The exponential map is a fundamental construction in the theory of Lie groups. It is a map from the Lie algebra of a Lie group to the group which allows one to completely recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras. The ordinary exponential function of mathematical analysis may be viewed as a special case of the exponential map when G is the multiplicative group of positive real ...

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Exponential map, Exponential map - Lie theory, Exponential map - Definition, Exponential map - Properties, Exponential map - Riemannian geometry, Exponential map - Definition, Exponential map - Properties, Exponential map - Relationships

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As explained in the article mathematical formulation of quantum mechanics, the physical state of a system may be characterized as a vector in an abstract Hilbert space (or, in the case of ensembles, as a countable sequence of vectors weighted by probabilities). Physically observable quantities are described by self-adjoint operators acting on these vectors. The quantum Hamiltonian H is the observable corresponding to the total energy of the system ...

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Hamiltonian quantum mechanics, Hamiltonian quantum mechanics - The quantum Hamiltonian, Hamiltonian quantum mechanics - Energy eigenket degeneracy symmetry and conservation laws, Hamiltonian quantum mechanics - Hamilton's equations

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Universal property - Existence and uniqueness. Defining a quantity does not guarantee its existence. Given a functor U and an object X as above, there may or may not exist a universal morphism from X to U (or from U to X). If, however, a universal morphism (A, φ) does exists then it is unique up to a unique isomorphism. That is, if (A′, φ′) is another such pair then there exists a unique isomorphism g : A → A′ such ...

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Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

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Here are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, there may be other ways to construct it, but they are all equivalent. 1. Suppose that X and Y as above are sets. Then if we write Z for their intersection, there are morphisms f : Z → X and g : Z → Y given by inclusion. The pushout of f and g is the union of X and Y t ...

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Pushout category theory, Pushout category theory - Universal property, Pushout category theory - Examples of pushouts, Pushout category theory - Construction via coproducts and coequalizers, Pushout category theory - Application: The Seifert-van Kampen theorem

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Any product of (even infinitely many) injective modules is injective. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules or infinite direct sums of injective modules need not be injective. In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: a left R-module Q is injective if and only if any homomorphism g : I → Q defined on a left ideal I of See also:

Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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The coequalizer is a special kind of colimit in category theory. Specifically it is the colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y. More explicity, the coequalizer can be defined as an object Q and a morphism q : Y → Q such that q O f = q O g. Moreover, the pair (Q, q) must be universal in the sense that given any other such pair (Q′, q′) there exists a unique morphism u : Q → Q′ ...

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Coequalizer, Coequalizer - Definition, Coequalizer - Examples, Coequalizer - Special cases

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Group algebra. Suppose G is a group. The group algebra KG is a unital associative algebra over K. It turns into a Hopf algebra if we define Δ : KG → KG ⊗ KG by Δ(g) = g⊗g for all g in G ε : KG → K by ε(g) = 1 for all g in G S : KG → KG by S(g) = g -1See also:

Hopf algebra, Hopf algebra - Examples, Hopf algebra - Quantum groups and non-commutative geometry, Hopf algebra - Related concepts

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In the category of sets, the exponential object ZY is the set of all functions from Y to Z. The map is just the evaluation map which sends the pair (f, y) to f(y). For any map the map is the curried form of g: In the category of topological spaces, the exponential object ZY exists provided that Y< ...

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Exponential object, Exponential object - Definition, Exponential object - Examples

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The Adventures of Ned Flanders was a short that appeared at the end of the episode "The Front". Here, the Flanders dedication to religion and perfect family niceness is yet again highlighted when Ned almost scolds his kids, who refuse to get ready for church, only to soon realise that it is Saturday. The short featured a theme song titled Everyone Loves Ned Flanders. Hens Love Roosters Geese Love Ganders Everyone else loves Ned Flanders Not me! (spoken by Hom ...

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Flanders family, Flanders family - Ned Flanders, Flanders family - Maude Flanders, Flanders family - Rod and Todd, Flanders family - Grandma Flanders, Flanders family - Other family members, Flanders family - Household, Flanders family - Other information, Flanders family - Relationship with the Simpsons, Flanders family - The Adventures of Ned Flanders, Flanders family - Everybody Hates Ned Flanders

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Explicitly, the pullback of the morphisms f and g consists of an object P and two morphisms p1 : P → X and p2 : P → Y for which the diagram commutes. Moreover, the pullback (P, p1, p2) must be universal with respect to this diagram. That is, for any other such set (Q, q1, q2) there must exist a unique u : Q → P making the following diagram commute: As with all universal constructions, th ...

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Pullback category theory, Pullback category theory - Universal property, Pullback category theory - Examples

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Referring to the above commutative diagram, one observes that every morphism h : A′ → A gives rise to a natural transformation Hom(h,–) : Hom(A,–) → Hom(A′,–) and every morphism f : B → B′ gives rise to a natural transformation Hom(–,f) : Hom(–,B) → Hom(–,B′) Yoneda's lemma asserts that every natural transformation between Hom functors is of this form. ...

