近端物理几何的两种形式。公理、区域缝合、区域类、对偶和平行纤维丛
Two Forms of Proximal Physical Geometry. Axioms, Sewing Regions Together, Classes of Regions, Duality, and Parallel Fibre Bundles
摘要：
本文介绍了Lenzen物理几何的两种近似形式，即建立在单纯复形上的\emph{axiomatized strongly proximal physical geometry}和\emph{axiomatized descriptive proximal physical geometry}，\emph{axiomatized strongly proximal physical geometry}具有来自弦几何的对偶性和缝合操作，\emph{axiomatized descriptive proximal physical geometry}基于空间区域的特征和区域之间的描述性近似来描述空间区域。这是物理空间的Lenzen几何的计算邻近方法。在这两种形式的物理几何中，区域都是基元。直觉上，区域是一组相连的子区域。此几何体中的图元是\emph{region}，而不是\emph{point}。具有$n$特征的区域的每个描述是从区域到$\mathbb{R}^n$中的特征向量的映射。在特征空间中，近似物理几何具有欧几里得几何、黎曼几何或非欧几里得几何的外观和感觉，因为我们可以自由地处理特征空间中点之间的关系。在这些新形式的几何学的重点是与自己的独特的功能，如形状，面积，周长和直径和非空区域之间的关系的各个区域之间的关系。给出了物理几何的公理和邻近物理几何的公理，并举例说明。平行类的区域，描述纤维丛和BreMiller-Sloyer层的结果。此外，基于区域的Borsuk-Ulam定理以及有线朋友定理的背景下，这两种形式的物理几何。
英文摘要：
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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