Sentence examples for formal series from inspiring English sources

The formal series (hat{u} epsilon )) formally solves ( 1 ) where the analytic forcing term (f^{mathfrak {d}_{p}} t,z,epsilon)) is replaced by its asymptotic expansion (hat{f} t,z,epsilon) inmathbb{F}left[hspace{-1.8pt}left[ epsilonright]hspace{-1.8pt}right]) of Gevrey order (1/k_{1}) (see Lemma  11 ).

Just as it becomes a formal series of male-female duets, it changes into something else.

As the only formal series of collected law cases available at the time, his reports formed the main source for the citation of cases for many years.

The festival will also present a number of other Iranian and Middle Eastern groups but has not linked them into any formal series.

It overlaps with the last part of the defined event for Hurricane Irene, and could not be included in the formal series.

Considering the stable dynamics of triangular libration points, the analytical solutions of the motion around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity of the primary, the long, short and vertical periodic amplitudes.

Moreover, the formal power series (hat{u} t,z,epsilon)) constructed in Theorem  4 formally solves equation (118).

This result is based on the existence of a common formal power series (hat{f} t,z,epsilon)) which is the q-Gevrey asymptotic expansion of order (1/k_{1}), seen as a formal power series in the perturbation parameter ϵ with coefficients in a certain Banach space, of every (f^{mathfrak{d}_{p}}) on (mathcal{E}_{p}).

This change is the formal power series expansion of the inverse of the normalization map.

In certain cases, elements of Lin(A, K) correspond to K-linear extensions of formal power series.

The existence and uniqueness of the ambient metric at the formal power series level is treated in detail.