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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 ).
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Given and arbitrary
Given unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church – Turing thesis.
Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).
Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that
Given that hearing impairments can vary by frequency and that audiograms are plotted with a logarithmic scale, the idea of a percentage of hearing loss is somewhat arbitrary, but where decibels of loss are converted via a recognized legal formula, it is possible to calculate a standardized " percentage of hearing loss " which is suitable for legal purposes only.
Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.
Given an inertial frame of reference and an arbitrary epoch ( a specified point in time ), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit.
Given Bluestein's algorithm, such a transform can be used, for example, to obtain a more finely spaced interpolation of some portion of the spectrum ( although the frequency resolution is still limited by the total sampling time ), enhance arbitrary poles in transfer-function analyses, etcetera.
Given an arbitrary ( n ; m ; p ) machine S, such that every two of its states are distinguishable from one another, then there exists an experiment of length
Given an arbitrary category C, a representation of G in the category C is a functor from G to C. Such a functor selects an object of C and a subgroup of automorphisms of that object.
Given an arbitrary series
Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.
Given an arbitrary window in a spatial file manager, it must be possible to determine with complete certainty which folder that window represents.
Given the precision of Brown's calculations, it must have come as a great disappointment to have to introduce this arbitrary adjustment.
Given an arbitrary direction z ( usually determined by an external magnetic field ) the spin z-projection is given by
Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over F, so the invariant factors a < sub > i </ sub >( x ) as given above must be used instead.
Given an arbitrary point on a torus, four circles can be drawn through it.
Given arbitrary complex numbers A, B, C, D such that AD − BC ≠ 0, define the quantities
Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g ′ compatible with the almost complex structure J in an obvious manner:
Given any Euclidean triangle ABC and an arbitrary point P let d ( P ) = PA + PB + PC.
Given and group
Given a group G, a factor group G / N is abelian if and only if ≤ N.
* 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 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 and G
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 and there
: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.
* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.
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 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 a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).
Given its different forms, there are various ways of representing uncertainty and modelling economic agents ' responses to it.
* Given any set X, there is an equivalence relation over the set of all possible functions X → X.
Given a left neutral element and for any given then A4 ’ says there exists an such that.
Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force.
Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.
Given that non-violence has priority, all other principles yield to it whenever there is a conflict.
Given this assumption, there are four categories in which a firm's profit may be considered to be.
Given the overall size of trade between Mexico and the United States, there are remarkably few trade disputes, involving relatively small dollar amounts.
Given the above commonalities there appear to be only two string theories: the heterotic string theory ( which is also the type I string theory ) and the type II theory.
Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.
Given the diversity of functions performed by neurons in different parts of the nervous system, there is, as expected, a wide variety in the shape, size, and electrochemical properties of neurons.
: Given any positive number ε, there is a sequence
Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.
Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!
Given the immense expanse of the entire Universe, it has been argued that there is a higher probability that there exists ( or has existed ) another Earth-like planet that has yielded life ( geogenesis ) than not.
# Given any two distinct points, there is exactly one line incident with both of them.
1.016 seconds.