An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

Bi-atomic classes of positive semirings

Semigroup Forum - John Meakin has had a distinguished career of over half a century in the theory of semigroups. This article gives a synopsis of his most important contributions. The author, a...     

Analysis of Pathogenic Bacterial and Yeast Biofilms Using the Combination of Synchrotron ATR-FTIR Microspectroscopy and Chemometric Approaches

Biofilms are assemblages of microbial cells, extracellular polymeric substances (EPS), and other components extracted from the environment in which they develop. Within biofilms, the spatial distribution of these components can vary. Here we present a fundamental characterization study to show differences between biofilms formed by Gram-positive methicillin-resistant Staphylococcus aureus (MRSA), Gram-negative Pseudomonas aeruginosa, and the yeast-type Candida albicans using synchrotron macro attenuated total reflectance-Fourier transform infrared (ATR-FTIR) microspectroscopy. We were able to characterise the pathogenic biofilms’ heterogeneous distribution, which is challenging to do using traditional techniques. Multivariate analyses revealed that the polysaccharides area (1200–950 cm⁻¹) accounted for the most significant variance between biofilm samples, and other spectral regions corresponding to amides, lipids, and polysaccharides all contributed to sample variation. In general, this study will advance our understanding of microbial biofilms and serve as a model for future research on how to use synchrotron source ATR-FTIR microspectroscopy to analyse their variations and spatial arrangements.     

Restricted Elasticity and Rings of Integer-Valued Polynomials Determined by Finite Subsets

Let D be an integral domain such that Int(D) ≠ K[X] where K is the quotient field of D. There is no known example of such a D so that Int(D) has finite elasticity. If E is a finite nonempty subset of D, then it is known that Int(E, D) = {f(X) ∈ K[X] | f(e) ∈ D for all e ∈ E} is not atomic. In this note, we restrict the notion of elasticity so that it is applicable to nonatomic domains. For each real number r ≥ 1, we produce a ring of integer-valued polynomials with restricted elasticity r. We further show that if D is a unique factorization domain and E is finite with |E| > 1, then the restricted elasticity of Int(E, D) is infinite.     

Die Bestimmung der Weins?ure

Ohne Zusammenfassung     

Problem-Solving Methodology Design on the Run

This paper discusses a risk analysis study which considered the best way to cross a river with a large diameter gas pipeline, one possibility being a bridge carrying a large diameter oil line. About 30 ‘riskateers’ were involved, over a 3-month period, most of whom were engineers or economists with no risk analysis, O.R. or systems experience. A large number of interested parties were involved, with different perspectives. The paper stresses methodology design as an organizational vehicle, and the implications of designing problem-solving methodology in the midst of a problem-solving process.