We select the patches according to: i) spatial connectivity: we sampled

1-connected and 1-isolated patch each month, so, we chose patches with this characteristic and minor distance to each other; ii) patch size: in each patch, we measured total area and core area (in hectares), mean of degree of isolation (in meters) and connectivity (presence/ absence).

That is expected because 2-connected is a stronger property than

1-connected. What's more, the difference among the three algorithms when k = 2 is in a greater range than when k = 1.

This correspondence yields an equivalence between the homotopy categories of rational

1-connected CW-complexes of finite type and

1-connected cdga's of finite type.

presented necessary and sufficient conditions for 1-covered,

1-connected sensor grid network.

Proof: By Lemma 4.2, if the graph [G.sub.1] [[??].sub.k] [G.sub.2] is [k.sub.0] +

1-connected, then either [G.sub.1] is [k.sub.0] +

1-connected and [G.sub.2] is the empty graph, or vice versa.

A 1-connected complex is usually called simply connected: any loop (closed path) can be continuously deformed to a point.

[R.sub.U](s) = [R.sub.p](s) [intersection] [R.sub.Q](s) is the single simplex {<P, (p, q)>, <Q, (p, q)>} reached in the execution in which both updates occurred before either scan (since updates commute, their order does not matter), [R.sub.U](s) is 1-connected. To show that P([S.sup.1]) is 1-connected, Theorem 2.38 implies that it is enough to show that both [R.sub.P](s) and [R.sub.Q](s) are 0-connected (connected in the graph-theoretic sense).

Throughout this section B denotes a 1-connected CW-complex with H*(B) = H*(B; Z/(p)) = Z/(p)[[x.sub.1],...,[x.sub.m]] ([absolute value of][x.sub.i] even) and we often write K(n) as E.

Let f : X [approaches] B and g : Y [approaches] B be CW-complexes over B (B-CW-complexes for short) with Y 1-connected. We assume that f : X [approaches] B is a fibration with H* (X; Z/(p)) concentrated in even dimensions and that g induces an epimorphism on mod p cohomology.

However, this is not a J-category and the Eckmann-Hilton dual of a partial version of the cube axiom is satisfied when restricting to

1-connected algebras [6, A.18].

Every node can be in one of the following three states: 0-connected,

1-connected and 2-connected.

It is easy to see that X is

1-connected and locally

1-connected.