Regulator

Regulator may refer to:

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Combined sewer

A combined sewer is a sewage collection system of pipes and tunnels designed to also collect surface runoff. Combined sewers can cause serious water pollution problems during combined sewer overflow (CSO) events when wet weather flows exceed the sewage treatment plant capacity. This type of sewer design is no longer used in building new communities (because current design separates sanitary sewers from runoff), but many older cities continue to operate combined sewers.

Background

Most sewage collection systems of the 19th and early to mid 20th century used single-pipe systems that collect both sewage and urban runoff from streets and roofs. This type of collection system is referred to as a combined sewer system. The rationale for combining the two was that it would be cheaper to build just a single system. Most cities at that time did not have sewage treatment plants, so there was no perceived public health advantage in constructing a separate "surface water sewerage" (the term used in the UK) or "storm sewer" (the term used in the U.S.) system.

Dirichlet's unit theorem

In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring O_K of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.

The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to

where r₁ is the number of real embeddings and r₂ the number of conjugate pairs of complex embeddings of K. This characterisation of r₁ and r₂ is based on the idea that there will be as many ways to embed K in the complex number field as the degree n = [K : Q]; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that

Note that if K is Galois over Q then either r₁ is non-zero or r₂ is non-zero, but not both.

Other ways of determining r₁ and r₂ are