Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero and non-unit x of R can be written as a product ( including an empty product ) of irreducible elements p < sub > i </ sub > of R and a unit u:

Formally, EMF is classified as the external work expended per unit of charge to produce an electric potential difference across two open-circuited terminals.

Formally, modified duration is a semi-elasticity, the percent change in price for a unit change in yield, rather than an elasticity, which is a percentage change in output for a percentage change in input.

Formally, the equation above defines the macroscopic neutron cross-section ( for reaction x ) as the proportionality constant between a neutron flux incident on a ( thin ) piece of material and the number of reactions that occur ( per unit volume ) in that material.

Formally the Prince-Electors elected a King of the Romans, who was elected in Germany but became Holy Roman Emperor only when crowned by the Pope.

Formally it was part of Arles, but in reality it was under the influence of the French king.

Formally named " United Press Associations " for incorporation and legal purposes, but publicly known and identified as United Press or UP, it was created by the 1907 uniting of three smaller news syndicates by the Midwest newspaper publisher E. W. Scripps.

Formally a census-designated place ( CDP ), Laguna West-Lakeside was annexed by Elk Grove in 2003.

In late 1900s the day that of the City was founded is Jakin School is a Private School is Formally Part of Jakin School Board was Grades One through Twelve there is no Football field on this school. In 1966 When Jakin School was Closed were Sent to Blakely-Union Elementary, Junior High, and High School ( which later Became Early County High, Middle, and Elementary School ). while Carver school was built for African-American students grades One through Twelve. although late 1960s when Carver school was closed along with Kestler School was based in Damascus were sent to Washington High & Elementary at Blakely. In addition to small farm agriculture, Jakin's early economic growth resulted from turpentine.

Formally, it was ceded to Russia in the Treaty of Nystad in 1721, together with Swedish Estonia and Swedish Ingria.

Formally declared on October 9, 1970, the Khmer Republic was a right-wing pro-United States military-led government headed by General Lon Nol and Prince Sisowath Sirik Matak that took power in the March 18, 1970 coup against Prince Norodom Sihanouk, then the country's head of state.

Formally it was part of Serbia, but it actually was a virtually separate autonomous region ruled by its German minority.

Formally known as Motion Picture Distribution LP, it was re branded and relaunched in 2007 due to the break-up of its preceding company, Alliance Atlantis, which was sold off piece by piece to Canwest Global, GS Capital Partners, along with several other smaller companies.

Formally, the Cisalpine Republic was an independent state allied with France, but the treaty of alliance established the effective subalternity of the new republic to France.

Formally known as Sportsdesk, it was re-vamped to closer resemble ESPN's own SportsCenter ( including the use of its theme music, logo, and opening ) in the Fall of 2001 as part of a corporate restructuring, closer aligning itself with new minority owner ESPN.

Formally known as the Kelowna Grizzlies, the club was purchased in

Formally speaking, the opportunity was extended to all classes to join China's project on its Leninist terms.

Formally known as the Inverurie Parish Church, a split in Church of Scotland over the appointment of ministers in 1843 ( The Disruption ) led to the creation of the West Parish Church ( known locally as " The West Kirk ") which was founded as a Free Church ( a church free from Edinburgh control ) thus causing the Inverurie Parish Church to change its name.

Formally, the group was called Studiengesellschaft für Geistesurgeschichte ‚ Deutsches Ahnenerbe e. V.

Formally a member of the country's collective head of state and, as president of parliament, holder of the third-highest office, he was left with little political influence.

Formally known as the " Long Tinh ", the flag was the official flag of the Nguyễn Court.

Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, their designation is the letter Ž and the number.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

Formally, the derivative of the function f at a is the limit

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

Formally, the base is known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).

Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.

Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

Formally, a bifunctor is a functor whose domain is a product category.

Formally, a set S is called finite if there exists a bijection

Formally, the system is said to have memory.

Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset

Formally, Φ = kx − ωt is the phase.