Sentence examples for following representation of from inspiring English sources

Denis et al. (2011) showed the following representation of the G-Brownian motion (cf. Theorem 52).

Use the following representation of the function σ S Open image in new window.

A second approach to generating bivariate P(IV) distributions makes use of the following representation of a P(IV) variable.

Based on the wavelet packet decomposition, we obtain the following representation of f0 t): f 0 t = ∑ n c − m, n ϕ − m, n t + ∑ j ∈ I ∑ n d j, n Ψ j, n t + ∑ j ∉ I ∑ n d j, n Ψ j, n t = f trend t + f dist t + e t, (6).

Then, in accordance with the introduced states of the coefficients from Equation 6, we obtain the following representation of geomagnetic field variations: f t = f trend t + ∑ j, n F 0 d j, n Ψ j, n t + ∑ j, n F 1 d j, n Ψ j, n t + ∑ j, n F 2 d j, n Ψ j, n t + e t, (8).

From [13], with another mixing condition AIM u n ) which is satisfied for functions of stationary Markov samples (e.g., degree samples) the following representation of EI holds, {lim}_{mathit{ntoinfty}}mathrm{P}{M_{1,p_{n}}le u_{n}|D_{1} >u_{n}}leq theta, (17).

One easily obtains the following representations of the Ioachimescu constant: ell= int_{0}^{infty}frac{1-x+lfloor x rfloor}{2(1+x)^{3/2}},dx and ell=2-sum_{k=1}^{infty}frac{1}{(sqrt {k}+sqrt{k-1})^{k}sqrt {k}}.

Von Heusinger and Egli (2000a) list, among others, the following: representations of questions (Reinhart, 1992), specific indefinites (Reinhart 1992 19977; Winter 1997), E-type pronouns (Hintikka and Kulas 1985; Slater 1986; Chierchia 1992, Egli and von Heusinger 1995) and definite noun phrases (von Heusinger, 1997, 2004).

Proof By applying the standard procedure of the Fourier method, we obtain the following representation for the solution of (1 - 3) for arbitrary p ( t ) ∈ C [ 0, T ] : u ( x, t ) = ∑ n = 1 ∞ [ φ n e − ( 2 π n ) 2 t − ∫ 0 t p ( s ) d s + ∫ 0 t F n e − ( 2 π n ) 2 ( t − τ ) − ∫ τ t p ( s ) d s d τ ] sin ( 2 π n x ).

Hence using (3.10) or (3.11), with the addition of the term ( ∑ k = 0 3 a k Re z k ) ( P ), we have the following representation for the solution u of problem (2.1), (2.2) at any P ∈ Π 01 u = S 4 u ˜ + ( ∑ k = 0 3 a k Re z k ) ( P ) + O ( h 4 ).

The following representation in terms of the spectral family is of interest in itself.