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Page "Formal group" ¶ 63
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Given and n-dimensional
Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.
Given constants C, D and V, there are only finitely many ( up to diffeomorphism ) compact n-dimensional Riemannian manifolds with sectional curvature | K | ≤ C, diameter ≤ D and volume ≥ V.
Given constants C, D and V, there are only finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V.
Given a collection of differential 1-forms α < sub > i </ sub >, i = 1, 2, ..., k on an n-dimensional manifold M, an integral manifold is a submanifold whose tangent space at every point p ∈ M is annihilated by each α < sub > i </ sub >.
Given a scheme X over a base scheme S or a complex manifold, a line bundle ( or in other words an invertible sheaf, that is, a locally free sheaf of rank one ) L on X is said to be very ample, if there is an immersion i: X → P < sup > n </ sup >< sub > S </ sub >, the n-dimensional projective space over S for some n, such that the pullback of the standard twisting sheaf O ( 1 ) on P < sup > n </ sup >< sub > S </ sub > is isomorphic to L:
Given a basis with n-dimensional integer coordinates, for a lattice L in R < sup > n </ sup > with, the LLL algorithm outputs an LLL-reduced ( short, nearly orthogonal ) lattice basis in time
Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional bloch sphere.

Given and formal
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 formal power series
Given a formal power series
Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences.
Given the UFA's formal adoption of the goal of replacing Stewart's Liberal government with a Farmer government, it remained surprisingly friendly towards the Premier.
Given that Ireland had escaped absorption into the Roman empire, this had time to develop into a highly sophisticated literature with well-documented formal rules and highly organised Bardic schools.
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 a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections.
Given of a monoid M of every strings over some alphabet, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating.
Given WSRP4J's incubator status, the project did not produce formal releases.
Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the precision of approximate equality might be the most obvious way to assess mathematical coincidences ; but there is no standard, and the strong law of small numbers is the sort of thing one has to appeal to with no formal opposing mathematical guidance.
Given such a choice of scale, a central representation of the Heisenberg group is a map of-algebras which is the formal way of saying that it sends the center to a chosen scale.
Given an arithmetic function and a prime, define the formal power series, called the Bell series of modulo as:
In a critical statement published in 1955, he asserted that " Given the modern conceptions of the beautiful in literature, given at the very least these essential conceptions, if a work is realistic it has many chances of being good ; if not, supposing even that it has formal qualities, it risks lacking resonance, profundity, that of which all literature has the greatest need -- the human ; from which it follows that it has much less chance of being good -- if only it had some -- than a realistic work.
Given this expansion, most people shorten the name to simply " The Kingdom " in common usage, though the formal name remains in some documents.
: Given any n formal power series f < sub > 1 </ sub >,..., f < sub > n </ sub > in tC < nowiki ></ nowiki > t < nowiki ></ nowiki > which are linearly independent over Q, then the field extension C ( t, f < sub > 1 </ sub >,..., f < sub > n </ sub >, exp ( f < sub > 1 </ sub >),..., exp ( f < sub > n </ sub >)) has transcendence degree at least n over C ( t ).
Given any desired positive integer c, this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by c. This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as schemes.
Given such a specification, it is possible to use formal verification techniques to demonstrate that a candidate system design is correct with respect to the specification.

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 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.

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