Sentence examples for are uniformly valid from inspiring English sources

He also resolved a long-standing problem concerning the theory of asymptotic solutions of ordinary differential equations (of higher order than 2), which are uniformly valid around turning points.

Thus, higher-order terms, when sensibly small based on scaling arguments, may be formally neglected in all regions of the flow, and the mathematical simplifications afforded by the film equation containing the critical point are uniformly valid.

Moreover, we show that the obtained approximate solutions are valid for the whole solution domain and the approximations are uniformly valid not only for small parameters, but also for very large parameters.

The terms retained in this process assure that for moderate Reynolds numbers the resulting equations are uniformly valid between the body surface and the shock, which may be treated as a thin discontinuity.

A solution which is uniformly valid throughout the duct is also derived.

This quite simple approach is uniformly valid in the whole range of subsonic Mach numbers and frequencies.

This primitive-variable formulation is uniformly valid in wet and dry regions with embedded wetting and drying fronts.

It is further proved that the result obtain herein is uniformly valid for any combination of rigid and dynamically free boundaries.

The expansion is shown to be uniformly valid over all angles of incidence and values of surface impedance, including the limiting cases of zero and infinite impedance.

Finally, the third-order asymptotic solution obtained is uniformly valid in contrast with early works containing resonant terms presented by Wiegel [1964. Oceanographical Engineering. Prentice-Hall, New Jersey, pp. 37 40] (Eqs. (B.1) and (B.2) in Appendix B) or Chen et al.[2006. Theoretical analysis of surface waves shoaling and breaking on a sloping bottom. Part 2 nonlinear waves.

Then for all and Also, is uniformly valid on.