Genus Comparisons in the Topological Analysis of RNA Structures

RNA folding prediction remains challenging, but can be also studied using a topological mathematical approach. In the present paper, the mathematical method to compute the topological classification of RNA structures and based on matrix field theory is shortly reviewed, as well as a computational software, McGenus, used for topological and folding predictions. Additionally, two types of analysis are performed: the prediction results from McGenus are compared with topological information extracted from experimentally-determined RNA structures, and the topology of RNA structures is investigated for biological significance, in both evolutionary and functional terms. Lastly, we advocate for more research efforts to be performed at intersection of physics-mathematics and biology, and in particular about the possible contributions that topology can provide to the study of RNA folding and structure.Comment: 11 pages, 8 figure

On projective spaces over local fields

Let $\mathcal{P}$ be the set of points of a finite-dimensional projective space over a local field $F$, endowed with the topology $\tau$ naturally induced from the canonical topology of $F$. Intuitively, continuous incidence abelian group structures on $\mathcal{P}$ are abelian group structures on $\mathcal{P}$ preserving both the topology $\tau$ and the incidence of lines with points. We show that the real projective line is the only finite-dimensional projective space over an Archimedean local field which admits a continuous incidence abelian group structure. The latter is unique up to isomorphism of topological groups. In contrast, in the non-Archimedean case we construct continuous incidence abelian group structures in any dimension $n \in \mathbb{N}$. We show that if $n>1$ and the characteristic of $F$ does not divide $n+1$, then there are finitely many possibilities up to topological isomorphism and, in any case, countably many.Comment: 22 page

Reasoning by Analogy in Mathematical Practice

The testimony and practice of notable mathematicians indicate that there is an important phenomenological and epistemological difference between superficial and deep analogies in mathematics. In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of 'suggesting' a conjecture in the psychological sense). The proposed conditions generalize the criteria put forward by Hesse (1963) in her influential work on analogical reasoning in the empirical sciences. By reference to several case-studies, we argue that the account proposed in this paper does a better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha (2009). Moreover, our proposal offers novel insights into the practice of extending to the infinite case mathematical properties known to hold in finite domains.Comment: 35 pages, 4 figures. Forthcoming in Philosophia Mathematic

What is so special about analogue simulations?

Contra Dardashti, Th\'ebault, and Winsberg (2017), this paper defends an analysis of arguments from analogue simulations as instances of a familiar kind of inductive inference in science: arguments from material analogy (Hesse, Models and Analogies in Science, Univ Notre Dame Press, 1963). When understood in this way, the capacity of analogue simulations to confirm hypotheses about black holes can be deduced from a general account - fully consistent with a Bayesian standpoint - of how ordinary arguments from material analogy confirm. The proposed analysis makes recommendations about what analogue experiments are worth pursuing that are more credible than Dardashti, Hartmann, Th\'ebault, and Winsberg's (2019). It also offers a more solid basis for addressing the concerns by Crowther, Linneman, and W\"utrich (2019), according to which analogue simulations are incapable of sustaining hypotheses concerning black hole radiation.Comment: 26 pages, 1 figur

Exact solutions for a Solow-Swan model with non-constant returns to scale

The Solow-Swan model is shortly reviewed from a mathematical point of view. By considering non-constant returns to scale, we obtain a general solution strategy. We then compute the exact solution for the Cobb-Douglas production function, for both the classical model and the von Bertalanffy model. Numerical simulations are provided.Comment: 10 pages, 4 figure