In mathematics, a projective bundle is a fiber bundle whose fibers are projective space s. Every vector bundle over a variety X gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way there is an obstruction in the cohomology group H sup 2 sup X ,O . References Citation last1 Elencwajg first1 G. last2 Narasimhan first2 M. S. title Projective bundles on a complex torus url http dx.doi.org 10.1515 crll.1983.340.1 doi 10.1515 crll.1983.340.1 id MR 691957 year 1983 journal Journal f r die reine und angewandte Mathematik issn 0075 4102 volume 340 pages 1 5 Category Algebraic topology Category Algebraic geometry ... more details
distinguish2 an optical fiber bundle File Roundhairbrush.JPG thumb A cylindrical hairbrush showing the intuition behind the term fiber bundle . This hairbrush is like a fiber bundle in which the base space ... topology , a fiber bundle or, in British English , fibre bundle is intuitively a space ... structure. Specifically, the similarity between the fiber bundle E and a product space B F is defined ... regions of B F to B . The map , called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle ... the projection from the product space to the first factor. This is called a trivial bundle . Examples ... bottle , as well as nontrivial covering space s. Fiber bundles such as the tangent bundle of a manifold and more general vector bundle s play an important role in differential geometry and differential topology , as do principal bundle s. Mappings which factor over the projection map are known as bundle maps , and the set of fiber bundles forms a category theory category with respect to such mappings. A bundle map from the base space itself with the identity mapping as projection to E is called a section fiber bundle section of E . Fiber bundles can be generalized in a number of ways, the most ... A fiber bundle consists of the data E , B , , F , where E , B , and F are topological spaces ... below. The space B is called the base space of the bundle, E the total space , and F the fiber . The map is called the projection map or bundle projection . We shall assume in what follows that the base .... The set of all U sub i sub , sub i sub is called a local trivialization of the bundle. Thus for any ... is and is called the fiber over p . Every fiber bundle E B is an open map , since projections ... bundle E , B , , F is often denoted math F longrightarrow E xrightarrow , pi B math that, in analogy ... as the map from total to base space. A smooth fiber bundle is a fiber bundle in the category mathematics ... more details
In conformal geometry , the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an group action effective group representation representation of the conformal group see associated bundle . The term tractor is a portmanteau of Tracey Thomas and twistor , the bundle having been introduced first by T. Y. Thomas as an alternative formulation of the Cartan connection Cartan conformal connection ref Thomas, T. Y., On conformal differential geometry , Proc. N.A.S. 12 1926 , 352 359 Conformal tensors , Proc. N.A.S. 18 1931 , 103 189. ref , and later rediscovered within the formalism of local twistor s and generalized to projective connection s by Michael Eastwood et al. in ref Bailey, T. N. Eastwood, M. G. Gover, A. R., Thomas s structure bundle for conformal, projective and related structures , Rocky Mountain J. 24 1994 , 1191 1217. ref References references Category Differential geometry Category Conformal geometry Category Vector bundles differential geometry stub ... more details
In mathematics , especially differential geometry , the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent space s at every point in the manifold. It may be described also as the dual bundle to the tangent bundle . The cotangent sheaf Smooth function Smooth Fiber bundle sections of the cotangent bundle are differential one form s. Definition of the cotangent sheaf Let M × M be the Cartesian product of M with itself. The diagonal mapping &Delta sends a point p in M ... of smooth functions of M . Thus it defines a vector bundle on M the cotangent bundle . The cotangent bundle as phase space Since the cotangent bundle X T M is a vector bundle , it can be regarded ... X is always an orientable manifold meaning that the tangent bundle of X is an orientable vector bundle . A special set of coordinates can be defined on the cotangent bundle these are called the canonical ... function on the cotangent bundle can be interpreted to be a symplectic vector space Hamiltonian thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out. The tautological one form main Tautological one form The cotangent bundle carries a tautological one ... as a manifold in its own right, there is a canonical Section fiber bundle section of the vector bundle ..., suppose that nowrap &pi T M &rarr M is the Projection mathematics projection of the bundle. Taking .... math That is, for a vector v in the tangent bundle of the cotangent bundle, the application of the tautological one form &theta to v at x , &omega is computed by projecting v into the tangent bundle ... one form is not a pullback of a one form on the base M . Symplectic form The cotangent bundle ... can be done by noting that being symplectic is a local property since the cotangent bundle ... the set of possible positions in a dynamical system , then the cotangent bundle math ,T M ... space looks like a cylinder. The cylinder is the cotangent bundle of the circle. The above symplectic ... more details
