Band 64, Heft 1-2

The results of some new research on a new class of triangular functions that unite the characteristics of the classical triangular functions are presented. Taking into consideration the great role played by triangular functions in geometry and physics, it is possible to expect that the new theory of the triangular functions will bring new results and interpretations in mathematics, biology, physics and cosmology. New traveling wave solutions of some nonlinear partial differential equations are obtained in a unified way. The main idea of this method is to express the solutions of these equations as a polynomial in the solution of the Riccati equation that satisfy the symmetrical triangular Fibonacci functions. We apply this method to the combined Korteweg-de Vries (KdV) and modified KdV (mKdV) equations, the generalized Kawahara equation, Ito’s 5th-order mKdV equation and Ito’s 7th-order mKdV equation.

The classical symmetry method and the modified Clarkson and Kruskal (C-K) method are used to obtain the Lie symmetry group of a nonisospectral Kadomtsev-Petviashvili (KP) equation. It is shown that the Lie symmetry group obtained via the traditional Lie approach is only a special case of the symmetry groups obtained by the modified C-K method. The discrete group analysis is given to show the relations between the discrete group and parameters in the ansatz. Furthermore, the expressions of the exact finite transformation of the Lie groups via the modified C-K method are much simpler than those obtained via the standard approach.

With the aid of the symbolic computation system Mathematica, many exact solutions for the Fitzhugh-Nagumo equation and the Klein-Gordon equation with a quadratic nonlinearity are constructed by an auxiliary equation method, the so-called ( G ' /G )-expansion method, where the new and more general forms of solutions are also obtained. Periodic and solitary traveling wave solutions capable of moving in both directions are observed.

By means of an extended tanh method, a new type of variable separation solutions with two arbitrary lower-dimensional functions of the (2+1)-dimensional nonlinear Schrödinger (NLS) equation is derived. Based on the derived variable separation excitation, some special types of localized solutions such as a curved soliton, a straight-line soliton and a periodic soliton are constructed by choosing appropriate functions. In addition, one dromion changes its shape during the collision with a folded solitary wave.

The homotopy analysis method (HAM) is used for solving the ordinary differential equations which arise from problems of the calculus of variations. Some numerical results are given to demonstrate the validity and applicability of the presented technique. The method is very effective and yields very accurate results.

Starting from a special conditional similarity reduction method, we obtain the reduction equation of the (2+1)-dimensional dispersive long-water wave system. Based on the reduction equation, some new exact solutions and abundant localized excitations are obtained.

The nonthermal and plasmon effects on elastic electron-ion collisions are investigated in hot quantum Lorentzian plasmas. The modified interaction model taking into account the nonthermal screening and plasmon effects is employed to represent the electron-ion interaction potential in hot quantum Lorentzian plasmas. The eikonal phase and differential collision cross-section are obtained as functions of the impact parameter, collision energy, spectral index, and plasma parameters by using the second-order eikonal analysis. It is shown that the plasmon effect suppresses the eikonal phase and collision cross-section for 0 < β (ћω 0 /k B T < 0 . 6) and, however, enhances it for 0 . 6 < β < 1, where ω 0 is the plasma frequency and T is the plasma temperature. It is also shown that the nonthermal character of the quantum Lorentzian plasma suppresses the elastic electron-ion collision cross-section.

The collective effects on the transition bremsstrahlung spectrum due to the polarization interaction between the electron and Debye shielding cloud of an ion are investigated in nonideal plasmas. The impact parameter analysis with the effective pseudopotential model taking into account the nonideal collective and plasma screening effects is applied to obtain the bremsstrahlung radiation cross-section as a function of the nonideality plasma parameter, Debye length, photon energy, and projectile energy. It is shown that the collective effects enhance the bremsstrahlung radiation cross-section and decrease with increasing impact parameter. It is also shown that the collective effect is the most significant near the maximum position of the bremsstrahlung cross-section. In addition, it is shown that the collective effect decreases with an increase of the radiation photon energy

A new isothermal equation of state (EOS) for solids is derived by starting from the theory of lattice potential and using an analytical function for the volume dependence of the short-range force constant. A critical analysis of the isothermal EOSs: Murnaghan EOS, Vinet EOS, and the new EOS derived here, is presented by investigating the pressure-volume data for rare gas solids, metals and minerals. It is found that the results obtained from the new EOS are in good accordance with the corresponding values obtained from the Vinet EOS and with experimental data for all the solids up to very large compressions. On the other hand, the Murnaghan EOS is less successful at high pressure in most cases.

