Dimensionality Reducing by Alpha-Dense Curves ... - RUA

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Example In the euclidean space , N�2 for each positive integer m , the unit cube is densified by the curve defined by It is easy to check that has density in where is a positive constant depending on N. For instance, Therefore their densities go to 0 as m�� . The curve is said to be the Cosines Curve of order m .
Example In , N�2 , for each positive integer m the unit cube is densified by the polynomial curve defined by where is the Chebyshev polynomial of the first kind of degree j , that is, The density of in the unit cube is ,where �(N) is a positive constant that depends on N (for instance, , etc.). Hence, also the densities tend to 0 as m�� . The curve is called the Chebyshev Curve of order m .
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Page 10 and 11: Definition Let , be the spaces of c
Page 12 and 13: Theorem Let A be an arbitrary const
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Page 16 and 17: Theorem Let f be a real nonnegative
Page 18 and 19: Theorem Let K be a densifiable comp
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Page 22 and 23: Definition (Filling condition ) We
Page 24 and 25: 6.2. �-Dense Curves of the Minima
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Page 28 and 29: Observe that for any normed space a
Page 30 and 31: 11. Kharazishvili, A.B., Strange Fu
Page 32: 30. Mora,G., Minimizing Multivariab

Dimensionality Reducing by Alpha-Dense Curves ... - RUA

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