The Neumann Problem for the Kohn-Laplacian on the Heisenberg Group ℍn$\mathbb H_{n}$

The existence and uniqueness of the solution of the Neumann problem for the Kohn-Laplacian relative to the Korányi ball on the Heisenberg group \(\mathbb {H}_{n}\) are discussed. Explicit representation for a Green’s type function (Neumann function) for the Korányi ball in \(\mathbb {H}_{n}\) for circular functions has been obtained. This function is then used on the above region in \(\mathbb {H}_{n}\) to solve the inhomogeneous Neumann boundary value problem for certain circular data.     

Application of Doehlert design in optimizing the determination of degraded products of nerve agents by ion-pair liquid chromatography electrospray ionization tandem mass spectrometry

A qualitative method was developed for the determination of degraded products of nerve agents by using ion-pair liquid chromatography electrospray ionization tandem mass spectrometry (IP-LC-ESI-MS(n)). Generally, alkylphosphonic acids (APAs) and O-alkyl alkylphosphonic acids (AAPAs) give deprotonated molecular ion [M-H](-) in negative mode. Interestingly, first time we obtained the molecular radical anion [M](.-) of phosphonic acids in negative mode by using tri-n-butyl amine as an ion-pairing agent. We interpreted this observation as an indication of electrochemical reduction of phosphonic acids in electrospray needle. Three variables such as sheath gas flow, electrospray needle voltage and pH of the mobile phase were investigated to enhance the molecular radical anion [M](.-) signal of each analyte. The Doehlert design was used to obtain the region in which the optimum value of such variables is simultaneously achieved. Limit of detection achieved was 0.5 microg mL(-1) for AAPAs and 10 microg mL(-1) for APAs. Excellent precision was observed with less than 8.61% RSD. Finally, the method was applied for the detection of ethyl methylphosphonic in aqueous extract of soil sample.     

In situ derivatization hollow fiber mediated liquid phase microextraction of alkylphosphonic acids from water

Alkylphosphonic acids (APAs), particularly the methyl-, ethyl-, isopropyl- and n-propyl-phosphonic acids are important markers of extremely toxic nerve agents. Hence, their detection and identification is of vital importance to verification of chemical weapons convention (CWC). Verification analysis of CWC requires development of fast, reliable, simple and reproducible sample preparation methods of water and soil samples. Present investigation is focused on the optimization of alkylation of APAs in water with subsequent extraction of alkylated acids by hollow fiber liquid phase microextraction (HF-LPME). This simple and sensitive sample preparation of APAs from water offered better recoveries in comparison to conventionally used extraction technique. Under optimized conditions, the APAs were detected at the concentration of 0.5-0.75 microg/mL with S/N ratio > or = 5, whereas the LODs for alkyl APAs (monobasic APAs) were achieved up to 0.1 microg/mL. The developed method was finally tested with water samples supplied in 19th official proficiency test conducted by the OPCW.     

On-line high-performance liquid chromatography–ultraviolet–nuclear magnetic resonance method of the markers of nerve agents for verification of the Chemical Weapons Convention

This paper details an on-flow liquid chromatography–ultraviolet–nuclear magnetic resonance (LC–UV–NMR) method for the retrospective detection and identification of alkyl alkylphosphonic acids (AAPAs) and alkylphosphonic acids (APAs), the markers of the toxic nerve agents for verification of the Chemical Weapons Convention (CWC). Initially, the LC–UV–NMR parameters were optimized for benzyl derivatives of the APAs and AAPAs. The optimized parameters include stationary phase C₁₈, mobile phase methanol:water 78:22 (v/v), UV detection at 268 nm and ¹H NMR acquisition conditions. The protocol described herein allowed the detection of analytes through acquisition of high quality NMR spectra from the aqueous solution of the APAs and AAPAs with high concentrations of interfering background chemicals which have been removed by preceding sample preparation. The reported standard deviation for the quantification is related to the UV detector which showed relative standard deviations (RSDs) for quantification within ±1.1%, while lower limit of detection upto 16 μg (in μg absolute) for the NMR detector. Finally the developed LC–UV–NMR method was applied to identify the APAs and AAPAs in real water samples, consequent to solid phase extraction and derivatization. The method is fast (total experiment time ∼2 h), sensitive, rugged and efficient.     

Critically indecomposable graphs

An undirected graph G=(V,E) with a specific subset X⊂V is called X-critical if G and G(X), induced subgraph on X, are indecomposable but G(V−{w}) is decomposable for every w∈V−X. This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191-205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the (n−2)-property, Theoretical Computer Science 132 (1994) 151-178], who deal with the case where X is empty.We present several structural results for this class of graphs and show that in every X-critical graph the vertices of V−X can be partitioned into pairs (a₁,b₁),(a₂,b₂),…,(a_m,b_m) such that G(V−{a_j1,b_j1,…,a_jk,b_jk}) is also an X-critical graph for arbitrary set of indices {j₁,…,j_k}. These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G(X), then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs (x₁,y₁),…,(x_t,y_t) such that x_i,y_i∈V−X. As an application of the commutative elimination sequence of an X-critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G(X) to entire G.     

