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to the area including the input from algebraic geometry, as well as that flowing from the work of
Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated
316:), bifurcations or catastrophes. Classifying the types of changes and characterizing sets of parameters which give rise to these changes are some of the main mathematical goals. Singularities can occur in a wide range of mathematical objects, from matrices depending on parameters to wavefronts.
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0). This means that the simple process of "lifting" a piece of string off itself, by the "obvious" use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general
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defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the properties undergo sudden change under a small variation of the parameters. These situations are called perestroika
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In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes.
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the stable. Whitney had shown that in low numbers of variables the stable structure of critical points is very restricted, in local terms. Thom built on this, and his own earlier work, to create a
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develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials. To compensate, only the
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An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of
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is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our
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phenomena are considered. One can argue that in nature, anything destroyed by tiny changes is not going to be observed; the visible
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333:); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include
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780:, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking,
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was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of
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is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of
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that states a
Knowledge editor's personal feelings or presents an original argument about a topic.
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terms the coordinates in the ambient space don't straightforwardly translate the geometry of the
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This article is about the mathematical discipline. For other geometric uses, see
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as a value at which a function is not defined. For that, see for example
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at the point. Intensive studies of such singularities led in the end to
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The theory mentioned above does not directly relate to the concept of
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meaning of a singular point isn't in question; it is just that in
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will cross itself in an approximate "X" shape. The points on the
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Historically, singularities were first noticed in the study of
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is also disallowed. A major advance was the introduction of
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it on the floor, and flattening it. In some places the flat
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The general position of singularities in algebraic geometry
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It was then a short step to define the general notion of a
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are qualitatively different, as is seen just by sketching.
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personal reflection, personal essay, or argumentative essay
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supposed to account for discontinuous change in nature.
716:. Applications, according to Arnold, are to be seen in
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609:manifold (for the strong topology, rather than the
163:. Unsourced material may be challenged and removed.
884:. Isaac Newton Institute for Mathematical Sciences
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621:At about the same time as Hironaka's work, the
340:Other ways in which singularities occur is by
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796:can degenerate; and these fields are linked.
68:Learn how and when to remove these messages
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978:
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241:Learn how and when to remove this message
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962:Foundations of Mechanics, Second Edition
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964:. Benjamin/Cummings Publishing Company.
528:; that is, to allow higher dimensions.
344:of manifold structure. The presence of
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526:singular point of an algebraic variety
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505:carried out a detailed study of all
161:adding citations to reliable sources
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356:Singularities in algebraic geometry
277:where it does this are one kind of
20:. For other mathematical uses, see
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960:R. Abraham and J. Marsden (1987).
617:The smooth theory and catastrophes
593:to acquire singular points on the
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939:E. Brieskorn; H. Knörrer (1986).
438:{\displaystyle y^{2}=x^{2}+x^{3}}
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24:. For non-mathematical uses, see
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589:: it is entirely typical for an
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57:or discuss these issues on the
1032:Differentiable/Smooth manifold
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491:{\displaystyle y^{2}=x^{3}\ }
361:Algebraic curve singularities
832:Stratification (mathematics)
696:. Technically this involves
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389:at (0, 0) of the curve
285:of the floor corresponds to
26:Singularity (disambiguation)
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1738:Classification of manifolds
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18:Singular point of a curve
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752:Other possible meanings
738:intersection cohomology
688:as the construction of
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1037:Differential structure
941:Plane Algebraic Curves
876:Arnold, V. I. (2000).
788:can degenerate, while
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637:. Roughly speaking, a
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536:Such singularities in
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943:. Birkhauser-Verlag.
842:Mixed Hodge structure
837:Intersection homology
822:Contact (mathematics)
812:Zariski tangent space
770:removable singularity
766:essential singularity
690:equivalence relations
517:must be counted with
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1469:Covariant derivative
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918:V.I. Arnold (1992).
878:"Singularity Theory"
762:isolated singularity
542:polynomial equations
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172:"Singularity theory"
157:improve this article
1503:Exterior derivative
1105:Atiyah–Singer index
1054:Riemannian manifold
922:. Springer-Verlag.
