- For other uses of "atlas", see Atlas (disambiguation).
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold.
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Charts [link]
The definition of an atlas depends on the notion of a chart. A chart for a topological space M is a homeomorphism Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi
from an open subset U of M to an open subset of Euclidean space. The chart is traditionally recorded as the ordered pair Failed to parse (Missing texvc executable; please see math/README to configure.): (U, \varphi)
.
Formal definition of atlas [link]
An atlas for a topological space M is a collection Failed to parse (Missing texvc executable; please see math/README to configure.): \{(U_{\alpha}, \varphi_{\alpha})\}
of charts on M such that
Failed to parse (Missing texvc executable; please see math/README to configure.): \bigcup U_{\alpha} = M . If the range of each chart is the n-dimensional Euclidean space, then M is said to be an n-dimensional manifold.
Transition maps [link]
A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)
To be more precise, suppose that Failed to parse (Missing texvc executable; please see math/README to configure.): (U_{\alpha}, \varphi_{\alpha})
and Failed to parse (Missing texvc executable; please see math/README to configure.): (U_{\beta}, \varphi_{\beta})
are two charts for a manifold M such that Failed to parse (Missing texvc executable; please see math/README to configure.): U_{\alpha} \cap U_{\beta}
is non-empty.
The transition map Failed to parse (Missing texvc executable; please see math/README to configure.): \tau_{\alpha,\beta}
is the map defined on the intersection
Failed to parse (Missing texvc executable; please see math/README to configure.): U_{\alpha} \cap U_{\beta}
by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.
Note that since Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi_{\alpha}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi_{\beta}
are both homeomorphisms, the transition map Failed to parse (Missing texvc executable; please see math/README to configure.): \tau_{\alpha, \beta}
is also a homeomorphism.
More structure [link]
One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Then one can unambiguously define the notion of tangent vectors and then directional derivatives.
If each transition function is a smooth map, then the atlas is called a smooth atlas. Alternatively, one could require that the transition maps have only k continuous derivatives in which case the atlas is said to be Failed to parse (Missing texvc executable; please see math/README to configure.): C^k .
Very generally, if each transition function belongs to a pseudo-group Failed to parse (Missing texvc executable; please see math/README to configure.): {\mathcal G}
of homeomorphisms of Euclidean space, then the atlas is called a Failed to parse (Missing texvc executable; please see math/README to configure.): {\mathcal G} -atlas.
References [link]
- Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
- Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8.
External links [link]
- Atlas by Rowland, Todd
https://wn.com/Atlas_(topology)
Atlas
An atlas is a collection of maps; it is typically a map of Earth or a region of Earth, but there are atlases of the other planets (and their satellites) in the Solar System. Furthermore, atlases of anatomy exist, mapping out the human body or other organisms. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographic features and political boundaries, many atlases often feature geopolitical, social, religious and economic statistics. They also have information about the map and places in it.
Etymology
The word atlas dates from 1636, first in reference to the English translation of Atlas, sive cosmographicae meditationes de fabrica mundi (1585) by Flemish geographer Gerhardus Mercator, who might have been the first to use this word in this way. A picture of the Titan Atlas holding up the world appeared on the frontispiece of this and other early map collections.
History
The first work that contained systematically arranged woodcut maps of uniform size, intended to be published in a book, thus representing the first modern atlas, was De Summa totius Orbis (1524–26) by the 16th-century Italian cartographer Pietro Coppo. Nonetheless, this distinction is conventionally awarded to the Flemish cartographer Abraham Ortelius who in 1570 published the collection of maps Theatrum Orbis Terrarum.
Atlas (computer)
The Atlas Computer was a joint development between the University of Manchester, Ferranti, and Plessey. The first Atlas, installed at Manchester University and officially commissioned in 1962, was one of the world's first supercomputers, considered to be the most powerful computer in the world at that time. It was said that whenever Atlas went offline half of the United Kingdom's computer capacity was lost. It was a second-generation machine, using discrete germanium transistors. Two other Atlas machines were built: one for British Petroleum and the University of London, and one for the Atlas Computer Laboratory at Chilton near Oxford.
A derivative system was built by Ferranti for Cambridge University. Called the Titan, or Atlas 2, it had a different memory organisation and ran a time-sharing operating system developed by Cambridge University Computer Laboratory. Two further Atlas 2s were delivered: one to the CAD Centre in Cambridge (later called CADCentre, then AVEVA), and the other to the Atomic Weapons Research Establishment (AWRE), Aldermaston.
Atlas (1951 automobile)
The Atlas was a mini-car made in France in 1951. Originally known as La Coccinelle, it used a single-cylinder engine of a mere 175 cc capacity. The fiberglass body seated two, the maximum speed said to be over 40 miles per hour (64 km/h).
Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.
Definition
The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.
Topology (musical ensemble)
Topology is an indie classical quintet from Australia, formed in 1997. A leading Australian new music ensemble, they perform throughout Australia and abroad and have to date released 12 albums, including one with rock/electronica band Full Fathom Five and one with contemporary ensemble Loops. They were formerly the resident ensemble at the University of Western Sydney. The group works with composers including Tim Brady in Canada, Andrew Poppy, Michael Nyman, and Jeremy Peyton Jones in the UK, and Terry Riley, Steve Reich, Philip Glass, Carl Stone, and Paul Dresher in the US, as well as many Australian composers.
In 2009, Topology won the Outstanding Contribution by an Organisation award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.
Members
Bernard studied viola at the Queensland Conservatorium (B.Mus 1987) and at Michigan State University (Master of Music 1993) with John Graham and Robert Dan. He studied in summer schools with Kim Kashkashian (Aldeborough), the Alban Berg Quartet and the Kronos Quartet. While in the US, he played with the Arlington Quartet, touring the US and UK. He was a violist in the Queensland Philharmonic Orchestra from 1994-2000, and is now Associate Principal Violist of the Queensland Orchestra, playing solo parts in works such as the sixth Brandenburg Concerto. He has directed several concerts for the Queensland Philharmonic’s Off the Factory Floor chamber series.
Topology (disambiguation)
Topology, the study of surfaces, is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these properties are the topological invariants.
Topology may also refer to:
- In geographic information systems and their data structures, topology and planar enforcement are the storing of a border line between two neighboring areas (and the border point between two connecting lines) only once. Thus, any rounding errors might move the border, but will not lead to gaps or overlaps between the areas.
- Also in cartography, a topological map is a greatly simplified map that preserves the mathematical topology while sacrificing scale and shape
- Topology is often confused with the geographic meaning of topography (originally the study of places). The confusion may be a factor in topographies having become confused with terrain or relief, such that they are essentially synonymous.
