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# Topic: Differentiable structure

###### In the News (Sun 16 Jun 19)

 Differentiable manifold - Wikipedia, the free encyclopedia The global differentiable structure is induced when it can be shown that the natural composition of the homeomorphisms between the cooresponding open Euclidean spaces are differentiable on overlaps of charts in the atlas. An alternate definition of a differentiable manifold is a topological space with a sheaf of functions, which is locally isomorphic to Euclidean space with the sheaf of differentiable functions. The differentiable structure of the manifolds ensures that the differential (which is a linear transformation on the respective tangent spaces) is independent of the choice of coordinates. en.wikipedia.org /wiki/Smooth_manifold   (1993 words)

 Unitary representation - Wikipedia, the free encyclopedia Note that if G is a Lie group, this representation is necessarily smooth (respectively real analytic) with respect to the differentiable structure (respectively real analytic structure) of the Lie group. A unitary representation is completely reducible, in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace. For example, it implies that finite dimensional unitary representations are always a direct sum of irreducible representations, in the algebraic sense. en.wikipedia.org /wiki/Unitary_representation   (474 words)

 [No title] These structures are observed at size ranges as diverse as the ones relevant for laboratory experiments\cite{sommerer,Gollub99,Gollub01}, atmospheric transport\cite{Balluch,tuck}, or temperature or chlorophyll patchiness in the ocean\cite{patchiness,Mackas85}. In most cases the structure of all the fields will be determined by the combined effect of their mutual coupling and of the flow. The corresponding structure functions are in Fig.~\ref{fig:structA02T10}, which shows that phytoplankton and zooplankton have the same behavior at small scales, different from the one of the carrying capacity. www.imedea.uib.es /physdept/publications/downfile.php?fid=2691   (5806 words)

 Wikinfo | Manifold Differentiable manifolds are used in mathematics to describe geometrical objects; they are also the most natural and general setting to study differentiability. In physics, differentiable manifolds serve as the phase space in classical mechanics and four dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity. Also, it is possible for two non-equivalent differentiable manifolds to be homeomorphic. www.wikinfo.org /wiki.php?title=Manifold   (1188 words)

 Diffeology - Introduction Birth of differential spaces I had the feeling that these structures, the axiomatic of differential groups, could be easily extended to any underlying set, not necessarily a group, and I remember a particular discussion about this question in the cafeteria of the Luminy's campus. The choice of the wording "differential spaces" or "differential groups" was not very happy, because "differential" is already used in math and has some kind of copyright, especially as "differential groups" are groups with an operation of derivation. The space of all differentiable forms on a diffeological space being himself a diffeological (vector)-space, there is no objection to talk about differentiable maps between space of differentiable forms. www.umpa.ens-lyon.fr /~iglesias/articles/Diffeology/dflg_Intro.html   (1742 words)

 Warwick Mathematics Institute – Research Areas Differential Geometry is the study of geometric structures on manifolds. The way in which coordinate charts are pieced together give the manifold a differentiable structure or a complex structure. Symplectic structure in which a closed skew-symmetric bilinear form is specified on the tangent space at each point of the manifold. www.maths.warwick.ac.uk /research/research_areas/diff_geom.html   (295 words)

 [No title] The case for the fractal and non-differentiable structure of the quantum space-time is argued further in Chapter 5. The challenge of chaos is that structures are very often observed in domains where chaos has developed, while ordinary methods fail to make prediction because of the presence of chaos itself. Already the new structure of space-time revealed by special motion relativity at the beginning of the century underlies in an inescapable way the Riemannian structure of Einstein's general relativity: Space-time is locally Minkowskian, so that all attempts at constructing a Riemannian theory of gravitation were condemned to fail in the absence of special relativity constraints. www.chez.com /etlefevre/rechell/ukliwo12.htm   (9267 words)

 [No title] In our case, the groups considered will be groups of differentiable diffeomorphisms of the sphere and the subgroups in which we factor will be groups of diffeomorphisms that commute with reflections across a plane and which are essentially one dimensional. The construction is somewhat delicate since we want to obtain a loss of differentiability that is independent of the dimension, in spite of the fact that the number of inductive steps has to grow with the dimension. The key is that the differentiability properties of functions can be read of from the size of analytic approximations on thin strips --see \cite{Mo}--. www.ma.utexas.edu /mp_arc/papers/98-10   (3333 words)

 PlanetMath: submanifold There are several conflicting definitions of what a submanifold is, depending on which author you are reading. All that agrees is that a submanifold is a subset of a manifold which is itself a manifold, however how structure is inherited from the ambient space is not generally agreed upon. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003. planetmath.org /encyclopedia/Submanifold.html   (311 words)

