On pure-strategy Nash equilibria in a duopolistic market share model

This paper develops a duopolistic discounted marketing model with linear advertising costs and advertised prices for mature markets still in expansion. Generic and predatory advertising effects are combined together in the model. We characterize a class of advertising models with some lowered production costs. For such a class of models, advertising investments have a no-free-riding strict Nash equilibrium in pure strategies if discount rates are small. We discuss the entity of this efficiency at varying of parameters of our advertising model. We provide a computational framework in which market shares can be computed at equilibrium, too. We analyze market share dynamics for an asymmetrical numerical scenario where one of the two firms is more effective in generic and predatory advertising. Several numerical insights on market share dynamics are obtained. Our computational framework allows for different scenarios in practical applications and it is developed, thanks to Mathematica software

The influence of fractional diffusion in Fisher-KPP equations

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the stan- dard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable L\'evy process, the front position is exponential in time. Our results provide a mathe- matically rigorous justification of numerous heuristics about this model

Complete solution of a constrained tropical optimization problem with application to location analysis

We present a multidimensional optimization problem that is formulated and solved in the tropical mathematics setting. The problem consists of minimizing a nonlinear objective function defined on vectors over an idempotent semifield by means of a conjugate transposition operator, subject to constraints in the form of linear vector inequalities. A complete direct solution to the problem under fairly general assumptions is given in a compact vector form suitable for both further analysis and practical implementation. We apply the result to solve a multidimensional minimax single facility location problem with Chebyshev distance and with inequality constraints imposed on the feasible location area.Comment: 20 pages, 3 figure

Geometric Properties of the Maxwell Set and a Vortex Filament Structure for Burgers Equation

The inviscid limit of the stochastic Burgers equation is discussed in terms of the level surfaces of the minimising Hamilton–Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. We examine the geometry of the Maxwell set in terms of the behaviour of the pre-Maxwell set, the pre-caustic and the pre-level surfaces. In particular, contrary to the ideas of Helmholtz and Lord Kelvin, we prove that even if initially the fluid flow is irrotational, in the inviscid limit, associated with the advent of the Maxwell set a non-zero vorticity vector forms in the fluid with vortex lines on the Maxwell set. This suggests that in quite general circumstances for small viscosity there is a vortex filament structure near the Maxwell set for both deterministic and stochastic Burgers equations

Max-plus convex geometry

Abstract. Max-plus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including max-plus versions of the separation theorem, existence of linear and non-linear projectors, max-plus analogues of the Minkowski-Weyl theorem, and the characterization of the analogues of “simplicial ” cones in terms of distributive lattices.