query: a bundle of sticks that starts with an f

pos: Faggot (unit) steel rods or bars totaling . Faggot (unit) A faggot, in the meaning of "bundle", is an archaic English unit applied to bundles of certain items. Alternate spellings in Early Modern English include "fagate, faget, fagett, faggott, fagot, fagatt, fagott, ffagott," and "faggat". Sometimes called a short faggot, a faggot of sticks equals a bundle of wood sticks or billets that is in length and in circumference. The measurement was standardised in ordinances by 1474. A small short faggot was also called a nicket. A brush-faggot (sometimes shortened to brush) was a bundle of similar size made of brushwood. A

pos: Faggot (unit) Faggot (unit) A faggot, in the meaning of "bundle", is an archaic English unit applied to bundles of certain items. Alternate spellings in Early Modern English include "fagate, faget, fagett, faggott, fagot, fagatt, fagott, ffagott," and "faggat". Sometimes called a short faggot, a faggot of sticks equals a bundle of wood sticks or billets that is in length and in circumference. The measurement was standardised in ordinances by 1474. A small short faggot was also called a nicket. A brush-faggot (sometimes shortened to brush) was a bundle of similar size made of brushwood. A long faggot of sticks equals a

neg: Fiber bundle construction theorem Fiber bundle construction theorem In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic. The theorem is important in the associated bundle construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same. Let "X" and "F" be topological spaces and let "G" be a topological group with a continuous left action on "F". Given an open

neg: Bundle of rights dominion over a thing, placing restrictions "on others" from interfering with the owner's property. "Bundle of rights," however, implies rules specifying, proscribing, or authorizing actions on the part of the owner. Ownership of land is a much more complex proposition than simply acquiring all the rights to it. It is useful to imagine a bundle of rights that can be separated and reassembled. A "bundle of sticks" - in which each stick represents an individual right – is a common analogy made for the bundle of rights. Any property owner possesses a set of "sticks" related directly to the land.

neg: Fiber bundle construction theorem left multiplication then one obtains the associated principal bundle. Fiber bundle construction theorem In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic. The theorem is important in the associated bundle construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same. Let "X" and "F" be topological spaces and let "G" be a topological group with

neg: Charles Deburau candle. "What a weird room! But it's warm: that's good. Ah, but I'm hungry." He lights the bundle of sticks, breaks the eggs, makes an omelet, and eats it with delight. But the flames burn the whale, which thrashes about convulsively. And that makes Pierrot seasick. "I must be in a boat," he says. The whale again starts to thrash about. Pierrot discovers a little chest at his feet. He opens it; it is full of gold, which the whale has swallowed during a shipwreck. He plays with the gold pieces; he is enchanted: now he can marry Columbine. But

neg: Associated bundle necessary, the "g" transform via the same coboundary.) Hence, by the fiber bundle construction theorem, this produces a fibre bundle "E"′ with fibre "F"′ as claimed. As before, suppose that "E" is a fibre bundle with structure group "G". In the special case when "G" has a free and transitive left action on "F"′, so that "F"′ is a principal homogeneous space for the left action of "G" on itself, then the associated bundle "E"′ is called the principal "G"-bundle associated to the fibre bundle "E". If, moreover, the new fibre "F"′ is identified with "G" (so that "F"′ inherits

neg: Fiber bundle construction theorem "G", we see that two fiber bundles with the same base, fiber, structure group, trivializing neighborhoods, and transition functions are isomorphic. A similar theorem holds in the smooth category, where "X" and "Y" are smooth manifolds, "G" is a Lie group with a smooth left action on "Y" and the maps "t" are all smooth. The proof of the theorem is constructive. That is, it actually constructs a fiber bundle with the given properties. One starts by taking the disjoint union of the product spaces "U" × "F" and then forms the quotient by the equivalence relation The total space

neg: Ehresmann connection thus a principal connection). It is "not always" possible for an Ehresmann connection to induce, in a natural way, a connection on an associated bundle. For example, a non-equivariant Ehresmann connection on a bundle of frames of a vector bundle may not induce a connection on the vector bundle. Suppose that "E" is an associated bundle of "P", so that "E" = "P" × "F". A "G"-connection on "E" is an Ehresmann connection such that the parallel transport map τ : "F" → "F" is given by a "G"-transformation of the fibers (over sufficiently nearby points "x" and "x"′ in

neg: Essentially finite vector bundle if there exists a finite formula_2-group scheme formula_7 and a formula_7-torsor formula_9 such that formula_4 becomes trivial over formula_11 (i.e. formula_12, where formula_13). Let "X" be a scheme and "E" a vector bundle on "X". For formula_14 an integral polynomial with nonnegative coefficients define A vector bundle "E" is called finite if there are two distinct polynomials "f, g" for which "f(E)" is isomorphic to "g(E)". A bundle is essentially finite if it's a subquotient of a finite vector bundle. Essentially finite vector bundle In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by

neg: Frame bundle denoted F"M" or O("M"), is the orthonormal frame bundle associated to the tangent bundle of "M" (which is equipped with a Riemannian metric by definition). If "M" is orientable, then one also has the oriented orthonormal frame bundle F"M". Given a Riemannian vector bundle "E", the orthonormal frame bundle is a principal O("k")-subbundle of the general linear frame bundle. In other words, the inclusion map is principal bundle map. One says that F("E") is a reduction of the structure group of F("E") from GL("k", R) to O("k"). If a smooth manifold "M" comes with additional structure it is often natural

neg: Connection (vector bundle) on "E" is an affine space for Ω(End "E"). Let "E" → "M" be a vector bundle of rank "k" and let F("E") be the principal frame bundle of "E". Then a (principal) connection on F("E") induces a connection on "E". First note that sections of "E" are in one-to-one correspondence with right-equivariant maps F("E") → R. (This can be seen by considering the pullback of "E" over F("E") → "M", which is isomorphic to the trivial bundle F("E") × R.) Given a section σ of "E" let the corresponding equivariant map be ψ(σ). The covariant derivative on "E" is

