Ramanujan — Fourier series and a theorem of Ingham

Given two arithmetical functions f, g, we derive, under suitable conditions, asymptotic formulas for the convolution sums ∑ _n≤N f (n) _g (n + h) for a fixed number h. To this end, we develop the theory of Ramanujan expansions for arithmetical functions. Our results give new proofs of some old results of Ingham proved by him in 1927 using different techniques.     

Ab initio and DFT benchmark study for the calculations of isotopic shifts of fundamental frequencies for 2,3-dihydropyran

Structural Chemistry - A systematic investigation has been carried out to assess the performance of various exchange–correlation energy density functionals coupled with various basis sets to...     

Crystal and molecular structure of 7-hydroxyflavone monohydrate, C14H10O3·H2O

The X-ray crystal structure of 7-hydroxyflavone monohydrate, C₁₄H₁₀O₃ · H₂O, is determined. The compound crystallizes in the monoclinic space group, P2₁/n with a = 3.801(3), b = 19.665(4), c = 16. 039(6), = 93.69(3)°, and = 0.68 mm^–1 for Z = 4. The phenyl ring of the flavone moiety is rotated 18.6(1)° out of the penzopyran plane, which is a typical value for flavones. In the crystal lattice, there are wide channels which are lined mainly by C–H groups. The water molecules enclosed in these channels are severely disordered.     

Studies on oxygen chemical surface exchange and electrical conduction in thin film nanostructured titania at high temperatures and varying oxygen pressure

We report on oxygen surface exchange studies in ～450-nm-thick nanocrystalline titania films with an average grain size of ～13 nm by electrical conductivity relaxation along with the conductivity measurements at varying temperatures and oxygen partial pressures (pO(2)s). By electrochemical impedance spectroscopy technique, the high temperature conductivity was measured in the pO(2) range from ～10(-16) to ～10(-6) Pa at temperatures from 973 to 1223 K and activation energy, ΔE(a), for conduction was estimated as ～3.23 eV at pO(2) ～10(-11) Pa. Under reducing atmosphere (pO(2) < 10(-6) Pa), two distinct n-type conduction regimes were observed and corresponding predominant defects are discussed while, at high pO(2) regime (pO(2) >10(-6) Pa), ionic conduction appears dominant leading to a conductivity plateau. The surface relaxation was observed to have two independent time constants likely originating from microstructural effects. The surface exchange coefficients are measured as ～10(-8)-10(-7) m∕s and ～10(-9)-10(-8) m∕s for each contribution with ΔE(a)s of 2.79 and 1.82 eV, respectively, without much pO(2) dependence across several orders of pO(2) range of ～10(-16)-10(-6) Pa in the temperature range between 973 and 1223 K. The results are of potential relevance to understanding the near-surface chemical phenomena in nanocrystalline titania which is of great interest for energy and environmental studies.     

Simple technique for alignment of a dispersive ring laser cavity

A simple technique for the alignment of a dispersive ring cavity consisting of a reflection-grating and two mirrors is described. Lasing on different rotational–vibrational transitions of CO₂ molecule is achieved in the ring configuration.     

Finite element study of free surface turbulent flow in rotating channels

Under certain conditions of liquid flow through rotating channels, the Coriolis force can induce a free surface to be formed. This problem is of practical importance in a Coriolis wear tester, which is used for determining the sliding wear coefficient of wear materials in slurry handling equipment. A deforming Galerkin finite element method is presented for predicting two‐dimensional turbulent free surface mean flow in rotating channels. Reynolds‐averaged Navier–Stokes (RANS) equations are cast into weak(algebraic) form using primitive variables (velocity and pressure). Eddy viscosity is determined via a mixing length model. Velocity is interpolated biquadratically, while pressure is interpolated bilinearly. The kinematic condition is used to form the Galerkin residual for the free surface. The free surface is represented by Hermite polynomials of zeroeth order for continuity of position and slope. Combined Newton's iteration is used to simultaneously solve for the free surface and the field variables. Results of velocity and pressure fields, as well as the free surface are shown to converge with mesh‐size refinement. There is excellent respect for mass conservation. Results are presented for various values of Rossby number (Ro) and height‐based Reynolds number (Re_H). Parameter continuation in Ro and Re_H space is used to compute solutions at higher values of flow rate and angular velocity. Copyright © 2002 John Wiley & Sons, Ltd.