Sentence examples for later purposes from inspiring English sources

We first introduce the angle between two vector subspaces below, which will be useful for later purposes.

For later purposes we state the following two simple estimates which can easily be deduced from Lemma 2.1(i) and (vi).

Such theories assume that people learn not only because of interest or a feeling of self-determination but also because they consider the learned content to be useful for later purposes or because they want to achieve.

For later purposes, we state the following function spaces: W^{2,2}_{n}(Omega)= biggl{ varphiin W^{2,2}( Omega Big|frac{partial varphi}{partial n}=0 text{ on } overline{Omega} biggr},qquad L^{2}_{0}(Omega)= biggl{ varphiin L^{2}( Omega Big| int_{Omega }varphi,dx=0 biggr}.

For later purposes, we want to note the two leading terms in the perturbations series begin{aligned} E(beta ) = E_0 + a_1beta +a_2beta ^2+text {O}(beta ^3) end{aligned} (2.6 of a simple eigenvalue, (E_0), of (A+beta B) with A and B Hermitian.

The remaining non-fat fraction of the milk samples was frozen and stored for possible later purposes.

Here's a passage I underlined for some unknown later purpose: I didn't just imagine them doing things.

For later purpose note the explicit formula for λ a h c.

For the later purpose, kinetic studies were carried out under different operating conditions (chromate concentration, adsorbent content, and temperature), and adsorption isotherm measurements were performed.

For later purpose, it is useful to express (M_{i}) in terms of the topology of the network: M_{i}=sum_{j=0}^{N-1}T_{ij}.

Temporarily it keeps h(PWD * ) and ue used for later purpose, and then derives ( {C}_1={E}_{bas{p}_{bk}}left CASID leftVert {E}_{bas{p}_{bk}}(UID rightVert right) ) using BAS public key basp bk and sends C1 to BAS.