There are some conditions for this, called positive-definiteness, but they are automatically satisfied by correlation functions obtained from experimental data.
The balance of angular momentum is automatically satisfied by a symmetric CAUCHY stress tensor, in other words, the flux of angular momentum is assumed to vanish.
This second condition is automatically satisfied, by using the geodesic itself, in the Euclidean space (and in every Hilbert space), since the function (xmapsto |x-y|^2) is quadratic, and its Hessian matrix is 2I at every point.
Note that constraint (21) is automatically satisfied by the construction of s, constraint (23) simply drops out since wk,n,s = 0 for k ∉ s and constraint (22) remains only for k ∈ s, but we write it as (27) because we are considering only users that belong to SDMA sets.
Fourth, we show that by representing the unknown axial velocity profile with a suitable Fourier Bessel series, the discharge constraint can be automatically satisfied.
In other words, once observation limitations are respected, limitations on control will be automatically satisfied.
(ii) If F and G are linear functions with respect to their variables, then the calmness condition (4.14) is automatically satisfied in this case by [15].
Secondly, we show that the desired estimate for the generating point set on S2 is automatically satisfied for m sufficiently large by a spherical n-design with m="O(n2) points (if such spherical designs exist) and by a spherical n-design with m="O(n3) points if slightly stronger assumptions are made on the smoothness of the weighted function space.
By design Eq. (8) is automatically satisfied, hence it cannot be used in the process of evaluating the model parameters.
Model., 2011, doi 10.1016/j.mcm.2011.01.036) by significantly weakening the contractive condition involved and by replacing the mixed monotone property with another property which is automatically satisfied in the case of a totally ordered space.
