There is a natural map F: X → X ′′ ( the dual of the dual

Because F ( x ) is a map from X ′ to K, it is an element of X ′′.

The map F: x → F ( x ) is thus a map X → X ′′.

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

is a linear map from V to X, and for any v in V the map

is a linear map from W to X.

A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has.

Combining the above, a map τ is said to be a measure-preserving transformation of X, if it is a map from X to itself, it is Σ-measurable, and is measure-preserving.

The canonical map ker: X ^ X → Con X, relates the monoid X ^ X of all functions on X and Con X. ker is surjective but not injective.

Power sets: The power set functor P: Set → Set maps each set to its power set and each function to the map which sends to its image.

The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, a, and the eccentricity, e. The forms given are, apart from notational variants, those in the standard reference for map projections, namely " Map projections — a working manual " by J. P. Snyder.

The proof follows by the fact that if f were indeed a map onto P ( S )), then we could find r in S, such that f (

Geo-political map of Africa divided for ethnomusicological purposes, after Alan P. Merriam, 1959.

This can be made concrete by defining a functor D: P → Set which maps each object x to and each arrow x → y to the inclusion map.

The challenge of a complete theory of population genetics is to provide a set of laws that predictably map a population of genotypes ( G < sub > 1 </ sub >) to a phenotype space ( P < sub > 1 </ sub >), where selection takes place, and another set of laws that map the resulting population ( P < sub > 2 </ sub >) back to genotype space ( G < sub > 2 </ sub >) where Mendelian genetics can predict the next generation of genotypes, thus completing the cycle.

Furthermore, the elements that are actually in P are precisely those whose reflection vanishes at P. So if we think of the map, associated to any element a of A:

* Heinrich, P. V., 2008, Loess map of Louisiana., Public Information Series.

An equivalent, but purely algebraic description of linear differential operators is as follows: an R-linear map P is a kth-order linear differential operator, if for any k + 1 smooth functions we have

P. F. Henshaw's work on decoding net-energy system construction processes termed " Natural Systems Theory ", uses various analytical methods to quantify and map them such as System Energy Assessment for taking true quantitative measures of whole complex energy using systems, and for anticipating their successions, such as Models Learning Change to permit adapting models to their emerging inverted designs.

Here π: P → X is required to be a smooth map between smooth manifolds, G is required to be a Lie group, and the corresponding action on P should be smooth.

The simple transitivity of the G action on the fibers of P guarantees that this map is a bijection, it is also a homeomorphism.

Global sections determine maps to projective spaces in the following way: Choosing r + 1 not all zero points in a fiber of L chooses a fiber of the tautological line bundle on P < sup > r </ sup >, so choosing r + 1 non-simultaneously vanishing global sections of L determines a map from X into projective space P < sup > r </ sup >.

Therefore, if the sections never simultaneously vanish, they determine a form: ...: s < sub > r </ sub > which gives a map from X to P < sup > r </ sup >, and the pullback of the dual of the tautological bundle under this map is L. In this way, projective space acquires a universal property.

In other words, a characteristic class associates to any principal G-bundle P → X an element c ( P ) in H *( X ) such that, if f: Y → X is a continuous map, then c ( f * P )

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

That is, every element α ∈ T < sub > x </ sub >< sup >*</ sup > M is a linear map

In either case, df < sub > x </ sub > is a linear map on T < sub > x </ sub > M and hence it is a tangent covector at x.

We can then define the differential map d: C < sup >∞</ sup >( M ) → T < sub > x </ sub >< sup >*</ sup > M at a point x as the map which sends f to df < sub > x </ sub >.

Since the map d restricts to 0 on I < sub > x </ sub >< sup > 2 </ sup > ( the reader should verify this ), d descends to a map from I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup > to the dual of the tangent space, ( T < sub > x </ sub > M )< sup >*</ sup >.

The image of an ellipse by any affine map is an ellipse, and so is the image of an ellipse by any projective map M such that the line M < sup >− 1 </ sup >( Ω ) does not touch or cross the ellipse.

From these two axioms, it follows that for every g in G, the function which maps x in X to g · x is a bijective map from X to X ( its inverse being the function which maps x to g < sup >− 1 </ sup >· x ).

functions f ( x ) thus map systematically to umbral finite-difference analogs involving f ( xT < sub > h </ sub >< sup >− 1 </ sup >).

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.

The linear map that has the matrix A < sup >− 1 </ sup > with respect to some base is then the reciprocal function of the map having A as matrix in the same base.

In this case, the adjoint map is given by Ad < sub > g </ sub >( x ) = gxg < sup >− 1 </ sup >.

It follows that the full pre-image ƒ < sup >− 1 </ sup >( q ) in M of a regular value q ∈ N under a differentiable map ƒ: M → N is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected.

Specifically, given g ∈ G, ρ ( g ) is the linear map on V determined by its action on the basis by right translation by g < sup >− 1 </ sup >, i. e.

A local section of a fiber bundle is a continuous map s: U → E where U is an open set in B and π ( s ( x )) = x for all x in U. If ( U, φ ) is a local trivialization of E, where φ is a homeomorphism from π < sup >− 1 </ sup >( U ) to U × F ( where F is the fiber ), then local sections always exist over U in bijective correspondence with continuous maps from U to F. The ( local ) sections form a sheaf over B called the sheaf of sections of E.

The unit map is given by sending x to one, multiplication is given by sending x to x ⊗ x, and the inverse is given by sending x to x < sup >− 1 </ sup >.

The map that sends x to x < sup >− 1 </ sup > is an example of a group antiautomorphism.

* The map that sends x to its inverse x < sup >− 1 </ sup > is an involutive antiautomorphism in any group.