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Hom functor, Hom functor - Formal definition, Hom functor - Yoneda's lemma

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More generally, one can construct the pullback map between tensor bundles of rank (0,n); the construction proceeds entirely analogously to that for a tensor. That is, by considering the cotangent space at point p in M, one defines the tensor space at point p as the n-fold tensor product . The pullback then proceeds analogously to the tensor space defined through an n-fold tensor product of . This definition applies as well to the exterior bundles ΛkT*N and ΛSee also:

Pullback, Pullback - Pullback on tensors, Pullback - Pullback of cotangent bundles, Pullback - Pullback on tensor bundles, Pullback - Pullback of diffeomorphisms

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We give a few worked examples to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction. Universal property - Tensor algebras. Let C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K (assumed to be unital and associative). Let U be the forgetful functor which assig ...

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Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

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Let U : D → C be a functor from a category D to a category C, and let X be an object of C. A universal morphism from X to U consists of a pair (A, φ) where A is an object of D and φ : X → U(A) is a morphism in C, such that the following universal property is satisfied: Whenever Y is an object of D and f : X → U(Y) is a morphism in C, then there exists a unique morphism g : A → < ...

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Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

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According to the U.S. Census Bureau, the county has a total area of 1,861 km² (718 mi²). 1,853 km² (716 mi²) of it is land and 7 km² (3 mi² or 0.39%) of it is water. Yamhill County Oregon - Adjacent Counties. Clackamas County, Oregon - (easternmost tip) Marion County, Oregon - (southeast) Polk County, Oregon - (south) Tillamook County, Oregon - (west)< ...

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Yamhill County Oregon, Yamhill County Oregon - Economy, Yamhill County Oregon - Geography, Yamhill County Oregon - Adjacent Counties, Yamhill County Oregon - Demographics, Yamhill County Oregon - History, Yamhill County Oregon - Cities and towns

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As of the census2 of 2000, there are 84,992 people, 28,732 households, and 21,376 families residing in the county. The population density is 46/km² (119/mi²). There are 30,270 housing units at an average density of 16/km² (42/mi²). The racial makeup of the county is 88.98% White, 1.47% Native American, 1.07% Asian, 0.85% Black or African American, 0.12% Pacific Islander, 5.08% from other races, and 2.42% from two or more races. 10.61% ...

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Yamhill County Oregon, Yamhill County Oregon - Economy, Yamhill County Oregon - Geography, Yamhill County Oregon - Adjacent Counties, Yamhill County Oregon - Demographics, Yamhill County Oregon - History, Yamhill County Oregon - Cities and towns

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The earliest known inhabitants of the area were the Yamhill Indians, who have inhabited the area for over 8000 years. They are one of the tribes incorporated into the Confederated Tribes of the Grand Ronde. In 1857 they were forced to migrate to the Grand Ronde Indian Reservation created in Oregon's Coastal Range two years earlier. The earliest non-native settlers were employees of the various fur companies operating in Oregon, who started settling there around 1814. But it was the establishment of the Oregon Tra ...

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Yamhill County Oregon, Yamhill County Oregon - Economy, Yamhill County Oregon - Geography, Yamhill County Oregon - Adjacent Counties, Yamhill County Oregon - Demographics, Yamhill County Oregon - History, Yamhill County Oregon - Cities and towns

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All of the above examples may be regarded as special cases of the following very general construction, which works in any category C satisfying: For any objects A and B of C, their coproduct exists in C; For any morphisms j and k of C with the same domain and target, the coequalizer of j and k exists in C. In this setup, we obtain the pushout of morphisms f : Z → X and g : Z → Y< ...

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Pushout category theory, Pushout category theory - Universal property, Pushout category theory - Examples of pushouts, Pushout category theory - Construction via coproducts and coequalizers, Pushout category theory - Application: The Seifert-van Kampen theorem

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Explicitly, the pushout of the morphisms f and g consists of an object P and two morphisms i1 : X → P and i2 : Y → P for which the following diagram commutes: Moreover, the pushout (P, i1, i2) must be universal with respect to this diagram. That is, for any other such set (Q, j1, j2) there must exist a unique u : P → Q making the following diagram commute: As with all universal constructions, th ...

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Pushout category theory, Pushout category theory - Universal property, Pushout category theory - Examples of pushouts, Pushout category theory - Construction via coproducts and coequalizers, Pushout category theory - Application: The Seifert-van Kampen theorem

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Given the Set (the category of sets), the product in the category theoretic sense is the cartesian product. Given a family of sets Xi the product is defined as with the canonical projections Given any set Y with a family of functions the the unique arrow f is defined as ...

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Product category theory, Product category theory - Definition, Product category theory - Examples, Product category theory - Discussion

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The pullback of smooth map f : M → N between differentiable manifolds is a smooth vector bundle morphism f* : T*N → T*M, for which the following diagram commutes: Here T*M and T*N are the cotangent bundles of M and N respectively, and πM and πN are the natural projections. Perhaps the easiest way to understand the pullback is in terms of the pushforward of f. Picking a point , the pushforward at p is a ...

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Pullback, Pullback - Pullback on tensors, Pullback - Pullback of cotangent bundles, Pullback - Pullback on tensor bundles, Pullback - Pullback of diffeomorphisms

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