Chaos and chaotic fluid mixing

Very simple mathematical equations can give rise to surprisingly complicated, chaotic dynamics, with behavior that is sensitive to small deviations in the initial conditions. We illustrate this with a single recurrence equation that can be easily simulated, and with mixing in simple fluid flows

Analysing the Factors that Influence Social Media Adoption Among SME's in Developing Countries

Social media penetration is on the rise in developing countries and is an important channel of growth for small and medium enterprises (SMEs). Many SMEs in developing countries use social media to connect their customers to their products and services. However, the factors that have led the existing SMEs in Africa to adopt or reject Social Media need to be clarified to understand the key contributing factors and influences at play. This paper adopts the learning-by-doing concept from economic theory to explore the factors that influence the adoption of social media. A primary survey follows this to examine the use of social media among firms in the commercial districts of Kenya and Nigeria. The preliminary surveys in both countries were combined into a single dataset to analyse the relationship between social media use and learning-by-doing. The results show that while small SMEs are limited in their social media capacity, medium size firms tend to invest in their social media presence, and larger-size firms still rely on traditional marketing channels

Barriers to front propagation in ordered and disordered vortex flows

We present experiments on reactive front propagation in a two-dimensional (2D) vortex chain flow (both time-independent and time-periodic) and a 2D spatially disordered (time-independent) vortex-dominated flow. The flows are generated using magnetohydrodynamic forcing techniques, and the fronts are produced using the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. In both of these flows, front propagation is dominated by the presence of burning invariant manifolds (BIMs) that act as barriers, similar to invariant manifolds that dominate the transport of passive impurities. Convergence of the fronts onto these BIMs is shown experimentally for all of the flows studied. The BIMs are also shown to collapse onto the invariant manifolds for passive transport in the limit of large flow velocities. For the disordered flow, the measured BIMs are compared to those predicted using a measured velocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems

Frozen reaction fronts in steady flows: a burning-invariant-manifold perspective

The dynamics of fronts, such as chemical reaction fronts, propagating in two-dimensional fluid flows can be remarkably rich and varied. For time-invariant flows, the front dynamics may simplify, settling in to a steady state in which the reacted domain is static, and the front appears "frozen". Our central result is that these frozen fronts in the two-dimensional fluid are composed of segments of burning invariant manifolds---invariant manifolds of front-element dynamics in $xy\theta$-space, where $\theta$ is the front orientation. Burning invariant manifolds (BIMs) have been identified previously as important local barriers to front propagation in fluid flows. The relevance of BIMs for frozen fronts rests in their ability, under appropriate conditions, to form global barriers, separating reacted domains from nonreacted domains for all time. The second main result of this paper is an understanding of bifurcations that lead from a nonfrozen state to a frozen state, as well as bifurcations that change the topological structure of the frozen front. Though the primary results of this study apply to general fluid flows, our analysis focuses on a chain of vortices in a channel flow with an imposed wind. For this system, we present both experimental and numerical studies that support the theoretical analysis developed here.Comment: 21 pages, 30 figure

Periodicity of joint co-tiles in Zd\mathbb{Z}^d

An old theorem of Newman asserts that any tiling of $\mathbb{Z}$ by a finite set is periodic. Few years ago Bhattacharya proved the periodic tiling conjecture in $\mathbb{Z}^2$. Namely, he proved that for a finite subset $F$ of $\mathbb{Z}^2$, if there exists $A \subseteq \mathbb{Z}^d$ such that $F \oplus A = \mathbb{Z}^d$ then there exists a periodic $A' \subseteq \mathbb{Z}^d$ such that $F \oplus A' = \mathbb{Z}^d$. The recent refutation of the periodic tiling conjecture in high dimensions due to Greenfeld and Tao motivates finding different generalizations of Newman's theorem and of Bhattacharya's theorem that hold in arbitrary dimension $d$. In this paper, we formulate and prove such generalizations. We do so by studying the structure of joint co-tiles in $\mathbb{Z}^d$. Our generalization of Newman's theorem states that for any $d \ge 1$, any joint co-tile for $d$ independent tiles is periodic. For a $(d-1)$-tuple of finite subsets of $\mathbb{Z}^d$ that satisfy a certain technical condition that we call property $(\star)$, we prove that any joint co-tile decomposes into disjoint $(d-1)$-periodic sets. Consequently, we show that for a $(d-1)$-tuple of finite subsets of $\mathbb{Z}^d$ that satisfy property $(\star)$, the existence of a joint co-tile implies the existence of periodic joint co-tile. Conversely, we prove that if a finite subset $F$ in $\mathbb{Z}^d$ admits a periodic co-tile $A$, then there exist $(d-1)$ additional tiles that together with $F$ are independent and admit $A$ as a joint co-tile, and $(d-2)$ additional tiles that together with $F$ satisfy the property $(\star)$. Combined, our results give a new necessary and sufficient condition for a subset of $\mathbb{Z}^d$ to tile periodically. We also discuss tilings and joint tilings in other countable abelian groups.Comment: Minor update