The aim of the present paper is to study certain integral properties of M-series [7], a general class of polynomials [9], Fox’s H-function [3] and H̄function [6] which contains a certain class of Feynman integrals, exact partition of a Gaussian model in statistical mechanics and several other functions as its particular cases. The main results of our paper are unified in nature and capable of yi...

We introduce the analytic functions f (z) = z + ∞ ∑ k=n+1 akz k ( n ∈ N) and p(z) = z + m ∑ s=1 bsj−s+1zsj−s+1 (j ≥ n + 1;n ∈ N) in the open unit disk U. By means of the subordination theorem of J.E. Littlewood, we shall investigate the integral means with coefficients inequalities of analytic functions f(z) and p(z). Some applications of the integral mean of f(z) are considered. 2000 Mathemati...

Hilbert’s proof, apart from the determination of the best possible constant π csc(π/p), was published by Weyl [7]. The calculation of the constant, and the integral analogue of Hilbert’s double series theorem (for p = 2) are due to Schur [6]. The generalizations to other p′s of both the discrete and integral versions of this result were discovered later on by Hardy and Riesz and published by Ha...

Given r > 2, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coefficients mapping the “cube” with real coordinates from [−r, r] into [−t, t]. This directly translates to a nice statement in logic (more specifically recursion theory) with a corresponding phase transition case of 2 being open. We ...

This paper shows the existence of asymptotically stable solutions to an integral equation. This is done by using a fixed point theorem, and without requiring that the solutions be bounded.

Let Γ̄ be the Picard modular group of an imaginary quadratic number field k and let D be the associated symmetric space. Let Γ ⊂ Γ̄ be a congruence subgroup. We describe a method to compute the integral cohomology of the locally symmetric space Γ\D. The method is implemented for the cases k = Q(i) and k = Q( √ −3), and the cohomology is computed for various Γ.

The effects of planktivorous fish on phytoplankton through nutrient recycling and zooplankton herbivory were experimentally separated and their relative importance quantified in a eutrophic humic lake. Natural phytoplankton assemblages were incubated in nutrient-permeable chambers placed inside enclosures with or without fish. Outside these chambers, phytoplankton were exposed to zooplankton he...