Image Moebiusstrip.png thumb 250px right The M bius strip is a line bundle over the 1 sphere S sup 1 ... , but the total bundle is different from S sup 1 sup × R which is a Cartesian product cylinder instead . In mathematics , a vector bundle is a topology topological construction that makes ... is then called a vector bundle over X . The simplest example is the case that the family of vector ... bundle X × V over X . Such vector bundles are said to be Fiber bundle Trivial bundle trivial . A more complicated and prototypical class of examples are the tangent bundle s of manifold ..., the tangent bundle of the two dimensional sphere is non trivial by the hairy ball theorem . In general, a manifold is said to be Parallelizable manifold parallelizable if and only if its tangent bundle ... they are examples of fiber bundle s. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle ... and first consequences A real vector bundle consists of topological spaces X base space and E total space a continuous function continuous surjection E X bundle projection for every x in X , the structure of a Hamel dimension finite dimensional real number real vector space on the Fiber bundle ... of the vector bundle. The local trivialization shows that locally the map looks like the projection ... of X , then k is called the rank of the vector bundle, and E is said to be a vector bundle of rank k . Often the definition of a vector bundle includes that the rank is well defined, so that k sub x sub is constant. Vector bundles of rank 1 are called line bundle s, while those of rank 2 are less commonly called plane bundle s. The Cartesian product X × R sup k sup , equipped with the projection X × R sup k sup X , is called the trivial bundle of rank k over X . Transition functions Given a vector bundle E X of rank k , and a pair of neighborhoods U and V over which the bundle ... more details
Infobox Anatomy Name Bundle of His Latin fasciculus atrioventricularis GraySubject GrayPage Image ConductionsystemoftheheartwithouttheHeart.png Caption Isolated Heart conduction system showing Bundle of His Image2 bundleofhis.png Caption2 Heart cut away showing Bundle of His Schematic representation of the atrioventricular bundle of His. The bundle, represented in red, originates near the orifice of the coronary sinus , undergoes slight enlargement to form the AV node . The AV node tapers down into the bundle of His, which passes into the ventricular septum and divides into two bundle branches ... DorlandsPre b 26 DorlandsSuf 12200650 The bundle of His , known as the AV bundle or atrioventricular bundle, is a collection of heart muscle cells specialized for electrical conduction that transmits ... medizinischen Klinik zu Leipzig, Jena, 1893 14 50. ref Function This bundle is an important part of the electrical .... The intrinsic rate of the Bundle of His is between 40 60 bpm. ref http sprojects.mmi.mcgill.ca cardiophysio anatomybundlehis.htm AnatomyBundleHis Bot generated title ref The bundle of His branches into the three bundle branch es the right, left anterior and left posterior bundle branches that run ... . These fibers distribute the impulse to the ventricular muscle. Together, the bundle branches and Purkinje ... to travel from the bundle of His to the ventricular muscle. Pathology If the Bundle of His is blocked ... of the right, left anterior, and left posterior bundle branch es. A third degree block is a very serious medical condition that will most likely require an artificial pacemaker . His Bundle Pacing Direct His Bundle pacing has produced synchronous ventricular depolarization and improved cardiac function ... His bundle pacing a novel approach to cardiac pacing in patients with normal His Purkinje activation ... needed date November 2010 See also Bundle of Kent References Reflist 2 External links GPnotebook 1469382599 UMichAtlas ht rt vent Right atrioventricular bundle branch, anterior view eMedicineDictionary ... more details
image ConductionsystemoftheheartwithouttheHeart.png right thumb 350px Image showing Bachmann s bundle Refimprove date December 2009 Bachmann s bundle , also known as the anterior interatrial band, is a broad band of atrial muscle that runs just behind the ascending aorta and connects the top of the right atrium with the top of the left atrium . Bachmann s bundle is, during normal sinus rhythm, the preferential path for electrical activation of the left atrium. It is therefore considered as part of the atrial conduction system of the heart . Interatrial conduction The normal cardiac rhythm originates in the sinoatrial node , which is located in the right atrium near the superior caval vein. From there, the electrical activation spreads over the right atrium. There are at least four different locations where the activation can pass to the left atrium. Apart from Bachmann s bundle these are the anterior interatrial septum, posterior interatrial septum, and the coronary sinus . ref Cite journal author Sakamoto, S I, et al. title Interatrial Electrical Connections The Precise Location and Preferential Conduction journal Journal of Cardiovascular Electrophysiology volume 16 pages 1077 1086 year 2005 doi 10.1111 j.1540 8167.2005.40659.x ref Because it originates close to the sinoatrial node and consists of long parallel fibers, Bachmann s bundle is, during sinus rhythm, the first of these connections to activate the left atrium. Bachmann s bundle and the atrial conduction system Besides Bachmann s bundle, the other three conduction tracts that constitute the atrial conduction system are known as the anterior , middle, and Posterior anatomy posterior tracts, which run from the sinoatrial node to the atrioventricular node , converging in the region near the coronary sinus . Atrial .... See also Electrical conduction system of the heart Bundle of His References Reflist Heart DEFAULTSORT Bachmann s Bundle Category Cardiac anatomy Circulatory stub id Berkas Bachmann nl Bundel van ... more details
