Sentence examples for notation as above from inspiring English sources

With notation as above, G ⊂ G θ | 1 ⊂ G θ | 2 ⊂ ⋯.

Using the same notation as above, we denote this reference point by (A^{Jm}).

Using the same notation as above, the discrete S-transform [24] can be represented as S n, k = ∑ m = 0 M - 1 z ( m ) k 2 π exp - ( m - n ) 2 k 2 2 exp ( - j 2 π k m / M ) (41).

In bra ket notation, as above, this equality implies the resolution of the identity: I = \sum |\phi_n\rangle\langle\phi_n|.

Using the same notation as above, we have the inequalities (3) t u 0 d > t { u 0, v 0 } ≥ t v 0 c ≥ t u 1 d > t { u 1, v 1 } ≥ … ≥ t v k − 1 c ≥ t u 0 d, which is a contradiction.

For this we use the same notation as above for the basic linear functions, namely (mathrm {i} psi _k) and (phi _k,).

With notations as above, assume that the set (C k)) of (k -rational points is infinite; (vert _{ell }mathrm{Br}(k -rationalur}}/iota _k(_{ell }mathrm{Br}(k)) vert =: M < infty ).

Notations as above, for any interior point P of (triangle ABC), we have R_{1}R_{2}R_{3} geq R-PO)^{2}cdot(R+PO), (8) and the equality holds only when P coincides with the circumcenter O.

Retaining the definitions and notations as above, we have begin{gathered} (1) quad|Q|=|T|^{2}; (2) quad T^{-1}=widetilde{T}^=widetilde{A}^,qquad widetilde {T}^{-1}=T^=A^; (3)quad|Q|=sup_{|x|=1}bigl{ big|(Qx,x) big|bigr}.

Indeed, in this region, equality of water potentials implies that M cell) iRT = M bath) i'RT (the same notations as above, for cell and bath, i and i' representing the respective Van't Hoff factors), and hence V (cell ) = i / i ′ × n (cell ) / M (bath ) + b.

Formally, the (restricted) input can be described as a six-tuple Q : = ( S, E, F, k I, co ), with the same notations as above.