* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given a function f of a real variable x and an interval of the real line, the definite integral

Given the tendency for real negotiations in Japan to be conducted privately ( in the nemawashi, or root binding, process of consensus building ), the shingikai often represented a fairly advanced stage in policy formulation in which relatively minor differences could be thrashed out and the resulting decisions couched in language acceptable to all.

Given m real values t < sub > i </ sub >, called knots, with

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )

: Given: a function f: A R from some set A to the real numbers

Given expectations about returns on fixed investment, every level of the real interest rate ( i ) will generate a certain level of planned fixed investment and other interest-sensitive spending: lower interest rates encourage higher fixed investment and the like.

Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition

Given that the Liber historiae Francorum admits four years for Chlothar's reign, it remains possible that Chlothar began to exert some real authority when his mentors deemed him an adult, 669.

Given an ordered sequence of real numbers: the first difference is defined as

Given that his predecessors were monarchists who tried without success to restore the French monarchy, Grévy is seen as the first real republican President of France.

Given the variety of approaches, Economic Geography has taken to many different subject matters, including: the location of industries, economies of agglomeration ( also known as " linkages "), transportation, international trade, economic development, real estate, gentrification, ethnic economies, gendered economies, core-periphery theory, the economics of urban form, the relationship between the environment and the economy ( tying into a long history of geographers studying culture-environment interaction ), and globalization.

Given the nature of this interrogation and the motive behind it, it is not known for certain what elements of Number Six's life so portrayed are real and which are fiction.

Given two normed vector spaces V and W ( over the same base field, either the real numbers R or the complex numbers C ), a linear map A: V → W is continuous if and only if there exists a real number c such that

Given a real dynamical system ( R, M, Φ ), I ( x ) is an open interval in the real numbers, that is.

* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that

Given an m × n matrix with real entries ( or entries from any other field ) and rank r, then there exists at least one non-zero r × r minor, while all larger minors are zero.

Given that bankruptcies and real estate prices did not fare as negatively in Central Canada as in the rest of Canada and the United States during the NEP, it is possible that the NEP had a positive effect in Central Canada.

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given that the Qur ' an is vague regarding the punishment of homosexual sodomy, Islamic jurists turned to the collections of the hadith and seerah ( accounts of Muhammad's life ) to support their argument for Hudud punishment ; these are perfectly clear but particularly harsh.

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

Given the immense expanse of the entire Universe, it has been argued that there is a higher probability that there exists ( or has existed ) another Earth-like planet that has yielded life ( geogenesis ) than not.

" As Shelly Kagan says, “ Given the parameters of the actual world, there is no question that …( maximally )… promoting the good would require a life of hardship, self-denial, and austerity … a life spent promoting the good would be a severe one indeed .”

Given how little we can know for sure, our focus should be on this earth and life ; beauty, justice, love.

Given the abundance of such optimization problems in everyday life, a positive answer to the " P vs. NP " question would likely have profound practical and philosophical consequences.

Given that many open source projects fail due to lack of developer support, exposure to such a large community of developers can continually breathe new life into a project.

On 24 July 2003, the Congregation for the Doctrine of the Faith stated, " Given the centrality of the celebration of the Eucharist in the life of a priest, one must proceed with great caution before admitting to Holy Orders those candidates unable to ingest gluten or alcohol without serious harm.

Given a pre-modern understanding of reason, rationalism is identical to philosophy, the Socratic life of inquiry, or the zetetic ( skeptical ) clear interpretation of authority ( open to the underlying or essential cause of things as they appear to our sense of certainty ).

Given that SAD impacts a wide variety of occupational performance areas in a person ’ s life as described in the aforementioned section, occupational therapists ( OTs ) play a key role in helping individuals cope with SAD.

Given this information, a timeline of roughly the years 15 – 98 CE can be established for his adult life.

Given a life sentence, he died in prison on May 29, 2011.

" Given the chance to relive his life by switching bodies with " a talentless 20 year-old wannabe rock star with all the looks and charm Rocky used to have ," the series deals with the subsequent soul-searching and reflection over such a decision.

When the measure was sent in May to the parliament they said " Given the damage it produces on those rules which allow the life in community, ensure the dignity of the person and equality between sexes, this practice, even if it is voluntary, cannot be tolerated in any public place ".

Given that humanity cannot exist except within a covenantal relationship with God, and all covenants use symbols to give us " the attestation of his grace ", he gives the tree, " not because it could confer on man that life with which he had been previously endued, but in order that it might be a symbol and memorial of the life which he had received from God.

Given FishBase's success, there was a demand for a database covering forms of aquatic life other than finfish.

Given this Scriptural principle, the UPCI teaches that one should live a life that demonstrates Christ's attributes.

Given Orr / Alex's desire to remain within the world of the Bridge, a world where he is well treated and lives a fairly pampered life, his attempts to stonewall and block the doctor's attempts to cure him are understandable.

Given that Constitutional reforms in 2005 abolished life senators starting in 2006, Frei ran for an elected Senate seat in the December 2005 parliamentary elections and was victorious.

Given that this parasite spends part of its life cycle in two different hosts it must use a proportion of its available resources within each host.