The Sherman equations as a nonlinear Perron eigenvalue problem

When a chemical sample composed of N elements is analyzed using sequential selective excitation by a tunable polyenergetic X-ray beam and selective measurement of the characteristic X-rays, the production of secondary fluorescence does not interfere with the measurements. This experimental situation leads to a particular case of the Sherman equations which can be written as a set of non-linear equations. The same kind of equations are also obtained when we excite a chemical sample with a polyenergetic X-ray beam and neglect the production of secondary fluorescence. This set of equations can be regarded as a non-linear eigenvalue problem. A non-linear extension of the Perron Frobenious theorem ensures that there is one and only one physically acceptable solution, and also leads to a method to obtaining it. The propagation off measurements errors of sample fluorescence to errors in the calculated sample concentrations, has been simulated, and the results show that the solution is well conditioned. The case of production of secondary fluorescence can not be treated, in general, as a nonlinear Perron eigenvalue problem, but it has been shown that it is rather plausible that Sherman equations corresponding to the actual chemical elements and that include the production of secondary fluorescence have one and only one physically acceptable solution. An exhaustive search could elucitate the existence and unicity of solutions for the equations corresponding to the actual chemical elements.