The phrase "and hence the solution" is correct and usable in written English.
It can be used to indicate a conclusion or result that follows logically from the preceding information.
Example: "The data indicates a significant increase in sales, and hence the solution is to expand our marketing efforts."
Alternatives: "therefore the solution" or "thus the solution".
The variable coefficients, the source term, and hence the solution itself and its derivatives may be discontinuous across the interfaces.
The PT content of the grown crystals, and hence the solution, plays a significant role in the growth mechanism and result.
However, the C-field is underresolved if the sub-filter Damköhler number is not much smaller than unity, and hence the solution of the C-equation cannot be expected to produce the correct flame propagation speed.
And not before time, in the view of experts who see something crazy about the way many food-aid efforts are now conceived and executed.One mistake, arguably, is the very idea of defining the main problem as massive hunger, and hence the solution as providing food by any means necessary.
The discrete effects in this lattice statics model are crucial for proper simulation of the influence of strain for thin film epitaxial growth, but the size of the atomistic systems of interest is in general quite large and hence the solution of the discrete elastic equations is a considerable numerical challenge.
and hence the solution reduces now to (24).
The balancing procedure gives M = 2 and N = 4, and hence the solutions of (5a - 5b) are of the form F ( z ) = A 0 + A 1 H + A 2 H 2, (7a).
In this method, unknowns of the problem were expanded in Maclaurin's series in the thickness coordinate and hence the solutions were obtained in terms of unknown initial functions on the reference plane (Vlasov 1957).
(14) Equation (14) is now in the form of the 2-dimensional wave equation in elliptic coordinates, and hence, the solutions (W_{n}) are in the form of the Mathieu and modified Mathieu equations as we mention in Section 2.2 for (q=-is_{n}=-inomegarho c^{2}{ 4mu}^{-1}).
Each node in the BP tree represents a newly generated solution, and hence the entire solution space comprises several sub-spaces defined by BP nodes distance measure, i.e. Chebyshev distance.
Since edge y is an outgoing edge of edge x in driving direction (see Fig. 2), this works for d = (x) and the tail solution (y), and hence the new solution (x,y) is added to the tentative set of solutions X.
