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

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

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 two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ.

Given that the very Arabic word " riwaya ", which is now used exclusively in reference to the " novel ", has traditionally conjured up a tangle of narrative genres [...], it might not be unfair to contend that the Arabic novel owes its early formation not only to the appropriation of the novel genre from Europe [...] but also, and more importantly, to the revival and transformation of traditional narrative genres in the wake of Napoleon's 1798 expedition into Egypt and the Arab world's firsthand encounter with industrialized imperial Europe.

Given a Fuchsian group, that is, a discrete subgroup Γ of PSL ( 2, R ), Γ acts on H via linear fractional transformation.

Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.

Given a natural transformation Φ: Hom ( A ,–) → F the corresponding element u ∈ F ( A ) is given by

Given a semiautomaton, let T < sub > a </ sub >: X → X, for a ∈ Σ, denote the transformation of X defined by T < sub > a </ sub >( x ) = T ( a, x ).

Given an initial configuration, and a twist, the homogeneous transformation to a new location and orientation can be computed with the following formula:

Given a-module, let denote the whose underlying-module is but where acts by the inverse covering transformation.

Given that the majority of the share of trade was made by Ottoman merchants, the Europeans, especially the British and the French, became irritated and unhappy with this trade arrangement and pushed for intervention and transformation of the economic policies of the Middle East.

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 ).

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 its general release in January 1931, City Lights proved to be a popular and financial success — eventually grossing over $ 5 million.

Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.

Given that the ColecoVision could produce near arcade-quality ports, industry magazines like Electronic Games were unanimous in their enthusiasm over the console.

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").

Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.

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.

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

Given the extraordinary competition and technological craftsmanship required for top search engine placement, the implication of the term " search engine optimization " has deteriorated over the last decade.

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

Given a function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is

The Fascist government had direct rule only over the Benadir territory Given the defeat of the Dervish movement in the early 1920s and the rise of fascism in Europe, on 10 July 1925, Mussolini gave the green light to De Vecchi to start the takeover of the north-eastern sultanates.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given a particular wing and its velocity through the air, debates over which mathematical approach is the most convenient to use can be misperceived by novices as differences of opinion about the basic principles of flight.

" Given their 3: 00 a. m. start time and non-inclusion ( at frontman John Fogerty's insistence ) in the Woodstock film, members have expressed bitterness over their experiences at the famed festival.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed ( maximum ) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK-at the same BER.

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that