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Brzdx119,k, Janusz,Jung, Soon-Mo Hindawi Publishing Corporation 2010 Journal of inequalities and applications Vol.2010 No.1
<P>We prove the Hyers-Ulam stability of a second-order linear functional equation in single variable (with constant coefficients) that is connected with the Fibonacci numbers and Lucas sequences. In this way we complement, extend, and/or improve some recently published results on stability of that equation.</P>
Barszcz, Bolesław,Kę,dzierski, Kamil,Oh, Won-Taek,Kim, Tae-Dong Elsevier 2018 Journal of luminescence Vol.203 No.-
<P><B>Abstract</B></P> <P>Two diketopyrrolopyrrole derivatives (DPPs) with spatial substituents (di-<I>tert</I>-butylbenzyl groups) and alkyl chains have been synthesized and investigated using DFT and TD-DFT calculations, vibrational (IR and Raman), electronic (absorption, fluorescence) spectroscopies, and electrochemical methods. Moreover, the creation of Langmuir layers was also investigated. The most of spectral investigations show only small differences between the investigated species as well as the electrochemical data. The electronic spectra show a small shift of the Soret and Q bands while the vibrational spectra in comparison with DFT calculation results suggest the presence of additional interactions involving the thiophene rings in one of the samples. The significant difference is observed in the formation of Langmuir layers of the mentioned molecules. The DPPs with spatial substituents form more densely packed layer in comparison with alkyl chain-substituted DPPs. Moreover, substitution with di-<I>tert</I>-butylbenzyl groups enhances the fluorescence quantum yields (up to 0.77) and slightly prolonging the fluorescence lifetimes (to over 5 ns). Spectroscopic studies of varied polarity solutions indicates J type aggregation of DPPs.</P>
Spectroscopic properties of diketopyrrolopyrrole derivatives with long alkyl chains
Barszcz, Bolesław,Kę,dzierski, Kamil,Jeong, Hee Yeon,Kim, Tae-Dong Elsevier 2017 Journal of luminescence Vol.185 No.-
<P><B>Abstract</B></P> <P>The diketopyrrolopyrrole derivatives with long alkyl chains were synthesised and spectrally characterised. Experimental spectroscopic techniques were supplemented by the DFT calculations of equilibrium geometries, normal mode frequencies and time-dependent DFT simulated UV–vis spectra. Comparison between the symmetrical and unsymmetrical substitution of alkyl chains resulted in some changes like bathochromic shift of absorption, excitation and emission spectra, significant reduction of molar absorption coefficient in UV–vis range, increase of fluorescence quantum yield, and slight increase of fluorescence lifetimes. The influence of the length of the alkyl chain on the properties of material was also discussed. The experimental spectroscopic data together with the calculated results show that presence of the alkyl substituent at the nitrogen atom influences more on the properties of the investigated DPPs than the length of attached chain.</P>
Hyers-Ulam Stability of the Delay Equation y'(t)=λy(t-τ)
Jung, Soon-Mo,Brzdx119,k, Janusz Hindawi Publishing Corporation 2010 Abstract and applied analysis Vol.2010 No.-
<P>We investigate the approximate solutions y:[-τ,∞)→R of the delay differential equation y'(t)=λy(t-τ)(t∈[0,∞)) with an initial condition, where λ>0 and τ>0 are real constants. We show that they can be “approximated” by solutions of the equation that are constant on the interval [-τ,0] and, therefore, have quite simple forms. Our results correspond to the notion of stability introduced by Ulam and Hyers.</P>