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Topological methods in astrophysics 天体物理学中的拓扑方法
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0846
M. Berger
{"title":"Topological methods in astrophysics","authors":"M. Berger","doi":"10.1098/rsta.2001.0846","DOIUrl":"https://doi.org/10.1098/rsta.2001.0846","url":null,"abstract":"Most objects in astrophysics are filled with highly conducting plasma and hence easily carry magnetic fields. The topological properties of these fields have important physical consequences. The atmospheres of the Sun, many types of stars, and accretion disks have magnetic fields rooted at the surface. The topological structure of the magnetic lines of force determines the possible equilibrium configurations of the field. Solar and stellar atmospheres are much hotter than expected given the surface temperature. A proposed model of heating involves tangled magnetic field lines, which release their energy in small flares. The degree of topological complexity of a magnetic field helps to determine how much energy it stores. Flares simplify the topology of the field and thereby release the stored energy. Topology is also important in understanding large–scale properties of the solar dynamo that generates the solar magnetic field. The magnetic helicity integral, which measures linking properties of the field, can be decomposed into contributions from different regions of the Sun and space. Transport of helicity from one region to another underlies many important processes in solar activity.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"37 4","pages":"1439 - 1448"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1098/rsta.2001.0846","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72464616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Homoclinic orbits and chaos in three– and four–dimensional flows 三维和四维流动中的同斜轨道和混沌
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0845
P. Holmes, A. Doelman, G. Hek, G. Domokos
{"title":"Homoclinic orbits and chaos in three– and four–dimensional flows","authors":"P. Holmes, A. Doelman, G. Hek, G. Domokos","doi":"10.1098/rsta.2001.0845","DOIUrl":"https://doi.org/10.1098/rsta.2001.0845","url":null,"abstract":"We review recent work in which perturbative, geometric and topological arguments are used to prove the existence of countable sets of orbits connecting equilibria in ordinary differential equations. We first consider perturbations of a three–dimensional integrable system possessing a line of degenerate saddle points connected by a two–dimensional manifold of homoclinic loops. We show that this manifold splits to create transverse homoclinic orbits, and then appeal to geometrical and symbolic dynamic arguments to show that homoclinic bifurcations occur in which ‘simple’ connecting orbits are replaced by a countable infinity of such orbits. We discover a rich variety of connections among equilibria and periodic orbits, as well as more exotic sets, including Smale horseshoes. The second problem is a four–dimensional Hamiltonian system. Using symmetries and classical estimates, we again find countable sets of connecting orbits. There is no small parameter in this case, and the methods are non–perturbative.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"359 1","pages":"1429 - 1438"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82343474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
A conjecture concerning the exponential map on Dμ(M) 关于Dμ(M)上指数映射的一个猜想
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0847
G. Misiołek
{"title":"A conjecture concerning the exponential map on Dμ(M)","authors":"G. Misiołek","doi":"10.1098/rsta.2001.0847","DOIUrl":"https://doi.org/10.1098/rsta.2001.0847","url":null,"abstract":"It is known that solutions of the Euler equations of hydrodynamics correspond to geodesics on the group of volume–preserving diffeomorphisms of a compact manifold. We conjecture that, regardless of the dimension of the manifold, the associated Riemannian exponential map on the group is nonlinear Fredholm of index zero. Such a result has been established for the Riemannian exponential maps of natural Sobolev metrics on loop spaces and loop groups.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"3 1","pages":"1469 - 1472"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80951643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Knotted solitons and their physical applications 结孤子及其物理应用
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0842
L. Faddeev
{"title":"Knotted solitons and their physical applications","authors":"L. Faddeev","doi":"10.1098/rsta.2001.0842","DOIUrl":"https://doi.org/10.1098/rsta.2001.0842","url":null,"abstract":"A nonlinear model in three–dimensional space allowing for the solitons localized in the vicinity of a loop is presented. Two possible applications in real physics are discussed.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"36 1","pages":"1399 - 1403"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74926165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
Homology of spaces of knots in any dimensions 任意维结点空间的同调性
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0838
V. Vassiliev
{"title":"Homology of spaces of knots in any dimensions","authors":"V. Vassiliev","doi":"10.1098/rsta.2001.0838","DOIUrl":"https://doi.org/10.1098/rsta.2001.0838","url":null,"abstract":"I shall describe the recent progress in the study of cohomology rings of spaces of knots in Rn, H*({knots in Rn}), with arbitrary n ⩾ 3. ‘Any dimensions’ in the title can be read as dimensions n of spaces Rn, as dimensions i of the cohomology groups Hi, and also as a parameter for different generalizations of the notion of a knot. An important subproblem is the study of knot invariants. In our context, they appear as zero–dimensional cohomology classes of the space of knots in R3. It turns out that our more general problem is never less beautiful. In particular, nice algebraic structures arising in the related homological calculations have equally (or maybe even more) compact description, of which the classical ‘zero–dimensional’ part can be obtained by easy factorization. There are many good expositions of the theory of related knot invariants. Therefore, I shall deal almost completely with results in higher (or arbitrary) dimensions.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"304 2","pages":"1343 - 1364"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1098/rsta.2001.0838","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72434102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 14
