Far-field Approximations to the Derivatives of the Green's Function for the Ffowcs Williams and Hawkings Equation

The surface correction to the quadrupole source term of the Ffowcs Williams and Hawkings integral in the frequency domain suffers from the computation of high-order derivatives of the Green's function. The far-field approximations to the derivatives of the Green's function have been used without derivation and verification in the previous work. In this work, we provide the detailed derivations of the far-field approximations to the derivatives of the Green's function. The binomial expansions for the derivatives of the Green's function and the far-field condition are employed during the derivations to circumvent the difficulties in computing the high-order derivatives. The approximations to the derivatives of the Green's function are systemically verified by using the benchmark two dimensional convecting vortex and the co-rotating vortex pair. In addition, we provide the derivations of the approximations to the multiple integrals of the Green's function by using the far-field approximations to the derivatives.

Integration

The concept of an integral on a general measure space is developed from first principles. Riemann–Stieltjes and Lebesgue–Stieltjes integrals are defined. The monotone convergence theorem, fundamental properties of integrals, and related inequalities are covered. Other topics include product measure and multiple integrals, Fubini’s theorem, signed measures, and the Radon–Nikodym theorem.

Elliptic modular graph forms. Part I. Identities and generating series

Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker-Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals over the torus of combinations of coefficients of the Kronecker-Eisenstein series, and may be assembled into generating series. These relations are exploited to derive holomorphic subgraph reduction formulas, as well as algebraic and differential identities between eMGFs and their generating series.

Refinements of Hermite–Hadamard Inequalities for Continuous Convex Functions via (p,q)-Calculus

In this paper, we present some new refinements of Hermite–Hadamard inequalities for continuous convex functions by using (p,q)-calculus. Moreover, we study some new (p,q)-Hermite–Hadamard inequalities for multiple integrals. Many results given in this paper provide extensions of others given in previous research.