The growth of the rank of Abelian varieties upon extensions

Number theory, Algebra and Geometr

Chabauty-Coleman experiments for genus 3 hyperelliptic curves

We describe a computation of rational points on genus 3 hyperelliptic curves $C$ defined over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in $C(\mathbb{Q})$.Comment: 18 page

Equilibrium states for potentials with \sup\phi - \inf\phi < \htop(f)

In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials $\phi$ with he `bounded range' condition \sup \phi - \inf \phi < \htop, first used by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of Perron-Frobenius operators. We demonstrate that this `bounded range' condition on the potential is important even if the potential is H\"older continuous. We also prove analyticity of the pressure in this context.Comment: Added Lemma 6 to deal with the disparity between leading eigenvalues and operator norms. Added extra references and corrected some typo

Complex maps without invariant densities

We consider complex polynomials $f(z) = z^\ell+c_1$ for $\ell \in 2\N$ and $c_1 \in \R$, and find some combinatorial types and values of $\ell$ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when $\ell$ sufficiently large and also for a class of `long-branched' maps of any critical order.Comment: Typos corrected, minor changes, principally to Section

Natural equilibrium states for multimodal maps

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials $-t \log|Df|$, for the largest possible interval of parameters $t$. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained

On the Lebesgue measure of Li-Yorke pairs for interval maps

We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure

A Review of Indigenous Food Crops in Africa and the Implications for more Sustainable and Healthy Food Systems

Indigenous and traditional foods crops (ITFCs) have multiple uses within society, and most notably have an important role to play in the attempt to diversify the food in order to enhance food and nutrition security. However, research suggests that the benefits and value of indigenous foods within the South African and the African context have not been fully understood and synthesized. Their potential value to the African food system could be enhanced if their benefits were explored more comprehensively. This synthesis presents a literature review relating to underutilized indigenous crop species and foods in Africa. It organizes the findings into four main contributions, nutritional, environmental, economic, and social-cultural, in line with key themes of a sustainable food system framework. It also goes on to unpack the benefits and challenges associated with ITFCs under these themes. A major obstacle is that people are not valuing indigenous foods and the potential benefit that can be derived from using them is thus neglected. Furthermore, knowledge is being lost from one generation to the next, with potentially dire implications for long-term sustainable food security. The results show the need to recognize and enable indigenous foods as a key resource in ensuring healthy food systems in the African continent

The relation between conscientiousness, empowerment and performance.

This study examined the relationship between conscientiousness, empowerment and job performance among information technology professionals. An Employee Empowerment Questionnaire (EEQ), a Conscientiousness Scale and a Social Desirability Scale were administered to 101 information technology customer service engineers. Managers completed a Performance Evaluation Questionnaire (PEQ) for each customer service engineer. The results indicated a significant relationship between conscientiousness and empowerment. A curvilinear relationship was found between empowerment and performance. The practical and theoretical implications of the findings are discussed

How does Europe Make Its Mind Up? Connections, cliques, and compatibility between countries in the Eurovision Song Contest

We investigate the complex relationships between countries in the Eurovision Song Contest, by recasting past voting data in terms of a dynamical network. Despite the British tendency to feel distant from Europe, our analysis shows that the U.K. is remarkably compatible, or 'in tune', with other European countries. Equally surprising is our finding that some other core countries, most notably France, are significantly 'out of tune' with the rest of Europe. In addition, our analysis enables us to confirm a widely-held belief that there are unofficial cliques of countries -- however these cliques are not always the expected ones, nor can their existence be explained solely on the grounds of geographical proximity. The complexity in this system emerges via the group 'self-assessment' process, and in the absence of any central controller. One might therefore speculate that such complexity is representative of many real-world situations in which groups of 'agents' establish their own inter-relationships and hence ultimately decide their own fate. Possible examples include groups of individuals, societies, political groups or even governments