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A C^infty function is a function that is differentiable for all degrees of differentiation. For instance, f(x)=e^(2x) (left figure above) is C^infty because its nth ...

The approximation of a piecewise monotonic function f by a polynomial with the same monotonicity. Such comonotonic approximations can always be accomplished with nth degree ...

y approx m+sigmaw, (1) where w = (2) where h_1(x) = 1/6He_2(x) (3) h_2(x) = 1/(24)He_3(x) (4) h_(11)(x) = -1/(36)[2He_3(x)+He_1(x)] (5) h_3(x) = 1/(120)He_4(x) (6) h_(12)(x) ...

Elliptic alpha functions relate the complete elliptic integrals of the first K(k_r) and second kinds E(k_r) at elliptic integral singular values k_r according to alpha(r) = ...

The third singular value k_3, corresponding to K^'(k_3)=sqrt(3)K(k_3), (1) is given by k_3=sin(pi/(12))=1/4(sqrt(6)-sqrt(2)). (2) As shown by Legendre, ...

Elliptic rational functions R_n(xi,x) are a special class of rational functions that have nice properties for approximating other functions over the interval x in [-1,1]. In ...

The finite volume method is a numerical method for solving partial differential equations that calculates the values of the conserved variables averaged across the volume. ...

x_(n+1) = 2x_n (1) y_(n+1) = alphay_n+cos(4pix_n), (2) where x_n, y_n are computed mod 1 (Kaplan and Yorke 1979). The Kaplan-Yorke map with alpha=0.2 has correlation exponent ...

The spherical harmonics form a complete orthogonal system, so an arbitrary real function f(theta,phi) can be expanded in terms of complex spherical harmonics by ...

In functional analysis, the term "Poincaré-Friedrichs inequality" is a term used to describe inequalities which are qualitatively similar to the classical Poincaré Inequality ...