In marked contrast, away from equilibrium, Equation 7 shows that every spanning tree rooted at that microstate is required.
Conversely, putting t := HC Y), by definition of HC Y) the graph G t (Y) is connected and every spanning tree T of G t (Y) is also a spanning tree of G* Y) with ω max (T) ≤ t = HC Y).
For every spanning tree T of G*(Y), that is, a subgraph of G* Y) that is a tree and that contains all vertices of G* Y), let ω max (T) be the maximum of the edge weights over all edges of T. We claim that Indeed, since every spanning tree T of G*(Y) is a connected subgraph of G* Y), we must have HC(Y) ≤ ω max (T).
Note that for non-repeated internal knots the support of a B-spline basis function of order p is always p + 1 knot spans, and every knot span is shared by p + 1 B-spline basis functions, see Fig. 1 where we plot B-spline basis functions for open, uniform knot vector { 0, 0, ⋯, 1 16, 1 8, ⋯, 7 8, 15 16, ⋯, 1, 1 } with order 2 and 8.
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This shows that every vector spanned by ∂ φ σ 2 Δ 1 t and ∂ φ Δ 2 s is a tangent vector of the curve Γ on S at P. □.
This shows that every vector spanned by ∂ φ Δ 1 t and ∂ φ σ 1 Δ 2 s is a tangent vector of the curve Γ on S at P. ii.
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