On the Gomory–Hu Inequality

It was proved by R. Gomory and T. Hu in 1961 that, for every finite nonempty ultrametric space (X, d), the inequaliy \( \left| {\mathrm{Sp}(X)} \right|\leq \left| X \right|-1 \), where Sp(X) = {d(x, y) : x, y ∈ X, x ≠ y} , holds. We characterize the spaces X for which the equality is attained by the structural properties of some graphs and show that the set of isometric types of such X is dense in the Gromov–Hausdorff space of the isometric types of compact ultrametric spaces.     

A note on scheduling to meet two min-sum objectives

We consider a single machine scheduling problem with two min-sum objective functions: the sum of completion times and the sum of weighted completion times. We propose a simple polynomial time (1+(1/γ),1+γ)-approximation algorithm, and show that for γ>1, there is no (x,y)-approximation with 1<x<1+(1/γ) and 1<y<1+(γ-1)/(2+γ).     

Influence of nonisothermal effects on gas production in northern regions

The influence of mathematical model parameters on the dynamics of pressure and temperature fields at nonisothermal gas filtration is investigated in a numerical experiment. A nonlinear system of partial differential equations obtained from the energy and mass conservation laws and the Darcy law are used to describe the process, and physical and caloric equations of state are used as closing relations. The boundary conditions correspond to a given pressure drop at the bottomhole. It is shown that the influence of the temperature field on such integral characteristics as cumulative gas production is most pronounced at moderate pressure drops. The size of the zone of possible hydrate formation in a gas reservoir is determined in a particular example.     

Synthesis,spatial structure and spectral properties of pyrylo‐4 (thio) squaraines variously substituted in cyclobutene moiety

Synthetic routes have been developed to a number of (thio) squaraine dyes containing the residues of CH‐acids at the central cyclobutene ring. The electronic and spatial structure as well as the chemical conversions and optical behaviour of the compounds obtained have been studied both theoretically and by X‐ray diffraction analysis, ¹H NMR and electronic spectroscopy. As shown, the electronic nature and sterical characteristics of the central ring substituents give rise to some general conformational features and crystal packing regularities and also govern the spectral position of the first π–π^* absorption band. The structure–property relationships established in the study provide guidance for the purposeful design of deeply coloured (thio) squaraines. Copyright © 2015 John Wiley & Sons, Ltd.     

Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials

A new generalized nonlinear Schr?dinger equation describing the propagation of ultrashort pulses in bulk media exhibiting frequency dependent dielectric susceptibility and magnetic permeability is derived and used to characterize wave propagation in a negative index material. The equation has new features that are distinct from ordinary materials (mu=1): the linear and nonlinear coefficients can be tailored through the linear properties of the medium to attain any combination of signs unachievable in ordinary matter, with significant potential to realize a wide class of solitary waves.     

Broadly wavelength- and pulse width-tunable high-repetition rate light pulses from soliton self-frequency shifting photonic crystal fiber integrated with a frequency doubling crystal

Soliton self-frequency shift (SSFS) in a photonic crystal fiber (PCF) pumped by a long-cavity mode-locked Cr:forsterite laser is integrated with second harmonic generation (SHG) in a nonlinear crystal to generate ultrashort light pulses tunable within the range of wavelengths from 680 to 1800?nm at a repetition rate of 20?MHz. The pulse width of the second harmonic output is tuned from 70 to 600?fs by varying the thickness of the nonlinear crystal, beam-focusing geometry, and the wavelength of the soliton PCF output. Wavelength-tunable pulses generated through a combination of SSFS and SHG are ideally suited for coherent Raman microspectroscopy at high repetition rates, as verified by experiments on synthetic diamond and polystyrene films.     

Cyanide Complexes Based on {Mo6I8}4+ and {W6I8}4+ Cluster Cores

Compounds based on new cyanide cluster anions [{Mo₆I₈}(CN)₆]^2–, trans-[{Mo₆I₈}(CN)₄(MeO)₂]^2– and trans-[{W₆I₈}(CN)₂(MeO)₄]²⁻ were synthesized using mechanochemical or solvothermal synthesis. The crystal and electronic structures as well as spectroscopic properties of the anions were investigated. It was found that the new compounds exhibit red luminescence upon excitation by UV light in the solid state and solutions, as other cluster complexes based on {Mo₆I₈}⁴⁺ and {W₆I₈}⁴⁺ cores do. The compounds can be recrystallized from aqueous methanol solutions; besides this, it was shown using NMR and UV-Vis spectroscopy that anions did not undergo hydrolysis in the solutions for a long time. These facts indicate that hydrolytic stabilization of {Mo₆I₈} and {W₆I₈} cluster cores can be achieved by coordination of cyanide ligands.     

Impact of the Oxygen Defects and the Hydrogen Concentration on the Surface of Tetragonal and Monoclinic ZrO2 on the Reduction Rates of Stearic Acid on Ni/ZrO2

The role of the specific physicochemical properties of ZrO₂ phases on Ni/ZrO₂ has been explored with respect to the reduction of stearic acid. Conversion on pure m‐ZrO₂ is 1.3 times more active than on t‐ZrO₂, whereas Ni/m‐ZrO₂ is three times more active than Ni/t‐ZrO₂. Although the hydrodeoxygenation of stearic acid can be catalyzed solely by Ni, the synergistic interaction between Ni and the ZrO₂ support causes the variations in the reaction rates. Adsorption of the carboxylic acid group on an oxygen vacancy of ZrO₂ and the abstraction of the α‐hydrogen atom with the elimination of the oxygen atom to produce a ketene is the key to enhance the overall rate. The hydrogenated intermediate 1‐octadecanol is in turn decarbonylated to heptadecane with identical rates on all catalysts. Decarbonylation of 1‐octadecanol is concluded to be limited by the competitive adsorption of reactants and intermediate. The substantially higher adsorption of propionic acid demonstrated by IR spectroscopy and the higher reactivity to O₂ exchange reactions with the more active catalyst indicate that the higher concentration of active oxygen defects on m‐ZrO₂ compared to t‐ZrO₂ causes the higher activity of Ni/m‐ZrO₂.