Modeling the survival of a resource-dependent population: Effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors

In this paper, a nonlinear mathematical model is proposed and analyzed to study the survival of a resource-dependent population. It is assumed that this population and its resource are affected simultaneously by a toxicant (pollutant) emitted into the environment from external sources as well as formed by precursors of this population. It is shown that as the cumulative rates of emission and formation of the toxicant into the environment increase, the densities of population and its resource settle down to lower equilibria than their initial carrying capacities, and their magnitudes decrease as rates of emission and formation of the toxicant increase. On comparing different cases, it is noted that when population is not affected directly by the toxicant but only its resource is affected, the possibility of its survival is greater than the case when both are affected simultaneously. But for large emission rate of toxicant, the affected resource may be driven to extinction under certain conditions and the population which wholly depends on it may not survive for long even if it is not affected directly by the toxicant.     

Synthesis of α‐Benzylthiobenzimidazoleacetonitriles and Their Chemoselective Reduction of the Double Bond with NaBH4

Condensation of 2‐((1H‐benzimidazol‐2‐yl)thio)acetonitre 1 with aromatic aldehydes in methanol containing piperidine gave the corresponding 2‐((1H‐benzimidazol‐2‐yl)thio)‐3‐arylacrylonitrile 2 , which on treatment with NaBH₄ in ethanol unexpectedly and chemoselectively gave 2‐((1H‐benzimidazol‐2‐yl)thio)‐3‐arylpropanenitrile 3 by the reduction of the double bond of 2 . 3 on methylation with dimethyl sulfate containing K₂CO₃ as a base and tetrabutylammonium bromide as PTC gave 2‐((1‐methylbenzimidazol‐2‐yl)thio)‐3‐arylpropanenitrile 6 . The latter could also be prepared in an alternative way by reaction of 1 with dimethyl sulfate giving the intermediary 2‐((1‐methylbenzimidazol‐2‐yl)thio)acetonitrile 4 , followed by condensation with aromatic aldehydes yielding 5 and subsequent reduction of 5 with NaBH₄ in methanol. 6 could be directly synthesized by treatment of 4 with benzyl chloride in DMF and triethylamine as a base at 60°C for 5 h.     

Synthesis,spectroscopic characterization and crystal structure of 5-bromo-1-(2-cyano-pyridin-4-yl)-1H-indazole-3-carboxylic acid diethylamide

The title compound 5-bromo-1-(2-cyano-pyridin-4-yl)-1H-indazole-3-carboxylic acid diethylamide, C₁₈H₁₆BrN₅O, is prepared from 5-bromoindazole-3-carboxylic acid methylester. N ¹-arylation is carried out with 4-chloro-2-cyanopyridine and the resulting product is converted to diethylamide by reacting with thionyl chloride and diethylamine. The structure is identified from its FT-IR, ¹H NMR, ¹³C NMR spectroscopy, elemental analysis data and unambiguously confirmed by single crystal X-ray diffraction studies. There are two symmetry independent molecules in the asymmetric unit with no significant differences in bond lengths and angles. The title compound crystallizes in the triclinic system, space group \(P\bar 1\) , with a = 11.2330(2); b = 11.6130(2); c = 15.4710(3) Å, α = 92.515(1)°; β = 109.956(1)°; γ = 107.199(1)°; V = 1788.45(6)ų and z = 4. An intramolecular C-H…N hydrogen bond forms an S(6) ring motif in one of the unique molecules. In the crystal, two molecules are linked about a center of inversion by C-H…O hydrogen bonded dimers generating an R ₂ ² (16) ring motif. The crystal packing is stabilized by C-H…N, C-H…O hydrogen bonds and π…π stacking interactions.     

Ionic liquid catalyzed synthesis of 7-phenyl-1,4,6,7-tetrahydro-thiazolo[5,4-d]pyrimidine-2,5-diones

A simple, green and catalyst free one pot synthesis of 7-phenyl-1,4,6,7-tetrahydro-thiazolo[5,4-d]pyrimidine-2,5-diones via a multicomponent reaction between thiazolidine-2, 4-dione (TZD), aromatic aldehyde and urea analogues is described. The ionic liquid has been used as a solvent as well as catalyst for this reaction. This reaction proceeded smoothly in good to excellent yields and offered several other advantages including short reaction time, simple experimental workup procedure and no by-products.