This paper is concerned with the application of principles of differential geometry in multigrid for the div-grad, curl-curl and grad-div equations. First, the discrete counterpart of the formulas for edge, face and volume elements are used to derive a sequence of a commuting edge, face and volume prolongator from an arbitrary partition of unity nodal prolongator. The implied coarse topology an...

1. SOBOLEV SPACES AND WEAK FORMULATIONS Let Ω be a bounded Lipschitz domain in R. We introduce the Sobolev spaces H(curl ; Ω) = {v ∈ L(Ω), curlv ∈ L(Ω)}, H(div; Ω) = {v ∈ L(Ω),div v ∈ L(Ω)} The vector fields (E,H) belong to H(curl ; Ω) while the flux (D,B) in H(div; Ω). We shall use the unified notation H(d; Ω) with d = grad , curl , or div. Note that H(grad ; Ω) is the familiar H(Ω) space. The...

Rick Bohan is a management consultant with Chagrin River Consulting, LLC. He has expertise in strategic planning and organizational development. He is a graduate of the Weatherhead School of Management at Case Western Reserve University. Married to a United Methodist pastor, he has a good sense of what is needed to get churches unstuck and has consulted with churches in need of developing strat...

The genetics underlying flower symmetry shifts between radial and bilateral symmetry has been intensively studied in the model Antirrhinum majus. Understanding the conservation or diversification of this genetic pathway in other plants is of special interest in understanding angiosperm evolution and ecology. Evidence from Antirrhinum indicates that TCP and MYB transcription factors, especially ...

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><div class="column"><p>Como consecuencia de la irrupción un evento tan crítico como Covid-19, las ciencias sociales y humanas han comenzado a repensar los conceptos e ideas que establecían complejidad del mundo en habitamos. Por ejemplo, reflexionando sobre el “pensar global”, expresión ac...