746:homological algebra
722:classical mechanics
718:symplectic geometry
605:, ending up with a
587:projective geometry
570:birational geometry
377:A curve with a cusp
31:Mathematical theory
1857:Singularity theory
1809:Secondary calculus
1763:Singularity theory
1718:Parallel transport
1486:De Rham cohomology
1125:Generalized Stokes
920:Catastrophe Theory
792:studies the way a
790:singularity theory
784:studies the way a
684:singularity theory
675:Christopher Zeeman
671:catastrophe theory
659:catastrophe theory
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538:algebraic geometry
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1758:Moving frame
1753:Morse theory
1743:Gauge theory
1535:Tensor field
1464:Closed/Exact
1443:Vector field
1411:Distribution
1352:Hypercomplex
1347:Quaternionic
1084:Vector field
1042:Smooth atlas
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886:. Retrieved
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786:covering map
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503:Isaac Newton
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213:October 2021
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155:Please help
150:verification
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111:October 2021
108:
89:
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51:Please help
48:
1703:Levi-Civita
1693:Generalized
1665:Connections
1615:Lie algebra
1547:Volume form
1448:Vector flow
1421:Pushforward
1416:Lie bracket
1315:Lie algebra
1280:G-structure
1069:Pushforward
1049:Submanifold
903:Arnold 1992
714:derivatives
312:перестройка
279:singularity
255:mathematics
1826:Stratifold
1784:Diffeology
1580:Associated
1381:Symplectic
1366:Riemannian
1295:Hyperbolic
1222:Submersion
1130:Hopf–Rinow
1064:Submersion
1059:Smooth map
911:References
776:theory of
702:Lie groups
513:that such
327:Projection
183:newspapers
54:improve it
1708:Principal
1683:Ehresmann
1640:Subbundle
1630:Principal
1605:Fibration
1585:Cotangent
1457:Covectors
1310:Lie group
1290:Hermitian
1233:manifolds
1202:Immersion
1197:Foliation
1135:Noether's
1120:Frobenius
1115:De Rham's
1110:Darboux's
1001:Manifolds
782:monodromy
774:monodromy
647:level set
627:René Thom
554:intrinsic
550:extrinsic
452:there of
350:orbifolds
263:manifolds
60:talk page
1851:Category
1804:Orbifold
1799:K-theory
1789:Diffiety
1513:Pullback
1327:Oriented
1305:Kenmotsu
1285:Hadamard
1231:Types of
1180:Geodesic
1005:Glossary
800:See also
794:manifold
579:collapse
448:and the
346:symmetry
335:caustics
267:dropping
1748:History
1731:Related
1645:Tangent
1623:)
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1570:Adjoint
1562:Bundles
1540:density
1438:Torsion
1404:Vectors
1396:Tensors
1379:)
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1360:,
1358:Pseudo−
1337:Poisson
1270:Finsler
1265:Fibered
1260:Contact
1258:)
1250:Complex
1248:)
1217:Section
807:Tangent
728:Duality
607:compact
306:Russian
197:scholar
97:Please
1713:Vector
1698:Koszul
1678:Cartan
1673:Affine
1655:Vector
1650:Tensor
1635:Spinor
1625:Normal
1621:Stable
1575:Affine
1479:bundle
1431:bundle
1377:Almost
1300:Kähler
1256:Almost
1246:Almost
1240:Closed
1140:Sard's
1096:(list)
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888:31 May
772:. The
651:stable
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385:. The
295:stable
271:string
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1821:Sheaf
1595:Fiber
1371:Rizza
1342:Prime
1173:Local
1163:Curve
1025:Atlas
864:Notes
694:germs
641:of a
275:floor
204:JSTOR
190:books
1688:Form
1590:Dual
1523:flow
1386:Tame
1362:Sub−
1275:Flat
1155:Maps
945:ISBN
924:ISBN
890:2016
706:jets
568:(in
450:cusp
331:eyes
176:news
1610:Jet
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