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...

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...

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.

A (k, ε)-biased sample space is a distribution over {0, 1}n that ε-fools every nonempty linear test of size at most k. Since they were introduced by Naor and Naor (SIAM J. Computing, 1993), these sample spaces have become a central notion in theoretical computer science with a variety of applications. When constructing such spaces, one usually attempts to minimize the seed length as a function ...

Let 1 ≤ t ≤ 7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that |A∩B∩C∩D| ≥ t holds for all A,B,C,D ∈ F . Then we have |F | ≤ (n−t k−t ) if | k n − 2 |< ε holds for some ε > 0 and all n > n0(ε). We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”

This example is studied using Floquet theory. A classic example is k(t) = 1 + ε cosωt, where one finds regions of instability in the ε, ω plane for which there are solutions which grow exponentially in time. The instability regions are open and with closures touching ε = 0 at critical frequencies (see [A], [MW]). For any t, the constant coefficient problems with k frozen at k(t) is conservative...

Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups. 1. Contractions, degenerat...

New high-order accurate finite difference schemes based on defect correction are considered for an initial boundary-value problem on an interval for singularly perturbed parabolic PDEs with convection; the highest space derivative in the equation is multiplied by the perturbation parameter ε, ε ∈ (0, 1]. Solutions of the well-known classical numerical schemes for such problems do not converge ε...