Characteristic value problem, generalization

Generalization of the characteristic value problem. The characteristic value problem can be formulated as the quest for the irreducible linear manifolds which are invariant under an operator. The principal result of the spectral theory of normal operators can be formulated, from this point of view, as the statement that all irreducible linear manifolds of normal operators are one-dimensional. Similarly, one can ask for irreducible closed linear manifolds which are invariant under a set of operators. Since a closed linear manifold which is invariant under a set of operators is also invariant under the group or algebra generated by these operators, one is naturally led in this way to a linear manifold which belongs to an irreducible representation of a group or an algebra. 

In general. Equation (1) will not admit of any time-independent solutions except the trivial one 0 = 0. To cast (1) into the form of the more usual characteristic value problem, we introduce a fictitious neutron multiplicity. 

Typical flow rates in FTA vary between 0.5 and 5.0 ml/min per channel, although higher values have also been used. Most of the published work on CL with HA is based on equal flow rates for every stream entering the manifold. Nevertheless, if different rates must be used, deterioration of repeatability and reproducibility might appear due to incomplete mixing and anomalous hydrodynamic characteristics. These problems can be avoided if the general rule that the ratio of the fastest to the slowest flow rate should not exceed the value of 3 or 4 is followed. 

The well-posedness of the two-fluid model has been a source of controversy reflected by the large number of papers on this issue that can be found in the literature. This issue is linked with analysis of the characteristics, stability and wavelength phenomena in multi -phase flow equation systems. The controversy originates primarily from the fact that with the present level of knowledge, there is no general way to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem. The mathematical theory of well posedness for systems of partial differential equations describing dispersed chemical reacting flows needs to be examined. 

---