neg: Frame bundle "E" and its frame bundle F("E") are associated bundles. Each one determines the other. The frame bundle F("E") can be constructed from "E" as above, or more abstractly using the fiber bundle construction theorem. With the latter method, F("E") is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as "E" but with abstract fiber GL("k", R), where the action of structure group GL("k", R) on the fiber GL("k", R) is that of left multiplication. Given any linear representation ρ : GL("k", R) → GL("V",F) there is a vector bundle associated to F("E") which is given

neg: Frame bundle on F("E") is the final topology coinduced by the inclusion maps π("U") → F("E"). With all of the above data the frame bundle F("E") becomes a principal fiber bundle over "X" with structure group GL("k", R) and local trivializations ({"U"}, {ψ}). One can check that the transition functions of F("E") are the same as those of "E". The above all works in the smooth category as well: if "E" is a smooth vector bundle over a smooth manifold "M" then the frame bundle of "E" can be given the structure of a smooth principal bundle over "M". A vector bundle

neg: Bundle map If π:"F"→ "N" is a fiber bundle over "N" and "f":"M"→ "N" is a continuous map, then the pullback of "F" by "f" is a fiber bundle "f""F" over "M" whose fiber over "x" is given by ("f""F").= "F". It then follows that a bundle map from "E" to "F" covering "f" is the same thing as a bundle map from "E" to "f""F" over "M". There are two kinds of variation of the general notion of a bundle map. First, one can consider fiber bundles in a different category of spaces. This leads, for example, to the notion of

neg: John Tewkesbury was ordered to carry a bundle of sticks at St. Paul's Church on the following Sunday. He was to carry a bundle of sticks other places on the following week and wear sticks embroidered on both sleeves. He was also ordered to enter the monastery of St. Bartholomew's on Whitsunday eve (30 May) and remain there until released by the Bishop. Following his incarceration at the monastery, he renounced his prior beliefs and was released. In close co-operation with Cuthbert Tunstall's successor, John Stokesley, Sir Thomas More, High Chancellor of England, arrested George Constantine, a Protestant book dealer, for heresy

neg: Pick-up sticks wood, bamboo, straw, reed, rush, yarrow, or plastics. Some Haida First Nation pick-up sticks are plain maple wood decorated with abalone shell and copper. Today, the most common pick-up sticks game is Mikado. It remains difficult in all variations. The object of the game is to pick up the most sticks. To begin the game, a bundle of sticks is randomly distributed so that they end up in a tangled pile. The more tangled the resulting disarray, the more challenging the game. In some versions of the game, any isolated sticks, or sticks lying alone, are removed. The players attempt

neg: Fiber bundle construction theorem "E" of the bundle is "T"/~ and the projection π : "E" → "X" is the map which sends the equivalence class of ("i", "x", "y") to "x". The local trivializations are then defined by Let "E" → "X" a fiber bundle with fiber "F" and structure group "G", and let "F"′ be another left "G"-space. One can form an associated bundle "E"′ → "X" with a fiber "F"′ and structure group "G" by taking any local trivialization of "E" and replacing "F" by "F"′ in the construction theorem. If one takes "F"′ to be "G" with the action of

neg: Associated bundle Associated bundle In mathematics, the theory of fiber bundles with a structure group formula_1 (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from formula_2 to formula_3, which are both topological spaces with a group action of formula_1. For a fibre bundle "F" with structure group "G", the transition functions of the fibre (i.e., the cocycle) in an overlap of two coordinate systems "U" and "U" are given as a "G"-valued function "g" on "U"∩"U". One may then construct a fibre bundle "F"′ as a new fibre bundle having

neg: Fiber bundle fiber bundle π : "E" → "B" is an open map, since projections of products are open maps. Therefore "B" carries the quotient topology determined by the map π. A fiber bundle ("E", "B", π, "F") is often denoted that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space. A smooth fiber bundle is a fiber bundle in the category of smooth manifolds. That is, "E", "B", and "F" are required to be smooth manifolds and all the functions above are

neg: Vector bundle "E" → "X" is given by a pair of continuous maps "f" : "E" → "E" and "g" : "X" → "X" such that Note that "g" is determined by "f" (because π is surjective), and "f" is then said to cover "g". The class of all vector bundles together with bundle morphisms forms a category. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a bundle map between

neg: Descent (mathematics) (gluing conditions). For example, the composition for transitivity (and choosing apt notation). The "f" should be identity maps and hence symmetry becomes formula_5 (so that it is fiberwise an isomorphism). These are indeed standard conditions in fiber bundle theory (see transition map). One important application to note is "change of fiber": if the "f" are all you need to make a bundle, then there are many ways to make an associated bundle. That is, we can take essentially same "f", acting on various fibers. Another major point is the relation with the chain rule: the discussion of the way there

neg: Spin structure of fiber bundle had been introduced; André Haefliger (1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and Max Karoubi (1968) extended this result to the non-orientable pseudo-Riemannian case. A spin structure on an orientable Riemannian manifold "(M,g)" is an equivariant lift of the oriented orthonormal frame bundle "F"("M") → "M" with respect to the double covering ρ: Spin("n") → SO("n"). In other words, a pair (P,"F") is a spin structure on the principal bundle π: "F"("M") → "M" when The principal bundle π: P → "M" is also called the bundle

neg: Bundle map formula_5. In other words, formula_3 is fiber-preserving, and "f" is the induced map on the space of fibers of "E": since π is surjective, "f" is uniquely determined by formula_3. For a given "f", such a bundle map formula_3 is said to be a bundle map "covering f". It follows immediately from the definitions that a bundle map over "M" (in the first sense) is the same thing as a bundle map covering the identity map of "M". Conversely, general bundle maps can be reduced to bundle maps over a fixed base space using the notion of a pullback bundle.

