Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two Forms of Proximal Physical Geometry. Axioms, Sewing Regions
  Together, Classes of Regions, Duality, and Parallel Fibre Bundles   This paper introduces two proximal forms of Lenzen physical geometry, namely,
an \emph{axiomatized strongly proximal physical geometry} that is built on
simplicial complexes with the dualities and sewing operations derived from
string geometry and an \emph{axiomatized descriptive proximal physical
geometry} in which spatial regions are described based on their features and
the descriptive proximities between regions. This is a computational proximity
approach to a Lenzen geometry of physical space. In both forms of physical
geometry, region is a primitive. Intuitively, a region is a set of connected
subregions. The primitive in this geometry is \emph{region}, instead of
\emph{point}. Each description of a region with $n$ features is a mapping from
the region to a feature vector in $\mathbb{R}^n$. In the feature space,
proximal physical geometry has the look and feel of either Euclidean,
Riemannian, or non-Euclidean geometry, since we freely work with the relations
between points in the feature space. The focus in these new forms of geometry
is the relation between individual regions with their own distinctive features
such as shape, area, perimeter and diameter and the relation between nonempty
sets of regions. The axioms for physical geometry as well as the axioms for
proximal physical geometry are given and illustrated. Results for parallel
classes of regions, descriptive fibre bundles and BreMiller-Sloyer sheaves are
given. In addition, a region-based Borsuk-Ulam Theorem as well as a Wired
Friend Theorem are given in the context of both forms of physical geometry.

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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
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The categories of this paper are: cs