We consider stability properties of equilibria in stochastic evolutionary dynamics. In particular, we study the stability of mixed equilibria in strategic form games. In these games, when the populations are small, all strategies may be stable. We prove that when the populations are large, the unique stable outcome of best-reply dynamics in 2 × 2 games with a unique Nash equilibrium that is com...

Necessary conditions of optimality in the form of a weak maximum principle are derived for optimal control problems with mixed constraints. Such conditions differ from previous work since they hold when a certain convexity assumption is replaced by an “interiority” assumption. Notably the result holds for problems with possibly nonsmooth mixed constraints and with additional pointwise set contr...

A minimal variance reconstruction scheme is derived using derivatives of the Gaussian as filters. A closed form mixed correlation matrix for reconstructions from multiscale points and their local derivatives up to the second order is presented. With the inverse of this mixed correlation matrix, a reconstruction of the image can be easily calculated. Some interesting results of reconstructions f...

The lme4 package uses sparse matrix technology and clever decompositions of the likelihood to fit linear, generalized, and nonlinear mixed-effects models. The amer package extends lme4’s scope to include generalized additive mixed models (GAMM). This vignette summarizes the main ideas behind additive models and their representation in the form of a mixed model, describes the modifications to lm...

Water flow into unsaturated porous media is governed by the Richards’ partial differential equation expressing the mass conservation law and the Darcy ́s law. This equation may be written in three standard forms: the “h-based” form, the “θ-based” form, and the “mixed” form. Numerical aproximations based on different forms of this one-dimensional nonlinear equation can lead to significantly diffe...

A mixed matrix pencil is a matrix pencil which has two kinds of nonzero coefficients: fixed constants that account for conservation laws and independent parameters that represent physical characteristics. In this paper, we characterize the indices of nilpotency of the Kronecker canonical form for a mixed matrix pencil in terms of matroids. As a byproduct, we provide an algorithm for computing t...

We provide a physical prescription based on interferometry for introducing the total phase of a mixed state undergoing unitary evolution, which has been an elusive concept in the past. We define the parallel transport condition that provides a connection form for obtaining the geometric phase for mixed states. The expression for the geometric phase for mixed state reduces to well known formulas...

Displacement finite element models of various beam theories have been developed using traditional finite element interpolations (i.e., Hermite cubic or equi-spaced Lagrange functions). Various finite element models of beams differ from each other in the choice of the interpolation functions used for the transverse deflection w, total rotation φ and/or shear strain γxz, or in the integral form u...