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 of "X" and takes values in "Y", written "T" : dom("T") ⊆ "X" → "Y". Sometimes this is abbreviated as "T" : "X" → "Y" when the context makes it clear that "X" might not be the set-theoretic domain of "T". 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