 [No title]   (Site not responding. Last check: 2007-10-21) Title : Mathematical Sciences: Noncommutative Differential Geometry and Topology Abstract : This project is mathematical research in the theory of operator algebras bearing on issues in geometry and topology. Differentiable structure can be introduced in the form of a C*-dynamical system, with a Lie group acting on the C*-algebra. Another problem is to characterize the smooth structures of manifolds (commutative or otherwise) in terms of certain subalgebras and cyclic homology theory. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a8805158.txt   (155 words)

 RESEARCH in DIFFERENTIAL GEOMETRY structure P' (which is always possible) on the orthogonal frame bundle, P over M, and constructs the determinant line bundle, K, of P'. ), is a differentiable invariant of the manifold. One may define the *-Ricci curvature tensor, Ric*, incorporating the almost Hermitian structure, J, on an almost Hermitian manifold in a manner analogous to that of the Ricci curvature tensor. facpub.stjohns.edu /~watsonw/diffgeom.htm   (1548 words)

 [No title] Alternatively, because of the fundamental nature of the differential $da$, we wish to pose differentiability properties of $a$ in terms of properties of $da$. The differentiable case for $a$ becomes the assumption that $da$ is a densely defined, bounded sesquilinear form on $\CH$, so $da$ uniquely determines a bounded, linear transformation in $\bh$. \ee This structure is familiar from ordinary geometry.\footnote{In the case that $\CH$ is the $L^2$-Hilbert space of differential forms on a smooth, compact manifold ${\cal M}$, the $\Z_2$-grading $\gamma$ may be taken to equal $(-1)^n$ on the subspace of forms of degree $n$. www.ma.utexas.edu /mp_arc/papers/97-485   (9847 words)

 [No title]   (Site not responding. Last check: 2007-10-21) Another consequence is that as in the case of Lie groups the smooth structures of smooth stable planes are uniquely determined by the underlying (topological) stable planes. This is used in order to show that the smooth structures of a smooth stable plane are determined by the smooth structures of the line pencils of a triangle and by the geometric operations. Another important result is the local product structure of the point and line manifolds, i.e.\ every point $p$ (or every line) admits a chart that is the product of charts of two given lines which intersect in $p$. servix.mathematik.uni-stuttgart.de /~boedi/Smoothstableplanes.sum.html   (3822 words)

 Summer Tutorials, 2003 A space can be endowed with various kinds of geometric structures: a real differentiable structure describes a manifold, a complex structure describes a complex manifold, and a Riemannian structure describes a manifold together with a metric. However, due to the less intuitive character of a symplectic structure, it took mathematicians a long time to answer some basic questions about symplectic manifolds: a famous example is the nonsqueezing theorem (one cannot squeeze a ball into a cylinder of smaller base radius by a symplectic transformation), which was proved by Gromov in 1985. In many important cases, a complex structure is "the same" as an algebraic structure, and this allows to apply differential methods to the study of algebraic varieties. www.math.harvard.edu /tutorials/2003.html   (1754 words)

 Open Questions: Mathematics The first step is to take a different point of view on differential equations and to regard the equation as defining an "operator" -- that is, a kind of transformation that applies to a set of functions rather than to a set of points -- a "function of functions". Differentiable manifolds have at each point what is known as a "tangent space", which is quite analogous to the tangent line to a smooth curve. "Differential topology" and "differential geometry" are often referred to as subfields of modern geometry (and/or topology). www.openquestions.com /oq-math.htm   (8934 words)

 [No title] Kijowski and A. SmolskiDarboux theorem and the linearization of ordinary differential equationsBull. Kijowski, G. Rudolph and A. ThielmannSymplectic structure of a non-abelian Higgs model on the latticeRep. Math. Kijowski, G. Rudolph and C. SliwaOn the Structure of the Observable Algebra for QED on the LatticeLett. www.cft.edu.pl /p_view/kijowski.lista.txt   (2177 words)

 [No title]   (Site not responding. Last check: 2007-10-21) To see how the structure arises, first note that the vector bundle in R^n is R^2n, since there is a canonical isomorphism between all the tangent spaces. Also, a vector field on M is a differentiable mapping v:M->TM which satisfies P(v(p)) = p, where P is the projection from TM onto M. Also, a vector field on M is a differentiable mapping > v:M->TM which satisfies P(v(p)) = p, where P is the projection from TM onto > M. The tangent bundle on a smooth manifold is just one of many examples of vector bundles. www.math.niu.edu /~rusin/known-math/99/vec_bundle   (654 words)

