Rank-based linkage I: triplet comparisons and oriented simplicial complexes

Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, \sigma)$ where $\sigma(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of out-ordered digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart`` the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Open combinatorial problems are presented in the last section.Comment: 37 pages, 12 figure

On the final sequence of a finitary set functor

AbstractA standard construction of the final coalgebra of an endofunctor involves defining a chain of iterates, starting at the final object of the underlying category and successively applying the functor. In this paper we show that, for a finitary set functor, this construction always yields a final coalgebra in ω2=ω+ω steps

Zeta-management: categorical and fractional differential approaches

The aim of this note is to introduce and justify the reasons why the traditional differential approach of complex systems, and more specifically non-additive systems, must be recognized as an epistemological failure (e.g. in economy, finance or limited in social agent models). The categorical character of the context proper to any type of irreversible exchange is analyzed. This approach underlines and explains the weaknesses of the set theory generally utilized. Beyond the mathematical concepts their application in project management provides an illustrative example allowing an easy understanding of the statements of problems attached to non-additive and irreversible complex system. We will see why it is necessary to shift the analysis from the set theory toward the theory of categories and why this choice very naturally introduces the use of non- integer order Differential Equations

Homotopy theory with bornological coarse spaces

We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the classification of coarse homology theories and the construction of examples. We show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The example of coarse K-homology will be discussed in detail.Comment: 220 pages (complete revision

Workshop Notes of the Sixth International Workshop "What can FCA do for Artificial Intelligence?"

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