Factorization in dedekind domains with finite class group

LetD be a Dedekind domain. It is well known thatD is then an atomic integral domain (that is to say, a domain in which each nonzero nonunit has a factorization as a product of irreducible elements). We study factorization properties of elements in Dedekind domains with finite class group. IfD has the property that any factorization of an elementα into irreducibles has the same length, thenD is called a half factorial domain (HFD, see [41]). IfD has the property that any factorization of an elementα into irreducibles has the same length modulor (for somer>1), thenD is called a congruence half factorial domain of orderr. In Section I we consider some general factorization properties of atomic integral domains as well as the interrelationship of the HFD and CHFD property in the Dedekind setting. In Section II we extend many of the results of [41], [42] and [36] concerning HFDs when the class group ofD is cyclic. Finally, in Section III we consider the CHFD property in detail and determine some basic properties of Dedekind CHFDs. IfG is any Abelian group andS any subset ofG−[0], then {G, S} is called a realizable pair if there exists a Dedekind domainD with class groupG such thatS is the set of nonprincipal classes ofG which contain prime ideals. We prove that for a finite abelian groupG there exists a realizable pair {G, S} such that any Dedekind domain associated to {G, S} is CHFD for somer>1 but not HFD if and only ifG is not isomorphic toZ ₂,Z ₂,Z ₂ ⊕Z ₂, orZ ₃ ⊕Z ₃. The first author received support under the John M. Bennett Fellowship at Trinity University and also gratefully acknowledges the support of The University of North Carolina at Chapel Hill.     

Structural and magnetic properties of FIB-irradiated microstructures on a Pt/Co multilayer

We present a study of the structural and magnetic effects induced by focused 30 keV Ga⁺ ion beam irradiation on a Pt(2.8 nm)/[Pt(0.6 nm)/Co(0.3 nm)]₆/Pt(6.5 nm) ferromagnetic multilayer, using transmission electron microscopy, optical and magneto-optical microscopy. The work is of relevance to high-density data-storage applications, where media planarity is crucial for device reliability.     

Synthesis and pH‐Dependent Spectroscopic Behavior of 2,4,6‐Trisubstituted Pyridine Derivatives

Seven 2,4,6‐trisubstituted pyridine derivatives with N,N‐diethylaniline substituents at the 4‐position were synthesized, and their spectroscopic properties in the absence and presence of acid were studied. The spectral effects of protonation, molar absorptivities, pK_a values, and the structural origins of the observed spectral behavior were ascertained. The pyridine nitrogen was found to be more basic than the diethylamino nitrogen atom. Protonation of the pyridine ring nitrogen is associated with the appearance of a red‐shifted intramolecular charge transfer peak in the UV‐visible spectra. Favorable color indicating properties result from electron‐donating substitution at the 2 and 6 positions of pyridine, which provide a greater absorptivity of the red‐shifted peak associated with protonation of the pyridine nitrogen. These findings will assist in the design and optimization of these compounds for ion‐indicating and pH‐sensing applications.     

Surface and Structural Investigation of a MnOx Birnessite‐Type Water Oxidation Catalyst Formed under Photocatalytic Conditions

Catalytically active MnO_x species have been reported to form in situ from various Mn‐complexes during electrocatalytic and solution‐based water oxidation when employing cerium(IV) ammonium ammonium nitrate (CAN) oxidant as a sacrificial reagent. The full structural characterization of these oxides may be complicated by the presence of support material and lack of a pure bulk phase. For the first time, we show that highly active MnO_x catalysts form without supports in situ under photocatalytic conditions. Our most active ⁴MnO_x catalyst (～0.84 mmol O₂ mol Mn^?1 s^?1) forms from a Mn₄O₄ bearing a metal–organic framework. ⁴MnO_x is characterized by pair distribution function analysis (PDF), Raman spectroscopy, and HR‐TEM as a disordered, layered Mn‐oxide with high surface area (216 m²g^?1) and small regions of crystallinity and layer flexibility. In contrast, the ^SMnO_x formed from Mn²⁺ salt gives an amorphous species of lower surface area (80 m²g^?1) and lower activity (～0.15 mmol O₂ mol Mn^?1 s^?1). We compare these catalysts to crystalline hexagonal birnessite, which activates under the same conditions. Full deconvolution of the XPS Mn2p_3/2 core levels detects enriched Mn³⁺ and Mn²⁺ content on the surfaces, which indicates possible disproportionation/comproportionation surface equilibria.     

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.