Abstract algebra - Wikipedia

Abstract algebra - Wikipedia: "Groups
Examples involving several operations include:

Rings"



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Module (mathematics) - Wikipedia

Module (mathematics) - Wikipedia: "A module over a ring is a generalization of the notion of vector space over a field, "



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Algebra over a field - Wikipedia

Algebra over a field - Wikipedia: "and scalar multiplication by elements of the underlying field"



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One-dimensional space - Wikipedia

One-dimensional space - Wikipedia: "A field k is a one-dimensional vector space over itself."

#Thus a field is not a torsor, and can underpin some structure as in the next entry.

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Metaglossary.com - Definitions for "affine space"

Metaglossary.com - Definitions for "affine space": "a torsor for a vector space
math.ucr.edu"



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Key to frequency | Oxford English Dictionary

Key to frequency | Oxford English Dictionary: "The following table shows the frequency range for each band, and the percentage of non-obsolete OED entries assigned to each band:

"



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torsors

torsors: "velocities don't live in just any old group - they live in a special sort of group called a "vector space". "



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torsors

torsors: "So, now you are treating the plane as a torsor!"



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Purdue OWL: Count and Noncount Nouns

Purdue OWL: Count and Noncount Nouns: "Uncountable Sense Countable Sense"



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Space (mathematics) - Wikipedia

Space (mathematics) - Wikipedia: "The modern approach defines the three-dimensional Euclidean space more algebraically, via vector spaces and quadratic forms, namely, as an affine space whose difference space is a three-dimensional inner product space."



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Help - Cambridge Dictionary

Help - Cambridge Dictionary: "Uncountable or singular noun: a noun that has no plural."



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Memento mori - Wikipedia

Memento mori - Wikipedia: "his triumphal procession, a victorious general would have "



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Automatically delete Mac OS X dot files from MS-DOS filesystems - Mac OS X Hints

Automatically delete Mac OS X dot files from MS-DOS filesystems - Mac OS X Hints:



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General topology - Wikipedia

General topology - Wikipedia: "The fundamental concepts in point-set topology are continuity, compactness, and connectedness:
"



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Synesis - Wikipedia

Synesis - Wikipedia: " an agreement of words with the sense,"



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Tangent space - Wikipedia

Tangent space - Wikipedia: "The elements of the tangent space at {\displaystyle x} are called the tangent vectors at {\displaystyle x} . This is a generalization of the notion of a bound vector in a Euclidean space. "



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Topological space - Wikipedia

Topological space - Wikipedia: "often topological spaces must be Hausdorff spaces where limit points are unique."



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Discrete space - Wikipedia

Discrete space - Wikipedia: "the finest topology that can be given on a set, i.e., it defines all subsets as open sets."



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Topological space - Wikipedia

Topological space - Wikipedia: "define a subset U of X to be open if U is a neighbourhood of all points in U."



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Irrational number - Wikipedia

Irrational number - Wikipedia: "When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself."



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Symmetric space - Wikipedia

Symmetric space - Wikipedia: "a symmetric space is the quotient G/H of Lie group G by a Lie subgroup H, "



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Symmetry breaking - Wikipedia

Symmetry breaking - Wikipedia:



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Topological group - Wikipedia

Topological group - Wikipedia: "the topology on G be Hausdorff; it is equivalent to assume that the identity element 1 is a closed subset of G."



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Quotient space (topology) - Wikipedia

Quotient space (topology) - Wikipedia: "In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space."



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Orbit - Encyclopedia of Mathematics

Orbit - Encyclopedia of Mathematics: "The orbits of any two points from either do not intersect or coincide"



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Special linear group - Wikipedia

Special linear group - Wikipedia: "this corresponds to the interpretation of the determinant as measuring change in volume and orientation"



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Equivariant map - Wikipedia

Equivariant map - Wikipedia: "Equivariant maps generalize the concept of invariants"



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Groups for Dummies

Groups for Dummies: "an n x n matrix has n eigenvalues, and they are invariant under a similarity transformation. ... The determinant is the product of all n eigenvalues, and the trace is their sum."



