Background. When studying the microscale behavior (beyond the reach of numerical solution methods) of physical systems, one is naturally lead to the concept of homogenization, i.e., the theory of the convergence of sequences of partial differential equations. The homogenization of periodic structures using the two-scale convergence technique is well-established due to the pioneering work by Gab...

Scale-free networks arise from power-law degree distributions. Due to the finite size of real-world networks, the power law inevitably has a cutoff at some maximum degree ∆. We investigate the relative size of the giant component S in the large-network limit. We show that S as a function of ∆ increases fast when ∆ is just large enough for the giant component to exist, but increases ever more sl...

Passive scalar motion in a family of random Gaussian velocity fields with longrange correlations is shown to converge to persistent fractional Brownian motions in long times.

Theorem 4.3 Let the initialization u0 be sufficiently close to the solution ū of P. Then the iterates uk of Algorithm 1 converge superlinearly to ū in L2(Ω). Moreover, the corresponding states yk converge superlinearly to ȳ in H1 0 (Ω). Proof To apply Theorem 4.1, it remains to show that the generalized derivative (4.7) is invertible and that the norms of the inverse linear mappings are bounded...

Keywords: Density Statistical convergence Weighted statistical convergence Positive linear operator Korovkin type approximation theorem a b s t r a c t Korovkin type approximation theorems are useful tools to check whether a given sequence ðL n Þ nP1 of positive linear operators on C½0; 1Š of all continuous functions on the real interval ½0; 1Š is an approximation process. That is, these theore...

The integration of Distributed Energy Resources (DER) in the distribution network has plenty of advantages if their allocation and Penetration Level (PL) are done appropriately. Hence, the challenge of finding the best allocation and PL of DERs in large distribution networks is an important but intricate problem. This paper proposes a novel methodology to simultaneously determine the optimal lo...