Periods of generalized Fermat curves

Let $k,n\ge 2$ be integers. A generalized Fermat curve of type $(k,n)$ is a compact Riemann surface S that admits a subgroup of conformal automorphisms $H\le \text{Aut}(S)$ isomorphic to ${\mathbb{Z}}_k^n$, such that the quotient surface $S/H$ is biholomorphic to the Riemann sphere $\stackrel{?}{\mathbb{C}}$ and has $n+1$ branch points, each one of order k. There exists a good algebraic model for these objects, which makes them easier to study. Using tools from algebraic topology and integration theory on Riemann surfaces, we find a set of generators for the first homology group of a generalized Fermat curve. Finally, with this information, we find a set of generators for the period lattice of the associated Jacobian variety.     

An Introduction to Influence Theory: Kinematics and Dynamics

Influence theory is a foundational theory of physics that is not based on traditional empirically defined concepts, such as positions in space and time, mass, energy, or momentum. Instead, the aim is to derive these concepts, and their empirically determined relationships, from a more primitive model. It is postulated that there exist things, which are call particles, that influence one another in a discrete and directed fashion resulting in a partially ordered set of influence events. The problem of consistent quantification of the influence events is considered. Observers are modeled as particle chains (observer chains) as if an observer were able to track a particle and quantify the influence events that the particle experiences. From these quantified influence events, consistent quantification of the universe of events based on the observer chains is studied. Herein, the kinematics and dynamics of particles from the perspective of influence theory are both reviewed and further developed.