| When thinking about and solving some problems, it can give ones the new revelation and open a new situation from the view of inversion. Combining some practical physics problems with inversion will open many interesting research field and can propose many new physics problems. One of objects of this thesis is to apply some inversion methods into variable of practical physics problems. One side some new inversion methods are proposed, on the other side, these methods are used to solve many practical problems and some meaning and valuable results. Because in the application, the inversion methods will offer meet some difficulties that sometimes can't be eliminated, for example the existence and uniqueness of inversion solution, instability, unintegrality of information, errors from extracting information and so on, it is impossible to totaly avoid getting some unsatisfying results. So it also is an attention of this thesis how to eliminate or relax these difficulties.In this thesis, the following class of inversion problems and relate topics are studied:1.The inverse problem of black-body radiation (BRI);Because the frontier problems in these inverse problems are the unique existence theorem, instability problem and the unification of the theories. We proposed the improved unique existence theorem and the asymptotic behavior control (ABC) condition. In this study, we developed the universal function set (UFS) method, based on found the asymptotic behavior control (ABC) condition and asymptotic behavior control (ABC)theory. Although one can not solve the instability (illness or ill-posed) problem, one can relax the difficulty due to instability of the solution. The numerical calculation is realized. Its unified theory is developed, derived Chen's formula from Dai's exact solution. Some concrete exact solutions are obtained by this modified Mobius inversion formula. The convergence, absolute and homogeneous convergence of the series solution were proved firstly.2.The inverse problem of transmissivity (TI):It is significant to propose some new inverse problems. In this study, We proposed a new type of inverse problem-the inverse problem of transmissivity (TI). Its unique existence theorem has been proved. Its inversion solution is also derived from Dai's formula by applying a new representation of inverse Laplace transform. Some concrete exact solutions are also obtained by this formula.3. The inverse problem of relaxation of flux:The studies of high Tc superconductivity are significant in both theories and applications. The study of flux dynamics and the flux relaxation are very interesting. There are many relaxation theories developed. How to distinguish and exam them in practice are of importance. It is very interesting to find the maximum current, and the characteristic structures of different theories in the super-early time. We developed an inversion method to study these problems. We finally found the maximum current and their interesting structures. When the time of the measurements becomes shorter, our theory has some predictions and can be used to check these theories.4. Bose-Einstein condensation in confined geometry studied by the thermodynamic mapping (inversion).Many years ago, an idea of thermodynamic mapping and a relation were proposed, which map the thermodynamic behavior at low temperature into that at high temperature. Recently the properties (such as the shift of A lines) of liquid helium in confined geometry and the studies on Bose-Einstein condensation become the hot fields, we try to study these problemsby our thermodynamic mapping relations. We obtained the thermodynamic properties by these relations, and it is shown that the mapping relations are powerful.5. The spacial and momentum distributions are the main signatures of BEC in modern experimental studies. In this thesis, the spacial and momentum distributions of Bose-Einstein condensation are studied by generalizing the Bloch sum. The results supported the experimental observations.
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