Mathematics

In this paper we study the connectivity of Fatou components for maps in a large family of singular perturbations. We prove that, for some parameters inside the family, the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and we determine precisely these connectivities. In particular, these results extend the ones obtained in [Can17, Can18].

The COVID-19 pandemic has proved to be one of the most disruptive public health emergencies in recent memory. Among non-pharmaceutical interventions, social distancing and lockdown measures are some of the most common tools employed by governments around the world to combat the disease. While mathematical models of COVID-19 are ubiquitous, few have leveraged network theory in a general way to explain the mechanics of social distancing. In this paper, we build on existing network models for heterogeneous, clustered networks with random link activation/deletion dynamics to put forth realistic mechanisms of social distancing using piecewise constant activation/deletion rates. We find our models are capable of rich qualitative behavior, and offer meaningful insight with relatively few intervention parameters. In particular, we find that the severity of social distancing interventions and when they begin have more impact than how long it takes for the interventions to take full effect.

We study sets of recurrence, in both measurable and topological settings, for actions of (N,?) and ( Q >0 ,?) . In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form {(an+b ) ??/(cn+d ) ??:n?�N} are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.

In this paper, we deal with reversing and extended symmetries of shifts generated by bijective substitutions. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how to check them. Moreover, we show that, for any finite group G and any subgroup P of the d -dimensional hyperoctahedral group, there is a bijective substitution which generates an aperiodic hull with symmetry group Z d ?G and extended symmetry group ( Z d ?�P)?G .

The ongoing Coronavirus disease 2019 (COVID-19) is a major crisis that has significantly affected the healthcare sector and global economies, which made it the main subject of various fields in scientific and technical research. To properly understand and control this new epidemic, mathematical modelling is presented as a very effective tool that can illustrate the mechanisms of its propagation. In this regard, the use of compartmental models is the most prominent approach adopted in the literature to describe the dynamics of COVID-19. Along the same line, we aim during this study to generalize and ameliorate many existing works that consecrated to analyse the behaviour of this epidemic. Precisely, we propose an SQEAIHR epidemic system for Coronavirus. Our constructed model is enriched by taking into account the media intervention and vital dynamics. By the use of the next-generation matrix method, the theoretical basic reproductive number R 0 is obtained for COVID-19. Based on some nonstandard and generalized analytical techniques, the local and global stability of the disease-free equilibrium are proven when R 0 <1 . Moreover, in the case of R 0 >1 , the uniform persistence of COVID-19 model is also shown. In order to better adapt our epidemic model to reality, the randomness factor is taken into account by considering a proportional white noises, which leads to a well-posed stochastic model. Under appropriate conditions, interesting asymptotic properties are proved, namely: extinction and persistence in the mean. The theoretical results show that the dynamics of the perturbed COVID-19 model are determined by parameters that are closely related to the magnitude of the stochastic noise. Finally, we present some numerical illustrations to confirm our theoretical results and to show the impact of media intervention and quarantine strategies.

An algorithm is proposed for finding numerical solutions of a kinetic equation that describes an infinite system of point articles placed in R d (d≥1) . The particles perform random jumps with pair wise repulsion, in the course of which they can also merge. The kinetic equation is an essentially nonlinear and nonlocal integro-differential equation, which can hardly be solved analytically. The derivation of the algorithm is based on the use of space-time discretization, boundary conditions, composite Simpson and trapezoidal rules, Runge-Kutta methods, adjustable system-size schemes, etc. The algorithm is then applied to the one-dimensional version of the equation with various initial conditions. It is shown that for special choices of the model parameters, the solutions may have unexpectable time behaviour. A numerical error analysis of the obtained results is also carried out.

We focus on the presence of almost automorphy in strongly monotone skew-product semiflows on Banach spaces. Under the C 1 -smoothness assumption, it is shown that any linearly stable minimal set must be almost automorphic. This extends the celebrated result of Shen and Yi [Mem. Amer. Math. Soc. 136(1998), No. 647] for the classical C 1,α -smooth systems. Based on this, one can reduce the regularity of the almost periodically forced differential equations and obtain the almost automorphic phenomena in a wider range.

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set {1,2 } N of directive sequences. For a given set C of two substitutions, we show that there exists a C -adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most 2n+1 and is 2n+1 if and only if the letter frequencies are rationally independent if and only if the C -adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that μ -almost every C -adic sequence is balanced, where μ is any shift-invariant ergodic Borel probability measure on {1,2 } N giving a positive measure to the cylinder [12121212] . We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure μ is negative.

We study the point spectrum of the linearisation of Euler's equation for the ideal fluid on the torus about a shear flow. By separation of variables the problem is reduced to the spectral theory of a complex Hill's equation. Using Hill's determinant an Evans function of the original Euler equation is constructed. The Evans function allows us to completely characterise the point spectrum of the linearisation, and to count the isolated eigenvalues with non-zero real part. In particular this approach also works in the case where complex eigenvalues appear.

We show that non-trivial chain recurrent classes for generic C 1 star flows satisfy a dichotomy: either they have zero topological entropy, or they must be isolated. Moreover, chain recurrent classes for generic star flows with zero entropy must be sectional hyperbolic, and cannot be detected by any non-trivial ergodic invariant probability. As a result, we show that C 1 generic star flows have only finitely many Lyapunov stable chain recurrent classes.