The homogeneous spectrum of a graded commutative ring

Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.

     

Rapidly swept, continuous-wave cavity ringdown spectroscopy with optical heterodyne detection: single- and multi-wavelength sensing of gases

Spectroscopic sensing of gases can be performed with high sensitivity and photometric precision by cavity ringdown (CRD) absorption spectroscopy. Our cavity ringdown spectrometer incorporates continuous-wave (cw) tunable diode lasers, fibre-optic coupling and standard photonics and optical telecommunications components. It comprises a rapidly swept optical cavity in a single-ended optical heterodyne transmitter–receiver configuration, enabling optical absorption of gases to be recorded either as single-frequency scanned spectra or as simultaneous, multi-wavelength tailored spectra. By measuring weak near-infrared rovibrational spectra of carbon dioxide gas (CO₂), with high resolution in the vicinity of 1.53 μm, we have realised a noise-limited absorption sensitivity of 2.5×10^-9 cm^-1 Hz^-1/2. Analytical sensitivity limits (both actual and projected) and prospective gas-diagnostic applications are discussed. Our approach to cw-CRD spectroscopy offers high performance in a relatively simple, low-cost, compact instrument that is amenable to chemical analysis of trace gases in medical, agricultural, industrial and environmental situations. Received: 16 May 2002 / Revised version: 3 June 2002 / Published online: 21 August 2002 RID="*" ID="*"Corresponding author. Fax: +61-2/9850-8313, E-mail: brian.orr@mq.edu.au     

Self-assembled calix[6]pyrrole capsules: solid-state encapsulation of different guests in preorganized calix[6]pyrrole capsules

meso-hexamethyl-meso-hexaphenyl calix[6]pyrrole assembles into well-defined dimeric capsules in the crystalline state; the preorganized capsule serves as an efficient host for different organic guests as well as for solvent molecules.     

Inclusion relations involving k-numerical ranges

Let W_k(A) denote the k-numerical range of an n × n matrix A. It is known that W_i(A) ? W_j(A) for 1 ? j? i? n. It this paper we derive more general inclusion relations of the form Σⁿ_iλ_iW_i(A) ? Σⁿ_iμ_iW_i(A), where λ_i, μ_i are real coefficients.     

An improved algorithm for solving communicating average reward Markov decision processes

This paper provides a policy iteration algorithm for solving communicating Markov decision processes (MDPs) with average reward criterion. The algorithm is based on the result that for communicating MDPs there is an optimal policy which is unichain. The improvement step is modified to select only unichain policies; consequently the nested optimality equations of Howard's multichain policy iteration algorithm are avoided. Properties and advantages of the algorithm are discussed and it is incorporated into a decomposition algorithm for solving multichain MDPs. Since it is easier to show that a problem is communicating than unichain we recommend use of this algorithm instead of unichain policy iteration.This research has been partially supported by NSERC Grant A-5527.