Adjoint polynomials of bridge–path and bridge–cycle graphs and Chebyshev polynomials

The chromatic polynomial of a simple graph G with n>0 vertices is a polynomial of degree n, where α_k(G) is the number of k-independent partitions of G for all k. The adjoint polynomial of G is defined to be , where is the complement of G. We find explicit formulas for the adjoint polynomials of the bridge–path and bridge–cycle graphs. Consequence, we find the zeros of the adjoint polynomials of several families of graphs.     

Ft-Raman Study of Three Synthetic Solid Solutions Formed by Orthorhombic Sulfates: Celestite-Barytes,Barytes-Anglesite and Celestite-Anglesite

Abstract

In the present work, three different solid solutions whose end-members are the orthorhombic sulfates celestite (SrSO₄), barytes (BaSO₄) and anglesite (PbSO₄) are studied using FT-Raman spectroscopy. Sulfate anion symmetric internal modes have been examined in detail by means of band-shape analysis and component fitting procedures. the symmetric stretching mode v₁(A₁) changes its wavenumber position linearly with the cationic composition of the samples which further confirms the ideal character of the solid solutions studied. the corresponding full-width at half-height is strongly increased in the central components of the different solid solutions which can be understood as an effect of the positional disorder induced by random cationic substitution. Similar results are observed in the symmetric bending mode, v² (E). the study of the low frequency spectral region permits one to differentiate translational modes of the sulfate anion, which changes its wavenumber position when the alkali-earth metal ion changes from rotational modes. This permits the tentative band assignment of the anglesite rotational Raman bands at 134 and 152 cm^?1, which were previously not assigned.     

Solid‐particle erosion in the tube end of the tube sheet of a shell‐and‐tube heat exchanger

Erosion is one of the major problems in many industrial processes, and in particular, in heat exchangers. The effects of flow velocity and sand particle size on the rate of erosion in a typical shell‐and‐tube heat exchanger were investigated numerically using the Lagrangian particle‐tracking method. Erosion and penetration rates were obtained for sand particles of diameters ranging from 10 to 500 µm and for inlet flow velocities ranging from 0.197 to 2.95 m/s. A flow visualization experiment was conducted with the objective of verifying the accuracy of the continuous phase calculation procedure. Comparison with available experimental data of penetration rates was also conducted. These comparisons resulted in a good agreement. The results show that the location and number of eroded tubes depend mainly on the particle size and velocity magnitude at the header inlet. The rate of erosion depends exponentially on the velocity. The particle size shows negligible effect on the erosion rate at high velocity values and the large‐size particles show less erosion rates compared to the small‐size particles at low values of inlet flow velocities. The results indicated that the erosion and penetration rates are insignificant at the lower end of the velocity range. However, these rates were found to increase continuously with the increase of the inlet flow velocity for all particle sizes. The particle size creating the highest erosion rate was found to depend on the flow velocity range. Copyright © 2005 John Wiley & Sons, Ltd.     

On the zeros of basic finite Hankel transforms

In this article we prove that the basic finite Hankel transform whose kernel is the third-type Jackson q-Bessel function has only infinitely many real and simple zeros, provided that q satisfies a condition additional to the standard one. We also study the asymptotic behavior of the zeros. The obtained results are applied to investigate the zeros of q-Bessel functions as well as the zeros of q-trigonometric functions. A basic analog of a theorem of G. Pólya (1918) on the zeros of sine and cosine transformations is also given.     

Kinetic spectrophotometric method for the determination of trace amounts of oxalate by an activation effect.

A new, simple and inexpensive kinetic catalytic spectrophotometric method for the determination of oxalate is described. The method is based on an activation effect of oxalate on a catalytic effect of iron(II) on the oxidation of iodide by bromate. The reaction is monitored by measuring the absorbance of triiodide ion at lambda max = 352 nm. A calibration graph was obtained from 0.10 - 7.0 microg cm(-3) of oxalate with a detection limit of 0.080 microg cm(-3). The standard deviations for ten replicate determinations of 0.50, 1.0 and 5.0 microg cm(-3) of oxalate were 4.0, 2.6 and 1.8%, respectively. The applicability of the method was demonstrated by the determination of oxalate ion in real samples.     

The stabilization mechanism and structure of turbulent hydrocarbon lifted flames

The structure and stabilization mechanism of turbulent lifted non-premixed hydrocarbon flames have been investigated using combined laser imaging techniques. The techniques include Rayleigh scattering, laser induced predissociation fluorescence of OH, LIF of PAH, LIF of CH₂O, and planar imaging velocimetry. The geometrical structure of multi-reaction zones and flow field at the stabilization region have been simultaneously measured in 16 hydrocarbon flames. The data reveal the existence of triple flame structure at the stabilization region of turbulent lifted flames. Increasing the jet velocity leads to an increase of the lift-off height and to a broadening of the lift-off region. Further analysis of the stabilization criterion at the lift-off height based on the premixed nature of triple-flame propagation and flow field data has been presented and discussed.     

Die optischen Konstanten des Kaliums im Wellenlängenbereich von 3650 bis 20000 Å

The optical constantsn andk of the alkali metal potassium have been measured in the wavelength region from 0·365μ to 2μ, in the solid and liquid state at temperatures between ?180° C and +85° C by the usual method of analysing intensity and polarisation of the monochromatic radiations reflected by the metals surface. The mirrorlike surfaces, free of any contamination and not distorted by any treatment, were prepared and measured in ultrahigh vacuum, the metal being purified before by careful repeated destinations in high vacuum. Values thus obtained forn, k, k ²?n ² and 2nk are listed in tables 1 and 2. Discussion of these values especially with regard to the Drude-theory of optical constants of metals is given in two subsequent communications.