Plane algebraic curves with many cusps,with an appendix by Eugenii Shustin

The maximum number \(k(d)\) of cusps on a plane algebraic curve of degree \(d\) is an open classical problem that dates back to the nineteenth century. A related open problem is the asymptotic value ( \(a.v.\) ) of the number of cusps on plane curves, that is \(a.v.=\limsup _{d\rightarrow \infty } k(d)/{d^2}\) . In this paper, we improve the best known lower bound for the asymptotic value by constructing curves with the largest known number of cusps for infinitely many degrees. Some particular curves of relatively low degree with many cusps are constructed too. The “Appendix” to this paper is devoted to the case of degree 11 and it is due to E. Shustin.     

On Vizing’s bound for the chromatic index of a multigraph

Two of the basic results on edge coloring are Vizing’s Theorem [V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25-30 (in Russian); V.G. Vizing, The chromatic class of a multigraph, Kibernetika (Kiev) 3 (1965) 29-39 (in Russian). English translation in Cybernetics 1 32-41], which states that the chromatic index χ^′(G) of a (multi)graph G with maximum degree Δ(G) and maximum multiplicity μ(G) satisfies Δ(G)≤χ^′(G)≤Δ(G)+μ(G), and Holyer’s Theorem [I. Holyer, The NP-completeness of edge-colouring, SIAM J. Comput. 10 (1981) 718-720], which states that the problem of determining the chromatic index of even a simple graph is NP-hard. Hence, a good characterization of those graphs for which Vizing’s upper bound is attained seems to be unlikely. On the other hand, Vizing noticed in the first two above-cited references that the upper bound cannot be attained if Δ(G)=2μ(G)+1≥5. In this paper we discuss the problem: For which values Δ,μ does there exist a graph G satisfying Δ(G)=Δ, μ(G)=μ, and χ^′(G)=Δ+μ.     

Analysis of aldehydes in beer by gas-diffusion microextraction: Characterization by high-performance liquid chromatography–diode-array detection–atmospheric pressure chemical ionization–mass spectrometry

In this work, a recently developed extraction technique for sample preparation aiming the analysis of volatile and semi-volatile compounds named gas-diffusion microextraction (GDME) is applied in the chromatographic analysis of aldehydes in beer. Aldehydes—namely acetaldehyde (AA), methylpropanal (MA) and furfural (FA)—were simultaneously extracted and derivatized with 2,4-dinitrophenylhydrazine (DNPH), then the derivatives were separated and analyzed by high-performance liquid chromatography with spectrophotometric detection (HPLC–UV). The identity of the eluted compounds was confirmed by high-performance liquid chromatography–atmospheric pressure chemical ionization–mass-spectrometry detection in the negative ion mode (HPLC–APCI–MS). The developed methodology showed good repeatability (ca. 5%) and linearity as well as good limits of detection (AA–12.3, FA–1.5 and MA 5.4 μg L⁻¹) and quantification (AA–41, FA–4.9 and MA 18 μg L⁻¹); it also appears to be competitive in terms of speed and cost of analysis.     

Solvent effects on solution enthalpies of adamantyl derivatives

Solution enthalpies of 1-bromoadamantane, 1-adamantanol, and 2-adamantanone in a large set of protic and aprotic solvents are reported at 298.15 K. Solvent effects on the solution processes of these solutes are analyzed in terms of a modified TAKA equation, involving δ_cav h ^s as the cavity term. The nature and magnitude of the major interactions which influence these processes are assessed and discussed in terms of the solutes’ characteristics. New insights on the solution processes under scrutiny are presented.     

Gas-diffusion microextraction

Gas-diffusion microextraction (GDME) is an innovative technique that combines the advantages of membrane-aided gas-diffusion with microextraction concepts. GDME makes uses of a novel portable and low-cost device that comprises a small, commercially available, semi-permeable membrane. Furthermore, if derivatization is integrated into the GDME concept, considerable enrichment factors can be obtained. It may become a powerful tool for any analyst who intends to quantify volatile and semi-volatile analytes in various kinds of matrices. The analysis of vicinal diketones in beer was used as a case study to show GDME applicability and capabilities. Vicinal diketones were derivatized with o-phenylenediamine and then determined by HPLC-UV. Obtained results showed good repeatability and precision with extraction periods at the minute time span.     

Kempe Equivalence of Edge‐Colorings in Subcubic and Subquartic Graphs

It is proved that all 4‐edge‐colorings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree Δ = 3 is a threshold for Kempe equivalence of (Δ+1)‐edge‐colorings, as such an equivalence does not hold in general when Δ = 4. One extra color allows a similar result in this latter case; however, namely, when Δ≤4 it is shown that all (Δ+2)‐edge‐colorings are Kempe equivalent. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Mechanistic study on the palladium-catalyzed (3 + 2) intramolecular cycloaddition of alk-5-enylidenecyclopropanes

The intramolecular (3 + 2) cycloaddition of alkenylidenecyclopropanes to alkenes under palladium catalysis provides a practical and stereoselective entry into a variety of interesting bicycles. The reaction outcome and stereoselectivity of the process are somewhat dependent on the characteristics of the substrate and of the palladium ligand, which is not easy to justify on the basis of the current mechanistic understanding. We therefore decided to study the different mechanistic alternatives from a theoretical point of view. The energies of the reaction intermediates and transition states for different possible pathways have been explored at DFT level in a model system, and using PH(3) and P(OMe)(3) as ligands. The results obtained suggest that the most favourable reaction pathway involves an initial oxidative addition of Pd(0) at the distal position of the cyclopropane to afford a palladacyclobutane intermediate. The evolution of this intermediate into the final cycloadduct can occur following different paths, the most favorable depending on the configuration and substitution of the alkene cycloaddition partner, and the number of ancillary ligands coordinated to Pd. The computational results are consistent with the experimental observations and provide the basis for proposing which would be the operative mechanistic pathway in different cases. The results also allow us to explain the stereochemical divergences observed in some of the reactions.     

Mechanical and Biological Performances of New Scaffolds Made of Collagen Hydrogels and Fibroin Microfibers for Vascular Tissue Engineering

A microstructured composite material made of collagen hydrogel (matrix) and silk fibroin microfibers (randomly oriented reinforcing fibers) is investigated in order to conjugate the mechanical resistance of fibroin with the suitable biological performance of collagen to design new scaffolds for vascular tissue engineering. Results show that fibroin microfibers and collagen fibrils have suitable interfacial adhesion, and the scaffold exhibits improved mechanical properties if compared with a pure collagen hydrogel. Furthermore, the overall biological performance is improved.