Group action - Wikipedia

Group action - Wikipedia: "Actions of groups on vector spaces are called representations of the group."



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Grammar Girl : Forensic Linguistics :: Quick and Dirty Tips ™

Grammar Girl : Forensic Linguistics :: Quick and Dirty Tips ™: "oil patterns on drinking glasses"



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Homogeneous coordinates - Wikipedia

Homogeneous coordinates - Wikipedia: "But a condition f(x, y, z) = 0 defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is homogeneous. "



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Rational function - Wikipedia

Rational function - Wikipedia: "In this case, one speaks of a rational function and a rational fraction over K."



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Field extension - Wikipedia

Field extension - Wikipedia: "Given a field extension L / K, the larger field L can be considered as a vector space over K. The elements of L are the "vectors" and the elements of K are the "scalars", with vector addition and scalar multiplication obtained from the corresponding field operations. The dimension of this vector space is called the degree of the extension and is denoted by [L : K].
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Finite field - Wikipedia

Finite field - Wikipedia: "According to Wedderburn's little theorem, any finite division ring must be commutative, and hence a finite field. This result shows that the finiteness restriction can have algebraic consequences."



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Complement (set theory) - Wikipedia

Complement (set theory) - Wikipedia: "In the LaTeX typesetting language, the command \setminus[5] is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered, the \setminus command looks identical to \backslash except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}. A variant \smallsetminus is available in the amssymb package.
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Dyadic rational - Wikipedia

Dyadic rational - Wikipedia: "The inch is customarily subdivided in dyadic rather than decimal fractions; similarly, the customary divisions of the gallon into half-gallons, quarts, and pints are dyadic. The ancient Egyptians also used dyadic fractions in measurement, with denominators up to 64.[1]
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Modular arithmetic - Wikipedia

Modular arithmetic - Wikipedia: "The ring of integers modulo n is a finite field if and only if n is prime. If n is a non-prime prime power, there exists a unique (up to isomorphism) finite field GF(n) with n elements, which must not be confused with the ring of integers modulo n, although they have the same number of elements.
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Function word - Wikipedia

Function word - Wikipedia: "For example, in some of the Khoisan languages, most content words begin with clicks, but very few function words do.[4] In English, very few words other than function words begin with voiced th-"[ð]"[citation needed] (see Pronunciation of English th);English function words may have fewer than three letters 'I', 'an', 'in' while non-function words usually have three or more 'eye', 'Ann', 'inn' (see three letter rule)."



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etymology - What does the word "symplectic" mean? - MathOverflow

etymology - What does the word "symplectic" mean? - MathOverflow: "as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic." "



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Functor - Wikipedia

Functor - Wikipedia: "There is a convention, now widely disparaged but still in use, which perversely refers to "vectors"—i.e, vector fields, elements of the space of sections {\displaystyle \Gamma (TM)} of a tangent bundle {\displaystyle TM} —as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections {\displaystyle \Gamma (T^{*}M)} of a cotangent bundle {\displaystyle T^{*}M} —as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as {\displaystyle x^{i}=\Lambda _{j}^{i}x^{j}} for {\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} } or {\displaystyle \omega _{i}=\Lambda _{i}^{j}\omega _{j}} for {\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{T}.} In this formalism it is observed that the coordinate transformation symbol {\displaystyle \Lambda _{i}^{j}} (representing the matrix {\displaystyle {\boldsymbol {\Lambda }}^{T}} ) acts on the basis vectors "in the same way" as on the "covector coordinates": {\displaystyle \mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}} —whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: {\displaystyle \mathbf {e} ^{i}\Lambda _{j}^{i}\mathbf {e} ^{j}} ). This terminology is perverse because it is the covectors that have pullbacks in general and are thus contravariant, whereas vectors in general are covariant since they can be pushed forward. See also Covariance and contravariance of vectors.
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Projective Geometry -- from Wolfram MathWorld

Projective Geometry -- from Wolfram MathWorld: "The most amazing result arising in projective geometry is the duality principle, which states that a duality exists between theorems such as Pascal's theorem and Brianchon's theorem which allows one to be instantly transformed into the other. More generally, all the propositions in projective geometry occur in dual pairs, which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line."

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Projective Geometry -- from Wolfram MathWorld

Projective Geometry -- from Wolfram MathWorld: "The most amazing result arising in projective geometry is the duality principle, which states that a duality exists between theorems such as Pascal's theorem and Brianchon's theorem which allows one to be instantly transformed into the other. More generally, all the propositions in projective geometry occur in dual pairs, which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line."

