All rings are commutative with identity and all modules are unitary. In this note we give some properties of a finite collection of submodules such that the sum of any two distinct members is multiplication, generalizing those which characterize arithmetical rings. Using these properties we are able to give a concise proof of Patrick Smith’s theorem stating conditions ensuring that the sum and ...

Vicinally deuterated sec-alkyl phenyl ethers, CH(3)(CH(2))(m)()CH(OPh)CHD(CH(2))(n)()CH(3), display significant differences in mass spectra between threo and erythro stereoisomers. MS/MS experiments, in which parent ions of a single mass are selected and their fragmentation patterns subsequently measured, show that alkene expulsion represents virtually the only decomposition pathway. Two types ...

Miniaturized ultra high field asymmetric waveform ion mobility spectrometry (ultra-FAIMS) combined with mass spectrometry (MS) has been applied to the analysis of standard and tryptic peptides, derived from α-1-acid glycoprotein, using electrospray and nanoelectrospray ion sources. Singly and multiply charged peptide ions were separated in the gas phase using ultra-FAIMS and detected by ion tra...

We calculate the eigenvalue ρ of the multiplication mapping R on the Cayley-Dickson algebra An. If the element in An is composed of a pair of alternative elements in An−1, half the eigenvectors of R in An are still eigenvectors in the subspace which is isomorphic to An−1. The invariant under the reciprocal transformation An × An ∋ (x, y) 7→ (−y, x) plays a fundamental role in simplifying the fu...

The discrete subgroup ∆(27) of SU(3) has the interesting multiplication rule 3×3 = 3̄ + 3̄ + 3̄, which is used to obtain near tribimaximal neutrino mixing. Using present neutrino oscillation data as input, this model predicts that the effective mass mee measured in neutrinoless double beta decay will be 0.14 eV. The non-Abelian discrete subgroup ∆(12) of SU(3) [1], more familiarly known as A4 [2, ...

Liquid matrix-assisted laser desorption/ionization (MALDI) allows the generation of predominantly multiply charged ions in atmospheric pressure (AP) MALDI ion sources for mass spectrometry (MS) analysis. The charge state distribution of the generated ions and the efficiency of the ion source in generating such ions crucially depend on the desolvation regime of the MALDI plume after desorption i...