* Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble

:: Given a recursive presentation P

Given a finite presentation P =

Given that many of the talented young artists had been raised in housing projects and were short on social and dress skills, this Motown department was not only necessary, it created an elegant style of presentation long associated with the label.

Given their role in antigen presentation, HSPs are useful as immunologic adjuvants in boosting the response to a vaccine.

Given the initial presentation of Guellen, Claire's power, the history between Claire and Ill, and the nature of Claire's decree, Ill's death appears almost inevitable.

Given a presentation of a group G by generators and relations and a subgroup H of G, the algorithm enumerates the cosets of H on G and describes the permutation representation of G on the space of the cosets.

Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.

Given the controversy on the etiology of PANDAS ( i. e., whether caused by strep throat ) and the apparent ambiguity in the definitions of the clinical presentation leading to different research results, a new definition was proposed that would clarify the presentation of symptoms to ensure that research studies were studying the same group of children.

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

Given its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

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 points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given the then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.

Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.

Given the preceding jurisprudence regarding due process, Chief Justice Taney was not entirely breaking ground in his Dred Scott opinion when, without elaboration, he pronounced the Missouri Compromise unconstitutional because an " act of Congress that deprived a citizen of his liberty or property merely because he came himself or brought his property into a particular territory of the United States, and who had committed no offence against the laws, could hardly be dignified with the name of due process of law.

Given that Ramesses VIII only reigned briefly some 25 years after his father ’ s death, it is hardly likely that the decoration of QV 52, with the mwt-nsw ( i. e. king's mother ) title intimately mixed with Tyti ’ s other titles, could have been delayed this late to refer to him.

Given his propensity for deception, this origin story can hardly be taken at face value.

“ Given that I was born a few months after Auschwitz was liberated, it is hardly surprising that I have a strong sense of the evil that humans – individually and collectively – do.

Given this, it would be hardly surprising if I could not be bothered to drag myself from my pit of sybaritic excess to a polling station.

Given the level of anti-communist sentiment at the time, it would hardly have seemed an auspicious name under which to launch such a magazine title.

Given how much trouble it was to deal with the Enforcers once they stopped being useful, since they could hardly be passed off as normal humans, they begin the project Adonai, creating a non-cyborg human who will have the qualities they want and be easier to control, not to mention kill if necessary.

Given its limited premise, it is hardly surprising that this series was never revived after its 12-issue run was up.

Given the rigorous admission process it is not surprising that CUNY BA boasts a 70 % graduation rate within an average of 2. 2 years and that 60 % graduate with academic honors.

Given the differences of opinion within the Lord Aberdeen cabinet over the direction of foreign policy with regard to relations between Britain and the French under Napoleon III, it is not surprising that debate raged within the government as Louis Bonaparte, now assuming the title of Emperor Napoleon III of France.

Given his record, it is not surprising that the other Party leaders were suspicious of Beria's motives.

Given a day mandated for seasonal rejoicing ( Shemini Atzeret ) with no specific obligations as to the focus of the rejoicing ( such as sukkah, lulav or etrog ), it is not surprising that the celebration of the conclusion of the annual reading was absorbed into the celebration of Shemini Atzeret.

Given the display of imperial power Henry III had inflicted on the Romans in intervening against Pope Gregory VI and installing Clement II, it is not surprising that on Christmas Day of 1047, an emissary was sent by the Roman people bringing news of Clement II's death to Henry III and asking him, in his position as Patricius of the Romans, to appoint a successor.

Given the circumstances under which Máel Morda had been appointed, it is not surprising that he launched an open rebellion against Brian's authority.

Given the reputation of oat grain to have invigorating properties and the obvious connection between plant seeds and human " seed ", it is not surprising that the meaning of the phrase shifted towards more or less explicitly referring to the destructive sexual liaisons of an unmarried young male, possibly resulting in unwanted children born out of wedlock.

Given Pasolini's well-known reputation as an atheist, a homosexual, and a Marxist, the reverential nature of his film was surprising, especially after the controversy of " La ricotta ".

Given that over 90 per cent of the area of kauri forest standing before 1000AD was destroyed by about 1900, it is not surprising that recent records are of smaller, but still very large trees.

Given that, it is not surprising growers named their new varieties with exalted titles.

Given the advantages that result from discussion, it is surprising that it is not used more often.

Given his expertise and background in understanding the voice and in understanding the wonders of the human voice, it is not surprising, maybe, that his approach in his early years was away from the theatrical side and more focused on singers.

Given this worldview, it is not surprising that John I led his sons and their assembled forces in an attack on the Muslim stronghold of Ceuta in 1415.

Given the complex macro-rheological behavior of blood, it is not surprising that a single equation fails to completely describe the effects of various rheological variables ( e. g., hematocrit, shear rate ).

Given the fondness of the Catalans for music, it may not be surprising to hear that Andorra has a Chamber Orchestra directed by the violinist Gérard Claret ; and that it also stages a famous international singing contest supported by the Spanish singer Montserrat Caballé.

Given Ialá's sharp hostility to Vieira in previous years, this endorsement was viewed as surprising by many, and there was reportedly significant dissatisfaction with the decision among Ialá's supporters.

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.

Given its setting, it wasn't surprising that this game was slow to desegregate, and in fact was once removed from the schedule of NBC for this reason.

Given the hydrological connection of the Humboldt River and Sink to the Carson Sink during flood years ( as recently as 1998 ), it is not surprising that beavers were historically extant on eastern Sierra watercourses.

Given the calculator's intended userbase of high school maths and science teachers and students, this is not surprising.

Given the conditions of the election, which were limited and not free under the watchful eye of the military-government, the MMA ’ s ascension does not seem as surprising.

Given the village connection with St. Edith it is perhaps surprising that the parish church, Saxon in origin, is however dedicated to St Mary the Virgin.

Given the size of the county it is perhaps just as surprising that there are so few castles, as that there are so many Abbeys up the Witham.

Given the similarity in terminology it is not surprising most casual observers would believe them to be closely related if not identical.