Covers counting via Feynman Calculus

Let $G$ be a finite group. In this paper we present a tool for counting the number of principle $G$-bundles over a surface. As an application, we express (non-standard) generating functions for double Hurwitz numbers as integrals over commutative Frobenius algebras, associated with symmetric groups.Comment: 15 pages, 5 figure

On mathematical theory of selection: Continuous time population dynamics

Mathematical theory of selection is developed within the frameworks of general models of inhomogeneous populations with continuous time. Methods that allow us to study the distribution dynamics under natural selection and to construct explicit solutions of the models are developed. All statistical characteristics of interest, such as the mean values of the fitness or any trait can be computed effectively, and the results depend in a crucial way on the initial distribution. The developed theory provides an effective method for solving selection systems; it reduces the initial complex model to a special system of ordinary differential equations (the escort system). Applications of the method to the Price equations are given; the solutions of some particular inhomogeneous Malthusian, Ricker and logistic-like models used but not solved in the literature are derived in explicit form.Comment: 29 pages; published in J. of Mathematical Biolog