Series solution for a steady flow of a third grade fluid between two porous walls is given by the homotopy analysis method (HAM). Comparison with the existing numerical solution is shown. It is found that, unlike the numerical solution, the present series solution holds for all values of the material parameter of a third grade fluid.

We discuss the relevance of soliton theory to the modeling of tsunami waves. Our analysis shows that for the two most devastating tsunamis of the last century, the 2004 Bay of Bengal and the 1960 Chile tsunami, the propagation distances were too short for soliton dynamics to apply. Thus the shallow water theory is appropriate for the modelling of tsunamis

Within a simplified atmospheric model the greenhouse effect is treated by analytical methods starting from physical first principles. The influence of solar radiation, absorption cross sections of the greenhouse molecules, and cloud formation on the earth’s temperature is shown and discussed explicitly by mathematical formulae in contrast to the climate simulations. The application of our analytical results on the production of 20 · 10 9 t of CO 2 per year yields an enlargement of the earth’s surface temperature of 2 . 3 · 10 −2 °C per year in agreement with other estimations

The generalized Maxwell model is formulated as a nonlinear relaxation equation for the symmetric traceless stress tensor. The relaxation term of the equation involves the derivative of a potential function with respect to the stress tensor. Two special cases for this potential referred to as “isotropic” and “anisotropic” are considered. In the first case, the potential solely depends on the second scalar invariant, viz. the norm of the tensor. In the second case, also a dependence on the third scalar invariant, essentially the determinant, is taken into account in analogy to the Landau-de Gennes potential of nematic liquid crystals. Rheological consequences of the model are presented for a plane Couette flow with an imposed shear rate. The non-Newtonian viscosity and the normal stress differences are analyzed for stationary solutions. The dependence on the model parameters is discussed in detail. In particular, the occurrence of a shear-thickening behaviour is studied. The possibility to describe substances with yield stress and the existence of non-stationary, stick-slip-like solutions are pointed out. The extension of the model to magneto-rheological fluids is indicated.

Cycle-to-cycle variations of maximum pressure in a diesel engine are studied by using the methods of recurrence plots and recurrence quantification analysis. The pressure variations are found to exhibit strong periodicities in low frequency bands and intermittent oscillations at higher frequencies. The results are confirmed by wavelet analysis.

We report on a comparative study of the nuclear quadrupole interaction of the nuclear probes 180 m Hf and 181 Hf(β − ) 181 Ta in HfF 4 ・ HF ・ 2H 2 O using time differential perturbed angular correlations (TDPAC) at 300 K. For the first probe, assuming a Lorentzian frequency distribution, we obtained ω Q = 103(4) Mrad/s, an asymmetry parameter η = 0 . 68(3), a linewidth δ = 7 . 3(3 . 9)%, and full anisotropy within experimental accuracy. For the second probe, assuming a Lorentzian frequency distribution, we obtained three fractions: (1) with 56 . 5(7)%, ω Q = 126 . 64(4) Mrad/s and η = 0 . 9241(4) with a rather small distribution δ = 0 . 40(8)% which is attributed to HfF 4 ・ HF ・ 2H 2 O; (2) with 4 . 6(4)%, ω Q = 161 . 7(3) Mrad/s and η = 0 . 761(4) assuming no line broadening which is tentatively attributed to a small admixture of Hf 2 OF 6 ・ H 2 O; (3) the remainder of 39 . 0(7)% accounts for a rapid loss of anisotropy and is modelled by a perturbation function with a sharp frequency multiplied by an exponential factor exp( − λ t ) with λ = 0 . 55(2) ns −1 . Whereas the small admixture of Hf 2 OF 6 ・ H 2 O escapes detection by the 180 m Hf probe, there is no rapid loss of roughly half the anisotropy as is the case with 181 Hf(β − ) 181 Ta. This loss could in principle be due to fluctuating electric field gradients originating from movements of nearest neighbour HF adducts and/or H 2 O molecules after nuclear transmutation to the foreign atom Ta which are absent for the isomeric probe. Alternatively, paramagnetic Ta ions could lead to fluctuating magnetic dipole fields which, when combined with fluctuating electric field gradients, could also lead to a rapid loss of anisotropy. In any case, Ta is not an “innocent spy” in this compound. Although 180 m Hf is not a convenient probe for conventional spectrometers, the use of fast digitizers and software coincidences would allow to use all γ -quanta in the stretched cascade which would greatly improve the efficiency of the spectrometer. 180 m Hf could also serve as a Pu analogue in toxicity studies.

