Each equivalence relation has a canonical projection map, the surjective function from to given by.

However, due to Fergus ' inability to pay taxes, it is presumed that this map is not canonical to The Life and Times of Scrooge McDuck, unless such treasure was obtained later in his life.

* It is the exponential map of a canonical left-invariant affine connection on G, such that parallel transport is given by left translation.

* It is the exponential map of a canonical right-invariant affine connection on G. This is usually different from the canonical left-invariant connection, but both connections have the same geodesics ( orbits of 1-parameter subgroups acting by left or right multiplication ) so give the same exponential map.

The ( pseudo ) Riemannian metric determines a canonical affine connection, and the exponential map of the ( pseudo ) Riemannian manifold is given by the exponential map of this connection.

Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, the map on cotangent bundles induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a Lie Group on a coadjoint orbit.

If is the twisting sheaf of Serre on, we let denote the pullback of to ; that is, for the canonical map

Let ( C, F ) be a concrete category ( i. e. F: C → Set is a faithful functor ), let X be a set ( called basis ), A ∈ C an object, and i: X → F ( A ) a map between sets ( called canonical injection ).

More precisely: the canonical map h: L → U ( L ) is always injective.

The homology groups are natural in the sense that if ƒ is a continuous map from X < sub > 1 </ sub > to X < sub > 2 </ sub >, then there is a canonical pushforward map ƒ < sub >∗</ sub > of homology groups ƒ < sub >∗</ sub >: H < sub > k </ sub >( X < sub > 1 </ sub >) → H < sub > k </ sub >( X < sub > 2 </ sub >), such that the composition of pushforwards is the pushforward of a composition: that is,.

The map depends on two parameters, a and b, which for the canonical Hénon map have values of a = 1. 4 and b = 0. 3.

For the canonical values the Hénon map is chaotic.

For the canonical map, an initial point of the plane will either approach a set of points known as the Hénon strange attractor, or diverge to infinity.

Numerical estimates yield a correlation dimension of 1. 25 ± 0. 02 and a Hausdorff dimension of 1. 261 ± 0. 003 for the attractor of the canonical map.

As a dynamical system, the canonical Hénon map is interesting because, unlike the logistic map, its orbits defy a simple description.

For the canonical values of a and b of the Hénon map, one of these points is on the attractor:

In mathematics, if is a subset of, then the inclusion map ( also inclusion function, insertion, or canonical injection ) is the function that sends each element, of to, treated as an element of:

or the ( possibly infinite ) Cartesian product of the topological spaces X < sub > i </ sub >, indexed by, and the canonical projections p < sub > i </ sub >: X → X < sub > i </ sub >, the product topology on X is defined to be the coarsest topology ( i. e. the topology with the fewest open sets ) for which all the projections p < sub > i </ sub > are continuous.

The X. Org implementation serves as the canonical implementation of X.

We say that a formula or a set X of formulas is canonical with respect to a property P of Kripke frames, if

* for any normal modal logic L which contains X, the underlying frame of the canonical model of L satisfies P.

When the canonical embedding J of X into the dual of is bijective, then X is said to be semi-reflexive.

Since Y is reflexive, the continuous dual of is equal to the image J ( X ) of X under the canonical embedding J, but the topology on X is not the strong topology, that is equal to the norm topology of Y.

If it does, however, it is unique in a strong sense: given another direct limit X ′ there exists a unique isomorphism X ′ → X commuting with the canonical morphisms.

* Let I be any directed set with a greatest element m. The direct limit of any corresponding direct system is isomorphic to X < sub > m </ sub > and the canonical morphism φ < sub > m </ sub >: X < sub > m </ sub > → X is an isomorphism.

Theorem The dual of G ^ is canonically isomorphic to G, that is ( G ^)^ = G in a canonical way.

As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization G < sup > ab </ sup > up to canonical isomorphism, whereas the explicit construction G → G / shows existence.

If a diagram F: J → C has a limit in C, denoted by lim F, there is a canonical isomorphism

Dually, if a diagram F: J → C has a colimit in C, denoted colim F, there is a unique canonical isomorphism

For a given diagram F: J → C and functor G: C → D, if both F and GF have specified limits there is a unique canonical morphism

p < sub > 1 </ sub >, where p < sub > 1 </ sub >: G × 1 → G is the canonical projection, and m ( e × id < sub > G </ sub >)

For each neighborhood U of x, the canonical morphism F ( U ) → F < sub > x </ sub > associates to a section s of F over U an element s < sub > x </ sub > of the stalk F < sub > x </ sub > called the germ of s at x.

In a category with finite products and coproducts, there is a canonical morphism X × Y + X × Z → X ×( Y + Z ), where the plus sign here denotes the coproduct.

If R is a ring and S is a subset, consider all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit.

If C has a zero object 0, given two objects X and Y in C, there are canonical morphisms f: 0 → X and g: Y → 0.

If K < sub > v </ sub > is a non-archimedean local field, there is a canonical isomorphism inv < sub > v </ sub >: Br ( K < sub > v </ sub >) → Q / Z constructed in local class field theory.

But it turns out that ( if A is " nice " enough ) there is one canonical way of doing so, given by the right derived functors of F. For every i ≥ 1, there is a functor R < sup > i </ sup > F: A → B, and the above sequence continues like so: 0 → F ( A ) → F ( B ) → F ( C ) → R < sup > 1 </ sup > F ( A ) → R < sup > 1 </ sup > F ( B ) → R < sup > 1 </ sup > F ( C ) → R < sup > 2 </ sup > F ( A ) → R < sup > 2 </ sup > F ( B ) → ....

It is also possible to read " G < sub > T </ sub > is true " in the formal sense that primitive recursive arithmetic proves the implication Con ( T )→ G < sub > T </ sub >, where Con ( T ) is a canonical sentence asserting the consistency of T ( Smoryński 1977 p. 840, Kikuchi and Tanaka 1994 p. 403 )</ ref > but not provable in the theory ( Kleene 1967, p. 250 ).

Because of the manner in which the definition of T * M relates to the differential topology of the base space M, X possesses a canonical one-form θ ( also tautological one-form or symplectic potential ).

After Serre we recognise l ( K − D ) as the dimension of H < sup > 1 </ sup >( D ), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H < sup > 1 </ sup >( D ) and H < sup > 0 </ sup >( KD *), and we are reading off dimensions ( notation: K is the canonical line bundle, D * is the dual line bundle, and juxtaposition is the tensor product of line bundles ).

One can also use canonical correlation analysis to produce a model equation which relates two sets of variables, for example a set of performance measures and a set of explanatory variables, or a set of outputs and set of inputs.

The " canonical approach " to studying visual rhetoric relates visual concepts to the canons of Western classical rhetoric ( Inventio, Dispositio, Elocutio, Memoria and Pronuntiatio ).

The adjunction formula relates the canonical bundles of X and D. It is a natural isomorphism