Calibration laws based on multiple linear regression applied to matrix-assisted laser desorption/ionization Fourier transform ion cyclotron resonance mass spectrometry

Operation of any mass spectrometer requires implementation of mass calibration laws to translate experimentally measured physical quantities into a m/z range. While internal calibration in Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR-MS) offers several attractive features, including exposure of calibrant and analyte ions to identical experimental conditions (e.g. space charge), external calibration affords simpler pulse sequences and higher throughput. The automatic gain control method used in hybrid linear trap quadrupole (LTQ) FT-ICR-MS to consistently obtain the same ion population is not readily amenable to matrix-assisted laser desorption/ionization (MALDI) FT-ICR-MS, due to the heterogeneous nature and poor spot-to-spot reproducibility of MALDI. This can be compensated for by taking external calibration laws into account that consider magnetic and electric fields, as well as relative and total ion abundances. Herein, an evaluation of external mass calibration laws applied to MALDI-FT-ICR-MS is performed to achieve higher mass measurement accuracy (MMA).     

A new proof of Ramanujan’s modular equation relating R(q) with R(q 5)

We give a new proof of Ramanujan’s modular identity relating R(q) with R(q ⁵), where R(q) is the famous Rogers–Ramanujan continued fraction. Our formulation is stronger than those of preceding authors; in particular, we give for the first time identities for the expressions appearing in the numerator and the denominator of Ramanujan’s identity. A related identity for R(q) that has partition-theoretic connections is also proved.     

Two modular equations for squares of the Rogers–Ramanujan functions with applications

Two modular identities of Gordon, McIntosh, and Robins are shown to be connected to the Rogers–Ramanujan continued fraction R(q), and in particular to Ramanujan’s parameter k:=R(q)R ²(q ²). Using this connection, we give new modular relations for R(q), and offer new and uniform proofs of several results of Ramanujan. In particular, we give a new proof of a famous and fundamental modular identity satisfied by the Rogers–Ramanujan continued fraction. We furthermore show that many analogous results hold for Ramanujan’s parameters μ:=R(q)R(q ⁴) and ν:=R ²(q ^1/2)R(q)/R(q ²). New proofs are offered for modular relations connecting R(q) to R(−q), R(q ²), and R(q ⁴), and new relations connecting R(q) at these arguments are offered. Eleven identities for the Rogers–Ramanujan functions are proved, including four new identities.