Alternative algebras are so named because they are precisely the algebras for which the associator is alternating.

Both of these identities together imply that the associator is totally skew-symmetric.

The associator of an alternative algebra is therefore alternating.

Conversely, any algebra whose associator is alternating is clearly alternative.

An alternating associator is always totally skew-symmetric.

This is equivalent to saying the associator vanishes for all such and.

That is, the associativity requirement is now taken over by the associator of the hom-category.

** is associative: there is a natural isomorphism, called associator, with components,

In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure.

For a nonassociative ring or algebra, the associator is the multilinear map given by

The associator is alternating precisely when is an alternative ring.

The associator is symmetric in its two rightmost arguments when is a pre-Lie algebra.

associator is the map defined by the equation

As with its ring theory analog, the quasigroup associator is a measure of nonassociativity of Q.

In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism

* Alternative rings: A nonassociative ring R is said to be an alternative ring if == 0, where = ( xy ) z-x ( yz ) is the associator.

Older techniques, such as bilinear and trilinear filtering, do not take into account the angle a surface is viewed from, which can result in aliasing or blurring of textures.

The fastest method is to use the nearest-neighbour interpolation, but bilinear interpolation or trilinear interpolation between mipmaps are two commonly used alternatives which reduce aliasing or jaggies.

The homothetic center ( which is a triangle center if and only if P is a triangle center ) is the point given in trilinear coordinates by

For two spatial dimensions, the extension of linear interpolation is called bilinear interpolation, and in three dimensions, trilinear interpolation.

* Letting x: y: z be a variable point in trilinear coordinates, an equation for the nine-point circle is

Let A, B, C denote the vertex angles of the reference triangle, and let x: y: z be a variable point in trilinear coordinates ; then an equation for the Euler line is

Although the main texture would still be used when the view is sufficient to render it in full detail, the renderer will switch to a suitable mipmap image ( or in fact, interpolate between the two nearest, if trilinear filtering is activated ) when the texture is viewed from a distance or at a small size.

If, for example, a scene is rendering this texture in a space of 40 × 40 pixels, then either a scaled up version of the 32 × 32 ( without trilinear interpolation ) or an interpolation of the 64 × 64 and the 32 × 32 mipmaps ( with trilinear interpolation ) would be used.

The obvious extension of bilinear interpolation to three dimensions is called trilinear interpolation.

There are several ways to arrive at trilinear interpolation, it is equivalent to 3-dimensional tensor B-spline interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators

The result of trilinear interpolation is independent of the order of the interpolation steps along the three axes: any other order, for instance along, then along, and finally along, produces the same value.

In practice, a trilinear interpolation is identical to three successive linear interpolations, or two bilinear interpolations combined with a linear interpolation:

Contains a very clever and simple method to find trilinear interpolation that is based on binary logic and can be extended to any dimension ( Tetralinear, Pentalinear, ...).

Similarly, a triality between three vector spaces over a field F is a nondegenerate trilinear map

Since it uses both larger and smaller mipmaps, trilinear filtering cannot be used in places where the pixel is smaller than a texel on the original texture, because mipmaps larger than the original texture are not defined.

Fortunately bilinear filtering still works, and can be used in these situations without worrying too much about abruptness because bilinear and trilinear filtering provide the same result when the pixel size is exactly the same as the size of a texel on the appropriate mipmap.

It is usually helpful to make a sketch map in the field, showing the size and location of the features of interest and to take photographs at the site.

Korzybski is remembered as the author of the dictum: " The map is not the territory ".

This was expressed by Korzybski's most famous premise, " the map is not the territory ".

* The map is not the territory

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** On every infinite-dimensional topological vector space there is a discontinuous linear map.

** In the category of Riemann surfaces, an automorphism is a bijective biholomorphic map ( also called a conformal map ), from a surface to itself.

Later, his published letters were the basis of Waldseemüller's 1507 map, which is the first usage of America.

" " Giving Directions " is the term for thinking and projecting an anatomically ideal map of how one's body may be used effortlessly.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

The map is oriented with south at the top.

A map of Cyprus in the later Bronze Age ( such as is given by J. L. Myres and M. O. Richter in Catalogue of the Cyprus Museum ) shows more than 25 settlements in and about the Mesaorea district alone, of which one, that at Enkomi, near the site of Salamis, has yielded the richest Aegean treasure in precious metal found outside Mycenae.

Celsius conducted many geographical measurements for the Swedish General map, and was one of earliest to note that much of Scandinavia is slowly rising above sea level, a continuous process which has been occurring since the melting of the ice from the latest ice age.

By definition a multilinear map is alternating if it vanishes whenever two of its arguments are equal.

More precisely, a binary operation on a non-empty set S is a map which sends elements of the Cartesian product S × S to S:

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

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 ′′.