Expert subject Mathematics date November 2008 In mathematics , a bundle gerbe is a geometry geometrical model of certain 1 gerbe s with connection mathematics connection , or equivalently of a 2 class in Deligne cohomology . Topology U 1 principal bundles over a space M see circle bundle are geometrical realizations of 1 classes in Deligne cohomology which consist of 1 form connection mathematics connections and 2 form curvatures. The topology of a U 1 bundle is classified by its Chern class , which is an element of H sup 2 sup M , the second integral cohomology of M . Gerbe s, or more precisely 1 gerbes, are abstract descriptions of Deligne 2 classes, which each define an element of H sup 3 ... sheaf mathematics sheaves of groupoid s over M . In 1994 Murray introduced bundle gerbes, which ... geometry. In fact, as their name suggests, they are fiber bundle s. This notion was extended to higher gerbes the following year. ref in http arxiv.org abs hep th 9511169 Higher Bundle Gerbes and Cohomology ... abs hep th 0106194 Twisted K theory and the K theory of Bundle Gerbes ref by Peter Bouwknegt , Alan ... Murray and Danny Stevenson ref the authors defined modules of bundle gerbes and used this to define a K theory for bundle gerbes. They then showed that this K theory is isomorphic to Rosenberg s twisted ... K theory Equivariant and Holomorphic Cases ref Relationship with field theory Bundle gerbes ... propagation on a Lie group group manifold as the connection mathematics connection of a bundle gerbe ... category 2006 10 bundle gerbes.html Bundle Gerbes General Idea and Definition http golem.ph.utexas.edu category 2006 10 bundle gerbes connections and.html Bundle Gerbes Connections and Surface Transport References http arxiv.org abs dg ga 9407015 Bundle gerbes , by Michael Murray. http arxiv.org abs hep th 0312154 Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory ...?level 1&index1 311272 Bundle gerbes on arxiv.org Notes Reflist Category Differential geometry de ... more details
In mathematics , the theory of fiber bundle s with a structure group math G math a topological group allows an operation of creating an associated bundle , in which the typical fiber of a bundle changes ... G math . For a fibre bundle F with structure group G , the transition functions of the fibre i.e. ... a fibre bundle F as a new fibre bundle having the same transition functions, but possibly a different ... in G . The associated bundle construction is just the observation that this data does just as well ... from a bundle with fiber math F math , on which math G math acts, to the associated principal bundle namely the bundle where the fiber is math G math , considered to act by translation on itself . For then we can go from math F 1 math to math F 2 math , via the principal bundle. Details in terms ... is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fibre, from a given fibre bundle. This then specializes to the case when the specified ... principal bundle. If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction ... 1951 page 36 ref Associated bundles in general Let &pi E &rarr X be a fibre bundle over a topological ... of the principal bundle associated to E . ref There is a locally trivial local trivialization of the bundle E consisting of an open cover U sub i sub of X , and a collection of bundle map fibre maps ... &prime be a specified topological space, equipped with a continuous left action of G . Then the bundle associated to E with fibre F &prime is a bundle E &prime with a local trivialization subordinate ... of the original bundle E . This definition clearly respects the cocycle condition on the transition ... the same coboundary. Hence, by the fiber bundle construction theorem , this produces a fibre bundle E &prime with fibre F &prime as claimed. Principal bundle associated to a fibre bundle As before ... more details
