Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts

The Ramanujan Journal - Recently, Andrews and Merca considered the number of even parts in all partitions of n into distinct parts and obtained new combinatorial interpretations for this number....     

Scatter profiling: A potentially useful technique for isomeric ion distinction and stability determination

A method to determine the extent of angular scattering of fragment-ion products of keV-collision-activated decomposition (CAD) reactions and, in particular, the collisional scatter incurred by the parent ions prior to their dissociation, is outlined. Since the half-widths of the collisional scatter profiles correlate approximately with the mean reaction endothermicities for some ‘test’ reactions, the method may, in principle, be used to estimate the stabilities of isomeric ion structures relative to a common fragmentation threshold level. For single-proton-loss CAD reactions of some [H₃, C, X]⁺ ions (X = F, Cl, OH) with either [H₃CX]⁺ or ylidion, [H₂CXH]⁺ structure, collisional scatter is found in each case to be greater for the isomeric ion with the more stable structure. The estimated magnitudes of the mean energy depositions occurring in the keV-collision-activation processes are generally much larger than the calculated minimum energy requirements, suggesting that survivable [M ? H]⁺ products can be formed with up to several eV of internal energy.     

Congruence and Conjunctivity of Matrices

Congruence of arbitrary square matrices over an arbitrary field is treated here by elementary classical methods, and likewise for conjunctivity of arbitrary square matrices over an arbitrary field with involution. Uniqueness results are emphasized, since they are largely neglected in the literature. In particular, it is shown that a matrix S is congruent [conjunctive] to S₀⊕S₁ with S₁ nonsingular, and that if S₁ here is of maximal size among all nonsingular matrices R₁ for which R₀⊕R₁ is congruent [conjunctive] to S, then the congruence [conjunctivity] class of S determines that of S₁. Partially canonical forms (most of them already known) are derived, to the extent that they do not depend on the field. Nearly canonical forms are derived for “neutral” matrices (those congruent or conjunctive with block matrices $\text{O}\text{N}\text{M}\text{O}$ with the two zero blocks being square). For a neutral matrix S over a field F,the F-congruence [F-conjunctivity] class of S is determined by the F-equivalence class of the pencil S+tS' [S+tS¹] and, if the pencil is nonsingular, by the F[t]-equivalence class of S+tS' [S+tS¹].     

Relative molecular mass information from aliphatic nitro compounds: A chemical ionization study

The mass spectra of a number of aliphatic nitro compounds have been studied using electron Ionization (EI) and a variety of chemical Ionization (CI) techniques in attempts to obtain relative molecular mass information. The use of positive ion ammonia chemical Ionization techniques gave very satisfactory results, providing abundant [M + NH4]⁺ ions, not only from both primary and secondary nitro compounds, but also from the much more labile tertiary nitro compounds. However, the use of methane and isobutane positive ion CI or EI conditions resulted in facile fragmentation with little relative molecular mass information being made available. Negative ion CI using methane, isobutane or ammonia as moderating gases all gave abundant [M ? 1]^? ions with primary and secondary nitro compounds but at much reduced sensitivity.