Testing Booleanity and the Uncertainty Principle

Let f:{-1,1}^n -> R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial. We say that it is Boolean if its image is in {-1,1}. We show that every function on the hypercube with a sparse Fourier expansion must either be Boolean or far from Boolean. In particular, we show that a multilinear polynomial with at most k terms must either be Boolean, or output values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its domain. It follows that given oracle access to f, together with the guarantee that its representation as a multilinear polynomial has at most k terms, one can test Booleanity using O(k^2) queries. We show an \Omega(k) queries lower bound for this problem. Our proof crucially uses Hirschman's entropic version of Heisenberg's uncertainty principle.Comment: 15 page

A Black Hole Attack Model for Reactive Ad-Hoc Protocols

Net-Centric Warfare places the network in the center of all operations, making it a critical resource to attack and defend during wartime. This thesis examines one particular network attack, the Black Hole attack, to determine if an analytical model can be used to predict the impact of this attack on ad-hoc networks. An analytical Black Hole attack model is developed for reactive ad-hoc network protocols DSR and AODV. To simplify topology analysis, a hypercube topology is used to approximate ad-hoc topologies that have the same average node degree. An experiment is conducted to compare the predicted results of the analytical model against simulated Black Hole attacks on a variety of ad-hoc networks. The results show that the model describes the general order of growth in Black Hole attacks as a function of the number of Black Holes in a given network. The model accuracy maximizes when both the hypercube approximation matches the average degree and number of nodes of the ad-hoc topology. For this case, the model falls within the 95% confidence intervals of the estimated network performance loss for 17 out of 20 measured scenarios for AODV and 7 out of 20 for DSR

System Identification for Nonlinear Control Using Neural Networks

An approach to incorporating artificial neural networks in nonlinear, adaptive control systems is described. The controller contains three principal elements: a nonlinear inverse dynamic control law whose coefficients depend on a comprehensive model of the plant, a neural network that models system dynamics, and a state estimator whose outputs drive the control law and train the neural network. Attention is focused on the system identification task, which combines an extended Kalman filter with generalized spline function approximation. Continual learning is possible during normal operation, without taking the system off line for specialized training. Nonlinear inverse dynamic control requires smooth derivatives as well as function estimates, imposing stringent goals on the approximating technique

A New Approach to Numerical Quantum Field Theory

In this note we present a new numerical method for solving Lattice Quantum Field Theory. This Source Galerkin Method is fundamentally different in concept and application from Monte Carlo based methods which have been the primary mode of numerical solution in Quantum Field Theory. Source Galerkin is not probabilistic and treats fermions and bosons in an equivalent manner.Comment: 10 pages, LaTeX, BROWN-HET-908([email protected]), ([email protected]), ([email protected]

Ant-Inspired Density Estimation via Random Walks

Many ant species employ distributed population density estimation in applications ranging from quorum sensing [Pra05], to task allocation [Gor99], to appraisal of enemy colony strength [Ada90]. It has been shown that ants estimate density by tracking encounter rates -- the higher the population density, the more often the ants bump into each other [Pra05,GPT93]. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides new tools for understanding complex dependencies in the collision probabilities of multiple random walks. We bound the strength of these dependencies using $local\ mixing\ properties$ of the underlying graph. Our results extend beyond the grid to more general graphs and we discuss applications to size estimation for social networks and density estimation for robot swarms