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Space (mathematics) - Wikipedia

Space (mathematics) - Wikipedia: " The open interval {\displaystyle (0,1)} is homeomorphic to the whole real line {\displaystyle (-\infty ,\infty )} but not homeomorphic to the closed interval {\displaystyle [0,1]} , nor to a circle."



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Form - Encyclopedia of Mathematics

Form - Encyclopedia of Mathematics: "they are called linear (for n=1n=1), quadratic (for n=2n=2), cubic (for n=3n=3), etc."



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Box topology - Wikipedia

Box topology - Wikipedia: "In particular, if all the component spaces are compact, the box topology on their Cartesian product will not necessarily be compact, although the product topology on their Cartesian product will always be compact."



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Bundle - Wikipedia

Bundle - Wikipedia: "Bundle (mathematics), a generalization of a fiber bundle dropping the condition of a local product structure
Fiber bundle, a topology space that looks locally like a product space
"



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Derivation (differential algebra) - Wikipedia

Derivation (differential algebra) - Wikipedia: "The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; "



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Detexify LaTeX handwritten symbol recognition

Detexify LaTeX handwritten symbol recognition:



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Tangent space - Wikipedia

Tangent space - Wikipedia: "A real-valued function {\displaystyle f:M\to \mathbf {R} } is said to belong to {\displaystyle {C^{\infty }}(M)} if and only if for every coordinate chart {\displaystyle \varphi :U\to \mathbf {R} ^{n}} , the map {\displaystyle f\circ \varphi ^{-1}:\varphi [U]\subseteq \mathbf {R} ^{n}\to \mathbf {R} } is infinitely differentiable. "



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Tangent space - Wikipedia

Tangent space - Wikipedia: "This map turns out to be bijective and may be used to transfer the vector-space operations on {\displaystyle \mathbf {R} ^{n}} over to {\displaystyle T_{x}M} , thus turning the latter set into an {\displaystyle n} -dimensional real vector space."



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Volume

Volume - Wikipedia: "Integrating the volume form gives the volume of the manifold according to that form."



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Volume form - Wikipedia

Volume form - Wikipedia:



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Multilinear form - Wikipedia

Multilinear form - Wikipedia: "Given a basis {\displaystyle (v_{1},\ldots ,v_{n})} for {\displaystyle V} and its dual {\displaystyle (\phi ^{1},\ldots ,\phi ^{n})} for dual vector space {\displaystyle V^{*}={\mathcal {A}}_{1}(V)} , the wedge products {\displaystyle \phi ^{i_{1}}\wedge \cdots \wedge \phi ^{i_{k}}} , with {\displaystyle 1\leq i_{1}<\cdots

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One-form - Wikipedia

One-form - Wikipedia: "Gravitation"



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Linear map - Wikipedia

Linear map - Wikipedia: "Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V).
If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.
"



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Linear map - Wikipedia

Linear map - Wikipedia: "If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.
"



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Generalized function - Wikipedia

Generalized function - Wikipedia: "A further way in which the theory has been extended is as generalized sections of a smooth vector bundle. This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have compact support. The most developed theory is that of De Rham currents, dual to differential forms. These are homological in nature, in the way that differential forms give rise to De Rham cohomology. They can be used to formulate a very general Stokes' theorem."



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Distribution (mathematics) - Wikipedia

Distribution (mathematics) - Wikipedia: "Distributions are a class of linear functionals that map a set of test functions (conventional and well-behaved functions) into the set of real numbers"



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Distribution (mathematics) - Wikipedia

Distribution (mathematics) - Wikipedia: "Distributions are a class of linear functionals that map a set of test functions (conventional and well-behaved functions) into the set of real numbers"



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Distribution (mathematics) - Wikipedia

Distribution (mathematics) - Wikipedia: "Distributions are a class of linear functionals that map a set of test functions (conventional and well-behaved functions) into the set of real numbers"



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Singleton (mathematics) - Wikipedia

Singleton (mathematics) - Wikipedia: "In the standard set-theoretic construction of the natural numbers, the number 1 is defined as the singleton { {\displaystyle \varnothing } }."



