Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.     

Combinatorial manifolds with complementarity

A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex. We show that if a d-dimensional combinatorial manifold M with n vertices satisfies complementarity then d=0, 2, 4, 8, or 16 with n=3d/2+3 and |M| is a “manifold like a projective plane”. Arnoux and Marin had earlier proved the converse statement.     

Engel type identities with derivations for right ideals in prime rings

Let R be a prime ring of char R≠2, d a non-zero derivation of R and ρ a non-zero right ideal of R such that [[d(x),d(y)]_n [y,x]_m] = 0 for all x,y ∈ ρ or [[d(x),d(y)]_n d[y,x]_m] = 0 for all x,y ∈ ρ, n, m ≥ 0 are fixed integers. If [ρ,ρ]ρ ≠ 0, then d(ρ)ρ = 0.     

Skew-commuting and Commuting Mappings in Rings with Left Identity

Our main object in this paper is to discuss some results on rings with left identity in which certain subsets satisfy some functional identities.     

Electron transfer reactions of nickel(III) and nickel(IV) complexes

This review narrates the electron transfer reactions of various nickel(III) and nickel(IV) complexes reported during the period 1981 until today. The reactions have been categorized mainly with respect to the type of nickel complexes. The reactivity of nickel(III) complexes of macrocycles, 2,2′-bipyridyl and 1,10-phenanthroline, peptides and oxime–imine, and of nickel(IV) complexes derived from oxime–imine, oxime and miscellaneous ligands with various organic and inorganic electron donors have been included. Kinetic and mechanistic features associated with such interactions have been duly analyzed. The relevance of Marcus cross-relation equations in the delineation of the electron transfer paths has also been critically discussed.     

A Two‐Dimensional Coordination Compound as a Zinc Ion Selective Luminescent Probe for Biological Applications

A 2D coordination compound {[Cu₂(HL)(N₃)]?ClO₄}_∞ ( 1 ; H₃L=2,6‐bis(hydroxyethyliminoethyl)‐4‐methyl phenol) was synthesized and characterized by single‐crystal X‐ray diffraction to be a polymer in the crystalline state. Each [Cu₂(HL)(N₃)]⁺ species is connected to its adjacent unit by a bridging alkoxide oxygen atom of the ligand to form a helical propagation along the crystallographic a axis. The adjacent helical frameworks are connected by a ligand alcoholic oxygen atom along the crystallographic b axis to produce pleated 2D sheets. In solution, 1 dissociates into [Cu₂(HL)₂(H₃L)]?2H₂O ( 2 ); the monomer displays high selectivity for Zn²⁺ and can be used in HEPES buffer (pH 7.4) as a zinc ion selective luminescent probe for biological application. The system shows a nearly 19‐fold Zn²⁺‐selective chelation‐enhanced fluorescence response in the working buffer. Application of 2 to cultured living cells (B16F10 mouse melanoma and A375 human melanoma) and rat hippocampal slices was also studied by fluorescence microscopy.     

Intramolecular nitrone cycloaddition reaction on carbohydrate-based precursors: application in the synthesis of spironucleosides and spirobisnucleosides

A simple synthesis of chiral spironucleosides and spirobisnucleosides is described. Intramolecular 1,3-dipolar nitrone cycloaddition reaction of d-glucose-derived precursors having olefin at C-3 and nitrone at C-5, C-1, or C-2 (in nor-series) furnished bisisoxazolidinospirocycles 4-7, 11, and 12 in good yields. Reductive ring opening of the isoxazolidine moieties in 4-6 followed by construction of a nucleoside base upon the generated amino groups smoothly yielded spirobisnucleosides 17 and 18 and spironucleosides 20 and 21.     

Reduction of octacyanomolybdate(V) by thioglycolic acid in aqueous media

In aqueous media at 25 degrees C [Mo(CN)(8)](3-) is reduced by thioglycolic acid (HSCH(2)COOH, TGA), and the reaction is strongly accelerated by the presence of trace amounts of copper ions. Dipicolinic acid (dipic) is an effective inhibitor of the copper catalysis. Both with and without dipic the reaction has the stoichiometry 2[Mo(CN)(8)](3-) + 2TGA --> 2[Mo(CN)(8)](4-) + RSSR, where RSSR is the disulfide derived from formal oxidative dimerization of TGA. In the presence of dipic, PBN (N-tert-butyl-alpha-phenyl-nitrone), and with a large excess of TGA the rate law for consumption of [Mo(CN)(8)](3-) is first order in both [TGA] and [Mo(CN)(8)(3-)]. The complex pH dependence is consistent with (-)SCH(2)CO(2)(-) being highly reactive (k = 1.8 x 10(4) M(-1) s(-1)), the monoanion being less reactive, and HSCH(2)CO(2)H being unreactive. A mechanism is proposed in which the dianion undergoes electron transfer to [Mo(CN)(8)](3-), thus generating the thiyl radical. Analysis of the electron-transfer rate constant in terms of Marcus theory yields an effective self-exchange rate constant for the thiolate/thiyl redox couple that is in reasonable agreement with the value derived previously from the reaction of TGA with [IrCl(6)](2-). When copper catalysis is inhibited, the two reactions differ substantially in that the yield of (-)O(3)SCH(2)CO(2)(-) is significant for [IrCl(6)](2-) but undetectable for [Mo(CN)(8)](3-).     

Kinetic studies on the electron transfer between sulphur(IV) and an iron(III) oxime–imine complex

Electron transfer between [Fe^III(L²)]⁺ and sulphur(IV) has been proposed to proceed via an inner-sphere mechanism involving formation of a transient hydrogen-bonded intermediate between the acidic proton of SO₂ · xH₂O/HSO₃ ^– and the oximato oxygen of the coordinated ligand, providing the ready availability of the proton for the reduced complex. In the case of SO₃ ^2–, this is not possible and the reaction is believed to proceed via an outer-sphere scheme.     

Constrained photophysics of 3-acetyl-4-oxo-6,7-dihydro-12H indolo-[2,3-a] quinolizine in micellar environments: a spectrofluorometric study

Photophysical properties of 3-acetyl-4-oxo-6,7-dihydro-12H indolo-[2,3-a] quinolizine (AODIQ) have been studied in different aqueous micellar environments using steady-state and time-resolved emission spectroscopy. The charge transfer (CT) fluorescence exhibits appreciable hypsochromic shift, along with an enhancement in the fluorescence intensity in all the micellar media. This is associated with an increase in the fluorescence anisotropy (r), which suggests that the fluorophore molecule experiences motionally restricted environments upon binding with the micelles. Fluorescence spectral position and fluorescence quenching studies suggest that the fluorescing moiety does not penetrate into the core of the micellar units; rather it binds at the micelle-water interfacial region. The binding constant and free energy change during probe-micelle binding have been evaluated from relevant fluorescence data. Light has been thrown on the mode of action of urea on micelle bound probes. The results are interpreted in terms of the model that urea displaces water molecules from the micellar interface and the consequent destabilization leads to the expulsion of the probe molecules from the interfacial region. Polarity and viscosity of the microenvironments around the probe have been determined in the micellar systems.