Algebraic methods for the solution of linear functional equations

The equation

$$\sum^{n}_ {i=0} a_{i}f(b_{i}x + (1 - b_{i})y) = 0$$

belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started a few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the earliest results is due to Z. Daróczy [1]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.

     

On the structure of water—alcohol and ammonia—alcohol protonated clusters

Collision-induced dissociation (CID) of protonated ammonia-alcohol and water-alcohol heteroclusters was studied using a triple quadrupole mass spectrometer with a corona discharge atmospheric pressure ionization source. CID results suggested that the ammonia-alcohol clusters had NH: at the core of the cluster and that hydrogen-bonded alcohol molecules solvated this central ion. In contrast, CID results in water-alcohol clusters showed that water loss was strongly favored over alcohol loss and that there was a preference for the charge to reside on an alcohol molecule. The results also indicated that a loose chain of hydrogen-bonded molecules was formed in the water-alcohol clusters and that there appeared to be no rigid protonation site or a fixed central ion. (J Am Soc Mass     

Determination of collision-induced dissociation mechanisms and cross-sections in organophosphorus compounds by atmospheric pressure ionization tandem mass spectrometry

Nineteen organophosphorus compounds (OPCs) of the generic classes (dialkyl alkyl′-phosphonates (twelve compounds) and tri-alkyl phosphates (seven compounds) were investigated by atmospheric pressure ionization (API) tandem mass spectrometry (MS/MS). All OPCs showed consistent and comparable collision-induced dissociation (CID) pathways allowing for a generalized scheme for dissociation. When the alkyl groups are equal to or larger than ethyl, CID mechanisms are dominated by McLafferty rearrangements and neutral losses are stable molecules in all cases except cleavage of P? C or O? C bonds to form alkyl fragment ions. CID cross-sections for the OPCs were determined with excellent reproducibility and a good correlation was found with a simple model. Inferences with respect to ionic structures were made from observed trends in the cross-section measurements. Limitations in API-MS/MS instruments of this design are found in the energy distribution of the ionization and ion sampling events.     

Recent advances in DWS-based micro-rheology

We have reviewed recent developments in diffusing wave spectroscopy (DWS) that have facilitated micro-rheological studies of soft solids and complex liquids. In particular, we have focussed on two novel ensemble-averaging methods, two-cell DWS and charged-coupled device DWS (CCD-DWS), that directly address problems of non-ergodicity and broken time–translation invariance in these materials. The applications of these methods to the study of slow evolutionary dynamics, mechanical aging and transient phenomena are also discussed.     

An inter-company cross-validation exercise on capillary electrophoresis testing of dose uniformity of paracetamol content in formulations

Summary Successful transfer of an MECC method for the determination of paracetamol between 7 companies is shown. The method was applied to the quantitative determination of paracetamol levels in capsules. The results obtained agreed well between companies and with both HPLC data and the label claim. Acceptable method performance in terms of relative migration time precision and response factor precision was reported by each company.     

Crystal-amorphous polymer interface studied by neutron and x-ray scattering on labeled binary ultralong alkanes

The recently discovered stable semicrystalline phase in binary ultralong alkanes is studied by small-angle neutron and x-ray scattering. Either the shorter or the longer alkane has (CD2)11CD3 ends. It is confirmed that chains are tilted in the crystal layer and found that ca. seven end carbons of the shorter alkane are amorphous. It is also established that the longer chains protrude from the crystal at both ends and that their end groups are preferentially located in the middle of the amorphous layer, suggesting gradual dissipation of orientational order of the chains exiting the crystal.     

Study of theS-matrix near bound states in the continuum

The behaviour ofS-matrix for potentials generating bound states in continuum in the neighbourhood of the positive bound state energies is studied. It is shown that unlike the case of usual negative energy bound states, theS-matrix does not have a pole at the positive bound state energy but becomes unity at the energy corresponding to bound states in continuum. Calculations ofS-waveS-matrix for a local potential constructed by Stillinger and Herrick and a separable nonlocal potential constructed by the present authors verify these results. Our results indicate that the bound states embedded in continuum constructedvia the von Neumann and Wigner procedure cannot be interpreted as resonances with zero width.     

Structure selective ion molecule interactions (SSIMI) in ion mobility spectrometry

The overall objective of this project was to develop an analytical method that utilizes structure selective ion molecule interactions (SSIMI) in ion mobility spectrometry (IMS) to shift the mobility of a targeted analyte through the addition of a gas phase modifier to the buffer gas. IMS is a sensitive, rapid method for the detection of harmful chemicals; however false alarm responses do occur and a reduction in their frequency decrease both the cost and time required for detection. The investigation reported here probed the effects of a series of buffer gas modifiers on the mobilities of chemical warfare agent simulants (CWAs), toxic industrial chemicals (TICs) and a known interference (butyl carbitol) found in fire extinguishing agents. The major finding of this research was that a modifier with a proton affinity similar to, but not greater than, the target analyte produced the greatest changes in mobilities due to the formation of an ion cluster between the neutral modifier and target analyte ion. Mass spectrometry was utilized to confirm the formation of ion-neutral clusters that caused the target ion to shift its mobility. While a number of modifiers were screened, acetonitrile and isobutyronitrile were found to have sufficiently selective SSIMI with the target compound. For example, in the presence of acetonitrile modifier, the protonated response ion of the CWA simulant DMMP, [DMMP]H⁺, had a mobility shift of 10.8 %, but the mobility was unchanged for the interferent, butyl carbitol. The mobility of the simulant DMMP decreased with the introduction of modifiers, while the mobility of the interference did not change, demonstrating the potential of the SSIMI technique for reducing false alarm rates.