Continuing the paper “Structural properties of ideals over ${𝒫}_{\kappa }\lambda$ I” we first show that for every ideal $I$ over ${𝒫}_{\kappa }\lambda$ there is a function $f$ such that $f_{*}\left(I\right)=\{X\subset {𝒫}_{\kappa }\lambda :f^{-1}[X]\in I\}$ and ${\mathrm{J}}_{\kappa ,\lambda }$ are coherent. On the other hand, ${\mathrm{J}}_{\kappa ,\lambda }\not\subset g_{*}({\mathrm{I}}_{\kappa ,\lambda }\upharpoonright X)$ for any $g$ and $X\in {\mathrm{I}}_{\kappa ,\lambda }^{+}$ under some conditions, e.g. .

We also study coherence and rigidity for ideals over ${𝒫}_{\kappa }\lambda$, and get characterizations of these properties similar to those by Baumgartner, Taylor and Wagon for ideals on $\kappa$.

A semilinear Newton equation of motion is derived from a semilinear Schrödinger equation in homogeneous and isotropic spacetimes by the Ehrenfest theorem. The Cauchy problem for the equation is considered, especially, on the existence of global solutions and nonexistence of global weak solutions. The effects of spatial expansion and contraction are studied.

The order topologies on ordered sets are well-known. For ordered groups or ordered rings, we investigate algebraic order topologies which are compatible with their operations. We consider these topologies on the products of ordered groups, the product extension rings of ordered rings, and also Archimedean ordered groups. We give examples related to algebraic order topologies, etc.

We extend Salem’s integral equation to the non-homogenous form, and offer the associated criteria for the Riemann Hypothesis. Explicit solutions for the non-homogenous case are given, which in turn give further insight into Salem’s criteria for the RH. As a conclusion, we show these results follow from a corollary relating the uniqueness of solutions of the non-homogenous form with Wiener’s theorem.

We investigate a semilinear equation and associate each almost Hermitian metric with a unique constant ${\gamma }_k$. Then, we show that there is no 1-st Gauduchon metric in the conformal class of an almost Hermitian metric $\omega$ satisfying that ${\gamma }_1\left(\omega \right)>0$ on a real 6-dimensional compact almost Hermitian manifold. Especially, we find that there is no pluriclosed metric in the conformal class of $\omega$.

We determine all the homogeneous structure tensors on ${\mathbf{S}}^2\times \mathbf{R}$ and ${\mathbf{H}}^2\times \mathbf{R}$. This work together with previous articles [1, 3, 4, 7, 8] yields a complete classification of all the homogeneous structure tensors on three-dimensional homogeneous Riemannian manifolds.