Plane curves, wavefronts and Legendrian knots 平面曲线，波前和勒让德结
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0837
V. Goryunov
{"title":"Plane curves, wavefronts and Legendrian knots","authors":"V. Goryunov","doi":"10.1098/rsta.2001.0837","DOIUrl":"https://doi.org/10.1098/rsta.2001.0837","url":null,"abstract":"We survey some of the recent results on Legendrian knots and links in the standard contact 3–space and solid torus. These include the description of finite–order invariants and estimates of the self–linking number coming from the classical polynomial link invariants. We also describe the combinatorial invariant introduced by Chekanov and Pushkar, which allowed them to prove Arnold's conjecture on the necessity of four–cusp curves in generic eversions of a circular front in the plane.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"29 1","pages":"1497 - 1510"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74804164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Global well–posedness for the Lagrangian averaged Navier–Stokes (LANS–α) equations on bounded domains 有界区域上lagrange平均Navier-Stokes (LANS -α)方程的全局适定性
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0852
J. Marsden, S. Shkoller
{"title":"Global well–posedness for the Lagrangian averaged Navier–Stokes (LANS–α) equations on bounded domains","authors":"J. Marsden, S. Shkoller","doi":"10.1098/rsta.2001.0852","DOIUrl":"https://doi.org/10.1098/rsta.2001.0852","url":null,"abstract":"We prove the global well–posedness and regularity of the (isotropic) Lagrangian averaged Navier–Stokes (LANS–α) equations on a three–dimensional bounded domain with a smooth boundary with no–slip boundary conditions for initial data in the set {u ∈ Hs ∩ H10| Au = 0 on ∂Ω, div u = 0}, s ∈ [3, 5), where A is the Stokes operator. As with the Navier–Stokes equations, one has parabolic–type regularity; that is, the solutions instantaneously become space–time smooth when the forcing is smooth (or zero). The equations are an ensemble average of the Navier–Stokes equations over initial data in an α–radius phase–space ball, and converge to the Navier–Stokes equations as α → 0. We also show that classical solutions of the LANS–α equations converge almost all in Hs for s ∈ 2.5, 3), to solutions of the inviscid equations (ν = 0), called the Lagrangian averaged Euler (LAE–α) equations, even on domains with boundary, for time–intervals governed by the time of existence of solutions of the LAE–α equations.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"8 1","pages":"1449 - 1468"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84158159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 167
Configurations of points 点的构型
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0840
M. Atiyah
{"title":"Configurations of points","authors":"M. Atiyah","doi":"10.1098/rsta.2001.0840","DOIUrl":"https://doi.org/10.1098/rsta.2001.0840","url":null,"abstract":"Berry & Robbins, in their discussion of the spin–statistics theorem in quantum mechanics, were led to ask the following question. Can one construct a continuous map from the configuration space of n distinct particles in 3–space to the flag manifold of the unitary group U(n)? I shall discuss this problem and various generalizations of it. In particular, there is a version in which U(n) is replaced by an arbitrary compact Lie group. It turns out that this can be treated using Nahm's equations, which are an integrable system of ordinary differential equations arising from the self–dual Yang-Mills equations. Our topological problem is therefore connected with physics in two quite different ways, once at its origin and once at its solution.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"66 6","pages":"1375 - 1387"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1098/rsta.2001.0840","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72424491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 40
Topological structures in string theory 弦理论中的拓扑结构
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0841
G. Segal
{"title":"Topological structures in string theory","authors":"G. Segal","doi":"10.1098/rsta.2001.0841","DOIUrl":"https://doi.org/10.1098/rsta.2001.0841","url":null,"abstract":"In string theory space–time comes equipped with an additional geometric structure called a B–field or ‘gerbe’. I describe this structure, mention its relationship with noncommutative geometry, and explain how to use the B–field to define a twisted version of the K–theory of space–time. String–theoretical space–time can contain topologically non–trivial dynamical structures called D–branes. These are simply accounted for in the framework of conformal field theory. In a highly simplified limiting casetopological field theory with a finite gauge group—the D–branes naturally represent elements of the twisted K–theory of space–time: the K–theory class is the ‘charge’ of the D–brane.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"16 1","pages":"1389 - 1398"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90246879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 68
Instantons and the 11th dimension 瞬变子和第11维
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Pub Date : 2001-07-15 DOI: 10.1098/rsta.2001.0843
Nikita Nekrasov
{"title":"Instantons and the 11th dimension","authors":"Nikita Nekrasov","doi":"10.1098/rsta.2001.0843","DOIUrl":"https://doi.org/10.1098/rsta.2001.0843","url":null,"abstract":"In this almost non–technical note, mostly aimed at mathematicians, we review the construction of instantons on the non–commutative R4 and explain how their existence is tied up with the existence of M–theory.","PeriodicalId":20023,"journal":{"name":"Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences","volume":"9 1","pages":"1405 - 1412"},"PeriodicalIF":0.0,"publicationDate":"2001-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81904377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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