A method for solving singularly perturbed systems containing singular manifolds

Summary A method of obtaining the asymptotic solution of a singularly perturbed system containing singular manifolds is presented. Some sufficient conditions for the convergence of the solutions to the stable singular manifolds are provided. An example from magneto-hydrodynamics, showing interesting properties of the solution, is given.     

On the Bose-Einstein condensation of free relativistic bosons with or without mass

The Bose-Einstein condensation of free relativistic particles [=(M ² c ⁴ +c ² p ² ) ^1/2 –Mc ² ] is studied rigorously. For massless bosons (=cp), the condensation transition of third (second) order occurs in2 (3) dimensions (D). The molar heat capacity follows the T ² (T ³ ) law below the condensation temperature T_c [k ^B T_c=(2 ² c ² n/1.645) ^1/2 [( ^{2 3} c ³ n/1.202) ^1/3 ], reaches4.38 (10.8) R at T=T_c, and approaches the high-temperature-limit value2 (3) R with no jump (a jump equal to6.75R) in2 (3)D. For finite-mass (M) bosons, the phase transition occurs only in3D with the condensation temperature T_c always smaller than that of the corresponding nonrelativistic bosons [=(2M) ^–1 p ² ]. If the mass M is reduced to zero, the condensation temperature T_c grows monotonically and reaches eventually that of massless relativistic bosons. This mass-dependence of T_c is therefore distinct from the case of nonrelativistic bosons, where T_c grows to infinity as M 0. A brief discussion is given for a possible connection with the normal-to-super transition of the independently moving Cooper pairs (bosons).     

Gauge and Bäcklund transformations for the generalized sine-Gordon equation and its η-dependent modified equation

We study the generalized sine-Gordon hierarchy and its associated-dependent modified sine-Gordon hierarchy. Two Bäcklund transformations for these two families are constructed. One of them is a generalization of the Bäcklund transformations of Wadatiet al. and the other one is new. Gauge transformations of a relevant AKNS system are employed to reduce the integration of these equations via the Bäcklund transformations to quadratures. Three generations of explicit solutions of the sine-Gordon equation are presented.     

Preparation and characterization of spherical V2O3 nanopowder

V₂O₃ nanopowder with spherical particles was prepared by reducing pyrolysis of the precursor, (NH₄)₅[(VO)₆(CO₃)₄(OH)₉]·10H₂O, in H₂ atmosphere. The thermolysis process of the precursor in a H₂ flow was investigated by thermogravimetric analysis and differential thermal analysis. The results indicate that pure V₂O₃ forms at 620°C and crystallizes at 730°C. The effects of various reductive pyrolysis conditions on compositions of V₂O₃ products were studied. Scanning electron micrographs show that the particles of the V₂O₃ powder obtained at 650°C for 1 h are spherical about 30 nm in size with more homogeneous distribution. Experiments show that nanopowder has larger adsorption capacity to gases and is more easily reoxidized by air at room temperature than micropowder. Differential scanning calorimetry experiment indicates that the temperature of phase transition of nano-V₂O₃ powder is −119.5°C on cooling or −99.2°C on heating. The transition heats are −12.55 J g⁻¹ on cooling and 11.42 J g⁻¹ on heating, respectively.