Additive Coloring of Planar Graphs

An additive coloring of a graph G is an assignment of positive integers \({\{1,2,\ldots ,k\}}\) to the vertices of G such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number k for which there exists an additive coloring of G is denoted by \({\eta (G)}\) . We prove that \({\eta (G) \, \leqslant \, 468}\) for every planar graph G. This improves a previous bound \({\eta (G) \, \leqslant \, 5544}\) due to Norin. The proof uses Combinatorial Nullstellensatz and the coloring number of planar hypergraphs. We also demonstrate that \({\eta (G) \, \leqslant \, 36}\) for 3-colorable planar graphs, and \({\eta (G) \, \leqslant \, 4}\) for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each \({r \, \geqslant \, 2}\) there is an r-chromatic graph G _r with no additive coloring by elements of any abelian group of order r.     

Characterizations of Some Free Random Variables by Properties of Conditional Moments of Third Degree Polynomials

We investigate Laha–Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Lévy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a \(q\) -Gausian variable.     

Liquid‐Crystalline Phases Made of Gold Nanoparticles

Spontaneous formation of smectic and columnar structures was observed when spherical gold nanoparticles were functionalized with mesogenic thiols (see layered structure and X‐ray pattern of a sample in smectic phase). The particle ordering is stimulated by softening of the interparticle potential and flexibility for deformation of the grafting layer.

     

Modelling of interactions between volatile anaesthetics (halothane,enflurane) and aromatic compounds,ab initio study

For many years halothane and enflurane have been used clinically as volatile anaesthetics, however, their mechanism of action is still not fully understood. Recently, it has been suggested that they can act by a direct bonding to neuroreceptors containing the aromatic groups. In this work, the halothane?benzene and enflurane?benzene complexes were studied by the ab initio MP2 and CCSD(T) methods. All possible structures of the complexes were calculated by means of the counterpoise CP-corrected gradient optimization technique. It has been found that among these species, the C–H?π hydrogen bonded complexes are the most stable. The CCSD(T)/CBS calculated stabilization energies for halothane and enflurane complexes are: −10.56 and −9.72 kcal mol⁻¹, respectively. The interaction energy is mainly dominated by the dispersion attraction. In the case of enflurane, the C–H bond shows a very small contraction (by −0.0008 Å) upon complexation. This change is accompanied by the blue-shift (20 cm⁻¹) of the C–H stretching frequency and an increase of the infrared intensity of the corresponding mode by 7 km mol⁻¹. Similar results were obtained for the halothane complex: a small contraction of the C–H bond; an increase of the C–H stretching frequency by 11 cm⁻¹ (blue-shift); and an increase of the infrared intensity by 37 km mol⁻¹. In order to explain the nature of these effects, the halothane and enflurane molecules were studied in the electric field generated by benzene atoms, and Natural Bond Orbital (NBO) analyses were performed. The molecular dipole moments of these molecules were calculated with respect to the C–H bond changes. The positive dipole moment derivative obtained for halothane is in agreement with the literature data, while, in the case of enflurane, an unusual effect is observed, the blue-shift of the C–H stretching frequency is accompanied by the positive dipole moment derivative for one C–H bond and the negative for the other C–H bond. The mechanisms responsible for contraction and strengthening of the C–H bonds are discussed.     

Engineering of an enantioselective tyrosine aminomutase by mutation of a single active site residue in phenylalanine aminomutase

By replacing a single active-site residue Cys107 with Ser in phenylalanine aminomutase (PAM), the enzyme gained tyrosine aminomutase (TAM) activity while retaining PAM activity and high enantioselectivity. This engineered enantioselective TAM also catalyzed formation of β-tyrosine from p-coumaric acid and may prove to be useful for the synthesis of enantiopure β-tyrosine and its derivatives.     

The weak hereditary class of a variety

We study the weak hereditary class S _w ( ) of all weak subalgebras of algebras in a total variety }. We establish an algebraic characterization, in the sense of Birkhoff’s HSP theorem, and a syntactical characterization of these classes. We also consider the problem of when such a weak hereditary class is weak equational.     

Photocaging of Carboxylic Acids: A Modular Approach

Photocaged compounds are important tools for studying and regulating multiple processes, including biological functions. Reported herein is the use of the Passerini multicomponent reaction for modular preparation of photocaged carboxylic acids. The reaction is compatible with several functionalities and proceeds smoothly both in water and dichloromethane. The choice of aldehyde determines the wavelength used for deprotection and enables formation of orthogonally protected products. The isocyanide component can be used for introduction of reactive tags and photosensitizers, as well as for immobilization on a solid support.     

Noncovalent Bonds between Tetrel Atoms

The pairing of TFH₃ with a TH₂CH₃⁻ anion, where T represents tetrel atoms C, Si, Ge, Sn, Pb, results in a strong direct interaction between the two T atoms. The interaction energy is sensitive to the nature of the two T atoms but can be as large as 90 kcal/mol. The noncovalent bond strength rises quickly as the basic T atom of the anion becomes smaller, or as the Lewis acid T grows larger, although there is less sensitivity to the latter atom. The electrostatic component makes up some 55–70 % of the total attraction energy. This term is well accounted for by simple combination of the maximum and minimum values of the molecular electrostatic potential of the Lewis acid and base units, respectively. The complexation induces a rearrangement in the TFH₃ molecule from tetrahedral to trigonal pyramidal. The associated deformation energy reduces the exothermicity of the complexation reaction. Electron density shift patterns reveal a density loss on the basic T atom, along with accompanying increases on the acidic T and its attached F atom.