In this paper, we extend the weak r -net theorem to a kinetic setting where the underlying set of points is moving polynomially with bounded description complexity. We establish that one can find a kinetic analog N of a weak r -net of cardinality O(r d(d+1) 2 log r) whose points are moving with coordinates that are rational functions with bounded description complexity. Moreover, each member of...

We prove global Lorentz estimates for variable power of the gradient of weak solution to linear elliptic obstacle problems with small partially BMO coefficients over a bounded nonsmooth domain. Here, we assume that the leading coefficients are measurable in one variable and have small BMO semi-norms in the other variables, variable exponents p(x) satisfy log-Hölder continuity, and the boundarie...

We find more self-tuning solutions by introducing a general form for Lagrangian of a 3-index antisymmetric tensor field AMNP in the RS II model. In particular, for the logarithmic Lagrangian, ∝ log(−H2), we obtained a closed form weak self-tuning solution.

E. Mossel and J. Neeman recently provided a heat flow monotonicity proof of Borell’s noise stability theorem. In this note, we develop the argument to include in a common framework noise stability, Brascamp-Lieb inequalities (including hypercontractivity), and even a weak form of Slepian inequalities. The scheme applies furthermore to families of measures with are more log-concave than the Gaus...

We present a survey of the cosmological applications of the next generation of weak lensing surveys, paying special attention to the computational challenges presented by the number of galaxies, N gal ∼ 10 5. We focus on optimal methods with no pixelization and derive a multigrid P 3 M algorithm that performs the relevant computations in O(N gal log N gal) time. We test the algorithm by studyin...

Given a finite point set P ⊂ R, we call a multiset A a one-sided weak ε-approximant for P (with respect to convex sets), if |P ∩ C|/|P | − |A ∩C|/|A| ≤ ε for every convex set C. We show that, in contrast with the usual (two-sided) weak ε-approximants, for every set P ⊂ R there exists a one-sided weak ε-approximant of size bounded by a function of ε and d.

A fundamental task for an autonomous robot is to plan its own motions. Exact approaches to the solution of this motion planning problem suffer from high worst-case running times. The weak and realistic low obstacle density (L.O.D.) assumption results in linear complexity in the number of obstacles of the free space [11]. In this paper we address the dynamic version of the motion planning proble...