We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers' equation). The finite element method is higher-order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. We first prove that if a sequence of finite element so...

Numerical methods for the optimal transport problem is an active area of research. Recent work Kitagawa and Abedin shows that solution a time-dependent equation converges exponentially fast as time goes to infinity problem. This suggests numerical algorithm computing maps; we investigate such here in 1-dimensional case. Specifically, use finite difference scheme solve carry out error analysis s...

This paper considers solving optimization problem for linear discrete time systems such that closed-loop discrete-time system is positive (i.e., all of its state variables have non-negative values) and also finite-time stable. For this purpose, by considering a quadratic cost function, an optimal controller is designed such that in addition to minimizing the cost function, the positivity proper...

Does the interaction graph of a finite dynamical system can force this system to have a “complex” dynamics ? In other words, given a finite interval of integers A, which are the signed digraphsG such that every finite dynamical system f : An → An with G as interaction graph has a “complex” dynamics ? If |A| ≥ 3 we prove that no such signed digraph exists. More precisely, we prove that for every...

The rotor-router model is a deterministic analogue of random walk invented by Jim Propp. It can be used to define a deterministic aggregation model analogous to internal diffusion limited aggregation. We prove an isoperimetric inequality for the exit time of simple random walk from a finite region in Zd, and use this to prove that the shape of the rotor-router aggregation model in Zd, suitably ...

We consider a U(1)-invariant nonlinear Klein-Gordon equation in dimension n ≥ 1, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges as t →±∞ to the two-dimensional set of all “nonlinear eigenfunctions” of the form φ(x)e−iωt . This global attraction is caused by ...

For a finite mean supercriticial Bellman-Harris process, let Zt be the number of particles at time t. There exist numbers χt (the Seneta constants) such that χtZt converges almost surely to a non-degenerate limit. Furthermore, χt ∝ e −βt L(e), where β is the Malthusian parameter, and L is slowly varying at zero. We obtain a characterisation of the slowly varying part of the Seneta constants und...

Stabilization of simple mechanical systems in finite time with continuous state feedback is considered here. The dynamics is represented by generalized (local) coordinates. A general methodology to construct control Lyapunov functions that are Hölder continuous and that can be used to show finite-time stability of the feedback controlled system, is presented. This construction also gives the fe...

For a finite mean supercriticial Bellman-Harris process, let Zt be the number of particles at time t. There exist numbers χt (the Seneta constants) such that χtZt converges almost surely to a non-degenerate limit. Furthermore, χt ∝ e −βt L(e), where β is the Malthusian parameter, and L is slowly varying at zero. We obtain a characterisation of the slowly varying part of the Seneta constants und...

In this paper we give an analysis of a bubble stabilized discontinuous Galerkin method (BSDG) for elliptic and parabolic problems. The method consists of stabilizing the numerical scheme by enriching the discontinuous finite element space elementwise by quadratic non-conforming bubbles. This approach leads to optimal convergence in the space and time discretization parameters. Moreover the dive...