Mercer’s conditions

In both the dual solution and decision function, only the inner product in the attribute space and the kernel function based on attributes appear, but not the elements of the very high dimensional feature space. The constraints in the dual solution imply that only the attributes closest to the hyperplane, the so-called SVs, are involved in the expressions for weights w. Data points that are not SVs have no influence and slight variations in them (for example caused by noise) will not affect the solution, provides a more quantitative leverage against noise in data that may prevent linear separation in feature space [42]. Imposing the requirement that the kernel satisfies Mercer s conditions (K(xj, must be positive semi-definite)... 

Now there is a further problem about the kernel function, i.e. given a function K, how to verify that it is a kernel. The answer is given by Mercer s condition [132], which will be discussed in next chapter. 

If the kernel fC is a symmetric positive definite function, which satisfies the Mercer s conditions... 

This is termed a memoryless non-linearity since the output is a function of only the present value of the input s[n The expression may be regarded as a power series expansion of the non-linear input-output relationship of the non-linearity. In fact this representation is awkward from an analytical point of view and it is more convenient to work in terms of the inverse function. Conditions for invertibility are discussed in Mercer [Mercer, 1993],... 

Gilbert, J., Castle, L., Jickells, S.M., Mercer, A.J.. Sharman, M., 1988, Migration from plastics into foodstuffs under realistic conditions of use. Food Add. Contam. 5 (Suppl. 1) 513-523. 

Mercer J. L., Zhao M., and Coleman S. M. (1999) Alkenone evidence of sudden changes in Chesapeake Bay conditions ca. 300 years ago. EOS 80 Suppl. S185. 

In this test a mixture of red and green direct dyes is used to compare the maturity of cotton fibre samples. Immature fibres dye red, and mature fibres green. Mercerization increases the fibre s affinity for green compound and causticization number can be assessed related to the strength of the green hue. Fabric treated with liquid ammonia under industrial mill condition dyes red. 

Ott was sensitive to the full range of industrial issues, as well as all the scientific ones. The nature of a cellulose sample was a strong function of its source and method of purification. It was also a strong function of the processing conditions and history. The crystal structure of mercerized cellulose is different than native fiber cellulose. Cellulose is often used as fibers. The morphology of cellulose fibers was one of Ott s specialties. 

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