Sentence examples for strictly strong from inspiring English sources

If (lambda _{0}>0) and (lambda_{i}geq0) for (i=1,2,ldots,s), then (sum_{i=0}^{s}lambda_{i}varphi_{i}) is strictly strong convex of order m at x̄.

Suppose that (varphi_{0}) is a strictly strong convex function of order m and (varphi_{1},ldotsi_{2},ldots, varphi_{s}) are strongly convex functions of order m at x̄.

(a) φ is strictly strong convex of order m at x̄ if there exists a constant (c>0) such that, for each (xin X) with (xneqbar{x}) and (xiinpartialvarphi(bar{x})), varphi(x -varphi(bar{x -varphilexi,x-bar{x}rangle+c|x-bar{x}| ^{m}.

(b) φ is strictly strong quasiconvex of order m at x̄ if there exists a constant (c>0) such that, for each (xin X) with (xneqbar{x}) and any (xiinpartialvarphi(bar{x})), varphi(x leqvarphi(bar{x}) quad Rightarrowquad langlexi,x-bar{x} rangle +c|x-bar{x}|^{m}< 0.  .

φ is strictly strong convex of order m at x̄ if there exists a constant (c>0) such that, for each (xin X) with (xneqbar{x}) and (xiinpartialvarphi(bar{x})), varphi(x -varphi(bar{x -varphilexi,x-bar{x}rangle+c|x-bar{x}| ^{m}.

If the functions (f_{i}), (i=1,2,ldots,p), are strongly convex of order m at x̄, and (sum_{tinhat{T}(bar{x})}beta_{t} g_{t}) is strictly strong quasiconvex of order m at x̄, then x̄ is a semi-strict minimizer of order m for (P).

The topology of is strictly stronger than the topology of induced by the river metric.

Z is strictly stronger than Mac Lane set theory; the former theory proves the consistency of the latter.

Example 3.14 Observe that, in general, ( F d L ) -convergence is strictly stronger than ( U O F ) -convergence on compact sets.

We now use Theorem 3.2 to show that, in general, ℱ-exhaustiveness istrictlyly stronger than weak ℱ-exhaustiveness.

The paper concludes with an example which demonstrates that the condition of being torsion in the Grothendieck module is strictly stronger than having rational Poincaré series.