A Canonical Directly Infinite Ring

Let be the set of nonnegative integers and the ring of integers. Let be the ring of N × N matrices over generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of yields that the subrings generated by them coincide. This subring is the sum of the ideal consisting of all matrices in with only a finite number of nonzero entries and the subring of generated by the identity matrix. Regular elements are also described. We characterize all ideals of , show that all ideals are finitely generated and that not all ideals of are principal. Some general ring theoretic properties of are also established.     

Clusters of varieties of completely regular semigroups

The class \({\mathcal{CR}}\) of completely regular semigroups equipped with the unary operation of inversion forms a variety whose lattice of subvarieties is denoted by \({\mathcal{L(CR)}}\). The variety \({\mathcal B}\) of all bands induces two relations \({\mathbf{B}^{\land}}\) and \({\mathbf{B}^{\lor} }\) by meet and join with \({\mathcal B}\). Their classes are intervals with lower ends \({\mathcal V_{B^{\land}}}\) and \({\mathcal V_{B^{\lor}}}\), and upper ends \({\mathcal V^{B^{\land}}}\) and \({\mathcal V^{B^{\lor}}}\). These objects induce four operators on \({\mathcal{L(CR)}}\).The cluster at a variety \({\mathcal V}\) is the set of all varieties obtained from \({\mathcal V}\) by repeated application of these four operators. We identify the cluster at any variety in \({\mathcal{L(CR)}}\).     

Simultaneous representations in categories of algebras

Hedrlin and Pultr proved that every small category is isomorphic to a full subcategory of the category Alg (Δ) of all algebras of type Δ whenever the sum ∑Δ of their arities satisfies ∑Δ>2. This article deals with simultaneous representation in categories of algebras, a generalization of the related question: given a subcategory k′ of a small category k, when does there exist an extension Δ′ of a type Δ with ∑Δ?2 such that k′ is a full subcategory of Alg(Δ′) while the Alg (Δ)-redacts of algebras representing k′ determine a category isomorphic to k? We characterize simultaneous representability by algebras and their reducts completely, and show that it is closely related to Isbell's dominion. A consequence of the main result states that algebraically representable pairs (k′, k) of one-object categories k′ k are exactly those for which k′ coincides with its dominion in k, and provides an alternative characterization of the dominion. Simultaneous representability by partial algebras is not subject to any such restriction.     

Continuous glucose monitoring by means of mid-infrared transmission laser spectroscopy in vitro

The continuous surveillance of glucose concentration reduces short-term risks and long-term complications for people with diabetes mellitus, a disorder of glucose metabolism. As a first step towards the continuous monitoring of glucose, reagent-free transmission spectroscopy in the mid-infrared region has been carried out in vitro using a quantum cascade laser and an optical silver halide fiber. A 30 μm gap in the fiber allowed for transmission spectroscopy of aqueous glucose solutions at a wavelength of 9.69 μm, which is specific to a molecular vibration of glucose. A noise-equivalent concentration as low as 4 mg/dL was achieved at an average power of 1.8 mW and an integration time of 50 s. This is among the most precise of glucose measurements using mid-infrared spectroscopy. Even with the very low average laser power of 0.07 mW the sensitivity of previous results (using a fiber optical evanescent field analysis) has been improved upon by almost one order of magnitude. Finally, the impact of potentially interfering substances such as other carbohydrates was analyzed.     

Chemo- and regioselective cyclohydrocarbonylation of alpha-keto alkynes catalyzed by a zwitterionic rhodium complex and triphenyl phosphite

alpha-Keto alkynes react with CO and H(2) in the presence of catalytic quantities of the zwitterionic rhodium complex (eta(6)-C(6)H(5)BPh(3))(-)Rh(+)(1,5-COD) and triphenyl phosphite affording either the 2-, 2(3H)-, or 2(5H)-furanones in 61-93% yields. The cyclohydrocarbonylation is readily accomplished using substrates containing alkyl, aryl, vinyl, and alkoxy groups at the acetylenic terminal, as well as a variety of primary, secondary, and tertiary alkyl, aryl, and heteroaryl groups connected to the ketone functionality. Structural and electronic properties present in the starting materials mediate the chemo- and regioselectivity of the reaction.