In mathematics , a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topological space topology . Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle E B with E and B Set mathematics sets . It is no longer true that the preimage s sup 1 sup x must all look alike, unlike fiber bundles where the fibers must all be isomorphic in the case of vector bundle s and homeomorphic . More generally, bundles or bundle objects can be defined in any category mathematics category in a category C , a bundle is simply an epimorphism E B . If the category is not concrete category concrete , then the notion of a preimage of the map is not necessarily available. Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do for instance, for a category with pullback category theory pullbacks and a initial object terminal object 1 the points of B can be identified with morphisms p 1 B and the fiber of p is obtained as the pullback of p and . Just as fiber bundles have section fiber bundle section s, a bundle can have a global section, which is a morphism s B E such that s id sub B sub . The category of bundles over B is therefore a subcategory of the slice category C B of objects over B , while the category of bundles without fixed base object is a subcategory of the comma category C C which is also the functor category C , the category of morphism s in C . The category of smooth vector bundles is a bundle object over the category of smooth manifolds in Cat , the category of small categories . The functor taking each manifold to its tangent bundle is an example of a section of this bundle object. References cite book last Goldblatt first Robert title Topoi ... DEFAULTSORT Bundle Mathematics Category Category theory Category Fiber bundles topology stub kk ... more details
File MobiusF.PNG 297px right thumb A M bius band is a non orientable I bundle. The dark line is the base for a set of transversal lines that are homeomorphic to the fiber and that each touch the edge of the band twice. File Moebiusband wikipedia.png right thumb An annulus is an orientable I bundle. This example is embedded in 3 space with an even number of twists 200px File MxS1.PNG right thumb this image represents the twisted I bundle over the 2 torus, which it is also fibered as a m bius strip times the circle. That is, this space it is also a circle bundle 200px In mathematics, an I bundle is a fiber bundle whose fiber is an interval and whose base is a manifold . Any kind of interval, open, closed, semi open, semi closed, open bounded, compact, even Line mathematics Ray ray s, can be the fiber. Two simple examples of I bundles are the Annulus mathematics annulus and the M bius band , the only two possible I bundles over the circle math scriptstyle S 1 math . The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product math scriptstyle S 1 times I math , and the M bius band is a non trivial or twisted bundle. Both bundles are 2 manifold s, but the annulus is an orientable manifold while the M bius band is a non orientable manifold . Curiously, there are only two kinds of I bundles when the base manifold is any surface but the Klein bottle math scriptstyle K math . That surface has three I bundles the trivial bundle math scriptstyle K times I math and two twisted bundles. Together with the Seifert fiber space s, I bundles are fundamental elementary building blocks for the description of three dimensional spaces. These observations are simple well known facts on elementary 3 manifold s. Line bundle s are both I bundles and vector bundle ... and not their possible vector properties, as we might be for line bundle s. References Scott, Peter ... on the orientable I bundle over K , by Maria Rita Casali, Dipartimento di Matematica Pura e Applicata ... more details
In mathematics , a pullback bundle or induced bundle ref cite book last Steenrod first Norman title The Topology of Fibre Bundles publisher Princeton University Press location Princeton year 1951 isbn 0 691 00548 6 page 47 ref ref cite book last Husemoller first Dale title Fibre Bundles publisher Springer edition Third location New York year 1994 isbn 978 0 387 94087 8 page 18 ref ref Cite book last1 ... in the theory of fiber bundle s. Given a fiber bundle &pi E &rarr B and a continuous topology continuous map f B &prime &rarr B one can define a pullback of E by f as a bundle f E over B &prime ... Let &pi E &rarr B be a fiber bundle with abstract fiber F and let f B &prime &rarr B be a continuous topology continuous map . Define the pullback bundle by math f E x,e in B times E mid f x pi e ... E where math psi x,e x, mbox proj 2 varphi e . , math It then follows that f sup sup E is a fiber bundle over B &prime with fiber F . The bundle f sup sup E is called the pullback of E by f or the bundle induced by f . The map math tilde f math is then a bundle morphism covering f . Properties Any section fiber bundle section s of E over B induces a section of f sup sup E , called the pullback section f sup sup s , simply by defining math f s s circ f math . If the bundle E &rarr B has structure ... sub i sub , &phi sub i sub then the pullback bundle f sup sup E also has structure group G . The transition ... bundle or principal bundle then so is the pullback f sup sup E . In the case of a principal bundle .... In the language of category theory , the pullback bundle construction is an example of the more ... of the pullback bundle can be carried out in subcategories of the category of topological spaces ... image of a sheaf of sections of a bundle is not in general the sheaf of sections of some direct image bundle, so that although the notion of a pushforward of a bundle is defined in some contexts for example ... encyclopedia PullbackBundle.html Pullback Bundle , PlanetMath Category Fiber bundles ru ... more details
