Extending modules which are direct sums of injective modules and semisimple modules

It is proved that, if R is a right Noetherian ringM ₁ is an injective right R-module and M ₂ is a semisimple right R-module, then the right R-module M ₁ + M ₂ is extending if and only if M ₂ is (M ₁/Soc(M ₁))-injective.     

Unified min-max and interlacing theorems for linear operators

There exist striking analogies in the behaviour of eigenvalues of Hermitian compact operators, singular values of compact operators and invariant factors of homomorphisms of modules over principal ideal domains, namely diagonalization theorems, interlacing inequalities and Courant–Fischer type formulae. Carlson and Sá [D. Carlson and E.M. Sá, Generalized minimax and interlacing inequalities, Linear Multilinear Algebra 15 (1984) pp. 77–103.] introduced an abstract structure, the s-space, where they proved unified versions of these theorems in the finite-dimensional case. We show that this unification can be done using modular lattices with Goldie dimension, which have a natural structure of s-space in the finite-dimensional case, and extend the unification to the countable-dimensional case.     

Quantification of ampicillin in bovine milk by coupled‐column ultrahigh‐performance liquid chromatography‐tandem mass spectrometry

This work reports the use of two‐dimensional (2D) liquid chromatography system coupled with a tandem mass spectrometry for the quantification of ampicillin in bovine milk. A restrict access media column (RAM‐BSA C₈, 50 × 2.1 mm, Luna, 10 μm, 100 Å) was used in the first dimension in order to exclude macromolecules, while an ACQUITY UPLC BEH C₁₈ (50 × 2.1 mm, 1.7 μm) column was used in the second dimension. Three different channels of selected reaction monitoring (SRM) were used: 350 > 106 m/z, 350 > 160 m/z, and 350 > 192 m/z. The first transition was used for the quantification (higher intensity), and latter two for confirmation. The developed method is simple and requires a total analysis time of only 14 min/sample. The sample treatment involved only a centrifugation step for 20 min. The validated method has been successfully applied to monitor AMP residues in raw milk samples. To our knowledge, this is the first study to report the use of ultrahigh‐performance liquid chromatography (UHPLC) in 2D configuration.     

Synthesis of 1,8–Naphthyridines and Their Application in the Development of Anionic Fluorogenic Chemosensors

Two 1,8-naphthyridines were synthesized and found to be fluorescent in solution. These compounds were studied in the presence of Cu(+) and Cu(2+) ions and it was verified that the metal causes the quenching of their fluorescence emission, due to the formation of complexes between the naphthyridine and the metal. A displacement assay was carried out in a DMSO-water mixture with the addition of various anions to the solutions of the complexes, and it was observed that these systems have a high capacity to selectively detect cyanide.     

Complex Dynamics in Pendulum-Type Equations with Variable Length

We prove the existence of complex dynamics for a generalized pendulum type equation with variable length. The solutions we find switch from an oscillatory behavior around the stable vertical position to a rotational type behavior crossing the unstable position with positive or negative velocity following any prescribed two-sided sequence of symbols. Moreover, to any periodic sequence of symbols corresponds a periodic solution of the equation. The proof is based on a topological approach and the results are robust with respect to small perturbations. In particular a small friction term can be added to the equation.     

CD(4) has bounded width

We prove that the constraint languages invariant under a short sequence of Jónsson terms (containing at most three non-trivial ternary terms) are tractable by showing that they have bounded width. This improves a previous result by Kiss and Valeriote and presents some evidence that the Larose–Zádori conjecture holds in the congruence-distributive case. The first author was supported by EPSRC grant EP/C54384X/1. The second author was supported by the MEC under grant TIN 2006-15387-C03-03, the EU PASCAL Network of Excellence IST-2002-506778, and the MODNET Marie Curie Research Training Network MRTN-CT-2004-512234. The third author was supported by grant no. 144011G of the Ministry of Science and Environment of Serbia. The fourth author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant nos. T 48809 and K 60148.     

Can one have significant deviations from leptonic 3 $$\times $$ 3 unitarity in the framework of type I seesaw mechanism?

We address the question of deviations from \(3\times 3\) unitarity of the leptonic mixing matrix showing that, in the framework of type I seesaw mechanism, one may have significant deviations from unitarity that can be detected at the next round of experiments while some of the heavy neutrino masses are sufficiently low to become within experimental reach. For that purpose we introduce a specially useful parametrisation that enables to control all deviations of unitarity through a single \(3 \times 3\) matrix, which we denote by X and which connects the mixing of the light and heavy neutrinos in the context of type I seesaw. We show that there is no need for the Yukawa couplings to be extremely suppressed. We present specific examples where deviations from \(3\times 3\) unitarity are sufficiently small to conform to all the present stringent experimental bounds.