A Continuous Extension that preserves Concavity, Monotonicity and Lipschitz Continuity

The following is proven here: let W : X × C −→ R, where X is convex, be a continuous and bounded function such that for each y ∈ C, the function W (·, y) : X −→ R is concave (resp. strongly concave; resp. Lipschitzian with constant M; resp. monotone; resp. strictly monotone) and let Y ⊇ C. If C is compact, then there exists a continuous extension of W , U : X × Y −→ £ infX×C W, supX×C W ¤, such that for each y ∈ Y , the function U (·, y) : X −→ R is concave (resp. strongly concave; resp. Lipschitzian with constant My ; resp. monotone; resp. strictly monotone)

A Predator Prey Model with Disease Dynamics

We propose a model to describe the interaction between a diseased fish population and their predators. Analysis of the system is performed to determine the stability of equilibrium points for a large range of parameter values. The existence and uniqueness of solutions is established and solutions are shown to be uniformly bounded for all nonnegative initial conditions. The model predicts that a deadly disease and a predator population cannot co-exist. Numerical simulations illustrate a variety of dynamical behaviors that can be obtained by varying the problem data

Optimal Transportation and Curvature of Metric Spaces

In this thesis we study the notion of non-negative Ricci curvature for compact metric measure spaces introduced by Lott and Villani in their article (2009): Ricci curvature for metric measure spaces via optimal transport. We also define and prove the required prerequisites concerning length spaces, convex analysis, measure theory, and optimal transportation