* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

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 two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given our formula φ, we group strings of quantifiers of one kind together in blocks:

Given that different groups in society have different beliefs, priorities, and interests, to which group would the media tailor its bias?

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 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 category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

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 ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

Given that France and Britain had been at war since early 1793, administering or making such oaths turned the society into something more than a liberal pressure group.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given that and record company pressure to record more accessible, radio-friendly material similar to their first album – something Lee, Lifeson and Peart were unwilling to do – the trio feared that the end of the group was near.

Given any group G, the group consisting of only the identity element is a trivial group and being a subgroup of G is called the trivial subgroup of G.

Given these orbital elements and the physical characteristics known so far, Ananke is thought to be the largest remnant of an original break-up forming the Ananke group.

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Given a relation, scaling the argument by a constant factor causes only a proportionate scaling of the function itself.

Given events, and, Bayes ' rule states that the conditional odds of given are equal to the marginal odds of multiplied by the Bayes factor:

Given that propensity to noble rot was a factor in Hungarian vineyard demarcations some 50 years before a messenger was supposedly mugged on his way to Schloss Johannisberg in Germany, the Hungarians probably have a better case.

Given Orff's previous lack of commercial success, the monetary factor of Carmina Buranas acclaim was significant to him.

Given that one motive for the operational approach is stability, systems that relax the operational factor can be problematic, for several reasons, as the operational is a means to manage complexity.

Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm.

; Factor ring or quotient ring: Given a ring R and an ideal I of R, the factor ring is the ring formed by the set R / I of cosets

Given the controversy that has long surrounded the use of feedback, some extend their distaste for negative feedback amplifier designs ( and so a high damping factor ) as a mark of poor quality.

Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra, which is a type II < sub > 1 </ sub > factor.

Given a model selection problem in which we have to choose between two models, on the basis of observed data D, the plausibility of the two different models M < sub > 1 </ sub > and M < sub > 2 </ sub >, parametrised by model parameter vectors and is assessed by the Bayes factor K given by

Given the complexities of her emotional life at the time, post-natal ( with two miscarriages ) and in relation to parental conflicts, there is reasonable doubt whether that was the single factor.

Given certain assumptions about how to treat the gravitational force, he argued that the bullet sinks with acceleration, where g is the acceleration due to gravity ( assumed to be uniform over the scale of the thought experiment ) and is the factor mentioned above.

Given the organization's focus on serving the lower-income residents of the city, the location's excellent transit connections were a factor in choosing this location.

Given the same 3: 2 aspect ratio as 35mm's 36mm x 24mm area, this is equivalent to the ratio of heights or ratio of widths ; the ratio of sensor areas is the square of the crop factor.

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.

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

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.