Vinny Pagano | Searcharxiv

arXiv2023

Bicategories, Biequivalence, and Bi-Interpretability

We make explicit the correspondence between syntax and syntactic categories for coherent first-order logic, providing a categorical characterization of bi-interpretability. This is done by creating a biequivalence between a bicategory of coherent theories and the (strict) bicategory of coherent categories. While the biequivalence concerns the stronger equality-preserving bi-interpretability, we use it to obtain a necessary and sufficient condition for two theories to be bi-interpretable in general, by relating the exact completions of their syntactic categories. These results extend analogously to familiar fragments of first-order logic, thereby clarifying the long-intuited relation between logical syntax and syntactic categories.

math.LO↗

arXiv2021

A $p$-adic Approach To Piecewise Polynomial Dynamical Systems

Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to \mathbb{Z}_+$ defined by $$\text{Col}(n) = \begin{cases} 3n + 1, & n \text{ is odd;} \\ n/2, & \text{otherwise.}\end{cases}$$ Proving the non-existence of non-trivial Collatz cycles would reduce the Collatz conjecture to whether the Collatz map can diverge to infinity.

math.DS↗

arXiv2021

Modeling Oyster Reef Reproductive Sustainability: Analyzing Gamete Viability, Hydrodynamics, and Reef Structure to Facilitate Restoration of $\textit{Crassostrea virginica}$

The eastern oyster is a keystone species and ecosystem engineer. However, restoration efforts of wild oysters are often unsuccessful, in that they do not produce a robust population of oysters that are able to successfully reproduce. Furthermore, the dynamics of wild oyster fertilization is not yet well understood. Through conducting an experiment predicated on quantifying the influence of elementary aspects of fertilization kinetics--sperm concentration, gamete age, and success rate--we found that, as stochastic as the mating process may seem, there are correlations which fundamentally serve as the framework for assessing long-term sustainability, reef structure, and hydrodynamic parameters in relation to fertilization. We then focused on mathematically defining a procedure which simulated a concentration distribution of a single sperm and egg release where there existed conditions necessary for breeding to take place. We found a very significant impact of both gamete age and sperm concentration on fertilization rate ($p < 0.0001$). Our hydrodynamic model demonstrates that distance can also drastically influence broadcast spawning. This could be used as a foundation for developing a flexible model for wild oyster fertilization based on placement, initial seawater conditions, and size of the starting population. The results of this research could be implemented into a more user-friendly program which would accept multiple variables as inputs and output the probability of fertilization given arbitrary values. By accounting for environmental deviations, this generalization would increase its compatibility with the public and actualize the project's intended purpose: enhance the planning of oyster reef restoration projects.

q-bio.QM↗