neg: Conic bundle the scalar "a". One defines the reciprocal polynomial by formula_16, and the conic bundle "F" as follows : formula_17 is the surface obtained as "gluing" of the two surfaces formula_18 and formula_19 of equations and along the open sets by isomorphisms One shows the following result : The surface "F" is a "k" clean and smooth surface, the mapping defined by by and the same on formula_26 gives to "F" a structure of conic bundle over "P". Conic bundle In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the

neg: Quindaro Townsite the founder who purchased land for the settlement, named it after his wife "Quindaro" (meaning "bundle of sticks" or "strength through numbers") Nancy Guthrie. She was a member of the Wyandot tribe and had persuaded them to sell land to her husband. Construction started in January 1857, and the town soon contained numerous stone houses and starts of several businesses. Its sawmill was the largest in Kansas. The lower townsite near the river was the commercial core, and most residences were higher on the bluff, at the upper townsite. In the first year there were 100 buildings completed, with many

neg: BRST quantization principal bundle. It is the latter sort of gauge transformation that enters into the BRST method. Unlike a passive transformation, it is well-defined globally on a principal bundle with any structure group over an arbitrary manifold. (However, for concreteness and relevance to conventional QFT, this article will stick to the case of a principal gauge bundle with compact fiber over 4-dimensional Minkowski space.) A principal gauge bundle "P" over a 4-manifold "M" is locally isomorphic to "U" × "F", where "U" ⊂ R and the fiber "F" is isomorphic to a Lie group "G", the gauge group of the field

neg: Associated bundle foliation to be defined as a reduction of the tangent bundle to a block matrix subgroup - but here the reduction is only a necessary condition, there being an "integrability condition" so that the Frobenius theorem applies. Associated bundle In mathematics, the theory of fiber bundles with a structure group formula_1 (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from formula_2 to formula_3, which are both topological spaces with a group action of formula_1. For a fibre bundle "F" with structure group "G", the transition functions of the

neg: Unit tangent bundle a natural projection which takes each point of the bundle to its base point. The fiber "π"("x") over each point "x" ∈ "M" is an ("n"−1)-sphere S, where "n" is the dimension of "M". The unit tangent bundle is therefore a sphere bundle over "M" with fiber S. The definition of unit sphere bundle can easily accommodate Finsler manifolds as well. Specifically, if "M" is a manifold equipped with a Finsler metric "F" : T"M" → R, then the unit sphere bundle is the subbundle of the tangent bundle whose fiber at "x" is the indicatrix of "F": If "M"

neg: Chern class algebraic topology: the Chern classes arise via homotopy theory which provides a mapping associated with "V" to a classifying space (an infinite Grassmannian in this case). For any vector bundle "V" over a manifold "M", there exists a mapping "f" from "M" to the classifying space such that the bundle "V" is equal to the pullback, by "f", of a universal bundle over the classifying space, and the Chern classes of "V" can therefore be defined as the pullback of the Chern classes of the universal bundle; these universal Chern classes in turn can be explicitly written down in terms

neg: Darboux frame equations for an adapted system of frames classifies arbitrary embedded submanifolds up to a Euclidean motion. In detail, the projection π : "F"("M") → "M" given by π("x"; "f") = "x" gives "F"("M") the structure of a principal bundle on "M" (the structure group for the bundle is O("p") × O("n" − "p").) This principal bundle embeds into the bundle of Euclidean frames "F"("n") by φ("v";"f") := (φ("v");"f") ∈ "F"("n"). Hence it is possible to define the pullbacks of the invariant forms from "F"("n"): Since the exterior derivative is equivariant under pullbacks, the following structural equations hold Furthermore, because some

neg: Vector bundle just the fiber over "f"("x") ∈ "Y". Hence, Whitney summing "E" ⊕ "F" can be defined as the pullback bundle of the diagonal map from "X" to "X" × "X" where the bundle over "X" × "X" is "E" × "F". Remark: Let "X" be a compact space. Any vector bundle "E" over "X" is a direct summand of a trivial bundle; i.e., there exists a bundle "E" such that "E" ⊕ "E" is trivial. This fails if "X" is not compact: for example, the tautological line bundle over the infinite real projective space does not have this property. Vector

neg: The F Word (South Park) old woman" and a bundle of sticks. Upset by the national attention, Mayor McDaniels wants to resolve the situation, and the boys suggest getting the official dictionary definition updated. The town invites the English Dictionary Officiates, led by the head editor, former child actor Emmanuel Lewis, to review the proposal and consider making the definition change official. As the town celebrates the arrival of Lewis and the Officiates, the bikers suddenly crash the event and begin to riot. After inflicting damage to the town and scaring off the citizens, they corner the boys in an alley. The bikers demand they

neg: The Old Man and his Sons and Bath with the title “The old man, his children, and the bundle of sticks”. There “A good old man, no matter where, Whether in York or Lancashire,” gives the lesson on his deathbed and the poem concludes with a Christian reflection. Over the following decades of the 19th century, it also appeared as a cheaply printed chapbook and in book-length collections of moralising works. The Old Man and his Sons The Old Man and his Sons, sometimes alternatively titled The Bundle of Sticks, is one of Aesop's Fables and is numbered 53 in the Perry Index. The actions described

neg: Associated bundle a fibre product construction. Let π : "E" → "X" be a fiber bundle over a topological space "X" with structure group "G" and typical fibre "F". By definition, there is a left action of "G" (as a transformation group) on the fibre "F". Suppose furthermore that this action is effective. There is a local trivialization of the bundle "E" consisting of an open cover "U" of "X", and a collection of fibre maps such that the transition maps are given by elements of "G". More precisely, there are continuous functions "g" : ("U" ∩ "U") → "G" such that

neg: Fiber bundle ("E", "B", π, "F"), where "E", "B", and "F" are topological spaces and π : "E" → "B" is a continuous surjection satisfying a "local triviality" condition outlined below. The space "B" is called the base space of the bundle, "E" the total space, and "F" the fiber. The map π is called the projection map (or bundle projection). We shall assume in what follows that the base space "B" is connected. We require that for every "x" in "E", there is an open neighborhood "U" ⊂ "B" of π("x") (which will be called a trivializing neighborhood) such that there

neg: Tensor product bundle End("L") is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space "X" forms an abelian group called the Picard group of "X". One can also define a symmetric power and an exterior power of a vector bundle in a similar way. For example, a section of formula_1 is a differential "p"-form and a section of formula_2 is a differential "p"-form with values in a vector bundle "E". Tensor product bundle In differential geometry, the tensor product of vector bundles "E", "F" is a vector bundle, denoted by "E" ⊗ "F", whose fiber over

neg: Jet bundle sections of a bundle, but it is possible to define jets of functions "f: M" → "N", where "M" and "N" are manifolds; the jet of "f" then just corresponds to the jet of the section ("gr" is known as the graph of the function "f") of the trivial bundle ("M" × "N", π, "M"). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π. Jet bundle In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given

