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Page "Congruence relation" ¶ 22
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If and ƒ
If ƒ is differentiable, this is equivalent to:
If is continuous in an open set Ω and the partial derivatives of ƒ with respect to x and y exist in Ω, and satisfies the Cauchy – Riemann equations throughout Ω, then ƒ is holomorphic ( and thus analytic ).
* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.
If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space.
If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.
If and are groups, a homomorphism from to is a function ƒ: such that
If the limit exists, we say that ƒ is complex-differentiable at the point z < sub > 0 </ sub >.
If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa.
If ƒ maps X to Y, then ƒ < sup >– 1 </ sup > maps Y back to X.
If ƒ is invertible, the function g is unique ; in other words, there can be at most one function g satisfying this property.
If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.
If ƒ is an invertible function with domain X and range Y, then
If an inverse function exists for a given function ƒ, it is unique: it must be the inverse relation.
If ƒ is real-valued, the polynomial function can be taken over R.
If D is a derivation at x, then D ( ƒ )
If ƒ and g are elements of a C *- algebra, f * and g * denote their respective adjoints.
If ƒ ( x )
* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

If and
If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X Y is denoted by B ( X, Y ).
The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y Z which has the following universal property: If T ′: X × Y Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z Z ′ with exists.
If this limit exists, then it may be computed by taking the limit as h 0 along the real axis or imaginary axis ; in either case it should give the same result.
* Scanning: If a is the next symbol in the input stream, for every state in S ( k ) of the form ( X α • a β, j ), add ( X α a • β, j ) to S ( k + 1 ).
If f: X Y morphism of pointed spaces, then every loop in X with base point x < sub > 0 </ sub > can be composed with f to yield a loop in Y with base point y < sub > 0 </ sub >.
Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.
Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C C which is covariant in both arguments.
If f: A < sub > 1 </ sub > A < sub > 2 </ sub > and g: B < sub > 1 </ sub > B < sub > 2 </ sub > are morphisms in Ab, then the group homomorphism Hom ( f, g ): Hom ( A < sub > 2 </ sub >, B < sub > 1 </ sub >) Hom ( A < sub > 1 </ sub >, B < sub > 2 </ sub >) is given by φ g o φ o f. See Hom functor.
If f: X < sub > 1 </ sub > X < sub > 2 </ sub > and g: Y < sub > 1 </ sub > Y < sub > 2 </ sub > are morphisms in C, then the group homomorphism Hom ( f, g ): Hom ( X < sub > 2 </ sub >, Y < sub > 1 </ sub >) Hom ( X < sub > 1 </ sub >, Y < sub > 2 </ sub >) is given by φ g o φ o f.
If f: X Y is a continuous map, x < sub > 0 </ sub > ∈ X and y < sub > 0 </ sub > ∈ Y with f ( x < sub > 0 </ sub >) = y < sub > 0 </ sub >, then every loop in X with base point x < sub > 0 </ sub > can be composed with f to yield a loop in Y with base point y < sub > 0 </ sub >.
If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K H such that f = h φ.
If f: M N is any function, then we have f id < sub > M </ sub >
If it does, however, it is unique in a strong sense: given any other inverse limit X ′ there exists a unique isomorphism X ′ X commuting with the projection maps.
If G and H are Lie groups, then a Lie-group homomorphism f: G H is a smooth group homomorphism.
If the lifetime of this transition, τ < sub > 21 </ sub > is much longer than the lifetime of the radiationless 3 2 transition τ < sub > 32 </ sub > ( if τ < sub > 21 </ sub > ≫ τ < sub > 32 </ sub >, known as a favourable lifetime ratio ), the population of the E < sub > 3 </ sub > will be essentially zero ( N < sub > 3 </ sub > ≈ 0 ) and a population of excited state atoms will accumulate in level 2 ( N < sub > 2 </ sub > > 0 ).
If R = Π < sub > i in I </ sub > R < sub > i </ sub > is a product of rings, then for every i in I we have a surjective ring homomorphism p < sub > i </ sub >: R R < sub > i </ sub > which projects the product on the i-th coordinate.
If φ: M N is a local diffeomorphism at x in M then dφ < sub > x </ sub >: T < sub > x </ sub > M T < sub > φ ( x )</ sub > N is a linear isomorphism.

If and B
If A is the major axis of an ellipsoid and B and C are the other two axes, the radius of curvature in the ab plane at the end of the axis Af, and the difference in pressure along the A and B axes is Af.
If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
If the valid element indices begin at 0, the constant B is simply the address of the first element of the array.
If the numbering does not start at 0, the constant B may not be the address of any element.
If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.
If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.
If and we have for all v, w in V, then we say that B is symmetric.
In addition to acting and occasionally directing, Campbell has become a writer, starting with an autobiography, If Chins Could Kill: Confessions of a B Movie Actor published on August 24, 2002.
If the sets A and B are equal, this is denoted symbolically as A = B ( as usual ).
If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).
* Indicative conditional, a conditional in the form of " If A then B " in natural languages
If an atom A is double-bonded to an atom B, A is treated as being singly bonded to two atoms: B and a ghost atom that has the same atomic number as B but is not attached to anything except A.
If the ideals A and B of R are coprime, then AB = AB ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.
If A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.
If two cofinal subsets of B have minimal cardinality ( i. e. their cardinality is the cofinality of B ), then they are order isomorphic to each other.

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