Note on vertex degrees of planar graphs

In this note the second moment of the vertex degree sequence of planar graphs is considered. We prove that

• 1 If G is an outerplanar graph of order n ? 3 then

• 2 If G is a planar graph of order n ? 4 then

where d₁,…,d_n is the vertex degree sequence of G. We exhibit all graphs for which these upper bounds are attained.     

Application of the ellipsoid method in an interactive procedure for multicriteria linear programming

The research reported in this paper is concerned with an application of the ellipsoid algorithm in the interactive multicriteria linear programming step method (STEM) byBenayoun et al. [1971]. Due to this application we eliminate some drawbacks of the original version of STEM and, moreover, we avoid extra computations connected with sensitivity analysis in every iteration. Specifically, we use the ellipsoid algorithm to minimize the Euclidean norm in the criterion space instead of the Chebyshev norm, which ensures that every solution submitted to the decision maker is efficient. As follows from a computational experiment, in comparison with the application of the simplex method, the proposed modification of STEM shows a smaller increase of the computational effort when the number of criteria increases. However, the absolute computation time becomes worse for problems of larger size.

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Zusammenfassung In dieser Arbeit wird über eine Anwendungsmöglichkeit der Khachiyan-Shor-Algorithmus (Ellipsoid-Algorithmus) im Rahmen des STEM-Verfahrens zur interaktiven Lösung linearer Vektoroptimierungsmodelle berichtet. Auf diese Weise können einige spezifische Nachteile des STEM-Verfahrens in seiner Originalversion vermieden werden. Durch die Verwendung der Euklidischen Norm anstelle der beim STEM-Verfahren üblichen Tschebyscheff-Norm wird garantiert, daß dem Entscheidungsträger nur effiziente Lösungen vorgeschlagen werden. Die numerischen Erfahrungen zeigen, daß der Lösungsaufwand der hier vorgeschlagenen Modifikation des STEM-Verfahrens mit steigender Anzahl von Zielfunktionen weniger stark zunimmt als bei der üblichen Version. Dies gilt jedoch nicht hinsichtlich der allgemeinen Problemgröße.

     

13C Chemical shifts of some azaindolizines versus electron charge distribution

Azaindolizines, which contain all possible combinations of nitrogen atoms within the five-membered ring moiety, are used as models for the investigation of a relationship between electron charge distribution and ¹³C shifts. A linear correlation is observed between the shifts and total rather than π-charge densities as calculated by the INDO-MO method. The average excitation energy (AEE) approximation in the theory of nuclear screening is shown to hold separately for the CH moieties and the carbon atoms at the ring junction in indolizines. An empirical correlation with charge densities is obtained from the AEE method, as a result of the compensation of effects within the local paramagnetic term and the prevailing contribution to the latter of the effective nuclear charge.¹³C shifts afford a reasonable measure of the total net charges at the carbon atoms of indolizines. The INDO calculations indicate that the π-charges follow the pattern suggested by simple resonance structures but the overall charge density depends heavily on σ-core polarization effects.     

Magnetic equation of state for thin films

The magnetization fluctuations are considered in thin films treated as a set of monoatomic layers parallel to the film surfaces. The correlation functions have been determined and their forms are discussed. The influence of inhomogeneity of magnetization on the equation of state for a magnetic system described by Valenta's model is studied. The behaviour of the spontaneous magnetization as well as the magnetic susceptibility near the Curie point is described.This work is partially supported by the Institute of Physics of the P.A.S.     

Triplet losses in dye lasers

The expression for losses due to triplet states in dye laser considered as a six-level system is given. It is shown that triplet losses depend on pumping parameters and photon number. Depending on molecular and cavity parameters two different behaviours of dye lasers are expected. Physical conditions are discussed in which triplet losses as well as photon number undergo a jump at the threshold.     

Bond energy, aromatic stabilization energy and strain in IPR fullerenes

Various models applied to DFT structures and energies of 2-D and 3-D aromatic molecules shed new light on the effects of strain and aromaticity in these systems. The cyclic pi electron delocalisation does not stabilize the fullerene C60 formation; and 5-6 and 6-6 CC bonds have near-identical bond stretch potentials.     

Intramolecular easily polarizable hydrogen bonds with anilides of 1-piperidine carboxylic acids

IR spectra are plotted from anilides of 1-piperidine carboxylic acids C₅H₁₀N(CH₂)_n CONHC₆H₄R in CHCl₃ and CDCl₃ solutions. In the cases of n = 1 and n = 4, weak intramolecular (NH?N) hydrogen bonds are formed. An asymmetrical energy surface occurs and the proton is present at the N of the anilide group. In the cases of n = 2 and n = 3, intramolecular proton transfer hydrogen bonds of the types N_BH?N_P ? ^?N_B?H⁺N_p are formed. In contrast to the intramolecular OH? N ? O^?1 ? H⁺N bonds with 1-piperidine carboxylic acids, these bonds to not cause IR continua but two bands: one in the region 3250–3190 and one in the region 2500–2450 cm^?1. The fact that, instead of IR continua, bands are observed is explained by the following: (1) these hydrogen bonds are relatively long; (2) they show only a narrow distribution of bond length; (3) the electrical fields at these bonds are small, since they are strongly screened.     

The application of triphenylmethane dyes to visualization of selected aliphatic compounds in thin-layer chromatography

Selected triphenylmethane dyes were used as new visualizing agents in thin-layer chromatography of higher fatty acids, higher fatty alcohols, and higher aliphatic amines.     

Positron annihilation in and compressibility of n-butanol in ethylene glycol, 1,2-butanediol and 1,4-butanediol: Do hydrophobic-like phenomena occur in non-aqueous, highly-associated solvents?

Experimental results of positron annihilation experiments in solutions of n-butanol in three diols are compared to those of ultrasonic velocity and absorption. Variations of the annihilation parameters with composition show that the effect of the hydrophobic solute on the original diol structure is rather limited and occurs only in the system where ethylene glycol is the solvent. Subtle similarities were found when compared it to the respective results for similar aqueous systems, suggesting that crystalline-like solvates are forming in ethylene glycol. However, no such evidences were found for the systems where butanediols are the solvents. Ultrasonic investigations do not confirm formation of any specific structures. This is caused, most possibly, by the fact that compressibility of solvates is close to those of the pure components and/or that of unstructured medium (regular solution).     

Separation of mixtures of permanent gases and light hydrocarbons on spherical carbon molecular sieves by gas-solid chromatography

Summary Experimental results are presented on the application of Carbosieve S (Supelco) and Spherocarb (Analabs) spherical carbon molecular sieves for the gas chromatographic separation of mixtures of permanent gases and C₁–C₃ hydrocarbons using a single column or two columns in series. At a programmed temperature of 35–300°C, good separation of the sample components was obtained when using helium as the carrier gas. When hydrogen was used as the carrier gas and the analysis was carried out under isothermal conditions the elution sequence of oxygen and nitrogen reversed as the temperature was increased. This behaviour was observed within a temperature range of 35–225°C for Carbosieve S, and within a temperature range of 35–300°C for Spherocarb.