Consider the singular perturbation problem for εu(t; ε) + u(t; ε) = Au(t; ε) + ∫ t 0 K(t− s)Au(s; ε) ds+ f(t; ε) , where t ≥ 0, u(0; ε) = u0(ε), u (0; ε) = u1(ε), and w(t) = Aw(t) + ∫ t 0 K(t− s)Aw(s)ds+ f(t) , t ≥ 0 , w(0) = w0 , in a Banach space X when ε → 0. Here A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and K(t) is a bounded linear opera...

Let ε > 0 and consider ε2u′′(t; ε) + u′(t; ε) = Au(t; ε) + ∫ t 0 K(t− s)Au(s; ε)ds+ f(t; ε), t ≥ 0, u(0; ε) = u0(ε), u ′(0; ε) = u1(ε), and w′(t) = Aw(t) + ∫ t 0 K(t− s)Aw(s)ds+ f(t), t ≥ 0, w(0) = w0, in a Banach space X when ε → 0. Here A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and K(t) is a bounded linear operator for t ≥ 0. With some conv...

l k=0 (−1) m−k l k m − k n 2k k − 2l + m = l k=0 l k 2k n n − l m + n − 3k − l. On the basis of this identity, for d ∈ {0, 1, 2,. .. } and ε ∈ {0, ±1} we construct explicit f ε (d) and g ε (d) such that for any prime p > d we have p−1 k=1 k ε C k+d ≡ f ε (d) if p ≡ 1 (mod 3), g ε (d) if p ≡ 2 (mod 3), where C n denotes the Catalan number 1 n+1 2n n ; for example, if p 5 is a prime then 0<k<p−4 ...

In this paper a new macroscopic k-ε model is developed and validated for turbulent flow through porous media for a wide range of porosities. The morphology of porous media is simulated by a periodic array of square cylinders. In the first step, calculations based on microscopic v2 − f model are conducted using a Galerkin/Least-Squares finite element formulation, employing equalorder bilinear ve...

In this work, we study the socially fair k-median/k-means problem. We are given a set of points P in metric space X with distance function d(.,.). There ℓ groups: P1,…,Pℓ⊆P. also F feasible centers X. The goal k-median problem is to find C⊆F k that minimizes maximum average cost over all groups. That is, C objective Φ(C,P)≡maxj⁡{∑x∈Pjd(C,x)/|Pj|}, where d(C,x) x closest center C. k-means define...

We present a study of the Hausdorff Core problem on simple polygons. A polygon Q is a k-bounded Hausdorff Core of a polygon P if P contains Q, Q is convex, and the Hausdorff distance between P and Q is at most k. A Hausdorff Core of P is a k-bounded Hausdorff Core of P with the minimum possible value of k, which we denote kmin. Given any k and any ε > 0, we describe an algorithm for computing a...

We study the following problem: given a real number k and integer d, what is the smallest ε such that any fractional (k+ε)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k + ε)coloring of the whole graph? The exact values of ε were known for k ∈ {2} ∪ [3,∞) and any d. We determine the exact values of ε for k ∈ (2, 3)...