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Positive solutions for nonlinear fractional differential equations with coefficient that changes sign. (English) Zbl 1152.34304

Summary: Let \(f: F^n_2\to F_2\) be a Boolean function. Let \(W_f(a)= \sum_{x\in F^n_2} f(x)(-1)^{\langle a,x\rangle}\) denote the Walsh-Fourier transform of \(f\), here \(a\in F^n_2\). By \(f^{-1}(1)\) denote the set of all binary vectors of length \(n\), on which \(f\) gets the value 1. The graph \(G_f= (F^n_2, E)\) with the set of edges given as \(E= \{(a,b)\mid a\oplus b\in f^{-1}(1)\}\) is the Cayley graph of \(F^n_2\) with respect to the set \(f^{-1}(1)\).
Using these Cayley graphs for the representation of Boolean functions, the author studies the value of special product of Walsh-Fourier coefficients for an arbitrary Boolean function. The formula for such a product is given. There are given also applications of the results to the extremal nonlinear Boolean functions and to the calculating covering radii of Reed-Muller codes.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26A33 Fractional derivatives and integrals
34C11 Growth and boundedness of solutions to ordinary differential equations
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