On a Boltzmann mean field model for knowledge growth

In this paper we analyze a Boltzmann type mean field game model for knowledge growth, which was proposed by Lucas and Moll. We discuss the underlying mathematical model, which consists of a coupled system of a Boltzmann type equation for the agent density and a Hamilton-Jacobi-Bellman equation for the optimal strategy. We study the analytic features of each equation separately and show local in time existence and uniqueness for the fully coupled system. Furthermore we focus on the construction and existence of special solutions, which relate to exponential growth in time - so called balanced growth path solutions. Finally we illustrate the behavior of solutions for the full system and the balanced growth path equations with numerical simulations.Comment: 6 figure

Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth

In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas et al [13] to model knowledge growth in an economy. Agents can either increase their knowledge level by exchanging ideas in learning events or by producing goods with the knowledge they already have. The existence of balanced growth path solutions implies exponential growth of the overall production in time. We proof existence of balanced growth path solutions if the initial distribution of individuals with respect to their knowledge level satisfies a Pareto-tail condition. Furthermore we give first insights into the existence of such solutions if in addition to production and knowledge exchange the knowledge level evolves by geometric Brownian motion

Cross-diffusion systems with excluded volume effects and asymptotic gradient flows

In this paper we discuss the analysis of a cross-diffusion PDE system for a mixture of hard spheres, which was derived by Bruna and Chapman from a stochastic system of interacting Brownian particles using the method of matched asymptotic expansions. The resulting cross-diffusion system is valid in the limit of small volume fraction of particles. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. We discuss local stability and global existence for the symmetric case using the gradient flow structure and entropy variable techniques. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. Finally we illustrate the behavior of the model with various numerical simulations

Lane formation by side-stepping

In this paper we study a system of nonlinear partial differential equations, which describes the evolution of two pedestrian groups moving in opposite direction. The pedestrian dynamics are driven by aversion and cohesion, i.e. the tendency to follow individuals from the own group and step aside in the case of contraflow. We start with a 2D lattice based approach, in which the transition rates reflect the described dynamics, and derive the corresponding PDE system by formally passing to the limit in the spatial and temporal discretization. We discuss the existence of special stationary solutions, which correspond to the formation of directional lanes and prove existence of global in time bounded weak solutions. The proof is based on an approximation argument and entropy inequalities. Furthermore we illustrate the behavior of the system with numerical simulations

The real spectrum compactification of character varieties: characterizations and applications

We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber length compactification and apply our results to the theory of maximal and Hitchin representations

Buffy\u27s a Superhero : Buffy and Her Place in Feminist History

Buffy the Vampire Slayer was a successful television show in the late 90s and early 2000s. Joss Whedon created a blonde, female heroine to combat the image of a weak female who runs down the alley and dies during horror movies. But does a simple role reversal create a feminist icon

Lived Experiences Of Freshman Students In Their First Semester Of College

Why do some college students enter, persist, and successfully move through the educational system, while others lose their direction and leave college prior to achieving their academic goals? As students graduate from high school and enter a post-secondary educational environment, some are prepared for the experiences they will encounter, while others struggle through their first semester of enrollment as they attempt to make meaning of their life situations and transitions. This qualitative research study used Schlossberg’s (1981) Transition Theory as well as Baxter Magolda’s (2001) concept of self-reflection as a framework for discovery and understanding in an attempt to gain insight into the lived experiences of freshman students as they transitioned through their first semester of college. Individual interviews were conducted at three separate timeframes in the fall semester with freshman students enrolled in aviation and undeclared focused sections of an introduction to university life course. The data that emerged from the interviews revealed that academic experiences, time management, personal independence, as well as relationships and involvement with others make a difference in how students navigate through and make meaning of their first semester of their freshman year in college