Some Results On Normal Homogeneous Ideals

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by all homogeneous elements of degree at least m and monomial ideals in a polynomial ring over a field. For ideals of the first trype we generalize a recent result of S. Faridi. We prove that a monomial ideal in a polynomial ring in n indeterminates over a field is normal if and only if the first n-1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals obtained by taking integral closures of m-primary ideals generated by powers of the variables. We obtain classes of normal monomial ideals and arithmetic critera for deciding when the monomial ideal is not normal.Comment: 19 page

Photo of A. G. Reid

Photo of A. G. Rei

Criteria for generalized macroscopic and mesoscopic quantum coherence

We consider macroscopic, mesoscopic and "S-scopic" quantum superpositions of eigenstates of an observable, and develop some signatures for their existence. We define the extent, or size $S$ of a superposition, with respect to an observable \hat{x}, as being the range of outcomes of \hat{x} predicted by that superposition. Such superpositions are referred to as generalized $S$-scopic superpositions to distinguish them from the extreme superpositions that superpose only the two states that have a difference $S$ in their prediction for the observable. We also consider generalized $S$-scopic superpositions of coherent states. We explore the constraints that are placed on the statistics if we suppose a system to be described by mixtures of superpositions that are restricted in size. In this way we arrive at experimental criteria that are sufficient to deduce the existence of a generalized $S$-scopic superposition. The signatures developed are useful where one is able to demonstrate a degree of squeezing. We also discuss how the signatures enable a new type of Einstein-Podolsky-Rosen gedanken experiment.Comment: 15 pages, accepted for publication in Phys. Rev.

On the Computation of Power in Volume Integral Equation Formulations

We present simple and stable formulas for computing power (including absorbed/radiated, scattered and extinction power) in current-based volume integral equation formulations. The proposed formulas are given in terms of vector-matrix-vector products of quantities found solely in the associated linear system. In addition to their efficiency, the derived expressions can guarantee the positivity of the computed power. We also discuss the application of Poynting's theorem for the case of sources immersed in dissipative materials. The formulas are validated against results obtained both with analytical and numerical methods for scattering and radiation benchmark cases

Letter from A. G. Reid

Letter concerning a position as football coach at Utah Agricultural College

Chromospheric Inversions of a Micro-flaring Region

We use spectropolarimetric observations of the Ca II 8542~\AA\ line, taken from the Swedish 1-m Solar Telescope (SST), in an attempt to recover dynamic activity in a micro-flaring region near a sunspot via inversions. These inversions show localized mean temperature enhancements of $\sim$1000~K in the chromosphere and upper photosphere, along with co-spatial bi-directional Doppler shifting of 5 - 10 km s$^{-1}$. This heating also extends along a nearby chromospheric fibril, co-spatial to 10 - 15 km s$^{-1}$ down-flows. Strong magnetic flux cancellation is also apparent in one of the footpoints, concentrated in the chromosphere. This event more closely resembles that of an Ellerman Bomb (EB), though placed slightly higher in the atmosphere than is typically observed.Comment: 9 pages, 9 figures, accepted in ApJ. Movies are stored here: https://star.pst.qub.ac.uk/webdav/public/areid/Microflare