Packing chromatic number, $$\mathbf (1, 1, 2, 2) $$ ( 1 , 1 , 2 , 2 ) -colorings,and characterizing the Petersen graph

The packing chromatic number $$\chi _{\rho }(G)$$     

The mathematical technique in Fermat's deduction of the law of refraction

This article attempts to explain Fermat's not quite obvious calculations connected with his deduction of the law of refraction in Analysis ad refractiones (1662), and to describe the development which led to these calculations. In 1657 Fermat tried to deduce a law of refraction based on the principle that light follows the quickest path between two given points. He did not succeed because he found that the calculations were too long and tedious. The calculations are indeed complicated, but if Fermat, in 1657, had been willing to accept Descartes' law of refraction he would probably also have seen that it solved his problem. However, Fermat was of the opinion that Descartes' law was wrong and, therefore, he did not expect that solution. Only in 1662, when he succeeded in reducing the calculations substantially, did he realize that they led to the sine law of Descartes.     

Restrictions of relatively Tauberian operators

For operators T with the property which in addition satisfy a spectral condition we consider conditions under which the restriction of T to a closed invariant subspace also satisfies the same inclusion on the subspace.     

Acoustic Measures and Self-reports of Vocal Fatigue by Female Teachers

This study investigated the relation of symptoms of vocal fatigue to acoustic variables reflecting type of voice production and the effects of vocal loading. Seventy-nine female primary school teachers volunteered as subjects. Before and after a working day, (1) a 1-minute text reading sample was recorded at habitual loudness and loudly (as in large classroom), (2) a prolonged phonation on [a:] was recorded at habitual speaking pitch and loudness, and (3) a questionnaire about voice quality, ease, or difficulty of phonation and tiredness of throat was completed. The samples were analyzed for average fundamental frequency (F0), sound pressure level (SPL), and phonation type reflecting alpha ratio (SPL [1-5 kHz]-SPL [50 Hz-1 kHz]). The vowel samples were additionally analyzed for perturbation (jitter and shimmer). After a working day, F0, SPL, and alpha ratio were higher, jitter and shimmer values were lower, and more tiredness of throat was reported. The average levels of the acoustic parameters did not correlate with the symptoms. Increase in jitter and mean F0 in loud reading correlated with tiredness of throat. The results seem to suggest that, at least among experienced vocal professionals, voice production type had little relevance from the point of view of vocal fatigue reported. Differences in the acoustic parameters after a vocally loading working day mainly seem to reflect increased muscle activity as a consequence of vocal loading.     

Novel Lipophilic Fluorophores with Highly Acidity-Dependent Two-Photon Response

Lipophilic fluorophores are widely implemented in nonlinear microscopy; however, few existing membrane-specific probes combine the high brightness of two-photon excited fluorescence (2PEF) with pH sensitivity. Herein we describe four novel two-photon excited fluorophores, based on a coumarin 151 core structure, where lipophilicity is induced by a covalently attached phosphazene moiety. Changing the environmental acidity using trifluoromethanesulfonic (triflic) acid leads to profound changes in the linear fluorescence and 2PEF characteristics, due to chromophores’ switching between neutral- and protonated forms. We characterize this dependence by measuring the two-photon absorption (2PA) spectra over the region λ_2PA=550–1000 nm, observing 2PA cross sections of σ_2PA=10–20 GM, with an associated 2PEF brightness of 10–13 GM, in neutral solutions of both acetonitrile and n-octanol. Although quantum chemical modelling and NMR measurements show that, at high chromophore concentrations, protonation may be accompanied by a dimerization process, these dimers likely do not form at the lower concentrations used in optical spectroscopy.     

Electron ionization mass spectra and tautomerism of substituted 2‐phenacylquinolines

Tautomerism has been studied conventionally in solutions or in the solid state. However, the importance of mass spectrometry in the gas phase was realized relatively late. 2‐Phenacylquinolines are known to undergo ketimine‐enaminone tautomerism. The ratio of tautomers is dependent on the nature of the phenyl ring substituent and the Hammett substituent constants σ. Theoretical calculations indicate the presence of ketimine and enaminone tautomers in the gas phase. The electron ionization mass spectra of eight 2‐phenacylquinolines (ketimine form) were recorded at 70 eV in order to determine the fragmentation routes and to screen for the presence of their enaminone tautomers, (Z)‐2‐benzoylmethylene‐1,2‐dihydroquinolines, in the gas phase. The relative abundances or total ion currents of some ions correlated with the Hammett substituent constants and Hammett‐Brown constants. The product ions [M–CO]^{+ .} and [M–HCO]⁺ were observed. A reaction mechanism is suggested for the formation of these ions, requiring skeletal rearrangements. The results furnish information relating to tautomerism in the gas phase. Copyright © 2009 John Wiley & Sons, Ltd.