Finite frames,P-frames and basically disconnected frames

Let \({\mathcal{P}}\) be an ideal of closed quotients of a completely regular frame L and \({\mathcal{R}_{\mathcal{P}}(L)}\) the collection of all functions in the ring \({\mathcal{R}(L)}\) whose support belong to \({\mathcal{P}}\) . We show that \({\mathcal{R}(L)}\) is a Noetherian ring if and only if \({\mathcal{R}(L)}\) is an Artinian ring if and only if L is a finite frame. Using this result, we next show that if \({\mathcal{P}}\) is the ideal of all compact closed quotients of L and L is \({\mathcal{P}}\) -continuous, then \({\mathcal{R}_{\mathcal{P}}(L)}\) is a Noetherian ring if and only if L is finite. Moreover, we show that L is a P-frame if and only if each ideal of \({\mathcal{R}(L)}\) is of the form \({\mathcal{R}_{\mathcal{P}}(L)}\) for some choice of \({\mathcal{P}}\) . We furnish equivalent conditions for \({\mathcal{R}_{\mathcal{P}}(L)}\) to be a prime ideal, a free ideal, and an essential ideal of \({\mathcal{R}(L)}\) separately in terms of the cozero elements of L. Finally, we show that L is basically disconnected if and only if \({\mathcal{R}(L)}\) is a coherent ring.     

Two valued measure and summability of double sequences

In this paper, following the methods of Connor [2], we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely [12]) to μ-statistical convergence and convergence in μ-density using a two valued measure μ. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure μ called the (APO₂) condition, inspired by the (APO) condition of Connor [3]. We mainly investigate the interrelationships between the two types of convergence, divergence and Cauchy criteria and ultimately show that they become equivalent if and only if the measure μ has the condition (APO₂).     

Similarity of L-fuzzy Relations Based on L-topologies Induced by L-fuzzy Rough Approximation Operators

This paper proposes similarity of L-fuzzy relations based on L-topologies induced by L-fuzzy rough approximation operators. First, the notion L-fuzzy rough set is generalized and the relationship between generalized L-fuzzy rough sets and L-topologies on an arbitrary universe is investigated. It shows that Alexandrov L-topologies can be induced by L-fuzzy relations without any preconditions. Second, the concept of similarity of L-fuzzy relations is introduced and variations of an L-fuzzy relation are investigated. Third, algebraic structures on similarity of L-fuzzy relations are obtained. Finally, we prove that the subset of the transitive L-fuzzy relations similar to a fixed L-fuzzy relation is a complete distributive lattice.     

Derivatization of Coumarins at the Benzenoid Ring in Aqueous Medium

Some 6‐disubstituted, 8‐disubstituted, and/6,8‐disubstituted compounds have been prepared from coumarin, 7‐methylcoumarin, and 3,4‐benzocoumarin. The Reimer–Tiemann reaction, Lederer–Manasse reaction, bromination using molecular bromine as well as 2,4,4,6‐tetrabromocyclohex‐2,5‐dien‐1‐one, Elbs reaction, and diazocoupling have been carried under controlled conditions to obtain various derivatives. Further, several reactions of aldehyde derivatives of these coumarins have been carried on to prepare important functional compounds including some heterocycles. It is noteworthy that these aldehydes behave as phenolic aldehydes under alkaline conditions to undergo the Dakin reaction. The reactions are mostly carried in aqueous media involving a dianionic intermediate and hence fulfill one important criterion of green chemistry.     

Photoactive Ru(II) -polypyridyl complexes that display sequence selectivity and high-affinity binding to duplex DNA through groove binding

The duplex-DNA binding properties of a nonintercalating polypyridyl ruthenium(II) complex that incorporates a linear extended ligand with a catechol moiety has been probed with a variety of photo- and biophysical techniques. These studies reveal that the complex groove binds to DNA sequences biphasically, and displays binding constants equivalent to those of high-affinity metallointercalators. The complex also displays preferential binding to AT-rich sequences. Changes in the structure of the coordinated catechol ligand and the incorporation of intercalating ancillary ligands into the complex were found to modulate both the optical-binding response and binding parameters of the system, which indicates that the catechol moiety plays a crucial role in the observed enhancement to binding affinities.     

Gold-catalyzed 1,2-difunctionalizations of aminoalkynes using only N- and O-containing oxidants

We report two viable routes for the 1,2-difunctionalization of aminoalkynes using only oxidants. In the presence of a gold catalyst, nitrones enable the oxoamination of aminoalkynes 1 to form 2-aminoamides 2. With a suitable gold catalyst, nitrosobenzenes implement an alkyne/nitroso metathesis of the same substrates to give 2-oxoiminylamides 3. These two novel oxidations also provide 1,2-aminoalcohols with opposite regioselectivity via NaBH(4) reduction in situ.     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

Organogold clusters protected by phenylacetylene

A new class of monolayer-protected Au clusters with Au-C covalent bonds (organogold clusters) was synthesized by ligating phenylacetylene (PhC≡CH) to PVP-stabilized Au clusters. Matrix-assisted laser desorption ionization mass spectrometry revealed for the first time a series of stable compositions of the organogold (Au:C(2)Ph) clusters.     

Solvent-dependent aggregation behavior of a new Ru(II)-polypyridyl based metallosurfactant

Variation of the solvent polarity leads to the formation of vesicles and reverse vesicles of a newly synthesized amphiphilic Ru(II)-polypyridyl complex.     

The continuous-flow cycloaddition of azomethine ylides to carbon nanotubes

This communication demonstrates a straightforward continuous-flow method for efficient exohedral functionalisation of carbon nanotubes which affords soluble samples in a much shorter time over conventional batch processing.