The space c0 lies at the heart of many constructions of higher order smooth functions on Banach spaces. To name a few, recall Torunczyk’s proof of the existence of C-smooth partitions of unity on WCG spaces ([12]) or Haydon’s recent constructions of C-bump functions on certain C(K) spaces ([7]). The crucial property of c0 that allows for those constructions is a rich supply of C -smooth functio...

In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipsc...

Univariate piecewise-smooth refinable functions (i.e., compactly supported solutions of the equation φ( 2 ) = ∑N k=0 ckφ(x−k)) are classified completely. Characterization of the structure of refinable splines leads to a simple convergence criterion for the subdivision schemes corresponding to such splines, and to explicit computation of the rate of convergence. This makes it possible to prove a...

This study presents a new approach for optimal allocation of distributed generation (DG) and energy storage system (ESS) in microgrids (MGs). The practical optimal allocation problems have non-smooth cost functions with equality and inequality constraints that make the problem of finding the global optimum difficult using any mathematical approaches. A dynamic capacity adjustment algorithm is i...