The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants.
We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce.
Evocatively, these gene pairs are likely shadows over the low-rank matrix and connect different rank areas.
The resulting form of the potential energy two stable wells connected by rank-1 saddle points—shows an analogy with resonance transitions in celestial mechanics or molecular reconfigurations in chemistry, whereas here the transition corresponds to switching between two stable structural configurations.
Thus, the output of DECOBRA (DEnsely COnnected Biclustering RAnked) results from subsequent filtering of the output of DECOB, as described below.
Our positional candidate, Cd44, was not a highly connected gene ranking 362 out of the 666 genes in the module (Additional file 1: Table S1).
Besides these non-verbal cues, we humans developed a conceptual metaphor, which connects social rank to the spatial domain (e.g., "high" vs "low status" [ Pinker, 1997]).
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In his masterful work, The Civilizing Process, Norbert Elias revealed how the "late medieval upper class" had not yet demanded, as later generations would, that "everything vulgar should be suppressed from life and therefore from pictures". For centuries now, defining incivility has been intimately connected with social rank, class status, political hierarchy, and relations of power.
In the value-based (rank-based) method, two genes are connected if the (rank-transformed) similarity between their expression profiles is above a certain threshold, which can be chosen in an ad hoc manner or semi-automatically.
