Every group is the maximal subgroup of a naturally occurring free idempotent generated semigroup

Gray and Ruskuc have shown that any group G occurs as the maximal subgroup of some free idempotent generated semigroup IG(E) on a biordered set of idempotents E, thus resolving a long standing open question. Given the group G, they make a careful choice for E and use a certain amount of well developed machinery. Our aim here is to present a short and direct proof of the same result, moreover by using a naturally occuring biordered set. More specifically, for any free G-act F_n(G) of finite rank at least 3, we have that G is a maximal subgroup of IG(E) where E is the biordered set of idempotents of End F_n(G). Note that if G is finite then so is End F_n(G)

Free idempotent generated semigroups and endomorphism monoids of free GG-acts

The study of the free idempotent generated semigroup $\mathrm{IG}(E)$ over a biordered set $E$ began with the seminal work of Nambooripad in the 1970s and has seen a recent revival with a number of new approaches, both geometric and combinatorial. Here we study $\mathrm{IG}(E)$ in the case $E$ is the biordered set of a wreath product $G\wr \mathcal{T}_n$, where $G$ is a group and $\mathcal{T}_n$ is the full transformation monoid on $n$ elements. This wreath product is isomorphic to the endomorphism monoid of the free $G$-act $F_n(G)$ on $n$ generators, and this provides us with a convenient approach. We say that the rank of an element of $F_n(G)$ is the minimal number of (free) generators in its image. Let $\varepsilon=\varepsilon^2\in F_n(G).$ For rather straightforward reasons it is known that if $\mathrm{rank}\,\varepsilon =n-1$ (respectively, $n$), then the maximal subgroup of $\mathrm{IG}(E)$ containing $\varepsilon$ is free (respectively, trivial). We show that if $\mathrm{rank}\,\varepsilon =r$ where $1\leq r\leq n-2$, then the maximal subgroup of $\mathrm{IG}(E)$ containing $\varepsilon$ is isomorphic to that in $F_n(G)$ and hence to $G\wr \mathcal{S}_r$, where $\mathcal{S}_r$ is the symmetric group on $r$ elements. We have previously shown this result in the case $r=1$; however, for higher rank, a more sophisticated approach is needed. Our current proof subsumes the case $r=1$ and thus provides another approach to showing that any group occurs as the maximal subgroup of some $\mathrm{IG}(E)$. On the other hand, varying $r$ again and taking $G$ to be trivial, we obtain an alternative proof of the recent result of Gray and Ru\v{s}kuc for the biordered set of idempotents of $\mathcal{T}_n.$Comment: 35 page

Ruddlesden-Popper Phase in Two-Dimensional Inorganic Halide Perovskites: A Plausible Model and the Supporting Observations.

A Ruddlesden-Popper (RP) type structure is well-known in oxide perovskites and is related to many interesting properties such as superconductivity and ferroelectricity. However, the RP phase has not yet been discovered in inorganic halide perovskites. Here, we report the direct observation of unusual structure in two-dimensional CsPbBr3 nanosheets which could be interpreted as the RP phase based on model simulations. Structural details of the plausible RP domains and domain boundaries between the RP and conventional perovskite phases have been revealed on the atomic level using aberration-corrected scanning transmission electron microscopy. The finding marks a major advance toward future inorganic halide RP phase synthesis and theoretical modeling, as well as unraveling their structure-property relationship

Facile synthesis and enhanced visible light photocatalytic activity of N and Zr co-doped TiO2 nanostructures from nanotubular titanic acid precursors

Zr/N co-doped TiO2 nanostructures were successfully synthesized using nanotubular titanic acid (NTA) as precursors by a facile wet chemical route and subsequent calcination. These Zr/N-doped TiO2 nanostructures made by NTA precursors show significantly enhanced visible light absorption and much higher photocatalytic performance than the Zr/N-doped P25 TiO2 nanoparticles. Impacts of Zr/N co-doping on the morphologies, optical properties, and photocatalytic activities of the NTA precursor-based TiO2 were thoroughly investigated. The origin of the enhanced visible light photocatalytic activity is discussed in detail.Comment: 8 pages, 7 figure

Enhancement of Visible-Light-Induced Photocurrent and Photocatalytic Activity of V and N Codoped TiO2 Nanotube Array Films

Highly ordered TiO2 nanotube arrays (TNAs) codoped with V and N were synthesized by electrochemical anodization in association with hydrothermal treatment. The samples were characterized by field emission scanning electron microscopy, X-ray diffraction and X-ray photoelectron spectroscopy. The photocurrent and photocatalytic activity of codoped TiO2 nanotube arrays were investigated under visible light irradiation. Moreover, the production of hydroxyl radicals on the surface of visible light-irradiated samples is detected by a photoluminescence technique using terephthalic acid (TA) as a probe molecule. It was found that the V+N co-doped TiO2 nanotube arrays showed remarkably enhanced photocurrent and photocatalytic activity than undoped sample due to the V and N codoping.Comment: 15 Pages, 6 figure

Solvability for second order nonlinear impulsive boundary value problems

In this paper, we are concerned with the solvability for a class of second order nonlinear impulsive boundary value problem. New criteria are established based on Schaefer's fixed-point theorem. An example is presented to illustrate our main result. Our results essentially extend and complement some previous known results

Existence of solutions for fourth order differential equation with four-point boundary conditions

AbstractIn this paper we investigate the existence of solutions of a class of four-point boundary value problems for a fourth order ordinary differential equation. Our analysis relies on a nonlinear alternative of Leray–Schauder type