Harmonic function for which the second dilatation is α-spiral

Second-harmonic phonon spectroscopy of α-quartz

We demonstrate mid-infrared second-harmonic generation as a highly sensitive phonon spectroscopy technique that we exemplify using α-quartz (SiO2) as a model system. A mid-infrared free-electron laser provides direct access to optical phonon resonances ranging from 350 cm−1 to 1400 cm−1. While the extremely wide tunability and high peak fields of an free-electron laser promote nonlinear spectro...

Harmonic Function Theory Second Edition

Galerkin Finite-Element Method for the Analysis of the Second Harmonic Generation in Wagon Wheel Fibers

The nonlinear effects of the second harmonic generation have been investigated for the propagation of light along the axis of fibers of wagon wheel cross sectional shape. Nodal finite element formulation is utilized to obtain discretized Helmholtz equations under appropriate boundary conditions. The hierarchical p-version nodal elements are used for meshing the cross section of wagon wheel fibe...

Modules for which every non-cosingular submodule is a summand

‎A module $M$ is lifting if and only if $M$ is amply supplemented and‎ ‎every coclosed submodule of $M$ is a direct summand‎. ‎In this paper‎, ‎we are‎ ‎interested in a generalization of lifting modules by removing the condition‎"‎amply supplemented‎" ‎and just focus on modules such that every non-cosingular‎ ‎submodule of them is a summand‎. ‎We call these modules NS‎. ‎We investigate some gen...

Rings for which every simple module is almost injective

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...