In mathematics , a principal bundle ref cite book last Steenrod first Norman title The Topology of Fibre ... group G . In the same way as with the Cartesian product, a principal bundle P is equipped with An group ... of the space. A common example of a principal bundle is the frame bundle F E of a vector bundle ... bases. Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle ... group G determine a unique principal G bundle from which the original bundle can be reconstructed. Formal definition A principal G bundle, where G denotes any topological group , is a fiber bundle P ... the fibers of P and acts freely and transitively on them. This implies that the fiber of the bundle ... definition of a principal G bundle is as a G bundle P X with fiber G where the structure group acts ... example of a smooth principal bundle is the frame bundle of a smooth manifold M , often denoted ... bundle over M . Variations on the above example include the orthonormal frame bundle of a Riemannian ... bundle if E is any vector bundle of rank k over M , then the bundle of frames of E is a principal GL k , R bundle, sometimes denoted F E . A normal regular covering space p C X is a principal bundle where the structure group math pi 1 X p pi 1 C math acts on the fibres of p via the Covering space Monodromy action monodromy action . In particular, the universal cover of X is a principal bundle ... normal subgroup normal . Then G is a principal H bundle over the left coset space G H . Here the action ... bundle is the associated bundle of the M bius strip . Besides the trivial bundle, this is the only principal Z sub 2 sub bundle over S sup 1 sup . Projective space s provide some more interesting examples ... O 1 bundle over RP sup n sup . Likewise, S sup 2 n 1 sup is a principal U 1 bundle over complex projective space CP sup n sup and S sup 4 n 3 sup is a principal Sp 1 bundle over quaternionic ... with the Euclidean metric . For all of these examples the n 1 cases give the so called Hopf bundle ... more details
In algebraic geometry , a Tango bundle is one of the indecomposable vector bundle s of rank n &minus 1 constructed on n dimensional projective space P sup n sup by harvtxt Tango 1976 References Citation last1 Tango first1 Hiroshi title An example of indecomposable vector bundle of rank n &minus 1 on P sup n sup url http projecteuclid.org euclid.kjm 1250522965 id MathSciNet id 0401766 year 1976 journal Journal of Mathematics of Kyoto University issn 0023 608X volume 16 issue 1 pages 137 141 Category Algebraic geometry Category Vector bundles ... more details
In mathematics , a line bundle expresses the concept of a line that varies from point to point of a space ... the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology a line bundle is defined as a vector bundle of rank 1. ref Hartshorne 1975 , Google books quote id eICMfNiDdigC page 7 text line bundle p. 7 ref One dimensional real line bundles as just ... of a circle . A real line bundle is therefore in the eyes of homotopy theory as good as a fiber bundle ... cover on a differential manifold indeed that s a special case in which the line bundle is the determinant bundle top exterior power of the tangent bundle. The M bius strip corresponds to a double ... line bundle, we are looking in fact also for circle bundle s. There are some celebrated ones, for example the Hopf fibration s of sphere s to spheres. The tautological bundle on projective space Main Tautological line bundle One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space . The projectivization P V of a vector space V over a field ... of k sup × sup can be assembled into a k sup × sup bundle over P V . k sup ... a line bundle on P V . This line bundle is called the tautological line bundle . This line bundle is sometimes ... math mathcal O 1 math . Maps to projective space Suppose that X is a space and that L is a line bundle ... neighborhood U in X in which L is trivial, the total space of the line bundle is the product of U ... chooses a fiber of the tautological line bundle on P sup r sup , so choosing nowrap begin r 1 nowrap ... P sup r sup . This map sends the fibers of L to the fibers of the dual of the tautological bundle ... sup r sup , and the pullback of the dual of the tautological bundle under this map is L . In this way ... when this procedure constructs a Lefschetz pencil . In fact, it is possible for a bundle to have no non zero global sections at all this is the case for the tautological line bundle. When the line ... more details
Orphan date February 2009 Bundle Day in Swiss German B ndelitag is an unofficial holiday in Switzerland labelling the last Saturday off school before the summer vacation summer break when bundles and suitcase are being packed and made up for the holidays. The term B ndelitag probably had been used only in the region of Basel , but is nowadays widely spread across the germanophone regions of Switzerland since the holidays begin on the same date in most cantons. Although in a wider sense the term is also used for Saturdays off school before holidays in general, it is nationally used by media and calendars only before the summer vacation, because it marks its beginning in 11 cantons and therefore is the main travelling day in Switzerland. Depending on the holidays date determination the Bundle Day falls either on the last Saturday in June or on the first Saturday in July. In other cantons Bundle Day takes place simultaneously, but isn t nationally labelled Bundle Day . Since the five day week has been introduced in most public schools the term has lost significance, because Saturdays are usually off school. Nonetheless Bundle Day as the first day of the summer vacation is still very popular and is often called an unofficial holiday, although it is not a public holiday . Category Swiss folklore de B ndelitag ... more details
