Let φ be a 3CNF formula with n variables and m clauses. A simple nonconstructive argument shows that when m is sufficiently large compared to n, most 3CNF formulas are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most such formulas proves that they are not satisfiable. A possible approach to refute a formula φ is: first, translate it into a...

Let M be a separable compact Hausdorff space with dim M ≤ 2 and θ : M → M be a homeomorphism with prime period p (p ≥ 2). Set Mθ = {x ∈ M | θ(x) = x} 6= ∅ and M0 = M\Mθ. Suppose that M0 is dense in M and H(M0/θ,Z) ∼= 0, H(χ(M0/θ),Z) ∼= 0. Let M ′ be another separable compact Hausdorff space with dim M ′ ≤ 2 and θ′ be the self–homeomorphism of M ′ with prime period p. Suppose that M ′ 0 = M ′\M ...

Firstly, a new state feedback model reference adaptive control approach is developed for uncertain systems with gain scheduled reference models in a multi-input multi-output (MIMO) setting. Specifically, adaptive state feedback for output tracking control problem of MIMO nonlinear systems is studied and gain scheduled reference model system is used for generating desired state trajectories. Usi...

M-estimators and their consistency This handout is adapted from Section 3.3 of 18.466 lecture notes on mathematical statistics, available on OCW. A sequence of estimators T n , one for each sample size n, possibly only defined for n large enough, is called consistent if for X 1 , X converges in probability as n → ∞ to a function g(θ) being estimated. We will consider consistency of estimators m...

In the first part of this paper, we let G be a finitely-generated amenable group such that G/[G,G] is torsion-free. We suppose that G acts by homeomorphisms homotopic to the identity on a manifold M , and give conditions on M which imply that such an action must lift to an action on the universal cover M̃ . The circle, all 2-manifolds except the open annulus, and most compact 3-manifolds satisfy...

On any strictly pseudoconvex CR manifold M, of dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM the Levi Gθ, such that norm Reeb vector field T (M, θ) is order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between manifolds M and A, where Θ on A. For every contraction gϵA GΘ, write...