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, the integral is the inner product of the luminosity function with the light spectrum.

Formally, the idea is to take the integral defining the ( inverse ) Fourier transform

Formally, the differential appearing under the integral behaves exactly as a differential: thus, the integration by substitution and integration by parts formulae for Stieltjes integral correspond, respectively, to the chain rule and product rule for the differential.

Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write

Formally, an analytic function ƒ ( z ) of the real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub > is transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ ( z ) are algebraically independent, i. e., if ƒ is transcendental over the field C ( z < sub > 1 </ sub >, …, z < sub > n </ sub >).

Formally, it is defined as the analytic signal corresponding to the real field.

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.

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

Formally, arbitrage transactions have negative skew – prices can get a small amount closer ( but often no closer than 0 ), while they can get very far apart.

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, they were part of the non-resident sections, but in fact constituted a separate division, largely autonomous in their activities.

Formally, the Congress serves two functions: to approve changes to the Party constitution regarding policy and to elect a Central Committee, about 300 strong.

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, 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, one has an implication, not an equivalence, so the solution set may get larger.

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 inaugurated in March 2004, the Global Leadership Foundation works to " promote good governance – democratic institutions, open markets, human rights and the rule of law – and to contribute to the prevention and resolution of conflict through mediation.

Formally given to bishop Philip of Senj, the permission to use the Glagolitic liturgy ( the Roman Rite conducted in Slavic language instead of Latin, not the Byzantine rite ), actually extended to all Croatian lands, mostly along the Adriatic coast.

Formally, the system is said to have memory.

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

Formally neutral and reluctant to become involved with the great powers except as a last resort, Kuwait turned to the United States, the Soviet Union and Britain for naval protection of its tanker fleet after twenty-one ships were attacked in the gulf since late 1986.

* 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.

Massiah: Formally Charged + Deliberate Elicitation.

*, see On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.

Formally, these failed when they were rejected by the Church of England's General Synod in 1972 ; conversations and co-operation continued, however, leading in 2003 to the signing of a covenant between the two churches.

Formally introduced in September 1979, Initial samples were released in February 1980, with production chips available over the counter in November.

* Nuclear bunker buster: Formally known as the Robust Nuclear Earth Penetrator ( RNEP ), this program aimed to modify an existing gravity bomb to penetrate into soil and rock in order to destroy underground targets.

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

Moreover, if ( 1 ) is construed as a Lebesgue integral, then ( 1 ) is also undefined, because ( 1 ) is then defined simply as the difference ( 2 ) between positive and negative parts.

However, if ( 1 ) is construed as an improper integral rather than a Lebesgue integral, then ( 2 ) is undefined, and ( 1 ) is not necessarily well-defined.

Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral.

The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory.

In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.

Measure theory provides the foundation for the modern notion of integral, the Lebesgue integral.

Some of the technical deficiencies in Riemann integration can be remedied by the Riemann – Stieltjes integral, and most of these disappear with the Lebesgue integral.

The Lebesgue integral with respect to the measure δ satisfies

As a result, the latter notation is a convenient abuse of notation, and not a standard ( Riemann or Lebesgue ) integral.

* Lebesgue – Stieltjes integral

Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations.

Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is " If the sequence f < sub > n </ sub >( x ) increases to the limit f ( x ), the integral of f < sub > n </ sub >( x ) tends to the integral of f ( x ).

" Lebesgue shows that his conditions lead to the theory of measure and measurable functions and the analytical and geometrical definitions of the integral.

In measure-theoretic analysis and related branches of mathematics, the Lebesgue – Stieltjes integral generalizes Riemann – Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.

Lebesgue integration has the property that every bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree.