We construct pseudorandom generators with improved seed length for several classes of tests. First we consider the class of read-once polynomials over GF(2) in m variables. For error ε we obtain seed length Õ(log(m/ε)) log(1/ε), where Õ hides lower-order terms. This is optimal up to the factor Õ(log(1/ε)). The previous best seed length was polylogarithmic in m and 1/ε. Second we consider produc...

The object of this thesis is the study of high density discrete systems as variational limit of low density discrete energies indexed by the number of nodes of the system itself. In this context the term discreteness should be understood rather broadly as inferring to different scales from crystal lattice to grain structure while the low-to-high density limit refers to the discrete-to-continuum...

We present an algorithm for the approximate decomposition of diagonal operators, focusing specifically on decompositions over the Clifford+T basis, that minimize the number of phase-rotation gates in the synthesized approximation circuit. The equivalent T -count of the synthesized circuit is bounded by k C0 log2(1/ε) +E(n, k), where k is the number of distinct phases in the diagonal n-qubit uni...

We consider the problem Max Sync Set of finding a maximum synchronizing set of states in a given automaton. We show that the decision version of this problem is PSPACEcomplete. We prove that, assuming P 6= NP , for any ε > 0 the Max Sync Set problem cannot be approximated in polynomial time within a factor of O(k) for monotonic kstate automata with an alphabet of linear size, within a factor of...

We present a variational reformulation of a class of doubly nonlinear parabolic equations as (limits of) constrained convex minimization problems. In particular, an ε−dependent family of weighted energy-dissipation (WED) functionals on entire trajectories is introduced and proved to admit minimizers. These minimizers converge to solutions of the original doubly nonlinear equation as ε → 0. The ...

In the polytope membership problem, a convex polytope K in R is given, and the objective is to preprocess K into a data structure so that, given a query point q ∈ R, it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting and assume that d is a constant. Given an approximation parameter ε > 0, the query can be answered either way if the distance...

Let F (uε) + ε(uε − w) = 0 (1) where F is a nonlinear operator in a Hilbert space H, w ∈ H is an element, and ε > 0 is a parameter. Assume that F (y) = 0, and F ′(y) is not a boundedly invertible operator. Sufficient conditions are given for the existence of the solution to (1) and for the convergence limε→0 ‖uε−y‖ = 0. An example of applications is considered. In this example F is a nonlinear ...

The summation formula n−1 i=0 ε i i! (i k + u k) = v k + ε n−1 n! A k−1 (n) A k−1 is a polynomial) is derived and its various aspects are considered. In particular, divisibility with respect to n is investigated. Infinitely many equivalents to Kurepa's hypothesis on the left factorial are found.

Abstract. We consider an asymptotic preserving numerical scheme initially proposed by F. Filbet & S. Jin [10] and G. Dimarco & L. Pareschi [9] in the context of nonlinear and stiff kinetic equations. Here, we propose a convergence analysis of such a scheme for the approximation of a system of transport equations with a nonlinear source term, for which the asymptotic limit is given by a conserva...