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.     

Dielectric properties of KDP-type ferroelectric crystals in the presence of external electric field

Considering external electric field as well as third- and fourth-order phonon anharmonic interaction terms in the pseudospin-lattice coupled mode (PLCM) model Hamiltonian for KDP-type ferroelectrics, expressions for field-dependent shift, width, renormalized soft mode frequency, Curie temperature, dielectric constant and dielectric loss are evaluated. For the calculation, method of statistical double-time temperature-dependent Green’s function has been used. By fitting model values of physical quantities, temperature and electric field dependences of soft mode frequency, dielectric constant and loss have been calculated which compare well with experimental results of Baumgartner [8] and Choi and Lockwood [9]. Both dielectric constant and loss decrease with electric field.     

On nonlinear programming with support functions

Optimality conditions are derived for a nonliear program in which a support function appears in the objective as well as in each constraint function. Wolfe and Mond-Weir type duals to this program are presented and various dualityresults are established under suitable convexity and generalized convexity assumptions. Special cases that often occur in the literature are those in which a support function is the square root of a positive semidefinite quadratic form or anLp norm. It is pointed out that these special cases can easily be generated from our results.     

Pollution-error in the h-version of the finite-element method and the local quality of a-posteriori error estimators

In this paper we study the pollution-error in the h-version of the finite element method and its effect on the local quality of a-posteriori error estimators. We show that the pollution-effect in an interior subdomain depends on the relationship between the mesh inside and outside the subdomain and the smoothness of the exact solution. We also demonstrate that it is possible to guarantee the quality of local error estimators in any mesh-patch in the interior of a finite-element mesh by employing meshes which are sufficiently refined outside the patch.     

Why chaos is rarely observed in natural populations

An attempt has been made to understand why chaotic dynamics have received poor evidential support from field studies. Our study opens up the possibility that the cause of failure might not be poor quality of data, as pointed out by earlier authors, but an ecological reality. We have designed two model food chains to examine whether there is a biological basis for the crisis. This investigation is effected with the help of a new method which we introduce at an appropriate place in the text. The fact that chaos exists in a narrow range of parametric values in both the model systems suggests that the crisis indeed has a biological origin.     

Electrochemical oxidation of ethyl 3-oxo-3-phenyl-2-phenylhydrazonopropionate

The electrochemical oxidation of ethyl 3-oxo-3-phenyl-2-phenylhydrazonopropionate has been studied in the pH range 3.0–11.0 at a pyrolytic graphite electrode by linear and cyclic sweep voltammetry, coulometry and spectral studies. The results indicate that the 2 e⁻, 2 H⁺ oxidation of this compound gives phenol and ethyl 3-phenyl-2,3-dioxopropionate as the major products of electrooxidation.     

Thin layer chromatographic studies of anils II. Separation,identification and determination

Thin layer chromatographic behaviour of ten anils was studied on silica gel adsorbent mixed with starch binder with ten solvent systems. Simultaneous separation, identification and determination of ternary mixtures of isomers was stressed. TheR _F increasing order was found to be dependent on the solvent and independent of the degree of saturation of the chamber. The correlations ofR _F with _max and (C=O) of isomers were established and used in their identification. Six anils were separated in methanol—benzene (11).     

Isoperthiocyanic acid (3-amino-5-thione-l, 2, 4-dithiazole) as a new reagent for the selective detection of nitrite ion

Summary A new, simple, and selective test is described for the detection of nitrite ion with isoperthiocyanic acid (I) as the reagent. The test is based on the reaction of a 5% solution ofI with the nitrite ion solution, a deep orange precipitate being formed. Conversely, the test can be used in the identification ofI itself. Presumably, the coloured product is 3-hydroxy-5-thione-1, 2, 4-dithiazole, formed by the action of nitrous acid (producedin situ by the action of sulphuric acid and nitrite) on the 3-amino group ofI. The limit of detection is 300g of NO₂ ^– in a drop (0.05 ml). Most anions (including IO₃ ^–) do not interfere. Only I^–, thiourea, and hydrogen peroxide interfere; S₂O₃ ^2– interferes only when present in large amounts.

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Zusammenfassung Ein neuer, einfacher und selektiver Nachweis von Nitrit mit Isoperthio cyansäure (I) wurde beschrieben. Er beruht auf der Umsetzung einer 5% igen Lösung von I mit Nitritlösung, wobei ein tief orange gefärbter Niederschlag entsteht. Umgekehrt kann die Reaktion auch zum Nachweis von I dienen. Vermutlich handelt es sich bei dem gefarbten Produkt um 3-Hydroxy-5-thion-1, 2, 4-dithiazol, das durch Einwirkung der salpetrigen Säure auf die 3-Aminogruppe von I entsteht. Die Erfassungsgrenze betragt 300g NO₂ ^–im Tropfen (0,05 ml). Die meisten Anionen (inklusive JO₃ ^–) stören nicht. Nur J^–, Thioharnstoff und H₂O₂ stören; Thiosulfat stört nur in großen Mengen.

     

Kinetics and mechanism of the ruthenium(III)-catalyzed oxidation of hydroxy-acids byN-bromosuccinimide

Summary The kinetics of the Ru^III-catalyzed oxidation of the hydroxy acids; lactic, tartaric, malic and citric acids byN-bromosuccinimide in HClO₄ and in the presence of Hg(OAc)₂ have been studied. The reactions exhibit a first order rate dependence with respect to the oxidant and zeroth order rate dependence with respect to substrate. The rate is retarded by [H⁺], accelerated at law Ru^III concentrations but independent of [Ru^III] at higher Ru^III concentrations. A mechanism consistent with the observed kinetic data is proposed.     

Alkyl–oxygen bond formation in the mass spectra of α-chloromethyl aryl sulphones

When subjected to electron-impact, chloromethyl aryl sulphones (II, X = Cl) fragment predominantly by the loss of CH₂Cl from the molecular ion followed by the loss of SO₂ and in most cases the appropriate metastable peaks are present to confirm the transitions. In addition, alkyl–oxygen bond formation in the molecular ion was revealed by the presence of prominent peaks corresponding to the [R? ?-SO]⁺ ions. In most spectra no evidence for aryl–oxygen bond formation could be found and thus the presence of the chlorine atom was able to effectively reverse the direction of skeletal rearrangement reported for aryl methyl sulphones.