Sentence examples for I admits from inspiring English sources
We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption.
But since (rho :M(I) rightarrow I) is a prefibration, the embedding (varepsilon (i):M(I _i rightarrow M(I)) of the fiber over the initial object i admits a right-adjoint functor (varepsilon (i ^dagger ), and the pullback functor ((varepsilon (i ^dagger )^*) is then left-adjoint to (varepsilon (i)^*).
Then I admits a global minimum.
Moreover, assume that the full embedding (gamma :I' rightarrow I) admits a left-adjoint functor (gamma _dagger :I rightarrow I'), with the adjunction map (a:mathsf{Id}rightarrow gamma circ gamma _dagger ) compatible with the prefibration (mathcal {C}' rightarrow I) in the sense of Definition 4.17.
It is clear that γ ( S t k r k ( k ) ) = k, so c k ≤ − ε k < 0. Finally, by Lemma 18 above, we can apply Theorem 17 to obtain that the functional I admits at least k pairs of distinct critical points, and since k is arbitrary, we obtain infinitely many critical points of I.
