Sentence examples for expanded by using a from inspiring English sources

The radial deflection of the shell is expanded by using a basis of seven linear modes.

Equation 11 was expanded by using the first-order VTS around an initial value n0 of n, and the mean of the training log-spectrum vector μ Tr = E{y Tr } to obtain the following equation: y Te = y Tr + G ( μ y Tr, n 0, n Tr ) + ∇ y Tr G ( μ y Tr, n 0, n Tr ) ( y Tr - μ y Tr ) + ∇ n G μ y Tr, n 0, n Tr ( n - n 0 ), (12).

The structure was solved by the direct methods, expanded by using Fourier transformation techniques [18] and refined by a full-matrix least-square calculation on F 2 with the aid of SHELXL97 program [19].

A possibility of the whey utilization would be expanded by using other microorganisms.

The power series can be further expanded by using Equation 0.314 in [24].

The theory of majorization was enriched and expanded by using these concepts.

The theory of majorization was enriched and expanded by using these concepts [5 15].

The validation range of the method is expanded by using piecewise lines to fit the exponential function.  .

The fundamental solution is expanded into the separable kernel by using the elliptic coordinates and the boundary densities are expanded by using the eigenfunction expansion.

Applications of active materials can be expanded by using robust, non-mammalian muscle cells, such as those of Manduca sexta.

The standard practice of testing these effects was further expanded by using the structural choice modelling (SCM) framework.