neg: Banach bundle construction to define a new Banach bundle on a second manifold. Specifically, let "π" : "E" → "N" be a Banach bundle and "f" : "M" → "N" a differentiable map (as usual, everything is "C"). Then the pull-back of "π" : "E" → "N" is the Banach bundle "f"*"π" : "f"*"E" → "M" satisfying the following properties: Banach bundle In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension. Let "M" be a Banach manifold of class "C" with "p" ≥ 0,

neg: Fiber bundle that the action of "G" on the fiber "F" is free and transitive). In this case, it is often a matter of convenience to identify "F" with "G" and so obtain a (right) action of "G" on the principal bundle. It is useful to have notions of a mapping between two fiber bundles. Suppose that "M" and "N" are base spaces, and formula_17 and formula_18 are fiber bundles over "M" and "N", respectively. A bundle map (or bundle morphism) consists of a pair of continuous functions such that formula_20. That is, the following diagram commutes: For fiber bundles with structure

neg: Clifford module bundle fact, such a bundle can be constructed if and only if "M" is a spin manifold. Let "M" be an "n"-dimensional spin manifold with spin structure "F"("M") → "F"("M") on "M". Given any "Cℓ"R-module "V" one can construct the associated spinor bundle where σ : Spin("n") → GL("V") is the representation of Spin("n") given by left multiplication on "S". Such a spinor bundle is said to be "real", "complex", "graded" or "ungraded" according to whether on not "V" has the corresponding property. Sections of "S"("M") are called spinors on "M". Given a spinor bundle "S"("M") there is a natural bundle

neg: Foliation called a "foliated bundle". Take as basepoint "x" ∈ "S" the equivalence class 0 + Z; so π("x") is the original cross section "N". For each loop "s" on "S", based at "x", the homotopy class ["s"] ∈ π("S","x") is uniquely characterized by deg "s" ∈ Z. The loop "s" lifts to a path in each flow line and it should be clear that the lift "s" that starts at "y" ∈ "N" ends at "f"("y") ∈ "N", where "k" = deg "s". The diffeomorphism "f" ∈ Diff("N") is also denoted by "h" and is called the "total holonomy" of

neg: Projective bundle obstruction in the cohomology group "H"("X",O*). In particular, if "X" is a compact Riemann surface, the obstruction vanishes i.e. "H"("X",O*)=0. The projective bundle of a vector bundle "E" is the same thing as the Grassmann bundle formula_3 of 1-planes in "E". The projective bundle P("E") of a vector bundle "E" is characterized by the universal property that says: For example, taking "f" to be "p", one gets the line subbundle "O"(-1) of "p""E", called the tautological line bundle on P("E"). Moreover, this "O"(-1) is a universal bundle in the sense that when a line bundle "L" gives a factorization "f"

neg: Vector bundle Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the category of vector spaces can also be performed on the category of vector bundles in a functorial manner. This is made precise in the language of smooth functors. An operation of a different nature is the pullback bundle construction. Given a vector bundle "E" → "Y" and a continuous map "f" : "X" → "Y" one can "pull back" "E" to a vector bundle "f*E" over "X". The fiber over a point "x" ∈ "X" is essentially

neg: The Old Man and his Sons The Old Man and his Sons The Old Man and his Sons, sometimes alternatively titled The Bundle of Sticks, is one of Aesop's Fables and is numbered 53 in the Perry Index. The actions described in it have been attributed to several later rulers and its political moral that there is strength in unity has been consistently commented on over the centuries. An old man has a number of sons who constantly quarrel with each other. As he nears death he calls them to him and gives them an object lesson in the need for unity. Having bound a bundle

neg: Darboux frame of functions "F"("n") → E: The projection operator "P" is of special significance. The inverse image of a point "P"("v") consists of all orthonormal bases with basepoint at "v". In particular, "P" : "F"("n") → E presents "F"("n") as a principal bundle whose structure group is the orthogonal group O("n"). (In fact this principal bundle is just the tautological bundle of the homogeneous space "F"("n") → "F"("n")/O("n") = E.) The exterior derivative of "P" (regarded as a vector-valued differential form) decomposes uniquely as for some system of scalar valued one-forms ω. Similarly, there is an "n" × "n" matrix of

neg: Yacatecuhtli Yacatecuhtli In Aztec mythology, Yacatecuhtli () was the patron god of commerce and travelers, especially business travelers. His symbol is a bundle of sticks. Merchants would carry an uttal cane as they moved from village to village peddling their wares, and at night-time would tie them together into a neat bundle before sprinkling them with blood from their ears. It was believed that this ritual in Yacatecuhtli's honor would guarantee success in future business ventures, not to mention protection from vicious beasts and robbers on their journeys. His name means "lord of the nose" (Nahuatl "yacatl", nose and "tecuhtli", lord).

neg: Bundle of rights For example, perfection of a mechanic's lien takes some, but not all, rights out of the bundle held by the owner. Extinguishing that lien returns those rights or "sticks" to the bundle held by the owner. In the United States (and under common law) the fullest possible title to real estate is called "fee simple absolute." Even the US federal government's ownership of land is restricted in some ways by state property law. Variations on the division between public and private property use can be found throughout the world. While the bundle of rights concept is strongly rooted in common

neg: Bundle map over a space "M". Then a bundle map from "E" to "F" over "M" is a continuous map formula_1 such that formula_2. That is, the diagram should commute. Equivalently, for any point "x" in "M", formula_3 maps the fiber "E" = "π"({"x"}) of "E" over "x" to the fiber "F" = "π"({"x"}) of "F" over "x". Let π:"E"→ "M" and π:"F"→ "N" be fiber bundles over spaces "M" and "N" respectively. Then a continuous map formula_1 is called a bundle map from "E" to "F" if there is a continuous map "f":"M"→ "N" such that the diagram commutes, that is,

neg: Frame bundle bundle of "E" is given by F("E") × (R)* where ρ* is the dual of the fundamental representation. Tensor bundles of "E" can be constructed in a similar manner. The tangent frame bundle (or simply the frame bundle) of a smooth manifold "M" is the frame bundle associated to the tangent bundle of "M". The frame bundle of "M" is often denoted F"M" or GL("M") rather than F("TM"). If "M" is "n"-dimensional then the tangent bundle has rank "n", so the frame bundle of "M" is a principal GL("n", R) bundle over "M". Local sections of the frame bundle of

