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Jordan Canonical Form
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
Jordan is best remembered today among analysts and topologists for his proof that a simply closed curve divides a plane into exactly two regions, now called the Jordan curve theorem. He also originated the concept of functions of bounded variation and is known especially for his definition of the length of a curve. These concepts appear in his Cours d’analyse de l’École Polytechnique first published in three volumes between 1882 and 1887.
Planarity and Kuratowski’s Theorem
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
The Jordan Curve Theorem is quite difficult to prove in full generality; in fact, Jordan himself did it incorrectly in 1887. The first correct proof was by Veblen in 1905. A proof for the greatly simplified case in which the closed curve consists entirely of straight-line segments, so that it is a closed polygon, is given by Courant and Robbins in What Is Mathematics?.
Place facets: a systematic literature review
Published in Spatial Cognition & Computation, 2020
Ehsan Hamzei, Stephan Winter, Martin Tomko
Location is one of the facets mentioned in all publications. Locational information of a place is the answer to where-questions that are asked in a wide range of situations, from our everyday life to human-machine interactions. It is therefore no surprise that locational information is the primary part of different place models. Localization (Scheider & Janowicz, 2014), footprint (Goodchild, 2011), and geographic location (Gieryn, 2000) are terms related to location. Localization is closely associated with another facet of place, i.e., boundary (Vasardani et al., 2016; Vasardani & Winter, 2016; Winter & Freksa, 2012). Based on the Jordan curve theorem, a boundary partitions space into three segments, the boundary itself, an inside, and an outside (Hales, 2007). However, unlike mathematical geometries, places in the geographical world do not necessarily have well-defined, crisp boundaries (Montello et al., 2003; Winter & Freksa, 2012). Unlike administrative places which are defined by crisp boundaries, people have vague and subjective perceptions of boundaries for the socially constructed places such as downtown (Hollenstein & Purves, 2010; Montello et al., 2003; Smith & Varzi, 2000), for natural places such as a mountain (Jones, 1959), and in natural communication where boundaries are often irrelevant.