 Why not SEDENIONS? Such a structure is represented by the design of the Temple of Luxor. The 28 new associative triple cycles of the sedenions are related to the 28-dimensional Lie algebra Spin(0,8), and to the 28 different differentiable structures on the 7-sphere S7 that are used to construct exotic structures on differentiable manifolds. It is exceptional, and is due to underlying octonion structures.) Then, as discussed by Gilbert and Murray in Clifford algebras and Dirac operators in harmonic analysis (Cambridge 1991), Dirac operators and related structures can be constructed and used to build the D4-D4-E6 physics model. www.valdostamuseum.org /hamsmith/sedenion.html   (5107 words)

 0pt   (Site not responding. Last check: 2007-10-21) The next thing to worry about is the idea of a differentiable structure, which is a (consistent) way of really doing calculus on the space. These equivalence classes of atlases are a'' smooth structure on M, or a differentiable structure on M. Given a manifold M, and here we should mean a given differentiable structure on a topological manifold, but we usually refer to it as `a manifold'', and the choice of a smooth structure is assumed. www.lehigh.edu /dlj0/yesterday/Desktop/dlj0/courses/423f96-lect2.html   (1062 words)

 Abstract   (Site not responding. Last check: 2007-10-21) The structure of globally hyperbolic spacetimes is investigated from the point of view of Connes' noncommutative geometry. The formula concerns the Lorentzian distance which determines the causal part of the Synge world function, satisfies an inverse triangular inequality and completely determines the topology, the differentiable structure, the metric tensor and the temporal orientation of a globally hyperbolic spacetime. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures. www.lqp.uni-goettingen.de /papers/02/04/02040800.html   (254 words)

 Tangent Spaces of Metric Spaces   (Site not responding. Last check: 2007-10-21) Subject Description: The Rademacher theorem says that a lipschitz function in R^n is differentiable almost everywhere, it is a basic ingredient in geometric measure theory. These development have shown that quite general metric spaces often have a kind of differentiable structure and generalized tangent spaces. Margulis, G. A.; Mostow, G. The differential of a quasi-conformal mapping of a Carnot-Carathéodory space. dmawww.epfl.ch /~troyanov/Seminaire/borel03.htm   (306 words)

 Outline of the course Geometry of SpaceTime Differentiable structure and differentiable manifold; (differentiable) functions and maps on/between manifolds; (differentiable) curves on manifolds. Differentiable partition of unity; existence of differentiable partitions of unity (statement without proof). Basics of the structural analysis of Einstein equations and their characterization with respect to the associated Cauchy problem. www-dft.ts.infn.it /~ansoldi/Didactics/Teaching/SpaceTimeCourse/Web/ProgEng.html   (1059 words)

 3.2.5 Topology It is for example not known if the three-sphere is characterized by its group invariants; this is the classical Poincaré conjecture, which is known to be true in every dimension except three. The Poincaré conjecture is no longer thought to be unreachable, and Thurston's structure theorems may even lead to a classification scheme for all compact three-manifolds. Robert Myers: The topology of 3-dimensional manifolds: structure and classification of compact 3-manifolds, group actions on these spaces, and their knot theory; generalizations to non-compact 3-manifolds. www.math.okstate.edu /grad/brief-hbk/3_2_5Topology.html   (498 words)

 The Klein Bottle   (Site not responding. Last check: 2007-10-21) The Klein bottle is simply a set on which we have placed a certain mathematical "structure." The Klein bottle is an example of a two-dimensional differentiable manifold. Let's just say that a differentiable manifold is a second countable Hausdorff topological space that has been endowed with a differentiable structure. Give the side of the bottle the same differentiable structure that it would have if the tube were not there. www.und.nodak.edu /instruct/lapeters/res/kb   (602 words)

 David Guarrera's Worldsheet: Exotic R^4 In order to do calculus (and hence, physics) on such spaces, we introduce something known as a differentiable structure on a space, making it into what's known as a differentiable manifold. The concept of a homeomorphism can then be applied to these spaces, and refined to a diffeomorphism--that is a homeomorphism that is somehow nicely differentiable. What is surprising (at least to me) is that there are spaces that you can put a differentiable structure on, which are homoeomorphic, but not diffeomorphic. web.mit.edu /guarrera/www/2005/10/exotic-r4.html   (451 words)

 articlestipus   (Site not responding. Last check: 2007-10-21) In particular, the set of the solutions is provided with a differentiable structure and a parametrization of all solutions is obtained through a coordinate atlas of the corresponding smooth manifold. More precisely we prove that the set of (C;A)-conditioned invariant subspaces having a fixed Brunovsky- Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. J. Ferrer, F. Puerta, X. Puerta Differentiable structure of the set of controllable (A,B)-invariant subspaces Linear Algebra and Appl. www.ioc.upc.es /usuaris/xavierpuerta/articlestipus.htm   (1456 words)

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