Sentence examples for by using the expansion from inspiring English sources

The eigenvalue equation was first derived analytically by using the expansion theorem and then the eigenvalues and eigenvectors were calculated numerically.

Finally, the sound power and radiation efficiency spectra, under the application of two simultaneously applied harmonic forces, are calculated by using the expansion procedure.

For promoting working efficiency by using the expansion energy of the compressed air, we proposed a new type of expansion energy used air-powered vehicle using pneumatic-hydraulic transmission.

We also present the computer program hosphe (v1.02), which solves the self-consistent equations by using the expansion of single-particle wave functions on the spherical harmonic oscillator basis.

By using the expansion of the unknown functions in Fourier series with respect to the angular variable of the natural cylindrical co-ordinates, the system of integral equations is reduced to an infinite set of systems of eight algebraic equations.

spring mass system by two massless equivalent springs with spring constants k v eq,iandk v eq,k, and then the foregoing natural frequencies and normal mode shapes for the bare" beam are in turn used to derive the equation of motion of the "loading" beam (i.e., the bare beam carrying any number of two-d.o.f. spring mass systems) by using the expansion theorem.

By using the expansions of the displacement components, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions that can be used to eliminate all but the four of lowest order and to express higher order expansion functions in terms of those of lowest orders.

In this paper, we present a new version of the iQIST software package, which is capable of solving various quantum impurity models by using the hybridization expansion (or strong coupling expansion) continuous-time quantum Monte Carlo algorithm.

By using the Chapman Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales are obtained.

By using the Taylor expansion for above equation, we have the Equation (11).

We interchange differentiation and the application of F, which can be justified by using the homogeneous expansion of the kernel.