Any monotone Boolean circuit computing the n-dimensional Boolean convolution requires at least n2 and-gates. This matches the obvious upper bound. The previous best bound for this problem was Ω(n4/3), obtained by Norbert Blum in 1981. More generally, exact bounds are given for all semi-disjoint bilinear forms.

In air pollution models, semi-Lagrangian methods are often used to solve the ad-vective part of the corresponding model equation. Interpolation is an essential part of these methods. In this note, ve diierent interpolation methods will be discussed and results of numerical experiments will be presented. To keep the concentration eld nonnegative, ltering techniques are used. Also a monotone inte...

In A Relative Value Iteration Algorithm for Nondegenerate Controlled Diffusions, [SIAM J. Control Optim., 50 (2012), pp. 1886–1902], convergence of the relative value iteration for the ergodic control problem for a nondegenerate diffusion controlled through its drift was established, under the assumption of geometric ergodicity, using two methods: (a) the theory of monotone dynamical systems an...

In order to ensure convergence to the viscosity solution, the standard method for discretizing HJB PDEs uses forward/backward differencing for the drift term. In this paper, we devise a monotone method which uses central weighting as much as possible. In order to solve the discretized algebraic equations, we have to maximize a possibly discontinuous objective function at each node. Nevertheless...

A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively. We call this process “the invelope” and show that it is an almost surely uniquely defined function of integrated Brownian motion. Its role is comparable to the role of the...

In A Relative Value Iteration Algorithm for Nondegenerate Controlled Diffusions, [SIAM J. Control Optim., 50 (2012), pp. 1886–1902], convergence of the relative value iteration for the ergodic control problem for a nondegenerate diffusion controlled through its drift was established, under the assumption of geometric ergodicity, using two methods: (a) the theory of monotone dynamical systems an...

Abstract. The ergodic control problem for a non-degenerate controlled diffusion controlled through its drift is considered under a uniform stability condition that ensures the well-posedness of the associated Hamilton–Jacobi– Bellman (HJB) equation. A nonlinear parabolic evolution equation is then proposed as a continuous time continuous state space analog of White’s ‘relative value iteration’ ...

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then app...

The depth hierarchy results for monotone circuits of Raz and McKenzie [5] are extended to the case of monotone circuits of semiunbounded fan-in. It follows that the inclusions NC ⊆ SAC ⊆ AC are proper in the monotone setting, for every i ≥ 1. Mathematics Subject Classification. 68Q17, 68Q15.

A parallel dynamic load balancing for 2-D and 3-D semiconductor device simulations is proposed. The hydrodynamic and drift diffusion models are discretized and solved with finite volume (box) and monotone iterative methods. Dynamic load balancing for parallel domain decomposition as well as parallel I-V point simulations are demonstrated to be efficient methods for multidimensional device simul...