Computer numerical control of machine tools - Second edition by G E THYER

By G E THYER

It is a accomplished textbook catering for BTEC scholars at NIII and better nationwide degrees, complex urban and Guilds classes, and the early years of measure classes. it's also perfect to be used in business retraining and post-experience programmes.

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Extra resources for Computer numerical control of machine tools - Second edition

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N k Note also that d ( x, f ° ( x )) ≤ d ( x, f ( x )) + d ( f ( x ), f ° ( x )) + ... + d ( f ° k 2 ≤ (1 + s + s + ... + s 1 ≤ d ( x, f ( x )), 1− s 2 k −1 ( k −1) ( x ), f ° ( x )) k )d ( x, f ( x )) where the final inequality follows from the series expansion of (1-s)-1 which is valid here because 0 ≤ s < 1. So, for example, if n < m, we have d ( f ° n ( x ), f ° m ( x )) ≤ sn d ( x, f ( x )). 1− s Since s < 1, the expression on the right approaches 0 as n,m → ∞. In other words, the sequence {f°n(x)} is a Cauchy sequence in (X,d).

3) holds. 3 Contraction mapping theorem for grayscale images Partition the unit square I2 into a collection of range cells {Ri} that tile I2: 46 Fractal Encoding of Grayscale Images I2 = UR , i Ri I R j = ∅ . ~ } be a PIFS such that Let {w i ~ :D → R w i i i for some collection of domains Di ⊂ I2 (the Di’s may overlap, and need not cover I2). Fig. 3 shows this configuration. ~ w i Di Ri ~ maps domain D to range R . The Fig. 3 Transformation w i i i domains may overlap, while the ranges tile the unit square.

Note that f(x), f°2(x), f°3(x),… forms a sequence in X. Suppose f is a contraction mapping with contractivity factor s. Note that d ( f ° ( x ), f ° n (n+k ) ( x )) ≤ s d ( f ° ( n −1) ( x ), f ° ( n + k− 1) ( x )) ≤ s d ( x, f ° ( x )). n k Note also that d ( x, f ° ( x )) ≤ d ( x, f ( x )) + d ( f ( x ), f ° ( x )) + ... + d ( f ° k 2 ≤ (1 + s + s + ... + s 1 ≤ d ( x, f ( x )), 1− s 2 k −1 ( k −1) ( x ), f ° ( x )) k )d ( x, f ( x )) where the final inequality follows from the series expansion of (1-s)-1 which is valid here because 0 ≤ s < 1.

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