Finitely generated module - Wikipedia: "A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.
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Sunday, April 30, 2017
Direct sum of modules - Wikipedia
Direct sum of modules - Wikipedia: "The subspace V × {0} of V ⊕ W is isomorphic to V and is often identified with V; similarly for {0} × W and W. (See internal direct sum below.) With this identification, every element of V ⊕ W can be written in one and only one way as the sum of an element of V and an element of W. The dimension of V ⊕ W is equal to the sum of the dimensions of V and W. One elementary use is the reconstruction of a finite vector space from any subspace W and its orthogonal complement:
{\displaystyle \mathbb {R} ^{n}=W\oplus W^{\perp }}
This construction readily generalises to any finite number of vector spaces."
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{\displaystyle \mathbb {R} ^{n}=W\oplus W^{\perp }}
This construction readily generalises to any finite number of vector spaces."
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Saturday, April 29, 2017
Tangent bundle - Wikipedia
Tangent bundle - Wikipedia: "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.
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Mapbox - Wikipedia
Mapbox - Wikipedia: "Mapbox is a large provider of custom online maps for websites such as Foursquare, Pinterest, Evernote, the Financial Times, The Weather Channel and Uber Technologies.[2] Since 2010, it has rapidly expanded the niche of custom maps, as a response to the limited choice offered by map providers such as Google Maps and OpenStreetMap.[2] Mapbox is the creator of, or a significant contributor to some open source mapping libraries and applications, including the MBTiles specification, the TileMill cartography IDE, the Leaflet JavaScript library, and the CartoCSS map styling language and parser.
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Klein bottle - Wikipedia
Klein bottle - Wikipedia: "Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, the Klein bottle is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however.
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Sinistral and dextral - Wikipedia
Sinistral and dextral - Wikipedia: "Chirality, however, is observer-independent: no matter how one looks at a right-hand screw thread, it remains different from a left-hand screw thread."
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Friday, April 28, 2017
Group action - Wikipedia
Group action - Wikipedia: "Actions of groups on vector spaces are called representations of the group."
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Wednesday, April 26, 2017
Representation of a Lie group - Wikipedia
Representation of a Lie group - Wikipedia: "If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group GL(n,C). This is known as a matrix representation.
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Symmetry (physics) - Wikipedia
Symmetry (physics) - Wikipedia: "A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group).
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Tuesday, April 25, 2017
Group homomorphism - Wikipedia
Group homomorphism - Wikipedia: "Types of group homomorphism[edit]
Monomorphism
A group homomorphism that is injective (or, one-to-one); i.e., preserves distinctness.
Epimorphism
A group homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain.
Isomorphism
A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements and are identical for all practical purposes.
Endomorphism
A homomorphism, h: G → G; the domain and codomain are the same. Also called an endomorphism of G.
Automorphism"
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Monomorphism
A group homomorphism that is injective (or, one-to-one); i.e., preserves distinctness.
Epimorphism
A group homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain.
Isomorphism
A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements and are identical for all practical purposes.
Endomorphism
A homomorphism, h: G → G; the domain and codomain are the same. Also called an endomorphism of G.
Automorphism"
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Sunday, April 23, 2017
Friday, April 21, 2017
Wednesday, April 19, 2017
Tuesday, April 18, 2017
Monday, April 17, 2017
Small seconds - Watch Wiki: The Best Watches and Watch Brands
Small seconds - Watch Wiki: The Best Watches and Watch Brands: "this dial arrangement is actually simpler, with fewer components used."
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Sunday, April 16, 2017
Isomorphism - Wikipedia
Isomorphism - Wikipedia: "is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse.["
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Differential structure - Wikipedia
Differential structure - Wikipedia: "In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If M is already a topological manifold, it is required that the new topology be identical to the existing one.
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Watch crystals | Europa Star Magazine
Watch crystals | Europa Star Magazine: "A watch crystal is a transparent cover that protects the watch face"
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Saturday, April 15, 2017
Portal:Manufacturers – Watch-Wiki
Portal:Manufacturers – Watch-Wiki: "The Portal Manufacturers gives an overview of the Manufacturers pages in Watch Wiki. New authors, who would like to help us with the extension of our data collection, are cordially welcome. Everything that is needed for helping you can find in the authors assistance Manufacturers.
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Thursday, April 13, 2017
Lie bracket of vector fields - Wikipedia
Lie bracket of vector fields - Wikipedia: "Conceptually, the Lie bracket [X,Y] is the derivative of Y along the flow generated by X. A generalization of the Lie bracket is the Lie derivative, which allows differentiation of any tensor field along the flow generated by X. The Lie bracket [X,Y] equals the Lie derivative of the vector Y (which is a tensor field) along X, and is sometimes denoted "
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Wednesday, April 12, 2017
Multivariable calculus - Wikipedia
Multivariable calculus - Wikipedia: "Fundamental theorem of calculus in multiple dimensions[edit]
In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus:[1]:543ff
Gradient theorem
Stokes' theorem
Divergence theorem
Green's theorem.
In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.[2]"
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In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus:[1]:543ff
Gradient theorem
Stokes' theorem
Divergence theorem
Green's theorem.
In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.[2]"
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Monday, April 10, 2017
Symmetric bilinear form - Wikipedia
Symmetric bilinear form - Wikipedia: "In other words, it is a bilinear function {\displaystyle B} that maps every pair {\displaystyle (u,v)} of elements of the vector space {\displaystyle V} to the underlying field such that {\displaystyle B(u,v)=B(v,u)} for every {\displaystyle u} and {\displaystyle v} in {\displaystyle V} ."
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Thursday, April 06, 2017
Direct product of groups - Wikipedia
Direct product of groups - Wikipedia: "This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G ⊕ H. "
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In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G ⊕ H. "
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Group theory - Wikipedia
Group theory - Wikipedia: "The first class of groups to undergo a systematic study was permutation groups. Given any set X and a collection G of bijections of X into itself (known as permutations) that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn; in general, any permutation group G is a subgroup of the symmetric group of X. An early construction due to Cayley exhibited any group as a permutation group, acting on itself (X = G) by means of the left regular representation.
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Sunday, April 02, 2017
Pullback (differential geometry) - Wikipedia
Pullback (differential geometry) - Wikipedia: "When the map φ is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from N to M or vice versa. In particular, if φ is a diffeomorphism between open subsets of Rn and Rn, viewed as a change of coordinates (perhaps between different charts on a manifold M), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches to the subject.
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Algebra over a field - Wikipedia
Algebra over a field - Wikipedia: "In mathematics, an algebra over a field (often simply called an algebra), is a vector space equipped with a bilinear product"
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Symplectic group - Wikipedia
Symplectic group - Wikipedia: "The name "symplectic group" is due to Hermann Weyl (details) as a replacement for the previous confusing names of (line) complex group and Abelian linear group, and is the Greek analog of "complex"."
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Group (mathematics) - Wikipedia
Group (mathematics) - Wikipedia: "Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups.
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