11 research outputs found

### Scaled-free objects II

This work creates two categories of "array-weighted sets" for the purposes of
constructing universal matrix-normed spaces and algebras. These universal
objects have the analogous universal property to the free vector space, lifting
maps completely bounded on a generation set to a completely bounded linear map
of the matrix-normed space.
Moreover, the universal matrix-normed algebra is used to prove the existence
of a free product for matrix-normed algebras using algebraic methods.Comment: 46 pages. Version 4 fixed a few minor typos. Version 3 added
matricial completion; fixed an arithmetic error in Example 3.5.10. Version 2
added a preliminaries section on weighted sets and matricial Banach spaces,
incorporating much of "Matricial Banach spaces" in summary; fixed a domain
issue in Lemma 3.3.2; simplified Examples 3.5.10 and 4.11; added more proofs
to Sections 4 and

### Matricial Banach spaces

This work performs a study of the category of complete matrix-normed spaces,
called matricial Banach spaces. Many of the usual constructions of Banach
spaces extend in a natural way to matricial Banach spaces, including products,
direct sums, and completions. Also, while the minimal matrix-norm on a Banach
space is well-known, this work characterizes the maximal matrix-norm on a
Banach space from the work of Effros and Ruan as a dual operator space.
Moreover, building from the work of Blecher, Ruan, and Sinclair, the Haagerup
tensor product is merged with the direct sum to form a Haagerup tensor algebra,
which shares the analogous universal property of the Banach tensor algebra from
the work of Leptin.Comment: 19 pages. This paper has been withdrawn as it has been merged with
arXiv:1405.711

### Incidence Hypergraphs: Box Products & the Laplacian

The box product and its associated box exponential are characterized for the
categories of quivers (directed graphs), multigraphs, set system hypergraphs,
and incidence hypergraphs. It is shown that only the quiver case of the box
exponential can be characterized via homs entirely within their own category.
An asymmetry in the incidence hypergraphic box product is rectified via an
incidence dual-closed generalization that effectively treats vertices and edges
as real and imaginary parts of a complex number, respectively. This new
hypergraphic box product is shown to have a natural interpretation as the
canonical box product for graphs via the bipartite representation functor, and
its associated box exponential is represented as homs entirely in the category
of incidence hypergraphs; with incidences determined by incidence-prism
mapping. The evaluation of the box exponential at paths is shown to correspond
to the entries in half-powers of the oriented hypergraphic signless Laplacian
matrix.Comment: 34 pages, 23 figures, 4 table

### 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