neg: Frame bundle also be closed. Frame bundle In mathematics, a frame bundle is a principal fiber bundle F("E") associated to any vector bundle "E". The fiber of F("E") over a point "x" is the set of all ordered bases, or "frames", for "E". The general linear group acts naturally on F("E") via a change of basis, giving the frame bundle the structure of a principal GL("k", R)-bundle (where "k" is the rank of "E"). The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle. Let "E"

neg: Frame bundle Frame bundle In mathematics, a frame bundle is a principal fiber bundle F("E") associated to any vector bundle "E". The fiber of F("E") over a point "x" is the set of all ordered bases, or "frames", for "E". The general linear group acts naturally on F("E") via a change of basis, giving the frame bundle the structure of a principal GL("k", R)-bundle (where "k" is the rank of "E"). The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle. Let "E" → "X" be

neg: Banach bundle be two Banach bundles. A Banach bundle morphism from the first bundle to the second consists of a pair of morphisms For "f" to be a morphism means simply that "f" is a continuous map of topological spaces. If the manifolds "M" and "M"′ are both of class "C", then the requirement that "f" be a morphism is the requirement that it be a "p"-times continuously differentiable function. These two morphisms are required to satisfy two conditions (again, the second one is redundant in the finite-dimensional case): One can take a Banach bundle over one manifold and use the pull-back

neg: Torsion tensor part of the connection, the torsion tensor determines the antisymmetric part. The torsion form, an alternative characterization of torsion, applies to the frame bundle F"M" of the manifold "M". This principal bundle is equipped with a connection form ω, a gl("n")-valued one-form which maps vertical vectors to the generators of the right action in gl("n") and equivariantly intertwines the right action of GL("n") on the tangent bundle of F"M" with the adjoint representation on gl("n"). The frame bundle also carries a canonical one-form θ, with values in R, defined at a frame (regarded as a linear function ) by where

neg: Stiefel–Whitney class the notion of classifying space. For any vector space "V", let "Gr"("V") denote the Grassmannian, the space of "n"-dimensional linear subspaces of "V", and denote the infinite Grassmannian Recall that it is equipped with the tautological bundle formula_11 a rank "n" vector bundle that can be defined as the subbundle of the trivial bundle of fiber "V" whose fiber at a point formula_12 is the subspace represented by "Ẃ". Let "f" : "X" → "Gr", be a continuous map to the infinite Grassmannian. Then, up to isomorphism, the bundle induced by the map "f" on "X" depends only on the

neg: Associated bundle space "F" (in the smooth category, we should have a smooth action on a smooth manifold). Without loss of generality, we can take this action to be effective. Define a right action of "G" on "P" × "F" via We then identify by this action to obtain the space "E" = "P" × "F" = ("P" × "F") /"G". Denote the equivalence class of ("p","f") by ["p","f"]. Note that Define a projection map π : "E" → "X" by π(["p","f"]) = π("p"). Note that this is well-defined. Then π : "E" → "X" is a fiber bundle with fiber "F"

neg: Fiber bundle A "G"-atlas for the bundle ("E", "B", π, "F") is a local trivialization such that for any two local trivializations for the overlapping charts ("U", φ) and ("U", φ) the function is given by where "t" : "U" ∩ "U" → "G" is a continuous map called a transition function. Two "G"-atlases are equivalent if their union is also a "G"-atlas. A "G"-bundle is a fiber bundle with an equivalence class of "G"-atlases. The group "G" is called the structure group of the bundle; the analogous term in physics is gauge group. In the smooth category, a "G"-bundle is a

neg: Muscogee but one is that they were named for the Muscogee tradition of carrying a bundle of sticks that mark the days until an event occurs. Sticks painted red symbolize war. The Creek War of 1813–1814, also known as the "Red Stick War", began as a civil war within the Muscogee Nation, only to become enmeshed within the War of 1812. Inspired by the Shawnee leader Tecumseh (to whom nineteenth-century writers attributed fiery speeches that he "must have said") and their own religious leaders, and encouraged by British traders, Red Stick leaders such as William Weatherford (Red Eagle), Peter McQueen, and

neg: Tensor product bundle Tensor product bundle In differential geometry, the tensor product of vector bundles "E", "F" is a vector bundle, denoted by "E" ⊗ "F", whose fiber over a point "x" is the tensor product of vector spaces "E" ⊗ "F". Example: If "O" is a trivial line bundle, then "E" ⊗ "O" = "E" for any "E". Example: "E" ⊗ "E" is canonically isomorphic to the endomorphism bundle End("E"), where "E" is the dual bundle of "E". Example: A line bundle "L" has tensor inverse: in fact, "L" ⊗ "L" is (isomorphic to) a trivial bundle by the previous example, as

neg: Pick-up sticks Pick-up sticks Pick-up sticks or pick-a-stick is a game of physical and mental skill in which a bundle of "sticks", between 8 and 20 centimeters long, are dropped as a loose bunch onto a table top, jumbling into a random pile. Each player in turn tries to remove a stick from the pile without disturbing any of the others. The game is known by several names including jackstraws (or jack straws), spellicans, and spillikins, and appears in a line of a nursery rhyme: "...five, six, pick-up sticks!" The sticks may be made of almost any material, such as ivory, bone,

neg: Hyperbolic set of local structural stability and has been much studied, cf. Axiom A. Let "M" be a compact smooth manifold, "f": "M" → "M" a diffeomorphism, and "Df": "TM" → "TM" the differential of "f". An "f"-invariant subset Λ of "M" is said to be hyperbolic, or to have a hyperbolic structure, if the restriction to Λ of the tangent bundle of "M" admits a splitting into a Whitney sum of two "Df"-invariant subbundles, called the stable bundle and the unstable bundle and denoted "E" and "E". With respect to some Riemannian metric on "M", the restriction of "Df" to "E"

neg: Fiber bundle not quite sufficient, and there are a variety of sufficient conditions in common use. If "M" and "N" are compact and connected, then any submersion "f" : "M" → "N" gives rise to a fiber bundle in the sense that there is a fiber space "F" diffeomorphic to each of the fibers such that ("E","B",π,"F") = ("M","N",ƒ,"F") is a fiber bundle. (Surjectivity of ƒ follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion ƒ : "M" → "N" is assumed to be a surjective proper map, meaning that

