Given any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

* Given any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

Given a group G, a factor group G / N is abelian if and only if ≤ N.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

It is described as " the ancient grayhound .. that had been with Lugh of the Mantles, / Given him by the sons of Tuireann Bicreann ;.."

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given the poorly understood nature of the endocannabinoid system, it will take many more generations of progress to even scratch the surface of the extraordinarily complex mechanisms that control neurological development / degeneration.

Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is p = 1 / Ω.

Given their vertical-cavity geometry, VCSOAs are resonant cavity optical amplifiers that operate with the input / output signal entering / exiting normal to the wafer surface.

Given the dimensions of the ellipsoid, the conversion from lat / lon / height-above-ellipsoid coordinates to X-Y-Z is straightforward — calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid.

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

Given that both Jonathan and Peleg Card were scheduled to be tried by the Vermont Committee of Safety ( Peleg died 6 Sep 1777, and it is unknown if the trial was actually held ) it is likely that Philo Card is, in reality, Peleg Card ( 1740 – 1777 ) --- and Stephen ( 1748 – 1798 ) -- ( Source: National Archives of Canada, WO 28 / 10, Reel B-2867, pp. 17 – 18.

Given the price of 1 / 32 Mark for six small fruit pies ( as seen in Dragonsinger ), it places the value at around $ 20 to the Mark.

Given that the majority of autoinducer – synthase / receptor occurs in tandem in bacterial genomes, it is also rare that they switch partners and so pairs tend to co-evolve.

) Given the diffusion of different Piers / Ploughman texts, it is usually not possible to be certain about what someone means to refer to when they mention " Piers Plowman " unless they provide specific identifying details — and most writers do not.

* Convex hull: Given a set of points, find the smallest convex polyhedron / polygon containing all the points.

* " The Property Regime in a Cyprus Settlement: A Reassessment of the Solution Proposed under the Annan Plan, Given the Performance of the Property Markets in Cyprus, 2003-2006 ( PDF )" Stelios Platis, Stelios Orphanides and Fiona Mullen, PRIO Report 2 / 2006, PRIO Cyprus Centre, November 2006

Given a strictly increasing integer sequence / function ( n ≥ 1 ) we can produce a faster growing sequence ( where the superscript n denotes the n < sup > th </ sup > functional power ).

Given a more realistic yield of 36 tonnes / hectare ( 3834 US gal / acre ) the area required is about 152, 000 square kilometers, or roughly equal to that of the state of Georgia or of England and Wales.

Given Western Electric / AT & T Unix licensing, DEC ( and others ) were restricted to selling binary-only licenses.

Given proper combustion chamber design, pre-ignition can generally be eliminated by proper spark plug selection, proper fuel / air mixture adjustment, and periodic cleaning of the combustion chambers.

The American Library Association issued a recommendation to libraries still holding Alms for Jihad: " Given the intense interest in the book, and the desire of readers to learn about the controversy first hand, we recommend that U. S. libraries keep the book available for their users.

Given a set of data and the desire to produce some kind of “ model ” of those data ( model, in this case, meaning a function fitted through the data ), there are a variety of functions that can be chosen for the fit.

Given Chris's resentment towards Tony about Adriana's death, and his continued desire to enter the movie business, he and Tony continue to drift apart.

Given that it was one of the few remaining structures, the authorities expressed the desire to see it rebuilt in part to cancel traumatic events from memory however rebuilding the tower would have required its total demolition and removing the fallen parts would have meant an unacceptable risk for the workers.

Given that the House of M reality was created by the Scarlet Witch and Charles Xavier combining their powers to give the assembled New Avengers and X-Men their heart's desires, it appears that Magneto's heart's desire was threefold.

Given investigators desire to choose larger margins to decrease the sample size needed to perform a trial, the chosen margin is sometimes larger than the effect size of the control compared to placebo.

The most recent measure of social dominance orientation ( see SDO-6 above ) focuses on the “ general desire for unequal relations among social groups, regardless of whether this means ingroup domination or ingroup subordination ” ( p. 312 ) Given these changes, Rubin and Hewstone believe that evidence for social dominance theory should be considered “ as supporting three separate SDO hypotheses, rather than one single theory ” ( p. 22 ).

Given his leader's power of mental persuasion, it is difficult to tell whether or not Off-Ramp actually had a homosexual desire for him, or if he was just being manipulated.

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 the magnitude of the disaster, reconstruction and economic recovery will remain the primary focus of the Salvadoran Government for some time to come.

Given that most French supporters of phrenology were liberal, left-wing or socialist, it was an objective of the social elite of France who held a restrained vision of social change that phrenology remain on the fringes.

Given the appropriate rites, she helped the deceased into afterlife as an underworld shade ( Di Manes ), else their spirit might remain to haunt the living, as a wandering, vengeful ghost.

Given that Rothko had known in advance about the luxury decor of the restaurant and the social class of its future patrons, the exact motives for his abrupt repudiation remain mysterious.

Given the borough's small size, as well as the county's small population, the presence of business and government remain limited.

Given a small angular displacement, the vessel may return to its original position ( stable ), move away from its original position ( unstable ), or remain where it is ( neutral ).

Given its heritage, MCC continues to participate in the administration of English cricket, and in 2010 offered Lord's as a neutral venue for Pakistan to stage a " home " Test match, as scheduled by the ICC, versus Australia ; although the outcome of that game proved controversial the club's initial offering was made with the intention that Pakistan, whose terrorist-stricken country had rendered it a no-go area for international cricket, could remain within the international cricketing fold.

Given that Hg ( CN )< sub > 2 </ sub > molecules remain undissociated in pure water and in basic solutions, it makes sense that dissociation would increase with increasing acidity.

Given the subversive nature of Lazarillo and its open criticism of the Catholic Church, it is likely that the author chose to remain anonymous out of fear of religious persecution.

Given the current economic climate, this is likely to remain in this state for at least another decade.

Given the editorial style of the books, it is not surprising that they were ( and to some extent remain ) controversial within the numerical analysis community.

The Government believes that it must remain so .” Given the prospect of " incineration without representation ", Canadians seemed to feel that the doctrine which most encouraged restraint was the strategically soundest one to support.

Given the sport's origins, Irish-bred and trained horses remain a dominant force in national hunt racing today.

Given that the only remaining humans ( after Zion's destruction ) remain in the Matrix, failure to enter the door to the Source will lead to an eventual catastrophic system crash of the Matrix, killing its human inhabitants and effectively annihilating humanity as a race.

Given the context of this conversation ( Blackadder is attempting to get transferred into the Royal Flying Corps ), and Darling's character in general, he has presumably pulled some strings to allow him to remain in the army without having to serve in battle.

Given numerous chances to escape antisemitic persecution ( as late as 1944 ), he chose to remain in Hungary, where his last novel, a Pirandellian fantasy about a king staging a coup against himself, then having to impersonate himself, Oliver VII, was published in 1942.

Given the total mass and the scalars r and v at a single point of the orbit, one can compute the specific orbital energy, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being " suborbital " ( a ballistic missile, for example ), having enough energy to be " orbital ", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and / or go to infinity ( as a meteor, for example ).

Given names that remain in modern use include:

Given the as-yet-undeciphered nature of the script, the precise significance of the post-firing addition of a Cypro-Minoan sign must remain uncertain.

Given the impossibility of sending a bacteria-free human being into space, forward-contamination will remain an issue for as long as manned missions continue.

Given such contradictions, these characters ' actual names and thus identities remain unclear.