Sentence examples for using the expansion of from inspiring English sources

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.

Also, using the expansion of different functions in Bernstein polynomials leads to good numerical results and has a high efficiency in convergence theorems.

and using the expansion of the exponential function as a power series, the sum in Equation (17) can be rewritten as ∑ c = 0 t - i t - i c ( - 1 ) c ∑ j = 0 ∞ ( t - i - c ) j j !

A [ s ℓ / ⋅ ) is the matrix obtained from A by taking rows whose indices are in s ℓ. Using the expansion of the permanent in the above identity, we get (1). □. Identity (1) can also be written in the form of an integral as follows.

In the case where (lambda_{i}^{n-1}vert a_{i}-y_{ell_{i}}vert ^{beta }geqfrac{delta}{2}), using the expansion of Proposition 3.2 and (4.3), we obtain bigllangle partial J u), X_{i} u) bigrrangle leq -c frac{vert a_{i}-y_{ell_{i}}vert ^{beta-1}}{lambda_{i}} + o biggl(frac {1}{lambda_{i}^{gamma}} biggr)+ O biggl(sum _{j neq i} varepsilon_{ij} biggr), (4.9) where γ is any real in ((n-1, min{beta, n})).

Using the expansions of Propositions 3.1, 3.3, and 3.5, for all (i = 2, ldots, p) and any (beta>n-2), we have frac{1}{lambda_{i}^{beta}}= o (varepsilon_{1i} ) quad mbox{as } lambda_{i}rightarrow+infty.

During both world wars millions of women used the expansion of munitions work as a means of escaping domestic service, which paid poor wages and often demanded a seven-day week.

The Bush administration signally failed to use the expansion of Medicare as a lever for introducing structural reforms, such as means-testing.

We thus used the expansion of gliosis and gliotic scar throughout areas devoid of RGC axons as a novel and objective measure of nerve degeneration severity.

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.