Narrow Results

Let $R$ be a commutative ring with identity, $X$ be an indeterminate over $R$, $R[X]$ be the polynomial ring over $R$, $R[[X]]$ be the power series ring over $R$, and $Q$ be a principal prime ideal of $R[X]$ with $(Q\cap R)[X]⊊Q$. It is well known that if $R$ is an integral domain, then $R{[X]}_Q$ is a DVR and $R[X]$ has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) $R_{Q\cap R}$ is a field, (ii) $R{[X]}_Q$ is a DVR, but (iii) there is a ring $R$ such that $R[X]$ has no principal prime ideal. We also study the maximal ideals of $R[[X]]$ that are principal.

A domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional nondiscrete valuation domain. If R ⊂ S is a minimal ring extension and S is a domain, then (R,S) is a residually algebraic pair. If S is a domain but not a field, a maximal non-ACCP subring extension R ⊂ S is a minimal ring extension if (R,S) is a residually algebraic pair and R is quasilocal. Results with a similar flavor are given for domains R ⊂ S sharing a nonzero ideal, with applications to rings R of the form A + XB[X] or A + XB[[X]]. If R ⊂ S is a minimal ring extension such that R is a domain and S is not (R-algebra isomorphic to) an overring of R, then R satisfies ACCP if and only if S satisfies ACCP.

We report exact quantum dynamics calculations of the eigenstate energy levels for the bound rovibrational states of the Ar₂Ne complex, across the range of J values for which such states are observed (J = 0–35). All calculations have been carried out using the ScalIT suite of parallel codes. These codes employ a combination of highly efficient methods, including phase-space optimized discrete variable representation, optimal separable basis, and preconditioned inexact spectral transform (PIST) methods, together with an effective massive parallelization scheme. The Ar₂Ne energy levels were computed using a pair-wise Aziz potential plus a three-body correction, in Jacobi co-ordinates. Effective potentials for the radial co-ordinates are constructed, which reveal important physical insight into the two distinct dissociation pathways, Ar₂Ne → NeAr + Ar and Ar₂Ne → Ar₂ + Ne. A calculation of the bound vibrational (J = 0) levels, computed using the Tang–Toennies potential, is also performed for comparison with results from the previous literature.

In this work we analyze the dynamics of a quantum particle subject to an asymmetric bistable potential and interacting with a thermal reservoir. We obtain the time evolution of the population distributions in both energy and position eigenstates of the particle, for different values of the coupling strength with the thermal bath. The calculation is carried out using the Feynman-Vernon functional under the discrete variable representation.