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In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
When N is set to an integer value, the position and momentum variables can be restricted to integers and the mapping becomes a mapping of a toroidial square grid of points onto itself. Such an integer cat map is commonly used to demonstrate mixing behavior with Poincaré recurrence utilising digital images. The number of iterations needed to ...
1838 map of pre-railroad cargo traffic in Ireland, one of the first thematic maps to use proportional symbols. The earliest known map to visually represent the volume of flow were two maps by engineer Henry Drury Harness, published in 1838 as part of a report on the potential for railroad construction in Ireland, showing the quantity of cargo traffic by road and canal.
The length of the line on the linear scale is equal to the distance represented on the earth multiplied by the map or chart's scale. In most projections, scale varies with latitude, so on small scale maps, covering large areas and a wide range of latitudes, the linear scale must show the scale for the range of latitudes covered by the map. One ...
To see how this number arises, consider the real one-parameter map =. Here a is the bifurcation parameter, x is the variable. The values of a for which the period doubles (e.g. the largest value for a with no period-2 orbit, or the largest a with no period-4 orbit), are a 1, a 2 etc. These are tabulated below: [7]
A bilinear map is a function: such that for all , the map (,) is a linear map from to , and for all , the map (,) is a linear map from to . In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.
More formally, an additive map is a -module homomorphism. Since an abelian group is a Z {\displaystyle \mathbb {Z} } - module , it may be defined as a group homomorphism between abelian groups. A map V × W → X {\displaystyle V\times W\to X} that is additive in each of two arguments separately is called a bi-additive map or a Z {\displaystyle ...
Another example is the application of conformal mapping technique for solving the boundary value problem of liquid sloshing in tanks. [ 19 ] If a function is harmonic (that is, it satisfies Laplace's equation ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} ) over a plane domain (which is two-dimensional), and is transformed via a conformal map to ...