In their last published paper [9], [16, pp. 310–321], G. H. Hardy and S. Ramanujan derived infinite series representations for the coefficients of certain modular forms of negative weight which are not analytic in the upper half-plane. In particular, they examined in detail the coefficients of the reciprocal of the Eisenstein series E6(τ). While confined to the sanitarium, Matlock House, in 191...

We develop a representation of the second kind for certain Hardy classes solutions to nonhomogeneous Cauchy–Riemann equations and use it show that boundary values in sense distributions these functions can be represented as sum an atomic decomposition error term. continuity Hilbert transform on this class Schwarz-type value problem constructed associated classes.

We develop further some aspects of the spectral theory of a class of Riemannian manifolds introduced by E. B. Davies; in particular we study the best constant in the Hardy Inequality which has become important in spectral theory. One application of this constant has been to a certain type of domain perturbation. This technique is useful when the domain has an irregular boundary and this is the ...

We give an intuitive proof of a path large-deviations result for a typed branching diffusion as found in Git, J.Harris and S.C.Harris [4]. Our approach involves an application of a change of measure technique involving a distinguished infinite line of descent, or spine, and we follow the spine setup of Hardy and Harris [5, 7, 6]. Our proof combines simple martingale ideas with applications of V...

We consider the hyperbolic Hardy class %Hp(B), 0 < p < ∞. It consists of φ holomorphic in the unit complex ball B for which |φ| < 1 and sup 0<r<1 ∫ ∂B {%(φ(rζ), 0)} dσ(ζ) < ∞, where % denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type g-function and the area function are defined in terms of the invariant gradient of B, and membership of %Hp(B) ...

Using a multivariable Faa di Bruno formula we give conditions on transformations τ : [0, 1] → X where X is a closed and bounded subset of R such that f ◦ τ is of bounded variation in the sense of Hardy and Krause for all f ∈ C(X ). We give similar conditions for f◦τ to be smooth enough for scrambled net sampling to attain O(n−3/2+ ) accuracy. Some popular symmetric transformations to the simple...

We develop further some aspects of the spectral theory of a class of Rie-mannian manifolds introduced by E. B. Davies; in particular we study the best constant in the Hardy Inequality which has become important in spectral theory. One application of this constant has been to a certain type of domain perturbation. This technique is useful when the domain has an irregular boundary and this is the...