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Page "Invariant (mathematics)" ¶ 27
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Formally and define
Formally, we define
Formally, the issue is that interfertile " able to interbreed " is not a transitive relation if A can breed with B, and B can breed with C, it does not follow that A can breed with C and thus does not define an equivalence relation.
Formally, we define indices inductively using
Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.
Formally however they define it as any variable that does not directly affect the fundamentals of the economy.

Formally and set
Formally, a binary operation on a set S is called associative if it satisfies the associative law:
Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).
Formally speaking, a collation method typically defines a total order on a set of possible identifiers, called sort keys, which consequently produces a total preorder on the set of items of information ( items with the same identifier are not placed in any defined order ).
Formally, one has an implication, not an equivalence, so the solution set may get larger.
Formally, a set S is called finite if there exists a bijection
Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies
Formally, according to the Constitution, citizens of Turkmenistan have the right to set up political parties and other public associations, acting within the framework of the Constitution and laws, and public associations and groups of citizens have the right to nominate their candidates in accordance with the election law.
Formally, the weighted mean of a non-empty set of data
Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.
Formally, Gaussian integers are the set
Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.
Formally, we are given a set of hypotheses and a set of manifestations ; they are related by the domain knowledge, represented by a function that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.
Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write
Formally, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω.
Formally, the movement is a rondo that acts as the theme for a set of eight variations, capped off by a dramatic coda.
Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.
Formally, starting with a linearly independent set of vectors
Formally, the discrete Hartley transform is a linear, invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes the set of real numbers ).
Formally, a constraint satisfaction problem is defined as a triple, where is a set of variables, is a domain of values, and is a set of constraints.

Formally and lines
Formally, a complex projective space is the space of complex lines through the origin of an ( n + 1 )- dimensional complex vector space.

Formally and plane
Formally, suppose U is an open subset of the complex plane C, a is an element of U and f: U
Formally, if is an open subset of the complex plane, a point of, and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on.
Formally, Aff ( V ) is naturally isomorphic to a subgroup of, with V embedded as the affine plane, namely the stabilizer of this affine plane ; the above matrix formulation is the ( transpose of ) the realization of this, with the ( n × n and 1 × 1 ) blocks corresponding to the direct sum decomposition.
Formally, a frieze group is a class of infinite discrete symmetry groups for patterns on a strip ( infinitely wide rectangle ), hence a class of groups of isometries of the plane, or of a strip.
Formally speaking, the curves must be differentiable curves in the Euclidean plane.

Formally and P
Formally, a decision problem is P-complete ( complete for the complexity class P ) if it is in P and that every problem in P can be reduced to it by using an appropriate reduction.
Formally, let P be an expression in which the variable x is free.
Formally, let P and Q be abelian categories, and let
Formally, given a partially ordered set ( P, ≤), then an element g of a subset S of P is the greatest element of S if
Formally, a partially ordered set ( P, ≤) is bounded complete if the following holds for any subset S of P:
Formally known as Thornhill, a gift from C. P.
Formally, a product term P in a sum of products is an implicant of the Boolean function F if P implies F. More precisely:
Formally, P ( with <) will be a ( strict ) partially ordered set, or poset.
Formally, P is a symmetric polynomial, if for any permutation σ of the subscripts 1, 2, ..., n one has P ( X < sub > σ ( 1 )</ sub >, X < sub > σ ( 2 )</ sub >, …, X < sub > σ ( n )</ sub >) = P ( X < sub > 1 </ sub >, X < sub > 2 </ sub >, …, X < sub > n </ sub >).
Formally, let P be a poset ( partially ordered set ), and let F be a filter on P ; that is, F is a subset of P such that:

Formally and L
Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset
* Ishpeming Middle School ( Formally C. L.
Formally, if there is a utility function that describes preferences over L commodities, the expenditure function

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