The class of piecewise linear homeomorphisms (PLH) provides a convenient functional representation for many applications wherein an approximation to data is required that is invertible in closed form. In this paper we introduce the graph intersection (GI) algorithm for ”learning” piecewise linear scalar functions in two settings we term ”approximation” (where an ”oracle” outputs accurate functi...

The traditional conservation of mass equation is derived using a first-order Taylor series to represent flux change in a control volume, which is valid strictly for cases of linear changes in flux through the control volume. We show that using higher-order Taylor series approximations for the mass flux results in mass conservation equations that are intractable. We then show that a fractional T...

In the article [17] we consider only the case when n and m are non-negative integer points determined by: n ≥ 1 is the multiplicity of the root x = a, otherwise n = 0 if x = a is not the root; and m ≥ 1 is the multiplicity of the root x = b, otherwise m = 0 if x = b is not the root. In this case, if for the function f(x) at the point x = a there is an approximation of the function by Taylor pol...

In this thesis we analyze a schematic Mode-Coupling Theory in two time variables for small amplitude oscillatory shear with frequency ω. We derive the second order Taylor approximation for the density correlator and formulas for its long-time behavior. We also numerically solve the resulting equations, thus confirming the analytic results. We find that the solutions for the perturbation of the ...

In this paper, we present a proof in ACL2(r) of Taylor’s formula with remainder. This important theorem allows a function f with n + 1 derivatives on the interval [a, b] to be approximated with a Taylor series of n terms centered at a. Moreover, the formula allows the error in the approximation to be bounded by a term involving the (n + 1)st derivative of f on (a, b). The results in this paper ...