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In this paper, we prove that the degree of regularity of square systems, a subfamily of the HFE systems, over a prime finite field of odd characteristic q is exactly q and, therefore, prove that inverting square systems algebraically using Gröbner basis algorithm is exponential, when q = \Omega(n), where n is the number of variables of the system.

Robustness of numerical methods has attracted an increasing interest in the community of computational fluid dynamics. One mathematical aspect of robustness for numerical methods is the positivity-preserving property. At high Mach numbers or for flows near vacuum, solving the conservative Euler equations may generate negative density or internal energy numerically, which may lead to nonlinear instability and crash of the code. This difficulty is particularly profound for high order methods, for multi-material flows and for problems with moving meshes, such as the Lagrangian methods. In this paper, we construct both first order and uniformly high order accurate conservative Lagrangian schemes which preserve positivity of physically positive variables such as density and internal energy in the simulation of compressible multi-material flows with general equations of state (EOS). We first develop a positivity-preserving approximate Riemann solver for the Lagrangian scheme solving compressible Euler equations with both ideal and non ideal EOS. Then we design a class of high order positivity-preserving and conservative Lagrangian schemes by using the essentially non-oscillatory (ENO) reconstruction, the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter which can be proven to maintain conservation and uniformly high order accuracy and is easy to implement. One-dimensional and two-dimensional numerical tests for the positivity-preserving Lagrangian schemes are provided to demonstrate the effectiveness of these methods.

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We develop mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we formulate a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behavior. Our approach utilizes mixed-type, multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy BBGKY-like equation hierarchies.