The bound becomes an equality when the decoder corrects all combinations of errors up to and including t errors, but no combinations of errors greater than t (i.e., bounded distance decoder)[29, 30].
The previous derivation forces the last inequality involved to become an equality.
Using the four inequalities of Equations 28, 29, 31, and 32 above, we have: -(ν max m -1-ν min m -1)ν i p 0 i ≤ [(W m - V m ) p o ] i ≤ (ν max m -1-ν min m -1)ν i p 0 i, Or equivalently: |[(W m - V m ) p o ] i | ≤ (ν max m -1-ν min m -1)ν i p 0 i, where the inequality above becomes an exact equality when all bins have the same neutrality.
According to (71), in order to satisfy the condition (73a), the beamforming matrix V n ⋆ should be rank one for all n ∈ N, i.e., when the first inequality of (71) becomes an equality.
When for all i, α i =1/n, Eq. 29 becomes an equality.Observing that αmax≤1, the following corollary is immediate.
Inequality (2.1) or (2.2) becomes an equality for all (x,yin mathbb{R}^) if (t=-1), (-1/2), or 0; otherwise inequality (2.1) or (2.2) becomes an equality only when (x=y).
Their cause became an equality of grievance.
For norm resolvent continuous operator functions we show that the variational inequality becomes an equality.
Obviously, inequality (15) becomes an equality for k = 1.
