The number of Hamiltonian decompositions of regular graphs

A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph K _n on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended by Kühn and Osthus. They proved that every r-regular n-vertex graph Γ with even degree r = cn for some fixed c > 1/2 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating H(Γ), the number of such decompositions of Γ. Our main result is that H(Γ) = r ^(1+o(1))nr/2. In particular, the number of Hamiltonian decompositions of K _n is \({n^{\left( {1 + o\left( 1 \right)} \right){n^2}/2}}\).     

Lewis Acid Catalyzed Synthesis of Cyanidophosphates

Salts containing new cyanido(fluorido)phosphate anions of the general formula [PF_6?n(CN)_n]^? (n=1–4) were synthesized by a very mild Lewis‐acid‐catalyzed synthetic protocol and fully characterized. All [PF_6?n(CN)_n]^? (n=1–4) salts could be isolated on a preparative scale. It was also possible to detect the [PF(CN)₅]^? but not the [P(CN)₆]^? anion. The best results with respect to purity, yield, and low cost were obtained when the F^?/CN^? substitution reactions were carried out in ionic liquids.     

Supertropical algebra

We develop the algebraic polynomial theory for “supertropical algebra,” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of “ghost elements,” which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polynomials, leading to an analog of the Hilbert Nullstellensatz.Particular attention is paid to factorization of polynomials. In one indeterminate, any polynomial can be factored into linear and quadratic factors, and although unique factorization may fail, there is a “preferred” factorization that is explained both geometrically and algebraically. The failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper.     

Der Effekt der Viscosit?t eines Peaks in der gas-chromatographischen S?ule auf die quantitative Analyse

Ohne Zusammenfassung     

5. Toxikologische Analyse

Ohne Zusammenfassung     

A three-dimensional, noninterpenetrating metal--organic framework with the moganite topology: a simple (4(2)x6(2)x8(2))(4x6(4)x8)(2) net containing two kinds of topologically nonequivalent points

Self assembly of Cu(2+) with the multifunctional ligand 2-(4-pyridyl)thiazole-4-carboxylic acid (Pytac) affords the neutral 3D coordination polymer [Cu(3)(Pytac)(6)](H(2)O)(14) (hereafter, SZL-1), which has the rare moganite topology. The mineral moganite has a topology that is closely related to the well-known quartz topology, but the two topologies are differentiated by the number of topologically inequivalent nodes. Whereas only one kind of node is present in quartz, two types of topologically inequivalent nodes are present in moganite. The title compound, which has three vertices in its repeat unit, has two types of topologically inequivalent nodes with the overall vertex symbol (4(2)x6(2)x8(2))(4x6(4)x8)(2) corresponding to the moganite net. Prior to this report, few metal-organic framework materials (MOFs) have been found to contain more than one type of node, and SZL-1 is the first MOF with the moganite topology.     

Contribution a la determination potentiometrique des constantes de stabilite des complexes 1:2 avec les acides aminopolyacetiques MeHnY2

An extension of the potentiometric method based on pH measurement is applied to the determination of the stability constants of addition complexes. On the one hand, the polybasic chelating agent H_nY, and on the other hand the same acid in presence of an equimolar concentration of the entity to be complexed, are titrated with standard sodium hydroxide in a medium of constant ionic strength. By means of a least-square computer program, the stepwise stability contants of the addition complexes are obtained. This method is applied to the 1:2 complexes formed between several trivalent lanthanides and hydroxyethylethylenediaminotriacetic acid (HEDTA).     

Semigroup identities in the monoid of triangular tropical matrices

We show that the submonoid of all n×n triangular tropical matrices satisfies a nontrivial semigroup identity and provide a generic construction for classes of such identities. The utilization of the Fibonacci number formula gives us an upper bound on the length of these 2-variable semigroup identities.     

Supertropical matrix algebra II: Solving tropical equations

We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of A, which provides the unique right (resp. left) quasi-inverse maximal with respect to the right (resp. left) quasi-identity matrix corresponding to A; this provides a unique maximal (tangible) solution to supertropical vector equations, via a version of Cramer’s rule. We also describe various properties of this tangible adjoint, and use it to compute supertropical eigenvectors, thereby producing an example in which an n × n matrix has n distinct supertropical eigenvalues but their supertropical eigenvectors are tropically dependent.