We consider a scheduling problem where the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The objective is to find a resource allocation and a schedule that minimizes the make...

It is shown that the energy (ε) and momentum (k) dependences of the electron selfenergy function Σ(k, ε + i0) ≡ Σ(k, ε) are, ImΣ(k, ε) = −aε|ε − ξk| −γ(k) where a is some constant, ξk = ε(k)−μ, ε(k) being the band energy, and the critical exponent γ(k), which depends on the curvature of the Fermi surface at k, satisfies, 0 ≤ γ(k) ≤ 1. This leads to a new type of electron liquid, which is the Fe...

Let F (u ε) + ε(u ε − 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 (1.1) and for the convergence lim ε→0 u ε − y = 0. An example of applications is considered. In this example F is a non...

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...