about Banach bundles in differential geometry Banach bundles in non commutative geometry Banach bundle non commutative geometry In mathematics , a Banach bundle is a vector bundle each of whose fibres ... dimension. Definition of a Banach bundle Let M be a Banach manifold of class C sup p sup with p 0, called ... is said to determine the structure of a Banach bundle on E M . If all the spaces X sub i sub ... X . In this case, E M is said to be a Banach bundle with fibre X . If M is a connected space ... in an obvious way to V itself. The tangent bundle T V of V is then a Banach bundle with the usual projection math pi mathrm T V to V math math x, v mapsto x. math This bundle is trivial in the sense ... mathrm T V to V times V math math x, v mapsto x, v . math If M is any Banach manifold, the tangent bundle T M of M forms a Banach bundle with respect to the usual projection, but it may not be trivial. Similarly, the cotangent bundle T M , whose fibre over a point x M is the Dual space Continuous dual ... also forms a Banach bundle with respect to the usual projection onto M . There is a connection between ... × H sup 1 sup , which as a Cartesian product also has the structure of a Banach bundle ... X are section fiber bundle cross section s of the bundle Y of some specified regularity L , in fact . If the differential geometry of the problem in question is particularly relevant, the Banach bundle ... M &prime be two Banach bundles. A Banach bundle morphism from the first bundle to the second consists ... Banach Bundle Morphism.png commutes, and, for each x &isin M , the induced map math f x E x to E ... p sup . Pull back of a Banach bundle One can take a Banach bundle over one manifold and use the Pullback bundle pull back construction to define a new Banach bundle on a second manifold. Specifically, let E N be a Banach bundle and f M N a differentiable map as usual, everything is C sup p sup . Then the pull back of E N is the Banach bundle f f E M satisfying the following properties for each ... more details
Merge from jet group date August 2011 In differential geometry , the jet bundle is a certain construction which makes a new smooth manifold smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equation s on Fiber bundle Sections section s of a fiber bundle ... vector field induced on the corresponding bundle e.g. , the geodesic spray on Finsler manifold ..., the jet bundle is now recognized as the correct domain for a covariant classical field theory ... M math be a fiber bundle in a category of manifold s and let math p in mathcal M math , with math ... mathematics submersion s. File Jet Bundle Image FbN.png 500px center A coordinate system on math .... In particular, if math mathcal E , pi, mathcal M , math is a fiber bundle, the triple math J r pi, pi r , mathcal M , math defines the math r th , math jet bundle of math pi , math . If math W subset ... pi , math is the trivial bundle math mathcal M times mathbb R , pr 1 , mathcal M math , then there is a canonical diffeomorphism between the first jet bundle math J 1 pi , math and math T mathcal ... , math is the trivial bundle math mathbb R times mathcal M , pr 1 , mathbb R math , then there exists ... bundle, and may be coordinated by math x,u,u 1 , math , where align right math x j 1 p sigma math align ... partial u alpha , math and math psi in Gamma T mathcal E , math . The jet bundle math J r pi , math ... equations Let math mathcal E , pi, mathcal M math be a fiber bundle. An math r th , math order partial ... bundle math mathbb R 2 times mathbb R , pr 1 , mathbb R 2 , math with global coordinates math x 1 , x ..., mathcal E math defines the first jet bundle, and may be coordinated by math x,u,u 1 , math , where ... algebra, which can be assumed to be the smooth functions algebra over the geometric object math ... structure it is a filtered commutative algebra. Roughly speaking, a concrete element math varphi ... is an ideal in the algebra math mathcal F k pi math , and hence in the direct limit math mathcal ... more details
Refimprove date September 2009 Bundle theory , originated by the 18th century Scottish philosopher David ... consists only of a collection bundle of properties, relations or trope philosophy tropes . According to bundle theory, an object consists of its properties and nothing more thus neither can there be an object without properties nor can one even conceive of such an object for example, bundle theory claims ... theory substance in which the properties inherence inhere . Arguments for the bundle theory ... properties is a common justification for bundle theory, especially among current philosophers in the Anglo ... of bare particulars leaves a bundle of properties and nothing more as the only possible conception of an object, thus justifying bundle theory. Objections to the bundle theory Objections to bundle theory concern the nature of the bundle of properties , the properties wikt compresence compresence ... on understanding reality. Compresence objection Bundle theory maintains that properties are bundled together in a collection without describing how they are tied together. For example, bundle theory ... . The apple is said to be a bundle of properties including redness, being four inches 100 mm wide, and juiciness. Critics question how bundle theory accounts for the properties wikt compresence ... if there is no substance in which they both inhere . Traditional bundle theory, according to Professor ... of a bundle of properties located on the table, one of which is the looks like an apple property .... The bundle theory of substance explains compresence . Specifically, it maintains that properties ... strata. The bundle theory of substance thus rejects the substance theories of Aristotle , Descartes ... reality objection Section OR date May 2012 The language reality objection to bundle theory relates ... that supports bundle theory. Per the objection, properties are synthetic constructions of language ... abstractions of experience. citation needed date August 2010 Bundle theory and Buddhism The Buddhist ... more details
