The "'real matrices "'are in one-one correspondence with the linear mappings of the two-dimensional Cartesian coordinate system into itself by the rule
22.
The fact that is well defined and continuous follows from H�lder's inequality . is a linear mapping which is an isometry by the extremal case of H�lder's inequality.
23.
D2 ( " U " ) by passing to the limit . ( This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous .)
24.
Instead of being a simple, linear mapping, this complex relationship between genotype and phenotype is not straightforward to emergent structure, such as a cake, that is not described explicitly in the recipe itself.
25.
When a bijective linear mapping exists between two vector spaces ( that is, every vector from the second space is associated with exactly one in the first ), we say that the two spaces are isomorphic.
26.
Since the LMIs realize a non-linear mapping between the vertexes in \ mathcal { S } and \ mathcal { F } we may find very different controllers for each z = 1 \ ldots Z.
27.
Related to autoencoders is the NeuroScale algorithm, which uses stress functions inspired by multidimensional scaling and Sammon mappings ( see below ) to learn a non-linear mapping from the high-dimensional to the embedded space.
28.
The request is " I did want something that would work for all the elements of the Clifford algebra . " We still need to define what is meant by a linear mapping from the Clifford algebra onto itself.
29.
If is smooth then the derivative is smooth, and also is smooth where denotes the space of all bounded linear mappings with the topology of uniform convergence on bounded subsets; see [ KM ], 3.18.
30.
Technically, the image of the ( mathematical ) cone over the simplex whose vertices are the spectral colors, by this linear mapping, is also a ( mathematical ) cone in "'R "'3 color.
