Interpolation formula between very low and intermediate-to-high damping Kramers escape rates for single-domain ferromagnetic particles

It is shown that the Mel'nikov-Meshkov formalism for bridging the very low damping (VLD) and intermediate-to-high damping (IHD) Kramers escape rates as a function of the dissipation parameter for mechanical particles may be extended to the rotational Brownian motion of magnetic dipole moments of single-domain ferromagnetic particles in nonaxially symmetric potentials of the magnetocrystalline anisotropy so that both regimes of damping, occur. The procedure is illustrated by considering the particular nonaxially symmetric problem of superparamagnetic particles possessing uniaxial anisotropy subject to an external uniform field applied at an angle to the easy axis of magnetization. Here the Mel'nikov-Meshkov treatment is found to be in good agreement with an exact calculation of the smallest eigenvalue of Brown's Fokker-Planck equation, provided the external field is large enough to ensure significant departure from axial symmetry, so that the VLD and IHD formulas for escape rates of magnetic dipoles for nonaxially symmetric potentials are valid.

Measuring the blame of each formula for inconsistent prioritized knowledge bases.

It is increasingly recognized that identifying the degree of blame or responsibility of each formula for inconsistency of a knowledge base (i.e. a set of formulas) is useful for making rational decisions to resolve inconsistency in that knowledge base. Most current techniques for measuring the blame of each formula with regard to an inconsistent knowledge base focus on classical knowledge bases only. Proposals for measuring the blames of formulas with regard to an inconsistent prioritized knowledge base have not yet been given much consideration. However, the notion of priority is important in inconsistency-tolerant reasoning. This article investigates this issue and presents a family of measurements for the degree of blame of each formula in an inconsistent prioritized knowledge base by using the minimal inconsistent subsets of that knowledge base. First of all, we present a set of intuitive postulates as general criteria to characterize rational measurements for the blames of formulas of an inconsistent prioritized knowledge base. Then we present a family of measurements for the blame of each formula in an inconsistent prioritized knowledge base under the guidance of the principle of proportionality, one of the intuitive postulates. We also demonstrate that each of these measurements possesses the properties that it ought to have. Finally, we use a simple but explanatory example in requirements engineering to illustrate the application of these measurements. Compared to the related works, the postulates presented in this article consider the special characteristics of minimal inconsistent subsets as well as the priority levels of formulas. This makes them more appropriate to characterizing the inconsistency measures defined from minimal inconsistent subsets for prioritized knowledge bases as well as classical knowledge bases. Correspondingly, the measures guided by these postulates can intuitively capture the inconsistency for prioritized knowledge bases.