On the explicit solutions of forms of the Sylvester and the Yakubovich matrix equations

In this paper, we consider the explicit solutions of two matrix equations, namely, the Yakubovich matrix equation V−AVF=BW and Sylvester matrix equations AV−EVF=BW,AV+BW=EVF and AV−VF=BW. For this purpose, we make use of Kronecker map and Sylvester sum as well as the concept of coefficients of characteristic polynomial of the matrix A. Some lemmas and theorems are stated and proved where explicit and parametric solutions are obtained. The proposed methods are illustrated by numerical examples. The results obtained show that the methods are very neat and efficient.     

Substituent effects of 1,2-dithiole groups on the electrochemical oxidation of some ferrocenyl-1,2-dithiole compounds

New ferrocenyl compounds substituted by sulfur containing groups were synthesized leading to ferrocenyl-3H-1,2-dithiole-3-thiones and related compounds. The substituent of the ferrocene was a [3-thioxo-3H-1,2-dithiol]-4 or 5-yl, a [3-oxo-3H-1,2-dithiol]-4 or 5- yl or a [3-methylsulfanyl-3H-1,2-dithiolium]-4 or 5-yl cation group. Their anodic behavior was studied by cyclic voltammetry at a Pt electrode in aprotic solvent. All synthesized ferrocenes exhibited a one-electron reversible oxidation leading to the corresponding ferricinium cation. At low potential scan, the irreversible oxidation of 5-ferrocenyl-3H-1,2-dithiole-3-thione was observed and attributed to a dimerization involving the dithiolethione group. Redox potential of the reversible oxidation allowed the determination of the electronic effect of the 1,2-dithiole groups. The Hammett σ_p constants of the dithiole substituents were obtained from linear correlation between oxidation potentials and electronic effects. The results showed that the [3-thioxo-3H-1,2-dithiol]-5-yl and the [3-methylsulfanyl-3H-1,2-dithiolium]-5-yl cation groups were strong inductive electron withdrawing substituents characterized by σ_p values of 0.55 and 0.97, respectively.     

Commute times of random walks on trees

In this paper we provide exact formula for the commute times of random walks on spherically symmetric random trees. Using this formula we sharpen some of the results presented in Al-Awadhi et al. to the form of equalities rather than inequalities.     

Synthesis and Spectroscopic Structural Elucidation of New Quinoxaline Derivatives

A quinoxaline‐2,3‐dione derivative was synthesized, and its chemical structure was determined through spectral analysis. Alkylation of this compound under phase transfer catalysis (PTC) conditions yielded monoalkylated and diakylated adducts. The monolalkylation process was shown to be regioselective occurring on the quinoxalic nitrogen atom rather than on its pyrazolic analogue. The full characterization of the synthesized compounds was studied by concerted use of NMR and MS techniques. Assignments of proton and carbon atoms were achieved through analysis of the 1D ¹H and ¹³C NMR spectra combined with homo‐ and hetero-nuclear 2D NMR experiments. Determination of the alkylation site was achieved through long‐range proton–carbon coupling correlations spectroscopy.     

Existence of solutions of fractional p-Laplacian systems with different critical Sobolev-Hardy exponents

In this paper, we investigate a fractional p-Laplacian system with different critical Sobolev-Hardy exponents. By variational methods and the Mountain-Pass lemma, positive minimizers of the related best Sobolev constants and the existence of positive solutions of the system are found.     

Approximation of an Inverse Initial Problem for a Biparabolic Equation

In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous and nonlinear biparabolic equation. The problem is severely ill-posed in the sense of Hadamard. First, we apply a general filter method to regularize the linear nonhomogeneous problem. Then, we also give a regularized solution and consider the convergence between the regularized solution and the sought solution. Under the a priori assumption on the exact solution belonging to a Gevrey space, we consider a generalized nonlinear problem by using the Fourier truncation method to obtain rigorous convergence estimates in the norms on Hilbert space and Hilbert scale space.     

The speciation of manganese in freshwaters

The extent of current knowledge regarding the speciation of manganese in freshwaters is delineated, and the analytical methods whereby such knowledge is obtained are discussed. Particular attention is paid in the review to the use of electron paramagnetic resonance spectroscopy.     

Determination of the optical constants of selenium films

The optical constants of evaporated Se films are calculated following Valeev’s method and applied to Se in the visible region. The calculated values are in good agreement with published data.