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By Kulikov A. S.

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In this case, we use the functional composition notation x∗ = x ◦ h. The function value x∗ (t) is indicated by (x ◦ h)(t). Moreover, in the same chapter, we will use the inverse function which results from solving the relation h(g) = t for g given t. This function, having values g(t), is denoted by h−1 . This does not mean, of course, the reciprocal of h, which we simply indicate as 1/h on the rare occasion that we need it. In fact, the functional compositions h ◦ h−1 and h−1 ◦ h satisfy (h ◦ h−1 )(t) = (h−1 ◦ h)(t) = t and, in functional composition sense, therefore h and h−1 cancel one another.

It is often clearer to use longer strings of letters in a distinctive font to denote quantities more evocatively than standard notation allows. For example, we use names such as • Temp for a temperature record, • Knee for a knee angle • LMSSE for a squared error fitting criterion for a linear model, and • RSQ for a squared correlation measure. 2 Derivatives and integrals Our notation for the derivative of order m of a function x is Dm x; this produces cleaner formulas than dm x/dtm . It stresses that differentiation is an operator that acts on a function x to produce another function Dx.

2 Data resolution and functional dimensionality This suggests the notation of the resolving power or resolution of a set of data. This is inversely related to the width of the narrowest event that can be estimated to our satisfaction. We mean by the phrase “high resolution data” that they can pin down small events. The resolution of a set of data can be a rather more useful concept than simply the number of observations taken. Resolution leads in turn to the notion of the dimensionality of a function.

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A 2E4-time algorithm for MAX-CUT by Kulikov A. S.


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