| 词条 | Negative-dimensional space |
| 释义 |
In topology, a discipline within mathematics, a negative-dimensional space is an extension of the usual notion of space, allowing for negative dimensions.[1] DefinitionSuppose that {{math|Mt0}} is a compact space of Hausdorff dimension {{math|t0}}, which is an element of a scale of compact spaces embedded in each other and parametrized by {{math|t}} ({{math|0 < t < ∞}}). Such scales are considered equivalent with respect to {{math|Mt0}} if the compact spaces constituting them coincide for {{math|t ≥ t0}}. It is said that the compact space {{math|Mt0}} is the hole in this equivalent set of scales, and {{math|−t0}} is the negative dimension of the corresponding equivalence class.[2] HistoryBy the 1940s, the science of topology had developed and studied a thorough basic theory of topological spaces of positive dimension. Motivated by computations, and to some extent aesthetics, topologists searched for mathematical frameworks that extended our notion of space to allow for negative dimensions. Such dimensions, as well as the fourth and higher dimensions, are hard to imagine since we are not able to directly observe them. It wasn’t until the 1960s that a special topological framework was constructed—the category of spectra. A spectrum is a generalization of space that allows for negative dimensions. The concept of negative-dimensional spaces is applied, for example, to analyze linguistic statistics.[3] See also
References1. ^{{cite conference|first1=Luke|last1=Wolcott|first2=Elizabeth|last2=McTernan|title=Imagining Negative-Dimensional Space|pages=637–642|book-title=Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture|year=2012|editor-first1=Robert|editor-last1=Bosch|editor-first2=Douglas|editor-last2=McKenna|editor-first3=Reza|editor-last3=Sarhangi|isbn=978-1-938664-00-7|issn=1099-6702|publisher=Tessellations Publishing|location=Phoenix, Arizona, USA|url=http://bridgesmathart.org/2012/cdrom/proceedings/65/paper_65.pdf|accessdate=25 June 2015}} 2. ^{{cite journal|url=https://static-content.springer.com/lookinside/art%3A10.1134%2FS0001434607010166/000.png|doi=10.1134/S0001434607010166|title=General notion of a topological space of negative dimension and quantization of its density|journal=Mathematical Notes|volume=81|pages=140|year=2007|last1=Maslov|first1=V. P.}} 3. ^{{cite arxiv|eprint=math/0612543 |last1= Maslov |first1= V. P. |title= Negative dimension in general and asymptotic topology |year= 2006 }} External links
6 : Dimension|Dimension theory|Descriptive set theory|Properties of topological spaces|Topology|Negative concepts |
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