Image Celery cross section.jpg thumb Cross section of celery stalk, showing vascular bundles, which include both phloem and xylem A vascular bundle is a part of the transport system in vascular plant s. The transport itself happens in vascular tissue , which exists in two forms xylem and phloem . Both these tissues are present in a vascular bundle, which in addition will include supporting and protective tissues. Also, it is a vein in the leaf that contains conducting tissues. The xylem typically lies wiktionary adaxial adaxial with phloem positioned wiktionary abaxial abaxial . In a stem or root this means that the xylem is closer to the centre of the stem or root while the phloem is closer to the exterior. In a leaf, the adaxial surface of the leaf will usually be the upper side, with the abaxial surface the lower side. This is why aphid s are typically found on the underside of a leaf rather than on the top, since the sugars manufactured by the plant are transported by the phloem, which is closer to the lower surface. The position of vascular bundles relative to each other may vary considerably see Stele biology stele . Bundle sheath cells Bundle sheath cells are photosynthetic cells arranged into tightly packed sheaths around the veins of a leaf. They form a protective covering on leaf veins, and consist of one or more cell layers, usually parenchyma . Loosely arranged mesophyll cells lie between the bundle sheath and the leaf surface. The Calvin cycle is confined to the chloroplasts of these bundle sheath cells in C4 carbon fixation C4 plants . External links Vascular bundles pictured in cross section, by http botweb.uwsp.edu anatomy primaryxylem.htm Curtis, Lersten, and Nowak and http www.sbs.utexas.edu mauseth weblab webchap8phloem chapter 8.htm Mauseth References Campbell, N. A. & Reece, J. B. 2005 . Photosynthesis. Biology 7th ed. . San Francisco Benjamin Cummings. Category Plant anatomy Category Plant physiology Category Tissues cs C vn svazek de Leitb nd ... more details
For the jazz album Bundle of Joy album Infobox Film name Bundle of Joy image Bundleofjoyposter.jpg image size caption Theatrical release poster director Norman Taurog producer Edmund Grainger writer Robert Carson screenwriter Robert Carson br Norman Krasna br Arthur Sheekman br Felix Jackson story starring Eddie Fisher singer Eddie Fisher br Debbie Reynolds br Adolphe Menjou music cinematography editing distributor RKO Pictures released start date 1956 12 12 runtime 98 minutes country Cinema of the United States United States language English budget gross Bundle of Joy 1956 in film 1956 is a musical remake of the comedy film Bachelor Mother 1939 . It stars Eddie Fisher singer Eddie Fisher , Debbie Reynolds , and Adolphe Menjou . An unmarried salesgirl at a department store finds and takes care of an abandoned baby. Much confusion results when her co workers assume the child is hers and that the father is the son of the store owner. Carrie Fisher tells the story, in her documentary Wishful Drinking of how Reynolds was pregnant with her when this movie was made. This movie is in color by Technicolor . Cast Eddie Fisher singer Eddie Fisher as Dan Merlin Debbie Reynolds as Polly Parish Adolphe Menjou as J. B. Merlin, Dan s father and Polly s ultimate boss Tommy Noonan as Freddie Miller Nita Talbot as Mary Una Merkel as Mrs. Dugan Melville Cooper as Adams Gil Stratton as Mike Clancy External links tcmdb title id 1647 title Bundle of Joy imdb title 0049034 Bundle of Joy Amg movie 7557 Bundle of Joy Category 1956 films Category 1950s musical films Category 1950s romantic comedy films Category Christmas films Category American musical comedy films Category American romantic comedy films Category American romantic musical films Category Film remakes Category Films directed by Norman Taurog Category RKO Pictures films 1950s comedy film stub ... more details
Wikify date October 2011 Orphan date February 2009 The Vote Bundle is the collection of paperwork given daily to each to Members of Parliament Member of the British House of Commons when the House is sitting., ref Cite web url http www.parliament.uk documents upload P16.pdf title The Vote Bundle date November 2008 work House of Commons Information Office, Factsheet P16 accessdate 2009 07 13 Dead link date October 2010 bot H3llBot ref These are considered the practical daily working papers of the house. ref Cite book last Rogers first Robert coauthors R. H. Walters. title How Parliament Works publisher Pearson Longman, location Harlow, England year 2004 isbn 9780582437449 ref This bundle usually includes Summary agenda Order of business Standing committee notices Future business Order of business in Westminster Hall The vote record of proceedings Notices of questions Private business Notice of motions Notice of amendments Other documents References Reflist Category Political terms in the United Kingdom UK poli stub ... more details
