Weak π-rings

In this paper we establish some characterizations for weak π-rings.     

On the inheritance of the strongly π-extending property

In this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.     

Armendariz group rings

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.     

Wedge Operations and Doubling Operations of Real Toric Manifolds

This paper deals with two things.First,the cohomology of canonical extensions of real topological toric manifolds is computed when coefficient ring G is a commutative ring in which 2 is unit in G.Second,the author focuses on a specific canonical extensions called doublings and presents their various properties.They include existence of infinitely many real topological toric manifolds admitting complex structures,and a way to construct infinitely many real toric manifolds which have an odd torsion in their cohomology groups.Moreover,some questions about real topological toric manifolds related to Halperin's toral rank conjecture are presented.     

Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with $p^4$ elements, (p a prime) are given. Second, the number and structure of γ-constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined.     

On realization of fusion rings from generalized Cartan matrices

The Casimir element of a fusion ring (R, B) gives rise to the so called Casimir matrix C of (R, B). This enables us to construct a generalized Cartan matrix D-C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D-C of finite (resp. affine) type. It turns out that there exists a fusion ring with D-C being of finite (resp. affine) type if and only if D-C has only the form A₂ (resp. A₁⁽¹⁾. We also realize all fusion rings with D-C being a particular generalized Cartan matrix of indefinite type.     

Solid-Phase Synthesis of an Insect Pyrokinin Analog Incorporating an Imidazoline Ring as Isosteric Replacement of a trans Peptide Bond

A facile solid-phase synthetic method for incorporating the imidazoline ring motif, a surrogate for a trans peptide bond, into bioactive peptides is reported. The example described is the synthesis of an imidazoline peptidomimetic analog of an insect pyrokinin neuropeptide via a cyclization reaction of an iminium salt generated from the preceding amino acid and 2,4-diaminopropanoic acid (Dap).     

Interplay of Hückel and Möbius Aromaticity in Metal-Metal Quintuple Bonded Complexes of Cr,Mo, and W with Amidinate Ligand: Ab initio DFT and Multireference Analysis**

The aromaticity of metal-metal quintuple bonded complexes of the type M₂L₂ (M=Cr, Mo, and W; L=amidinate) are studied employing gauge including magnetically induced ring current (GIMIC) analysis and electron density of delocalized bonds (EDDB). It is found that the complexes possess two types of aromaticity: i) Hückel aromaticity through delocalization of ligand π electrons with metal-metal δ-bond-forming 6 conjugated electrons (4π and 2δ) ring; ii) Craig-Möbius aromaticity through delocalization of π electrons of both the ligands with metal d-orbitals in Craig type orientation forming 10π electrons ring with a double twist. Extended transition state natural orbital chemical valence (ETS-NOCV) and canonical molecular orbital natural chemical shielding (CMO-NCS) analysis confirm the Craig-Möbius type arrangement of the orbitals. Furthermore, the unprecedented Hückel and Möbius type aromaticity is confirmed from the plot of the current pathways using 3D line integral convolution (3D-LIC) plots. The metal-metal bond order also increases down the group as justified from the complete active space self-consistent field (CASSCF) analysis. Due to an increase in the π and δ electron conjugation, both the Hückel and Möbius aromaticity increase down the group.     

Selective Construction of Trefoil knots and a Molecular Borromean Ring Induced by Steric Hindrance of Thioether Ligands

Two Cp*−Rh^III based trefoil knots were obtained in high yield under ambient conditions via the coordination-driven self-assembly of semi-rigid thioether dipyridyl ligand 1,4-bis[(pyridin-4-ylthio)methyl]benzene ( L₁ ), ligand chloranilic acid (H₂− CA ) and 6,11-dihydroxytetracene-5,12-dione (H₂- TtDo ) with Cp*Rh^III metal corner units, respectively. Furthermore, using the bulkier 4,4′-{[(2,5-dimethyl-1,4-phenylene)bis(methylene)]bis(sulfanediyl)}dipyridine ( L₂ ) in the place of ligand L₁ in the construction process resulted in the formation of a teranuclear metallacycle and a template-free Borromean ring in high yields thanks to significantly altered intermolecular forces between the constituent ligands induced by the sterically-hindering methyl groups of L₂ , as demonstrated via a detailed X-ray crystallographic analysis and NMR spectroscopy.     

Synthesis,chemical properties,and CO2 electroreduction of π-conjugated polymers bearing imidazolium and imidazolinium rings in the polymer backbone

π-Conjugated polymers (CPs) bearing imidazolium and imidazolinium rings in the polymer backbone were prepared from a previously reported monomer, 1,3-bis(2,3-dimethyl-4-bromophenyl)imidazolium chloride, and a newly synthesized monomer, 1,3-bis(2,3-dimethyl-4-bromophenyl)imidazolinium chloride, via Sonogashira and Suzuki–Miyaura coupling reactions. The corresponding model compounds were also synthesized via the same reactions and monomers. The absorption maxima and onset position of the UV–vis spectra of the polymers were observed at longer wavelengths than those of the model compounds, thus revealing the presence of extended π-conjugation along the polymer backbone chain. Both polymers and model compounds were photoluminescent in solution and exhibited solvatochromism. The cast films of the polymers on the Pt and Ag electrodes were demonstrated to be effective catalysts for CO₂ electroreduction.