neg: Frame bundle at each point "x" in the base space "X". It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank "k" Riemannian vector bundle "E" → "X" is a principal O("k")-bundle over "X". Again, the construction works just as well in the smooth category. If the vector bundle "E" is orientable then one can define the oriented orthonormal frame bundle of "E", denoted F("E"), as the principal SO("k")-bundle of all positively oriented orthonormal frames. If "M" is an "n"-dimensional Riemannian manifold, then the orthonormal frame bundle of "M",

neg: Horrocks–Mumford bundle Horrocks–Mumford bundle In algebraic geometry, the Horrocks–Mumford bundle is an indecomposable rank 2 vector bundle on 4-dimensional projective space "P" introduced by . It is the only such bundle known, although a generalized construction involving Paley graphs produces other rank 2 sheaves (Sasukara et al. 1993). The zero sets of sections of the Horrocks–Mumford bundle are abelian surfaces of degree 10, called Horrocks–Mumford surfaces. By computing Chern classes one sees that the second exterior power formula_1 of the Horrocks–Mumford bundle "F" is the line bundle "O(5)" on "P". Therefore, the zero set "V" of a general section of this bundle

neg: Chern class the new vector bundle: such that each fiber is the quotient of a fiber "F" of "E" by the line spanned by a nonzero vector "v" in "F" (a point of "B′" is specified by a fiber "F" of "E" and a nonzero vector on "F".) Then "E<nowiki>'</nowiki>" has rank one less than that of "E". From the Gysin sequence for the fiber bundle formula_10: we see that formula_12 is an isomorphism for "k" < 2"n" − 1. Let It then takes some work to check the axioms of Chern classes are satisfied for this definition. See also: The Thom

neg: Pick-up sticks more challenging pieces being worth more points. Players try to win points by removing jackstraws – in the shapes of various tools – from the pile, one at a time. At the end of the game, players total the point values of their jackstraws. The players with the most points wins. Pick-up sticks Pick-up sticks or pick-a-stick is a game of physical and mental skill in which a bundle of "sticks", between 8 and 20 centimeters long, are dropped as a loose bunch onto a table top, jumbling into a random pile. Each player in turn tries to remove a

neg: Fiber bundle × "F" would be a cylinder, but the Möbius strip has an overall "twist". Note that this twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space). A similar nontrivial bundle is the Klein bottle which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-torus, "S" × "S". A covering space is a fiber bundle such that the bundle projection is a local homeomorphism. It follows that the fiber is a discrete space. A special

neg: Fiber bundle required to be smooth maps. Let "E" = "B" × "F" and let π : "E" → "B" be the projection onto the first factor. Then "E" is a fiber bundle (of "F") over "B". Here "E" is not just locally a product but "globally" one. Any such fiber bundle is called a trivial bundle. Any fiber bundle over a contractible CW-complex is trivial. Perhaps the simplest example of a nontrivial bundle "E" is the Möbius strip. It has the circle that runs lengthwise along the center of the strip as a base "B" and a line segment for the

neg: Fiber bundle construction theorem cover {"U"} of "X" and a set of continuous functions defined on each nonempty overlap, such that the "cocycle condition" holds, there exists a fiber bundle "E" → "X" with fiber "F" and structure group "G" that is trivializable over {"U"} with transition functions "t". Let "E"′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions "t"′. If the action of "G" on "F" is faithful, then "E"′ and "E" are isomorphic if and only if there exist functions such that Taking "t" to be constant functions to the identity in

neg: Serre–Swan theorem bundle epimorphism from a trivial bundle formula_2 This can be done by, for instance, exhibiting sections "s"..."s" with the property that for each point "p", {"s"("p")} span the fiber over "p". When "M" is connected, the converse is also true: every finitely generated projective module over C("M") arises in this way from some smooth vector bundle on "M". Such a module can be viewed as a smooth function "f" on "M" with values in the "n" × "n" idempotent matrices for some "n". The fiber of the corresponding vector bundle over "x" is then the range of "f"("x"). If "M"

neg: Frame bundle "M" are called smooth frames on "M". The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in "U" in "M" which admits a smooth frame. Given a smooth frame "s" : "U" → F"U", the trivialization ψ : F"U" → "U" × GL("n", R) is given by where "p" is a frame at "x". It follows that a manifold is parallelizable if and only if the frame bundle of "M" admits a global section. Since the tangent bundle of "M" is trivializable over coordinate neighborhoods of "M" so is the frame bundle.

neg: Normally hyperbolic invariant manifold perturbations. Thus, in problems involving perturbation theory, invariant manifolds exist with certain hyperbolicity properties, which can in turn be used to obtain qualitative information about a dynamical system. Let "M" be a compact smooth manifold, "f": "M" → "M" a diffeomorphism, and "Df": "TM" → "TM" the differential of "f". An "f"-invariant submanifold "Λ" of "M" is said to be a normally hyperbolic invariant manifold if the restriction to "Λ" of the tangent bundle of "M" admits a splitting into a sum of three "Df"-invariant subbundles, one being the tangent bundle of formula_1, the others being the stable bundle and

neg: Spinor field "F") be a spin structure on a Riemannian manifold ("M, g") that is, an equivariant lift of the oriented orthonormal frame bundle formula_3 with respect to the double covering formula_4 One usually defines the spinor bundle formula_1 to be the complex vector bundle associated to the spin structure P via the spin representation formula_7 where U(W) denotes the group of unitary operators acting on a Hilbert space W. A spinor field is defined to be a section of the spinor bundle S, i.e., a smooth mapping formula_8 such that formula_9 is the identity mapping id of "M". Spinor field In

neg: Differentiable manifold in more precise terms, a tangent frame), is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of R to this tangent space. A moving tangent frame is an ordered list of vector fields that give a basis at every point of their domain. One may also regard a moving frame as a section of the frame bundle F("M"), a principal bundle made up of the set of all frames over "M". The frame bundle is useful because tensor fields on "M" can be regarded as equivariant vector-valued functions on F("M"). On a manifold