for the rank 2 bundle on 4 dimensional projective space Horrocks Mumford bundle In algebraic geometry , Horrocks bundles are certain indecomposable rank 3 vector bundle s locally free sheaves on 5 dimensional projective space, found by harvtxt Horrocks 1978 . References Citation last1 Ancona first1 Vincenzo last2 Ottaviani first2 Giorgio title The Horrocks bundles of rank three on P sup 5 sup doi 10.1515 crll.1995.460.69 id MathSciNet id 1316572 year 1995 journal Journal f r die reine und angewandte Mathematik issn 0075 4102 volume 460 pages 69 92 Citation last1 Horrocks first1 G. author1 link Geoffrey Horrocks title Examples of rank three vector bundles on five dimensional projective space doi 10.1112 jlms s2 18.1.15 id MathSciNet id 502651 year 1978 journal The Journal of the London Mathematical Society. Second Series issn 0024 6107 volume 18 issue 1 pages 15 27 Category Algebraic geometry Category Vector bundles ... more details
Image Bundle adjustment sparse matrix.png right thumb A sparse matrix obtained when solving a modestly sized bundle adjustment problem. This is the sparsity pattern of a 992× 992 normal equations i.e. approximate Hessian matrix. Black regions correspond to nonzero blocks. Given a set of images depicting a number of 3D points from stereoscopy different viewpoints , bundle adjustment can be defined as the problem of simultaneously refining the 3D Coordinate system coordinates describing the scene ... problem corresponding image projections of all points. Bundle adjustment is almost always used ... is zero mean Gaussian noise Gaussian , then bundle adjustment is the Maximum likelihood Maximum Likelihood ... categorical bundle mathematics bundle seems a pure coincidence . Bundle adjustment was originally conceived ... researchers during recent years. Bundle adjustment boils down to minimizing the reprojection error ... equations . When solving the minimization problems arising in the framework of bundle adjustment, the normal .... Mathematical definition Bundle adjustment amounts to jointly refining a set of initial camera and structure ... math by a vector math mathbf b i math . Bundle adjustment minimizes the total reprojection error with respect ... . Clearly, bundle adjustment is by definition tolerant to missing image projections and minimizes a physically meaningful criterion. Software http www.ics.forth.gr lourakis sba sba A Generic Sparse Bundle ... Matlab http www.inf.ethz.ch personal chzach opensource.html ssba Simple Sparse Bundle Adjustment ... projects mcba mcba Multi Core Bundle Adjustment CPU GPU . https github.com dkogan libdogleg ... equation References cite conference title Bundle Adjustment A Modern Synthesis author B. Triggs ... Bundle Adjustment author M.I.A. Lourakis and A.A. Argyros journal ACM Transactions on Mathematical ... pubs 2000 TMHF00 Triggs va99.pdf B. Triggs, P. McLauchlan, R. Hartley and A. Fitzgibbon, Bundle Adjustment ... rbf CVonline LOCAL COPIES ZISSERMAN bundle bundle.html Bundle adjustment . CV Online ... more details
In differential geometry , given a spin structure on a math n math dimensional Riemannian manifold math M, g , , math one defines the spinor bundle to be the complex vector bundle math pi mathbf S colon mathbf S to M , math associated to the corresponding principal bundle math pi mathbf P colon mathbf P to M , math of spin frames over math M math and the spin representation of its structure group math mathrm Spin n , math on the space of spinor s math Delta n. , math . A section of the spinor bundle math mathbf S , math is called a spinor field . Formal definition Let math mathbf P ,F mathbf P math be a spin structure on a Riemannian manifold math M, g , , math that is, an equivariant lift of the oriented orthonormal frame bundle math mathrm F SO M to M math with respect to the double covering math rho colon mathrm Spin n to mathrm SO n . , math The spinor bundle math mathbf S , math is defined ref citation last1 Friedrich first1 Thomas title Dirac Operators in Riemannian Geometry publisher American Mathematical Society year 2000 isbn 978 0 8218 2055 1 page 53 ref to be the complex vector bundle math mathbf S mathbf P times kappa Delta n , math associated to the spin structure math mathbf P math via the spin representation math kappa colon mathrm Spin n to mathrm U Delta n , , math where math mathrm U mathbf W , math denotes the Group mathematics group of unitary operator s acting on a Hilbert space math mathbf W . , math It is worth noting that the spin representation math kappa math is a faithful and unitary representation of the group math mathrm Spin n math ref citation last1 Friedrich first1 Thomas title Dirac Operators in Riemannian Geometry publisher American Mathematical Society year 2000 isbn 978 0 8218 2055 1 pages 20 and 24 ref . See also Orthonormal frame bundle Spinor Spin manifold Spinor representation Spin geometry Notes Reflist Books Cite book last1 Lawson first1 H. Blaine last2 Michelsohn first2 Marie Louise title Spin Geometry publisher Princeton U ... more details
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