We derive a beautiful series expansion for Spearman’s rho, ρ(θ) of the Ali-Mikhail-Haq (AMH) copula with paramater θ which is also called α or θ. Further, via experiments we determine the cutoffs to be used for practically fast and accurate computation of ρ(θ) for all θ ∈ [−1, 1].

We show how to find series expansions for π of the form π = ∞ n=0 S(n) mn pn a n , where S(n) is some polynomial in n (depending on m, p, a). We prove that there exist such expansions for m = 8k, p = 4k, a = (−4) k , for any k, and give explicit examples for such expansions for small values of m, p and a.

We prove that a two-variable p-adic lq-function has the series expansion lp,q s, t, χ 2 q/ 2 F ∑F a 1, p,a 1 −1 a χ a qa/〈a pt〉 ∑∞ m 0 −s m F/〈a pt〉 mE∗ m,qF which interpolates the values lp,q −n, t, χ E∗ n,χn,q pt − pχn p 2 q/ 2 qp E∗ n,χn,q t , whenever n is a nonpositive integer. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given in this ...

The generalized Taylor expansion including a secret auxiliary parameter h which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of h can’t be understood in the frame of the homotopy analysis method. This is a serious shortcoming of Liao’s method. We solve the problem. Through a detailed study of a simple ...

We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of series coefficients, and apply the inverse of this operator to the rational functions. We obtain new rational functions which seem to have only positive coefficients, whose positivity would imply posit...

The projection conditions for the quotient gauge theory of the D-brane world volume theory are studied for C/Z3 × Z3.The series expansion solutions which satisfy these conditions are constructed and they are shown to be incompatible with the F-flatness conditions for the discrete torsion case.

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random...

We have investigated skewed parton distributions in coordinate space. We found that their evolution can be described in a simple manner in terms of non-local, conformal operators introduced by Balitsky and Braun. The resulting formula is given by a Neumann series expansion. Its structure resembles, for all values of the asymmetry parameter, the well-known solution of the ERBL equation in the mo...

Abstract: In this paper, a perturbing nonlinear homogeneous Schrodinger equation is studied under limited time interval, complex initial conditions and zero Neumann conditions. The perturbation and Picard approximation methods together with the eigenfunction expansion and variational parameters methods are used to introduce an approximate solution for the perturbative nonlinear case for which a...