neg: Magnetic monopole separately. A "connection" on a -bundle tells you how to glue fibers together at nearby points of . It starts with a continuous symmetry group that acts on the fiber , and then it associates a group element with each infinitesimal path. Group multiplication along any path tells you how to move from one point on the bundle to another, by having the element associated to a path act on the fiber . In mathematics, the definition of bundle is designed to emphasize topology, so the notion of connection is added on as an afterthought. In physics, the connection is

neg: Equivariant sheaf equivariant vector bundle is a pair consisting of a vector bundle and the lifting of the action formula_35 to that of formula_36 so that the projection formula_37 is equivariant. Just like in the non-equivariant setting, one can define an equivariant characteristic class of an equivariant vector bundle. Equivariant sheaf In mathematics, given an action formula_1 of a group scheme "G" on a scheme "X" over a base scheme "S", an equivariant sheaf "F" on "X" is a sheaf of formula_2-modules together with the isomorphism of formula_3-modules that satisfies the cocycle condition: writing "m" for multiplication, On the stalk level, the

neg: Fiber bundle is a homeomorphism φ: π("U") → "U" × "F" (where "U" × "F" is the product space) in such a way that π agrees with the projection onto the first factor. That is, the following diagram should commute: where proj : "U" × "F" → "U" is the natural projection and φ : π("U") → "U" × "F" is a homeomorphism. The set of all {("U", φ)} is called a local trivialization of the bundle. Thus for any "p" in "B", the preimage π({"p"}) is homeomorphic to "F" (since proj({"p"}) clearly is) and is called the fiber over "p". Every

neg: Making an American Citizen a group of emigrants who invite Ivan to join them. When Ivan, together with his wife, arrives in America, "the land of freedom," he learns four "lesson[s] in Americanism" that turn him into a respectful husband with regard to his wife. The first lesson takes place upon landing. Ivan has his wife carry a huge bundle on her back. When she succumbs under the weight of the bundle, Ivan starts prodding her with a stick, threatening to beat her. A gentleman approaches them, picks up the bundle and puts it on Ivan's back. The gentleman takes the stick from Ivan,

neg: Fiber bundle smooth fiber bundle where "G" is a Lie group and the corresponding action on "F" is smooth and the transition functions are all smooth maps. The transition functions "t" satisfy the following conditions The third condition applies on triple overlaps "U" ∩ "U" ∩ "U" and is called the cocycle condition (see Čech cohomology). The importance of this is that the transition functions determine the fiber bundle (if one assumes the Čech cocycle condition). A principal "G"-bundle is a "G"-bundle where the fiber "F" is a principal homogeneous space for the left action of "G" itself (equivalently, one can specify

neg: Kaluza–Klein theory noncommutative space. The construction can be outlined, roughly, as follows. One starts by considering a principal fiber bundle "P" with gauge group "G" over a manifold M. Given a connection on the bundle, and a metric on the base manifold, and a gauge invariant metric on the tangent of each fiber, one can construct a bundle metric defined on the entire bundle. Computing the scalar curvature of this bundle metric, one finds that it is constant on each fiber: this is the "Kaluza miracle". One did not have to explicitly impose a cylinder condition, or to compactify: by assumption, the

neg: Descent (mathematics) "X". With this language, "descent" implies a vector bundle on "Y "(so, a bundle given on each "X"), and our concern is to 'glue' those bundles "V", to make a single bundle "V" on X. What we mean is that "V" should, when restricted to "X", give back "V", up to a bundle isomorphism. The data needed is then this: on each overlap intersection of "X" and "X", we'll require mappings to use to identify "V" and "V" there, fiber by fiber. Further the "f" must satisfy conditions based on the reflexive, symmetric and transitive properties of an equivalence relation

neg: Clifford bundle that is, when ("E", "g") is a Riemannian or pseudo-Riemannian vector bundle. For concreteness, suppose that ("E", "g") is a Riemannian vector bundle. The Clifford bundle of "E" can be constructed as follows. Let "Cℓ"R be the Clifford algebra generated by R with the Euclidean metric. The standard action of the orthogonal group O("n") on R induces a graded automorphism of "Cℓ"R. The homomorphism is determined by where "v" are all vectors in R. The Clifford bundle of "E" is then given by where "F"("E") is the orthonormal frame bundle of "E". It is clear from this construction that the

neg: Noncommutative geometry smooth Riemannian manifold "M" is a topological space with a lot of extra structure. From its algebra of continuous functions "C(M)" we only recover "M" topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple. It is constructed from a smooth vector bundle "E" over "M", e.g. the exterior algebra bundle. The Hilbert space "L(M,E)" of square integrable sections of "E" carries a representation of "C(M)" by multiplication operators, and we consider an unbounded operator "D" in "L(M,E)" with compact resolvent (e.g. the signature operator), such that the commutators "[D,f]" are bounded whenever "f" is smooth. A

neg: Hamiltonian vector field be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions "f" and "g" on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of "f" and "g". Suppose that is a symplectic manifold. Since the symplectic form is nondegenerate, it sets up a "fiberwise-linear" isomorphism between the tangent bundle and the cotangent bundle , with the inverse Therefore, one-forms on a symplectic manifold may be identified with vector fields and every differentiable function determines a unique vector field , called the "Hamiltonian vector

neg: The Old Man and his Sons was certainly given to the fable in, among others, Edward Garrett’s new edition of Aesop's fables in the 19th century. The political motto "Unity is strength" was definitely associated with the fable by the trade union organisations that adopted it. A depiction of a man kneeling over a bundle of sticks on the ground was used, often accompanied by the motto on, for example, the badge of the Nottinghamshire Miners Association, on a Durham trade union banner, and on a trade token of the Worcestershire Co-operative Society. Some of these show a man crouched with one knee on a bundle

neg: Fasci Siciliani autonomous organizations with their own insignia (red rosettes), uniforms and sometimes even musical bands, and their own local halls for reunions and congresses. They were called Fasci ("Fascio" literally means bundle) because everyone can break a single stick, but no one can break a bundle of sticks. While many of the leaders were of socialist or anarchist leanings, few of their supporters were revolutionaries. Nevertheless, the peasants who assembled into the Fasci were eager for social justice and convinced that a new world was about to be born. A crucifix hung beside the red flag in many of their meeting-places,

