Let \(f\) denote the density function of a crowd in a given space \(X\), and let \(\nabla f\) denote the gradient vector field. If I am at a given point \(x\in X\), compute the integral curve \(c_{\nabla f}(t)\) such that \(c_{\nabla f}(0)=x\) and \( c_{\nabla f}'(t) = -\nabla f(c_{\nabla f}(t)) \). You just computed the trajectory in which I will move.

When I am explaining persistence to Engineers, I start from dendrograms for hierarchical clustering, as most of them are familiar with that concept, and then present persistence diagrams and barcodes as analogies for "higher order topological properties".

I just realized that I had the wrong impression (because I never thought about it) that 0-persistence as a special case, has some properties which makes dendrograms possible as opposed their "coarser" counterparts, barcodes and persistence diagrams, for higher persistence. But, this is only because we implicitly assume a canonical basis for \( C_0 \) in single linkage clustering, or 0-persistence.

It is not always the case that a natural canonical basis (which makes some sort of sense) exists for higher dimensional chain spaces. However, in the case of Rips filtration based on Euclidean metric for point clouds, especially in \(\mathbb{R}^2\) and \(\mathbb{R}^3\), we can use the Alexander duality to define a canonical basis, and thus actually obtain a dendrogram. More generally, whenever we have a meaningful canonical basis for our chain spaces, we can have dendrograms for persistence.

1. Unfalsifiable hypothesis:

   $$ P(\mathcal{H}|\mathcal{O}) = 1, $$ where \(\mathcal{O}\) is the set of all possible observations.

2. Impossible hypothesis:

   $$ P(\mathcal{H}|\mathcal{O}) = 0. $$

3. Untestable hypothesis:

   $$ P(\mathcal{O}|H) = P(\mathcal{O}) $$

The unfalsifiable hypothesis is designed such that no matter what observation we make, the hypothesis cannot be rejected. For example, "whatever happens, I made that happen". Go ahead, prove it wrong. Equivalently, such hypotheses are true by definition. They can only function as axioms.

Impossible hypotheses are usually those which are logically inconsistent, or in other words, self contradictory.

Untestable hypotheses are those for which it is impossible to determine how they might affect any particular outcome or experiment. Needless to say, these are also useless, unless we have underestimated the size of \(\mathcal{O}\).