Sentence examples for of equal dimension to from inspiring English sources

We then simulated a random vector of equal dimension to Φ ⃗ from a LogN (m = − (s Φ (0 ) ) 2 / 2, s = s Φ (0 ) ) distribution where s Φ (0 ) represents the initial value of s Φ and controls the standard deviation of Φ. Next, these random Φ ⃗ variates were rank ordered and assigned to the corresponding gene of the same SCUO rank.

In matrix notation this can be written as: (Z+F e=x (7)In Eq. (7), e denotes a summation vector of dimension (ktimes 1) and for k countries, matrix Z of dimension (ktimes k) includes total domestic and international intermediate demand C&T-flows and matrix F of equal dimension contains total domestic and international C&T-deliveries to final demand.

Recall that P 2 = 1 and the eigenspaces to the eigenvalues +1 and −1 are indeed of equal dimension, as there are exactly 2 2 d − 1 even operators which map the vacuum state Ω into the +1 eigenspace of P. Note that P e q 1 ∧ e q 2 ∧ ⋯ ∧ e q ℓ = ( − 1 ) ℓ e q 1 ∧ e q 2 ∧ ⋯ ∧ e q ℓ.

Let G1and G2be connected, simply connected, nilpotent Lie groups of equal dimension.

[[13], Theorem 3.1] Let G1, G2be oriented, connected, simply connected, nilpotent Lie groups of equal dimension.

Let M1 and M2 be oriented, closed, connected manifolds of equal dimension.

[[14], Theorem 4.9] Let M1and M2be closed oriented infra-nilmanifolds of equal dimension and f, g : M1 → M2continuous maps.

Let f, g : M1 → M2 be continuous maps between closed oriented manifolds M1, M2 of equal dimension.

Let M1and M2be orientable infra-nilmanifolds of equal dimension modeled on connected, simply connected, nilpotent Lie groups G1and G1andpectively.

The following theorem gives a formula for the Lefschetz coincidence number of a pair of continuous maps between nilmanifolds of equal dimension.

We prove practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of continuous maps between oriented infra-nilmanifolds of equal dimension.