Extended Abstracts Spring 2014

The idea that geometry and physics are intimately related made its way in human thought during the early part of the nineteenth century. Gauss measured the angles of a triangle formed by three mountain peaks near Göttingen, Germany, apparently hoping to learn whether the universe has positive or negative curvature, but the inevitable observational errors rendered his results inconclusive [3]. In the 1830s, Bolyai and Lobachevsky took these investigations further. They independently addressed the connection between geometry and physics by seeking a natural extension of the gravitational law from Euclidean to hyperbolic space. Their idea led to the study of the Kepler problem and the 2-body problem in spaces of nonzero constant Gaussian curvature, κ ≠ 0, two fundamental problems that are not equivalent, unlike in Euclidean space. A detailed history of the results obtained in this direction since Bolyai and Lobachevsky can be found in [3, 5, 6].

The study of central configurations is very popular for producing the simplest solutions of the planar n-body problems (cf., [1, 2, 4]). In this paper, we study the central configuration of the isosceles trapezoidal five-body problem where four of the masses are placed at the vertices of the isosceles trapezoid and the fifth body can take various positions on the axis of symmetry. We identify regions in the phase space where it is possible to choose positive masses which will make the configuration central. A similar approach was adopted by Shoaib et al. in [3] for the rhomboidal five-body problem.

One of the most challenging problems in agronomy is to obtain plants that are resistant to the infection of pathogens. Not only this, but also that resistance must be as durable as possible. Unfortunately, most, if not all, strategies to generate such resistant plants have been overcome by the tremendous evolutionary potential of viral pathogens. In recent years, a new strategy based on the transgenic expression of artificial micro-RNAs (amiRs), designed to target viral genomes and induce their degradation, has been developed. This resistance has proven to be highly effective and sequence-specific against several plant viruses infecting Arabidopsis thaliana [7]. However, before these transgenic plants can be deployed in the field, it was important to evaluate the likelihood of the emergence of resistance-breaking mutants [2, 5]. Two issues were of particular interest: (1) whether such mutants can arise in non-transgenic plants that may act as reservoirs for the viral populations and (2) whether a suboptimal expression level of the transgene, resulting in sub-inhibitory concentrations of the amiR, would favor the emergence of escape mutants.

Viral populations are extremely plastic [5]. They maintain and steadily generate high levels of genotypic and phenotypic diversity that result in the coexistence of several different viral types in quasi-species, and eventually constitute a powerful tool to deploy different adaptive strategies. The interest in understanding and formally describing viral populations has steadily increased. At present, there are major unknown factors that difficult the construction of realistic models of viral evolution, as the way in which mutations affect fitness [19] or, in a broader scenario, which is the statistical nature of viral fitness landscapes. Our understanding of viral complexity is however improving thanks to new techniques as deep sequencing [17] or massive computation, and to systematic laboratory assays that reveal that, as other complex biological systems (e.g., cancer or ecosystems) the term virus embraces a dissimilar collection of populations with a remarkable ensemble of evolutionary strategies. New empirical data and improved models of viral dynamics are clearing up the role played by neutral networks of genotypes [21], by defective and cooperative interactions among viral mutants [13], by co-evolution with immune systems [22], or by changes in host populations [10], to cite but a few examples. Models of viral evolution are steadily improving their accuracy and becoming more competent from a conceptual and a predictive viewpoint [11, 12]. Here, we review some examples were well-motivated models of viral evolution succeed at capturing experimentally described features of those populations. Such are the relationship between intra-species competition and the geometry of the propagating substrate of a viral infection [3], the origin of bipartite viral genomes [8], and the adaptation to multi-drug therapies [9, 16].

Systems with many components (individuals or local populations as cities, or metropolitan areas, or regions, …) connected by non-trivial associations or relationships can be statistically described by means of the formalism of complex networks which is based on descriptors like degree distributions, degree-degree correlations, etc. In the last years, many researchers from different fields have been using different approaches to model processes taking place on complex networks.