Die Sechseck-n-Waben in der Ebene,welche von n Kreisbüscheln erzeugt werden

In this paper, the problem of determining the pencils of circles which form a hexagonal n-web in E₂, is completely solved. It is well-known that n pencils of circles orthogonal to a fixed circle form a hexagonal n-web. Therefore, the main problem is the determination of all circle pencils which form a hexagonal n-web and which do not cut a fixed circle orthogonally. In this connection the following results have been obtained: The number of hexagonal 4-webs is six, whereas the number of hexagonal 5-webs is two.Finally, after having proved that the number of hexagonal 6-webs is one, it is shown that, for n7, there exist no circle pencils forming a hexagonal n-web without being orthogonal to a fixed circle.     

New instanton and meron solutions of a generally covariant four-dimensional sigma model

We construct new solutions of the fourdimensional sigma model coupled to the metric tensor field and having an internalO invariance. Our solutions interpolate continuously between the known instanton and meron solutions depending upon a parameterf. We find that the typical domain for the instantons is 2<f≦3 while for the merons is 0≦f≦2.     

Anomalous NMR behavior of meso compounds with remote stereogenic centers on addition of chiral shift reagent or chiral solvating agent

A problem has arisen in using chiral shift reagents (CSR) and chiral solvating agents (CSA) to determine meso and racemic forms of diastereoisomers in which the stereogenic centers of the molecules are separated by achiral spacers. It is found that NMR signals of both meso and racemic forms of diastereoisomers may exhibit doubling on addition of CSR/CSA, which means that unequivocal assignments cannot be made without characterizing the effects for separate meso and racemic forms; this is particularly important for additions of CSR/CSA at relatively low concentrations, which always result in the splitting of some NMR signals of diastereoisomers. The phenomenon is demonstrated in the (31)P NMR spectra of meso and racemic forms of three spermine-bridged gem-disubstituted cyclotriphosphazatrienes, 1a-c, and compared with analogous achiral molecules, the per-substituted spermine-bridged cyclotriphosphazatrienes 2a-d. As expected, only one set of (31)P NMR signals was observed for the achiral compounds 2a-d, even on addition of CSA. Two sets of (31)P NMR ABX multiplets corresponding to meso and racemic diastereoisomers were observed for compounds 1a-c; on addition of CSA, the signals of at least one of the multiplets for each compound separated into more than the expected groups of three lines with an intensity distribution of 2:1:1. To understand this phenomenon, the meso and racemic forms of 1a and 1b and the meso form of 1c have been separated and characterized by X-ray crystallography. On addition of CSA to the racemic forms of 1a and 1b, the (31)P NMR spectrum shows the expected doubling of signals, but, unexpectedly, the same is observed for each of the meso forms of 1a-c. Analogous results using both CSA and CSR have been obtained for the meso and racemic forms of the diastereoisomeric piperazine-bridged macrocyclic-phosphazene compound, 3, whereas no effect was observed for the two meso forms of the doubly bridged macrocyclic-phosphazene compound 4. The phenomenon of doubling of the (31)P NMR signals of the meso form of singly bridged cyclotriphosphazatrienes, 1a-c and 3, is explained by consideration of the equilibrium in solution of independent complexation of a chiral ligand with molecules that have two chiral cyclophosphazene moieties separated by an achiral spacer group. The results show that the stereogenicity of such diastereoisomeric molecules in solution cannot be characterized unequivocally by NMR measurements on addition of either CSR or CSA.     

Alcohol-soluble functional vic-dioxime bearing alcoholhexylsulfanyl moieties: preparation, spectroscopy and electrochemistry

A new alcohol soluble functionalized vic-dioxime, bis-[(1-hydroxyhexyl)-(8,9-hydroxyimino)-7,10-dithiahexacosane (LH₂), and its alcohol-soluble mono and dinuclear complexes (Ni^II, Cu^II, Co^II, Mn^II, Pd^II and UO ₂ ^II ) have been prepared from 6-mercapto-1-hexanol and (E,E)-dichloroglyoxime under high dilution basic conditions. Reactive polyalcohol moieties appended at the periphery of the oxime containing two different heteroatoms (S-, O-), serve as a weak exocyclic binding sites for Pd^II and Ag^I metal ions and also provide solubility for the vic-dioxime complexes in low molecular-weight alcohols. Both mono-nuclear (LH)₂M and homodinuclear (LH)₂(UO₂)₂(OH)₂ and heterotrinuclear (LH)₂MM ₂ ^′ Xn, where M = Co^II M′ = Pd^II, X = Cl⁻, n = 4 and Ag^I X = NO ₃ ⁻ , n = 2) complexes have been obtained with a 1:2, 2:2, 3:2 metal/ligand ratio, respectively. Electronic spectra of the modified vic-dioximes exhibit monitorable changes in UV. All mono and dinuclear-complexes are soluble in common organic solvents. The elemental analysis, ¹H-n.m.r, i.r., u.v–vis, and f.a.b.–m.s data and by cyclic and differential pulse voltammetry measurements are presented.     

Variational Operators,Symplectic Operators,and the Cohomology of Scalar Evolution Equations

For a scalar evolution equation u_t = K(t, x, u, u_x, . . . , u_2m+1) with m ≥ 1, the cohomology space H^1,2() is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for u_t = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H^1,2() and subsequently can be naturally identified with the space of variational operators. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The variational operator (or symplectic) nature of the potential form of a bi-Hamiltonian evolution equation is also presented in order to generate examples of interest.     

On the difference spaces of almost convergent and strongly almost convergent double sequences

In this paper, we study the difference spaces \({\mathcal {F}}(\varDelta )\), \({\mathcal {F}}_0(\varDelta )\), \({\mathcal {[F]}}(\varDelta )\) and \({\mathcal {[F]}}_0(\varDelta )\) of double sequences obtained as the domain of four-dimensional backward difference matrix \(\varDelta \) in the spaces \({\mathcal {F}}\), \({\mathcal {F}}_{0}\), \({\mathcal {[F]}}\) and \({\mathcal {[F]}}_{0}\) of almost convergent, almost null, strongly almost convergent and strongly almost null double sequences; respectively. We examine general topological properties of those spaces and give some inclusion theorems. Furthermore, we deal with their dual spaces.

     

Investigation of spectroscopic studies and DFT calculations on glucofuranose derivatives of dithiophosphonates

Abstract

2,4-Bis(4-methoxyphenyl)-1,3,2,4-dithiadiphosphetane-2,4-disulfide (Lawesson’s reagent: LR) was reacted with 1,2:5,6-di-O-cyclohexylidene-α-D-glucofuranose and 1,2:5,6-di-O-isopropylidene-α-D-glucofuranose in toluene and gave rise to crude dithiophosphonic acids. The crude dithiophosphonic acids were treated with excess triethylamine and the triethylammonium salts of dithiophosphonic acids (1) and (2) were isolated by crystallization method. Compounds 1 and 2 were characterized by elemental analysis, IR, and NMR (¹H-, ¹³C- and ³¹P-) spectroscopies and mass spectrometry. In addition, NMR spectra, the electronic properties (the electronic chemical potential, electronegativity, chemical hardness and the global electrophilicity index) and NBO analysis of compound 2 have been calculated by using the Gaussian 16?W program. The electronic properties were calculated using Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) energies by density functional method (DFT/B3LYP) at 6-31G(d,p) level. The HOMO and LUMO energies are ?5.9 and ?0.72?eV, respectively. The HOMO–LUMO energy difference is 5.18?eV and this value shows that compound 2 has a high stability and low reactivity.