

We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to the action of a non-stationary force field with potential $U(q,t) = f(t)V(q)$. It is assumed that the factor $f(t)$ tends to $\infty$ as $t\to \pm\infty$ and vanishes at a unique point $t_{0}\in \mathbb{R}$. Let $X_{+}$, $X_{-}$ denote the sets of isolated critical points of $V(x)$ at which $U(x,t)$ as a function of $x$ attains its maximum for any fixed $t> t_{0}$ and $t

The present paper provides a method for finding partial differential equations satisfied by the Feynman integrals for diagrams of various types, using the Griffiths theorem on the reduction of poles of rational differential forms. As an application, an algorithm for computing partial differential equations satisfied by Feynman integrals for diagrams of a ladder type is described., Comment: 10 pages, 1 figure

We describe new irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(3)$ of semistable rank 2 coherent sheaves with Chern classes $c_1=0,\ c_2=3,\ c_3=0$ on $\mathbb{P}^3$, general points of which correspond to sheaves whose singular loci contain components of dimensions both 0 and 1. These sheaves are produced by elementary transformations of stable reflexive rank 2 sheaves with $c_1=0,\ c_2=2,\ c_3=2$ or 4 along a disjoint union of a projective line and a collection of points in $\mathbb{P}^3$. The constructed families of sheaves provide first examples of irreducible components of the Gieseker-Maruyama moduli scheme such that their general sheaves have singularities of mixed dimension., Comment: 16 pages, 1 table

The Dotsenko-Fateev integral is an analytic function of one complex variable expressing the amplitude in the 4-point correlator of the 2D conformal field theory. Dotsenko-Fateev found ODE of third order with Fuchsian singularities satisfied by their integral. In the present paper, this work is extended to generalized Dotsenko-Fateev integrals, in particular those associated to arbitrary multipoint correlators, and Pfaff systems of PDE of Fuchsian type are constructed for them. The ubiquity of the Fuchsian systems is in that they permit to obtain local expansions of solutions in the neighborhoods of singularities of the system.