The Singularity of Kontsevich’s Solution for QH<Superscript>*</Superscript>(CP<Superscript>2</Superscript>)

In this paper we study the nature of the singularity of the Kontsevichs solution of the WDVV equations of associativity. We prove that it corresponds to a singularity in the change of two coordinates systems of the Frobenius manifold given by the quantum cohomology of CP². Mathematics Subject Classifications (2000) 53C99, 32D99, 14N35, 88A99.     

An Example of Embedded Singular Continuous Spectrum for One-Dimensional Schrödinger Operators

We present a new example of a potential such that the corresponding Schrödinger operator in the halfaxis has singular continuous spectrum embedded in the absolutely continuous spectrum. The singular part is supported in an essentialy dense set. This generalizes a result of C. Remling [3].Mathematics Subject Classification: 34L40, 81Q10.     

New Soliton-like Solutions for (2+1)-Dimensional Breaking Soliton Equation

The (2+1)-dimensional breaking soliton equation describes the interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. In this paper, with the aid of symbolic computation, six kinds of new special exact soltion-like solutions of (2+1)-dimensional breaking soliton equation are obtained by using some general transformations and the further generalized projective Riccati equation method.     

Exact Solutions for a Nonisospectral and Variable-Coefficient KdV Equation

The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilinear transformation from its Lax pairs and find solutions with the help of the obtained bilinear transformation.     

New Explicit Rational Solitary Wave Solutions for Discretized mKdV Lattice Equation

In this letter, we study discretized mKdV lattice equation by using a new generalized ansatz. As a result,many explicit rational exact solutions, including some new solitary wave solutions, are obtained by symbolic computation code Maple.     

Thermodynamic properties of the mixtures of some ionic liquids with alcohols using a simple equation of state

In the present work, we have used a simple equation of state called the GMA EoS to calculate the density of three ionic liquid mixtures including 1-butyl-3-methylimidazolum hexafluorophosphate, [BMIM] [PF₆] + methanol, 1-butyl-3-methylimidazolum tetrafluoroborate, [BMIM] [BF₄] + methanol, and [BMIM] [BF₄] + ethanol at different temperatures, pressures, and compositions. The isothermal compressibility, excess molar volumes, and excess Gibbs molar energy of these mixtures have been computed using this equation of state. The values of statistical parameters show that the GMA EoS can predict these thermodynamic properties very well within the experimental errors. The results show that isothermal compressibility of ionic liquids is lower than alcohols and the effect of temperature and pressure on the isothermal compressibility of ionic liquids is lower than alcohols. The excess molar volumes and excess molar Gibbs energy for these ionic liquid mixtures with alcohols are all negative at various temperatures and pressures over the whole composition range. The results have been interpreted in terms of intermolecular interactions and structural factors of the ionic liquids and alcohols.     

Annular billiards with position-dependent mass

We present exact solutions for the position-dependent mass Schrödinger equation (PDMSE) in an annular billiard with impenetrable walls where the mass is allowed to depend on the radial position, i.e., μ=μ(r)$\mu =\mu \left(r\right)$. Exact wavefunctions are calculated explicitly for four mass distributions.     

An Exact Solution to the Two-Particle Boltzmann Equation System for Maxwell Gases

An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equation is also given.     

Bound states of the Klein－Gordon and Dirac equations for scalar and vector harmonic oscillator potentials

We give the exact bound states of the Klein－Gordon and Dirac equations with equal scalar and vector harmonic oscillator potentials.