In this paper, we investigate the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation, ẍ + [δ + ε(cosω1t + cosω2t)]x + αx3 = 0. (1) We assume that ε 1 and that the coefficient of the nonlinear term, α, is positive but not necessarily small. We utilize Lie transform perturbation theory with elliptic functions – rather than the usual trigonometric functions – to ...

We consider the boundary value problem ∆u+ε k(x) e = 0 in a bounded, smooth domain Ω in R with homogeneous Dirichlet boundary conditions. Here ε > 0, k(x) is a non-negative, not identically zero function. We find conditions under which there exists a solution uε which blows up at exactly m points as ε→ 0 and satisfies ε ∫ Ω keε → 8mπ. In particular, we find that if k ∈ C(Ω̄), infΩ k > 0 and Ω is...

Let k[ε]2 := k[ε]/(ε2). The single valued real analytic n-polylogarithm Ln : C → R is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in Ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms lin:k[ε]2 → k and prove that they satisfy functional equations analogous to those of Ln . Unde...

We consider a second order damped-vibration equation Mẍ + D(ε)ẋ + Kx = 0, where M, D(ε), K are real, symmetric matrices of order n. The damping matrix D(ε) is defined by D(ε) = Cu + C(ε), where Cu presents internal damping and rank(C(ε)) = r, where ε is dampers’ viscosity. We present an algorithm which derives a formula for the trace of the solution X of the Lyapunov equation AX + XA = −B, as a...

k-Anonymity and ε-differential privacy are two mainstream privacy models, the former introduced to anonymize data sets and the latter to limit the knowledge gain that results from the inclusion of one individual in the data set. Whereas basic k-anonymity only protects against identity disclosure, t-closeness was presented as an extension of k-anonymity that also protects against attribute discl...

For every d ≥ 3 and k ∈ {2} ∪ [3,∞), we determine the smallest ε such that every fractional (k + ε)-precoloring of vertices at mutual distance at least d of a graph G with fractional chromatic number equal to k can be extended to a proper fractional (k + ε)-coloring of G. Our work complements the analogous results of Albertson for ordinary colorings and those of Albertson and West for circular ...

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ∥u(t, ·) − u(t, ·) ∥∥ L = O(1)(1 + t) · √ε| ln ε| on the distance between an exact BV solution u and a viscous approximation u, letting the viscosity coefficient ε → 0. In the proof, starting from u we construct an approximation of the viscous solution u by taking a mollification u ∗ φ√ ε a...

Let X be a Banach space with closed unit ball B. Given k ∈ N, X is said to be k-β, repectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0, there exists δ, 0 < δ < 1, so that for every x ∈ B, and every ε-separated sequence (xn) ⊆ B, there are indices (ni) k i=1, respectively, (ni) k+1 i=1 , such that 1 k+1 ‖x + ∑k i=1 xni‖ ≤ 1 − δ, respectively, 1 k+1 ‖ ∑k+1 i=1 xni‖ ≤ 1−...

For a graph G with n vertices and m edges, we give a randomized Las Vegas algorithm that approximates a small balanced vertex separator of G in almost linear time. More precisely, we show the following, for any 2 3 ≤ α < 1 and any 0 < ε < 1 − α: If G contains an α-separator of size K, then our algorithm finds an (α + ε)-separator of size O(ε−1K2 log n) in time O(ε−1K3m log n) w.h.p. In particul...