a cycle is a path such that the first node appears twice and the other nodes appear exactly once; and a simple cycle is a cycle such that the first and last nodes are the same.
A cycle is a path such that the first vertex of the path corresponds to the last one.
This graph presents several advantages: (i) its tree shape is a very simple structure easy to manipulate and visualize, and (ii) every node is connected by a path such that we can access to the time order relation between each gene.
Let (alpha : [0,1] rightarrow {varphi ne 0}) be a continuous path such that (alpha (0) = x) and (alpha (1) = y).
Let (alpha : [0,1] rightarrow {varphi ne 0}) be a continuous path such that (alpha (0) = x) and (alpha (1) = z).
Let (alpha : [0,1] rightarrow {varphi ne 0}) be a continuous path such that (alpha (0) = z) and (alpha (1) = y).
A spacetime is said to be singular if it possesses a path such that the specified parameter associated with that path cannot increase without bound as one traverses the entirety of the maximally extended path.
If c 0 = c 0 ∗, then γ ( t ) = t t 0 y ϵ, with 0 ≤ t ≤ 1, is a path in Γ such that max t ∈ [ 0, 1 ] F ( γ ( t ) ) = c 0. Consequently, either F ′ ( t ¯ y ϵ ) = 0 and we are done, or γ can be deformed to a path γ ˜ ∈ Γ with max t ∈ [ 0, 1 ] F ( γ ˜ ( t ) ) < c 0 and we get a contradiction.
We have to show that which is fulfilled if there exists a path such that (5.4).
Finally, we prove that y 0 ≠ 0. It is equivalent to proving that there is a path h ∈ Γ such that for all x ∈ Q, φ ( h ( x ) ) < 0 = φ ( 0 ).