Residual stresses were determined in magnetron-sputtered Ag thin films of 400 nm thickness by asymmetric Bragg scattering. The corresponding cos 2 α sin 2 ψ plots were nonlinear which indicates a strong residual gradient along the depth of the samples. The in-plane stress was highly compressive at the sample surface and became tensile at the interface. The out-plane stress was compressive and reached its maximum at the sample interface. The stress gradient changed significantly with postannealing temperature. A Young’s modulus of E = 83 GPa and a Poisson ratio of ν = 0 . 3 were measured by surface acoustic wave dispersion.

Cu 2+ -doped K 3 H(CO 3 ) 2 , Rb 2 CO 3 , and Rb 2 KH(CO 3 ) 2 single crystals were investigated by electron paramagnetic resonance (EPR) spectroscopy. The EPR spectrum of K 3 H(CO 3 ) 2 indicates two different sites for Cu 2+ at room and at low temperatures. But the signals for the two sites overlap at 318 K which is attributed to a disorder phase transition. Each of the other compounds exhibits one site. The Cu 2+ ion seems to substitute the K + and Rb + ions and the charge compensations are fulfilled by the proton vacancies in K 3 H(CO 3 ) 2 , and another K + and Rb + in the other compounds. The spin Hamiltonian parameters g and A for each compound are determined and discussed.

An FT-IR spectrum of trimethylphenyl ammonium chloride (TMPAC) has been recorded in the region 4000 - 400 cm −1 . The optimized geometry and vibrational spectrum TMPAC in the ground state have been calculated by using ab initio Hartree-Fock (HF) calculations and the density functional method B3LYP with the 6-31G (d) basis set. The obtained vibrational frequencies and optimized geometric parameters (bond lengths and bond angles) were in very good agreement with the experimental data. The comparison of the observed and calculated results for the vibrational frequencies of TMPAC exhibited that the scaled B3LYP method is superior compared to the scaled HF method. Furthermore the calculated infrared and Raman intensities are also reported.

The semilocalized approach to chemical reactivity suggested recently is overviewed with respect to both theory and applications. The principal attention is paid to formulation of a common selection rule for organic reactions and to demonstration of its validity to various heterolytic (i. e. nucleophilic and electrophilic) and pericyclic processes. The total energy of the whole reacting system ( E ) is represented in this approach in the form of power series with respect to all interfragmental interactions (fragments coincide with individual chemical bonds, phenyl rings, etc.). For any reaction, a certain decisive k -th-order energy correction E ( k ) may be revealed, the sign of which depends on the actual way of the process. The allowed and forbidden reactions are then defined as those described by negative (stabilizing) and positive (destabilizing) corrections E ( k ) , respectively. The condition which ensures the negative sign of E ( k ) resolves itself into a universal requirement of coinciding signs of the principal direct and indirect interactions of basis orbitals localized on separate fragments (e. g. bond orbitals). This result forms the basis for the above-mentioned selection rule. Allowed (forbidden) ways of heterolytic reactions are exemplified by the back (frontal) attack of a nucleophile upon a substituted alkane in the S N 2 process, the meta ( ortho , para ) attacks of electrophile upon the pyridine molecule, the addition of electrophile to the C β (C α ) atom of a donor-containing derivative of ethene (D-C α H=C β H 2 ), the trans ( cis )-β -elimination processes of substituted alkanes, etc. Application of the same rule to pericyclic reactions is demonstrated to yield predictions coinciding with those of other approaches including the famous Woodward-Hoffmann rule.

The approximate and/or exact solutions of the generalized Klein-Gordon- and sine-Gordon-type equations are obtained. We introduce a new type of initial conditions to extend the class of solvable problems