Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length.
There exist minimal period-two solutions ( p, q ) ≤ se ( x ¯, y ¯ ) ≤ se ( r, s ) such that ( p, q ) and ( r, s ) lie at the north-west and south-east corners of B ∗, respectively, and ( x ¯, y ¯ ) lies in its interior.
For example, we prove that the difference of any two functions in R c, c ≥ 0, retains its sign on [ 0, T ), and that there exist minimal and maximal solutions v c, min, v c, max ∈ R c for each c ≥ 0, cf. Theorem 5.
(iii) There exist minimal period-two solutions ( p, q ) ≤ se ( x ¯, y ¯ ) ≤ se ( r, s ) such that ( p, q ) and ( r, s ) lie at the north-west and south-east corners of B ∗, respectively, and ( x ¯, y ¯ ) lies in its interior. .
Our main result is that for some subspaces there exist minimal-volume shadows that are far from parallelepipeds with respect to the Banach Mazur distance.
Proof ( a ) Since F is upper semicontinuous, following the idea of Lemma 2.2 in [5], we can easily obtain that there exists one minimal essential set of F ( u ) for each u ∈ H. Now, for each minimal essential set of F ( u ), as Yang and Yu did in [8], we prove that each minimal essential set of F ( u ) is connected.
As presented below the algorithm Minimize, when called on the output of FindAll, returns, if exists, a minimal hyperpath linking a given target to the source.
1. G F is a minimal triangulation of G. 2. Let (T, B) be a clique tree of G F. There exists a minimal separator F ∈ F if and only if there exist two adjacent vertices x and y in T where B x)∩ B y)= F. 3. △ H is a maximal set of pairwise parallel minimal separators of G and G △ H = H.
Hence, for every D ∈ F, there exists a minimal element A ∈ F such that A ⊂ D. We claim that if ∏β∈ΓA β is minimal, then each A β is a singleton.
By Lemma 2.3, there exists a minimal such that.
