Functional equations and the Cauchy mean value theorem

The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy mean value theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.     

Sharp spectral bounds for complex perturbations of the indefinite Laplacian

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real potentials. For $L^1$-potentials, we obtain optimal spectral enclosures which accommodate also embedded eigenvalues, while our result for $L^p$-potentials yield sharp spectral bounds on the imaginary parts of eigenvalues of the perturbed operator for all $p\in [1,\infty )$. The sharpness of the results are demonstrated by means of explicit examples.     

The removal of methylene blue as a remedy of dye-based marine pollution: a photocatalytic perspective

Research on Chemical Intermediates - The effluents containing the discarded water from the textile industry are graded as one of the foremost pollutants in all industrial sectors. The wide...