neg: Sheaf of modules A, any coherent sheaf on a Stein manifold is spanned by global sections. (cf. Serre's theorem A below.) In the theory of schemes, a related notion is ample line bundle. (For example, if "L" is an ample line bundle, some power of it is generated by global sections.) An injective "O"-module is flasque (i.e., all restrictions maps "F"("U") → "F"("V") are surjective.) Since a flasque sheaf is acyclic in the category of abelian sheaves, this implies that the "i"-th right derived functor of the global section functor formula_3 in the category of "O"-modules coincides with the usual "i"-th sheaf cohomology

neg: K3 surface line bundle "L" like this is called a K3 surface of genus "g". A K3 surface may have many different line bundles making it into a K3 surface of genus "g" for many different values of "g". The space of sections of the line bundle has dimension "g"+1, so there is a morphism of the K3 surface to projective space of dimension "g". There is a moduli space "F" of K3 surfaces with a primitive ample line bundle "L" with "c"("L") =2"g"-2, which is nonempty of dimension 19 for "g"≥ 2. showed that this moduli space "F" is unirational if

neg: Fiber bundle total space of the fiber bundle, "B" as the base space, and "F" the fiber. In the "trivial" case, "E" is just "B" × "F", and the map π is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles. Mappings between total spaces of fiber

neg: Coherent sheaf → "Y" be a morphism of ringed spaces (for example, a morphism of schemes). If "F" is a quasi-coherent sheaf on "Y", then the inverse image "O"-module (or pullback) "f"*"F" is quasi-coherent on "X". For a morphism of schemes "f": "X" → "Y" and a coherent sheaf "F" on "Y", the pullback "f""F" is not coherent in full generality (for example, "f"*"O" = "O", which might not be coherent), but pullbacks of coherent sheaves are coherent if "X" is locally Noetherian. An important special case is the pullback of a vector bundle, which is a vector bundle. If "f": "X"

neg: Grothendieck–Riemann–Roch theorem relative tangent sheaf of "f", defined as the element "TX" − "f""TY" in "K"("X"). For example, when "f" is a smooth morphism, "T" is simply a vector bundle, known as the tangent bundle along the fibers of "f". Using A-homotopy theory, the Grothendieck–Riemann–Roch theorem has been extended by to the situation where "f" is a proper map between two smooth schemes. Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination ch(—)td("X") and to the non-proper case by considering cohomology with compact support. The arithmetic Riemann–Roch theorem extends the Grothendieck–Riemann–Roch theorem

neg: Bundle map the vector space Hom("E","F") (also denoted "L"("E","F")) of linear maps from "E" to "F". Bundle map In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some

neg: Frame bundle "X" which sends ("x", "p") to "x". The group GL("k", R) acts on F("E") on the right as above. This action is clearly free and the orbits are just the fibers of π. The frame bundle F("E") can be given a natural topology and bundle structure determined by that of "E". Let ("U", φ) be a local trivialization of "E". Then for each "x" ∈ "U" one has a linear isomorphism φ : "E" → R. This data determines a bijection given by With these bijections, each π("U") can be given the topology of "U" × GL("k", R). The topology

neg: Integral curve the vector F(x("t")), and so the curve x("t") is tangent at each point to the vector field F. If a given vector field is Lipschitz continuous, then the Picard–Lindelöf theorem implies that there exists a unique flow for small time. Let "M" be a Banach manifold of class "C" with "r" ≥ 2. As usual, T"M" denotes the tangent bundle of "M" with its natural projection "π" : T"M" → "M" given by A vector field on "M" is a cross-section of the tangent bundle T"M", i.e. an assignment to every point of the manifold "M" of a tangent vector

neg: Jet (mathematics) that of "k"-th-order contact between curves at "p". We now define the k"-jet of a curve "f" through "p" to be the equivalence class of "f" under formula_35, denoted formula_71 or formula_72. The k"-th-order jet space formula_73 is then the set of "k"-jets at "p". As "p" varies over "M", formula_73 forms a fibre bundle over "M": the "k"-th-order tangent bundle, often denoted in the literature by "T""M" (although this notation occasionally can lead to confusion). In the case "k"=1, then the first-order tangent bundle is the usual tangent bundle: "T""M" = "TM". To prove that "T""M" is in fact

neg: Torsor (algebraic geometry) coefficient) group formula_27. A trivial torsor corresponds to the identity element. Conversely, it is easy to see any class in formula_27 defines a "G"-torsor on "X", unique up to an isomorphism. If "G" is a connected algebraic group over a finite field formula_29, then any "G"-bundle over formula_30 is trivial. (Lang's theorem.) Most of constructions and terminology regarding principal bundles in algebraic topology carry over in verbatim to "G"-bundles. For example, if formula_31 is a "G"-bundle and "G" acts from the left on a scheme "F", then one can form the associated bundle formula_32 with fiber "F". In particular, if

neg: Roswell UFO incident color and scattered over an area about 200 yards [180 m] in diameter. When the debris was gathered up, the tinfoil, paper, tape, and sticks made a bundle about three feet [1 m] long and 7 or 8 inches [18 or 20 cm] thick, while the rubber made a bundle about 18 or 20 inches [45 or 50 cm] long and about 8 inches [20 cm] thick. In all, he estimated, the entire lot would have weighed maybe five pounds [2 kg]. There was no sign of any metal in the area which might have been used for an engine,

neg: Tangent bundle only if the tangent bundle "TM" is stably trivial, meaning that for some trivial bundle "E" the Whitney sum is trivial. For example, the "n"-dimensional sphere "S" is framed for all "n", but parallelizable only for (by results of Bott-Milnor and Kervaire). One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if "f" : "M" → "N" is a smooth function, with "M" and "N" smooth manifolds, its derivative is a smooth function "Df" : "TM" → "TN". The tangent bundle comes equipped with a

neg: Associated bundle Now let "F"′ be a specified topological space, equipped with a continuous left action of "G". Then the bundle associated to "E" with fibre "F"′ is a bundle "E"′ with a local trivialization subordinate to the cover "U" whose transition functions are given by where the "G"-valued functions "g"("u") are the same as those obtained from the local trivialization of the original bundle "E". This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of "G"-valued functions. (Using another local trivialization, and passing to a common refinement if