Sentence examples for the formulations below from inspiring English sources

(The literature on supervenience contains several other variations; see the entry on supervenience for an excellent introduction, from which this entry adopts some of the formulations below.

The formulation below is directly applicable and extensible to a wide range of groundwater dewatering and remediation problems.

The formulations are shown below.

If links (_mathrm{new}) is less than the total number of nodes existing in the network at the time of introduction of a node, then the existing nodes are chosen for a link probabilistically according to the formulation explained below.

For each grid the probabilities are calculated via the formulation given below by Eqs. 1, 2, 3, and 4. For the 1st grid 20 binding energies are available, while for the remaining grid pairs 400 binding energies are available; the corresponding statistical weight matrix is given in Eq.1 and 2, respectively.

A similar observation can be made in relation to the formulations of the theorems given below.

The robust formulation below tolerates noise in the sample measurements m i and subsequently in the reduced representations x i, improving the stability of these estimates.

The last equality is a consequence of the LS theory and will be used in the NLS formulation below.

The formulations and details of the model are in the methods section below.

The formulations are different.

The formulations are comparable.