An overview of heterogeneous photocatalysis for the degradation of organic compounds: A special emphasis on photocorrosion and reusability

Heterogeneous photocatalysis has been extensively investigated for the degradation of organic pollutants from wastewater. The remarkable advantages of the heterogeneous photocatalysis process depend upon its ability to produce reactive oxygen species under visible/UV/solar light irradiation. However, the long-term stability and reuse potential of these catalysts are of great concern these days, yet understudied. This review aims to systematically present a state of the art understanding of such catalysts' reuse potential. Various important surface characteristics of the photocatalysts for improving the photostability and activity of the catalyst are discussed. Besides, the synergistic effect of different surface modified materials, composite materials and their surface characteristics for their enhanced activity are also covered. Finally, a discussion on various regeneration processes used for such catalysts is also presented, identifying some vital research needs in this field.     

Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups

Young’s orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young’s orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software.     

Better characterization of surface organometallic catalysts through resolution enhancement in proton solid state NMR spectra

Delayed-acquisition methods, namely, echo and constant-time-acquisition approaches, allow a significant improvement in resolution in the proton solid state NMR spectra of surface organometallic catalysts such as [syn-(SiO)Mo(=NAr)(=CH(t)Bu)(CH2(t)Bu)] and [(SiO)Re(C(t)Bu)(=CH(t)Bu)(CH2(t)Bu)] (syn/anti ratio = 1:1). This enables the observation of all of the proton resonances, which is not possible with the simple proton single-pulse technique under magic-angle spinning. For example, the methylene protons of the neopentyl ligands, buried in the large peak associated with all of the methyls in the 1H MAS spectrum, can easily be identified by recording a delayed-acquisition spectrum (resolution enhancement of a factor of 3 is obtained). Moreover, combining constant-time acquisition with heteronuclear carbon-proton correlation spectroscopy also improves the resolution of the 2D HETCOR spectra.     

Ultrametric logarithm laws, II

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock–Margulis and obtain related results in metric diophantine approximation.     

Inductive Algebras for Finite Heisenberg Groups

A characterization of the maximal abelian sub-algebras of matrix algebras that are normalized by the canonical representation of a finite Heisenberg group is given. Examples are constructed using a classification result for finite Heisenberg groups.     

Inductive Algebras and Homogeneous Shifts

Inductive algebras for the irreducible unitary representations of the universal cover of the group of unimodular two by two matrices are classified. The classification of homogeneous shift operators is obtained as a direct consequence. This gives a new approach to the results of Bagchi and Misra.     

Locally compact abelian groups with symplectic self-duality

Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer).     

Similarity Classes of 3 × 3 Matrices Over a Local Principal Ideal Ring

In this article similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.