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Projective module - Wikipedia

Projective module - Wikipedia: "if the ring R is a local ring. This fact is the basis of the intuition of "locally free = projective". This fact is easy to prove for finitely generated projective modules. In general, it is due to Kaplansky (1958).
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Ring (mathematics) - Wikipedia

Ring (mathematics) - Wikipedia: "Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if a3 − 4a + 1 = 0 then a3 = 4a − 1, a4 = 4a2 − a, a5 = −a2 + 16a − 4, a6 = 16a2 − 8a + 1, a7 = −8a2 + 65a − 16, and so on; in general, an is going to be an integral linear combination of 1, a, and a2."



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Differential form - Wikipedia

Differential form - Wikipedia: "One of the main reasons the cotangent bundle rather than the tangent bundle is used in the construction of the exterior complex is that differential forms are capable of being pulled back by smooth maps, while vector fields cannot be pushed forward by smooth maps unless the map is, say, a diffeomorphism. The existence of pullback homomorphisms in de Rham cohomology depends on the pullback of differential forms.
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Homogeneous polynomial - Wikipedia

Homogeneous polynomial - Wikipedia: "A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form.[3] A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form.
Homogeneous polynomials are ubiquitous in mathematics and physics.[4] They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials.
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Homogeneous function - Wikipedia

Homogeneous function - Wikipedia: "Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by (1).
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Heads will roll | Define Heads will roll at Dictionary.com

Heads will roll | Define Heads will roll at Dictionary.com: " Expand
heads will roll
sentence

People will be dismissed, punished, ruined, etc : If eventually the authorities catch up with you, no heads will roll/ I promise you: if this package is not delivered on time, heads will roll

[1930+; the source is a quotation from Adolf Hitler]"



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Differential form - Wikipedia

Differential form - Wikipedia: "The object df can be viewed as a function on U, whose value at p is not a real number, but the linear map dfp."



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Pushforward (differential) - Wikipedia

Pushforward (differential) - Wikipedia: "the differential of φ at a point x is, in some sense, the best linear approximation of φ near x."



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Division ring - Wikipedia

Division ring - Wikipedia: " Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”."



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Functional analysis - Wikipedia

Functional analysis - Wikipedia: "formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. "



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Function space - Wikipedia

Function space - Wikipedia: "Namely, if Y is a field, functions have inherent vector structure with two operations of pointwise addition and multiplication to a scalar. "



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Isomorphism - Wikipedia

Isomorphism - Wikipedia: "In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms "



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terminology - What are the differences between rings, groups, and fields? - Mathematics Stack Exchange

terminology - What are the differences between rings, groups, and fields? - Mathematics Stack Exchange: "They should feel similar! In fact, every ring is a group, and every field is a ring. A ring is a group with an additional operation, where the second operation is associative and the distributive properties make the two operations "compatible".

A field is a ring such that the second operation also satisfies all the group properties (after throwing out the additive identity); i.e. it has multiplicative inverses, multiplicative identity, and is commutative."



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Field extension - Wikipedia

Field extension - Wikipedia: "Q(√2) = {a + b√2 | a, b ∈ Q} is the smallest extension of Q that includes every real solution to the equation x2 = 2."



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Steel square - Wikipedia

Steel square - Wikipedia: "framing square. "



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Homogeneous polynomial - Wikipedia

Homogeneous polynomial - Wikipedia: " is not homogeneous, because the sum of exponents does not match from term to term. A polynomial is homogeneous if and only if it defines a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial.[2] A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis.
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Differentiable manifold - Wikipedia

Differentiable manifold - Wikipedia: "that is locally homeomorphic to a linear space, by a collection (called an atlas) of homeomorphisms called charts. "



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Derivative - Wikipedia

Derivative - Wikipedia: "Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space that can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R3. The derivative (or differential) of a (differentiable) map f: M → N between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses – see pushforward (differential) and pullback (differential geometry).
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Exterior derivative - Wikipedia

Exterior derivative - Wikipedia: "so d( f ∗ω) =  f ∗dω, where  f ∗ denotes the pullback of  f . This follows from that  f ∗ω(·), by definition, is ω( f∗(·)),  f∗ being the pushforward of  f . Thus d is a natural transformation from Ωk to Ωk+1.
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