Furthermore, a map f: A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings.

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

A null-homotopic map has degree zero, contradicting our only assumption, namely that f exists.

The differential of f at a point x is the map

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 >.

Just as every differentiable map f: M → N between manifolds induces a linear map ( called the pushforward or derivative ) between the tangent spaces

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Two manifolds M and N are diffeomorphic ( symbol usually being ) if there is a smooth bijective map f from M to N with a smooth inverse.

Model example: if U and V are two connected open subsets of R < sup > n </ sup > such that V is simply connected, a differentiable map f: U → V is a diffeomorphism if it is proper and if

* It is essential for U to be simply connected for the function f to be globally invertible ( under the sole condition that its derivative is a bijective map at each point ).

A morphism from ( X, x < sub > 0 </ sub >) to ( Y, y < sub > 0 </ sub >) is given by a continuous map f: X → Y with f ( x < sub > 0 </ sub >) = y < sub > 0 </ sub >.

Every continuous map f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ).

The identity element is the constant map at the basepoint, and the inverse of a loop f is the loop g defined by g ( t ) = f ( 1 − t ).

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 >.

A rectangular grid ( top ) and its image under a conformal map f ( bottom ).

Some of those algorithms will map arbitrary long string data z, with any typical real-world distribution — no matter how non-uniform and dependent — to a 32-bit or 64-bit string, from which one can extract a hash value in 0 through n − 1.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

* In particular, if V is a normed vector space and if z is any element of V, then there exists a continuous linear map with ψ ( z ) = || z || and || ψ || ≤ 1.

#: Consider a unit sphere placed at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis, see Euler angles.

The simplest construction is as the image of a sphere centered at the origin under the map f ( x, y, z ) = ( yz, xz, xy ).

The resulting map of the area z = f ( x, y ) represents the topography of the sample.

In the complex plane, the most obvious circle inversion map ( i. e., using the unit circle centered at the origin ) is the complex conjugate of the complex inverse map taking z to 1 / z.

Then, L < sub > 0 </ sub > is a Lie algebra, L < sub > 1 </ sub > is a linear representation of L < sub > 0 </ sub >, and there exists a symmetric L < sub > 0 </ sub >- equivariant linear map such that for all x, y and z in L < sub > 1 </ sub >,

The condition that the arc is a geodesic segment means that the map f above can be chosen to be an isometric embedding, that is it can be chosen so that for every z, t in we have d ( f ( z ), f ( t ))=| z-t | and that f ( a )= x, f ( b )= y.

In the above example, a connection with classical Galois theory can be seen by regarding as the profinite Galois group Gal (< span style =" text-decoration: overline "> F </ span >/ F ) of the algebraic closure < span style =" text-decoration: overline "> F </ span > of any finite field F, over F. That is, the automorphisms of < span style =" text-decoration: overline "> F </ span > fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the z < sup > n </ sup > map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n. Z of the fundamental group of the punctured disk.

The z z < sup > n </ sup > mapping shows this as a local pattern: if we exclude 0, looking at 0 < | z | < 1 say, we have ( from the homotopy point of view ) the circle mapped to itself by the n-th power map ( Euler-Poincaré characteristic 0 ), but with the whole disk the Euler-Poincaré characteristic is 1, n – 1 being the ' lost ' points as the n sheets come together at z = 0.

alt = A map showing a line of 13 ships, mostly dismasted and two on fire.

# d is a linear map: d ( af + bg ) =

alt = A map of Australia with the cane toad's distribution highlighted: The area follows the northeastern coast of Australia, ranging from the Northern Territory through to the north end of New South Wales.

alt = A map showing places in central southern England, including Gloucester, Cirencester, Bath, and Aylesbury

Even in the COBE map, it was observed that the quadrupole ( l = 2 spherical harmonic ) has a low amplitude compared to the predictions of the big bang.

The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0.

The Taylor series of the map is F ( x ) = J · x + O ( x < sup > 2 </ sup >), so a change of coordinates h can only be expected to simplify F to its linear part

alt = A world map with countries colored in different shades of orange

alt = Small, sepia-colored map of Edo in the 1840s

1 ) and thus p = 7, so no more than 7 colors are required to color any map on a torus.

alt = World map, with countries carrying terrestrial FNC in red and satellite providers in orange

The r = 4 case of the logistic map is a nonlinear transformation of both the bit shift map and the case of the tent map.

World map indicating literacy by country ( 2011 Human Development Report ) Grey = no data

We assume that A is an m-by-n matrix over either the real numbers or the complex numbers, and we define the linear map f by f ( x ) = Ax as above.

To define the vector space operations on T < sub > x </ sub > M, we use a chart φ: U → R < sup > n </ sup > and define the map ( dφ )< sub > x </ sub >: T < sub > x </ sub > M → R < sup > n </ sup > by ( dφ )< sub > x </ sub >( γ '( 0 )) = ( φ ∘ γ )'( 0 ).