Given the small population of the region ( compared with the Central Canadian provinces or the New England states ), the regional economy is a net exporter of natural resources, manufactured goods, and services.

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

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 ).

Given that the coiled texture is the natural state of some African Americans ' hair, or perceived as being more " African ", this simple style is now often seen as a sign of self-acceptance and an affirmation that the beauty norms of the ( eurocentric ) dominant culture are not absolute.

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 that the authors were the primary developers of the Unix operating system, it is natural that QED had a strong influence on the classic UNIX text editors ed, sed and their descendants such as ex and sam.

Given a natural transformation Φ from h < sup > A </ sup > to F, the corresponding element of F ( A ) is.

Given a morphism f: B → A the associated natural transformation is denoted Hom ( f ,–).

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

Given his socialist agenda in Egypt, the Ba ' ath should have been his natural ally, but Nasser was hesitant to share power.

Given the reputed humorous character of Valencians, it was natural that they began to burn figurines depicting people and events of the past year.

Given a stochastic process, the natural filtration for ( or induced by ) this process is the filtration where is generated by all values of up to time s

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

Given a series with values in a normed abelian group G and a permutation σ of the natural numbers, one builds a new series, said to be a rearrangement of the original series.

Given a natural number x, h outputs the index of the partial computable function that performs the following computation:

Given its central location, Athlone is a natural hub for transport, with frequent bus and rail schedules.

Given the finite supply of natural resources at any specific cost and location, agriculture that is inefficient or damaging to needed resources may eventually exhaust the available resources or the ability to afford and acquire them.

Given this difference, it is necessary that logistic regression take the natural logarithm of the odds ( referred to as the logit or log-odds ) to create a continuous criterion.

Given that the logit is not generally interpreted and that the inverse of the natural logarithm, the exponential function of the logit is generally interpreted instead, it is also helpful to examine this function ( denoted: ).

Given a presheaf, a natural question to ask is to what extent its sections over an open set U are specified by their restrictions to smaller open sets V < sub > i </ sub > of an open cover of U. A presheaf is separated if its sections are " locally determined ": whenever two sections over U coincide when restricted to each of V < sub > i </ sub >, the two sections are identical.

Given that some countries now ban docking, breeders are again attempting to select dogs with the genes for natural bob tails.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given its native distribution the species is also a potential natural vagrant to Western Europe and there have been records in areas such as Cornwall and the Isles of Scilly which some observers consider may relate to wild birds ; however, given the Wood Duck's popularity in captivity it would be extremely difficult to prove their provenance one way or another.

* Given a partial function f from the natural numbers into the natural numbers, f is a partial recursive function if and only if the graph of f, that is, the set of all pairs such that f ( x ) is defined, is recursively enumerable.

Given Chrétien's known tendency to create new stories and characters, being the first to mention the hero Lancelot and his love affair with Queen Guinevere for example, the name might also be entirely invented.

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 the tendency of political parties in India to make and break alliances frequently, the National Democratic Alliance does not have a formal governing structure in place, such as an executive board or politburo.

Given the Likert Scale's ordinal basis, summarizing the central tendency of responses from a Likert scale by using either the median or the mode is best, with ‘ spread ’ measured by quartiles or percentiles.

Given that Fildes is reported to have had a tendency to make " extravagant boasts ", and that the grenade modifications could have been aimed at making it lighter, the validity of the botulinum toxin theory has been questioned.

Given Aeschylus ' tendency to write connected trilogies, three plays attested in the catalogue of his work have been supposed to constitute the Achilleis: Myrmidons, Nereids and Phrygians ( alternately titled The Ransoming of Hector ).

Given the tendency of the legends of different seafarers – Greek, Norse, Irish, Arab and Iberian – to cross-fertilize and influence each other, the exact source of some legendary Atlantic islands – such as the mythical islands of Brasil and the Isle of Mam – are impossible to disentangle.

Given the tendency of birds that have become flightless to increase in size markedly in just a few million years, the probability of Hesperornithes deriving from an animal at least superficially similar and not too far related to Gansus is quite high.

Given the tendency of self-interest to overpower the pleasure felt through sympathy, the eventual accumulation of wealth necessitated the development of some form of government, initially somewhat monarchical, to ensure that the rules of justice were followed.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given India's historic position that the original intent of the line was to separate the two nations by the highest mountains in the world, in these locations India extended its forward posts northward to the ridges, regarding this move as compliant with the original border proposal, although the Simla Convention did not explicitly state this intention.

Given their original purpose, military marching bands typically march in a forward direction with ( no variation of ) straight lines.

: Given that desegregation has not produced the predicted leaps forward in black educational achievement, there is no reason to think that black students cannot learn as well when surrounded by members of their own race as when they are in an integrated environment.

Given the choice of reaching back for Sydney and reaching forward for the Horizon, Irina chose the Horizon and plunged through the skylight to her demise.

Given these factors, it was initially uncertain whether the offensive would go forward.

Given strong evidence that CIRP holds, the forward rate unbiasedness hypothesis can serve as a test to determine whether UIRP holds ( in order for the forward rate and spot rate to be equal, both CIRP and UIRP conditions must hold ).

: 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.