On the practical global uniform asymptotic stability of stochastic differential equations

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations (SDEs) by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of SDEs based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.     

Synthesis,Characterization, DFT and Photocatalytic Studies of a New Pyrazine Cadmium(II) Tetrakis(4-methoxy-phenyl)-porphyrin Compound

This study describes the synthesis, theoretical investigations, and photocatalytic degradational properties of a new (pyrazine)(meso-tetrakis(4-tert-methoxyphenyl)-porphyrinato)-cadmium (II) ([Cd(TMPP)-Pyz]) complex (1). The new penta-coordinated Cd^II porphyrin complex (1) was characterized by various spectroscopic techniques, including FT-IR, NMR, UV-visible absorption, fluorescence emission, and singlet oxygen, while its molecular structure was studied using single crystal X-ray diffraction. The UV–Vis spectroscopic study highlighted the redshift of the absorption bands after the insertion of the Cd(II) metal ion into the TMPP ring. The co-coordination of the pyrazine axial ligand enhanced this effect. A fluorescence emission spectroscopic study showed a significant blueshift in the Q bands, accompanied by a decrease in the fluorescence emission intensity and quantum yields of Φ_f = 0.084, Φ_f = 0.06 and Φ_f = 0.03 for H₂-TMPP free-base porphyrin, [Cd(TMPP)] and [Cd(TMPP)(Pyz)] (1) respectively. Singlet oxygen revealed that the H₂-TMPP porphyrin produced the most efficient singlet oxygen quantum yield of (Φ_Δ = 0.73) compared to [CdTMPP] (Φ_Δ = 0.57) and [Cd(TMPP)(Pyz)] (1) (Φ_Δ = 0.13). In the crystal lattice, the [Cd(TMPP)Pyz] was stabilized through non-covalent intermolecular interactions (NCI), such as the hydrogen bonds C-H···N and C-H···Cg. Additionally, crystal explorer software was then utilized to measure the quantitative analysis of the intermolecular interactions in the unit cell of the crystal structure and established that the C-H···π interaction dominated. The Natural bond orbital (NBO) analysis revealed that each molecule is stabilized by hyperconjugation and charge delocalization. As a photocatalyst, the coordination complex 1 showed excellent photocatalytic activity toward the degradation of Levafix Blue CA reactive dye (i.e., dye photo-degradation of 80%).