We will call a smooth path (f t ) in Ham (n ) joining the identity to some element f ∈ Ham (n ) a Hamiltonian isotopy.
In addition, interest in performing SciComm tasks significantly predicted intention to stay in a research career (path f = 0.13, p < 0.05).
We first show that if the forward path F reaches a dead end, then so does the reverse path R, and in that case, a cycle is closed (Lemma 1).
For example, h x, t) = (1 − t)f(x) + tg(x) is a homotopic function for paths f and g in part A of the figure; the points f(x) and g(x) are joined by a straight line segment, and for each fixed value of t, h x, t) defines a path joining the same two endpoints.
With declines in social competence, decreased productivity, and mental health, these can interact among themselves to further exacerbate these declines (Paths F).
From this, we get, as an example, a finite path in F 3 as ∞ → 1 1 → 4 3 → 11 8. Lemma 3 Let T be as above, then the sequence { T m ( 1 ) } is strictly monotone increasing and T ( 1 0 ) → T ( 1 1 ) → T 2 ( 1 1 ) → ⋯ → T m ( 1 1 ) → ⋯ is an infinite path in F 3 in increasing order.
Corollary 1 1 0 → 0 + 1 1 → 1 + 1 3 → 1 + 3 8 → T ⋯ → T 1 + a n b n → T 1 + b n 3 b n − a n → T ⋯ is an infinite path in F 3, and all vertices of the path are smaller than 5 − 5 2, and the natural numbers x and y in the vertex 1 + x y are such that 5 x 2 + 4 and 5 y 2 + 4 are square.
It is the length of the shortest path between f and f d in the Cayley graph for Σ T, where this path corresponds to σ1⊗⋯⊗ σ m.
On one-way ANOVA there was no difference between groups in path length (F = 1.99, P > 0.05), number of lines crossed (F = 1.59, P > 0.05), speed (F = 1.99, P > 0.05), exploration levels (F = 0.29, P > 0.05), grooming events (F = 0.85, P > 0.05), or the ratio of time spent in the centre of the arena to the periphery (F = 1.29, P > 0.05) (data not shown).
