Equation 44 · Part 3 · The Bend an Elevator Cannot Fake
=
=
What this part means
The expressions on both sides represent the same quantity under the stated assumptions.
Its job in the formula
The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.
Full expression→=→Article meaning
The passage around this formula
What r is actually sensitive to follows from the same argument in reverse. The null space of B has dimension equal to the number of connected components of the network graph; for a connected network it is exactly one-dimensional, spanned by the all-ones vector, because B = 0 forces = across every edge and connectivity propagates that equality to every node. By rank–nullity, a connected tree — with exactly |V|-1 edges for |V| nodes — has an incidence matrix of full row rank, meaning that for any data whatsoever a exists reproducing it exactly. A tree, in other words, cannot fail to close, on any data, ever; it carries zero redundant edges and therefore zero capacity to…
Learn the underlying idea
An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.
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