Northumberland role in the Industrial Revolution from the 18th century on. Many coal mines operated in Northumberland until the widespread closures in the 1970s and 1980s. Collieries operated at Ashington, Bedlington, Blyth, Choppington, Netherton, Ellington and Pegswood. The region's coalfields fuelled industrial expansion in other areas of Britain, and the need to transport the coal from the collieries to the Tyne led to the development of the first railways. Shipbuilding and armaments manufacture were other important industries before the deindustrialisation of the 1980s. Northumberland remains largely rural, and is the least-densely populated county in England. In recent years the county has
Densely defined operator Densely defined operator In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they "a priori" "make sense". A densely defined linear operator "T" from one topological vector space, "X", to another one, "Y", is a linear operator that is defined on a dense linear subspace dom("T")