and Applied Analysis 3 where A is an n × n constant matrix with n different eigenvalues λ1, λ2, . . . , λn and Q t is analytic quasi-periodic with respect to t with frequencies ω ω1, ω2, . . . , ωl . Here ε is a small perturbation parameter. Suppose that the following nonresonance conditions hold: ∣∣〈k,ω〉 √ −1 λi − λj ∣∣ ≥ α |k| , 1.8 for all k ∈ Z \ {0}, where α > 0 is a small constant and τ >...

We propose an algorithm to sample and mesh a k-submanifold M of positive reach embedded in R. The algorithm first constructs a crude sample of M. It then refines the sample according to a prescribed parameter ε, and builds a mesh that approximates M. Differently from most algorithms that have been developped for meshing surfaces of R, the refinement phase does not rely on a subdivision of R (su...

This paper investigates the following natural greedy procedure for clustering in the bi-criterion setting: iteratively grow a set of centers, in each round adding the center from a candidate set that maximally decreases clustering cost. In the case of k-medians and k-means, the key results are as follows. • When the method considers all data points as candidate centers, then selecting O(k log(1...

We consider a class of second order elliptic problems in a domain of RN , N > 2, ε-periodically perforated by holes of size r(ε) , with r(ε)/ε → 0 as ε → 0. A nonlinear Robin-type condition is prescribed on the boundary of some holes while on the boundary of the others as well as on the external boundary of the domain, a Dirichlet condition is imposed. We are interested in the asymptotic behavi...

Let F : X → X be a C 2 loc map in a Banach space X, and A be its Frèchet derivative at the element w := w ε , which solves the problem (*) ˙ w = −A −1 ε (F (w) + εw), w(0) = w 0 , where A ε := A + εI. Assume that A −1 ε ≤ cε −k , 0 < k ≤ 1, 0 < ε > ε 0. Then (*) has a unique global solution, w(t), there exists w(∞), and (* *) F (w(∞)) + εw(∞) = 0. Thus the DSM (Dynamical Systems Method) is just...

Proof of Observation 1. Before the proof, we remind that what we want to prove is the following statement: (∀ε ∈ R) ε > 0⇒ ((∃N ∈ N)(∀n ∈ N) n ≥ N ⇒ |an − x | < ε). To this end, choose arbitrary positive real ε > 0 and pick a positive rational ε′ with 0 < ε′ < ε by utilizing the Archimedean property. Then there exists N (depending on ε′ and hence on ε) such that j, k ≥ N ⇒ |a j − ak | < ε′. Now...

We give an efficient randomized algorithm for approximating an arbitrary element of SU(2) by a product of Clifford+T operators, up to any given error threshold ε > 0. Under a mild hypothesis on the distribution of primes, the algorithm’s expected runtime is polynomial in log(1/ε). If the operator to be approximated is a z-rotation, the resulting gate sequence has T -count K + 4 log 2 (1/ε), whe...

and Applied Analysis 3 where xi k , i 1, 2 stand for the density of the prey and the predator at time k, respectively. ui k , i 1, 2 are the control variables at time k. h2 is a positive constant, denoting the constant of capturing half-saturation. In this paper, we are concerned with the effects of the almost periodicity of ecological and environmental parameters and time delays on the global ...