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Binomial coefficient - Wikipedia

Binomial coefficient - Wikipedia: " "n choose k""



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Gravitation (book) - Wikipedia

Gravitation (book) - Wikipedia: "vectors or one-forms,"



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Linear map - Wikipedia

Linear map - Wikipedia: "A linear map from {\displaystyle \mathbf {V} } to {\displaystyle \mathbf {K} } (with {\displaystyle \mathbf {K} } viewed as a vector space over itself) is called a linear functional.[5]

"



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Continuous symmetry - Wikipedia

Continuous symmetry - Wikipedia: "an intuitive idea corresponding to the concept of viewing some symmetries as motions,"



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Schwarzschild radius - Wikipedia

Schwarzschild radius - Wikipedia: "every quantity "



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Algebraic variety - Wikipedia

Algebraic variety - Wikipedia: "The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry.

"



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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,"



n-forms share being homogeneous with their field.

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Hamiltonian constraint - Wikipedia

Hamiltonian constraint - Wikipedia: "Hamiltonian constraint
"



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Hamiltonian constraint - Wikipedia

Hamiltonian constraint - Wikipedia: "Hamiltonian constraint
"



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Metric tensor (general relativity) - Wikipedia

Metric tensor (general relativity) - Wikipedia: "The flat space metric (or Minkowski metric) is often denoted by the symbol η and is the metric used in special relativity."



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Symmetry group - Wikipedia

Symmetry group - Wikipedia: "The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant)."



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Functional (mathematics) - Wikipedia

Functional (mathematics) - Wikipedia: "Richard Feynman used functional integrals as the central idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space.
"



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Indefinite orthogonal group - Wikipedia

Indefinite orthogonal group - Wikipedia: "In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. The dimension of the group is n(n − 1)/2.
"



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Lagrangian (field theory) - Wikipedia

Lagrangian (field theory) - Wikipedia: "Often, a "Lagrangian density" is simply referred to as a "Lagrangian".
"



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–30– - Wikipedia

–30– - Wikipedia:



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Killing form - Wikipedia

Killing form - Wikipedia: "in many cases, the Killing form can be used as a metric tensor on a manifold,"



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Exponential map (Lie theory) - Wikipedia

Exponential map (Lie theory) - Wikipedia: " is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). "



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Representation theory of the Lorentz group - Wikipedia

Representation theory of the Lorentz group - Wikipedia: "Mathematically the Lorentz group is defined as the set of transformations preserving the bilinear form
{\displaystyle (ct_{1},x_{1},y_{1},z_{1})\cdot (ct_{2},x_{2},y_{2},z_{2})=-c^{2}t_{1}t_{2}+x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2},} "



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Weyl transformation - Wikipedia

Weyl transformation - Wikipedia:



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Adjoint representation - Wikipedia

Adjoint representation - Wikipedia: "linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint repre"



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Infinitesimal transformation - Wikipedia

Infinitesimal transformation - Wikipedia:



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Up First : NPR

Up First : NPR:



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Home - Quora

Home - Quora: "‘At’ is a preposition to do with location. It can be a location in space or time.

at = means the action takes place for the period of the time stated.

It can be at a time as referenced by a clock (9pm) or a period of time (night).

In is a preposition to do with containment. We use it with time to indicate an action that happens within a time period.

In other words:

‘At night’ IS the time period something happens

‘In the night’ CONTAINS the time period that something happens.
"



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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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Finitely generated module - Wikipedia

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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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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Metrization theorem - Wikipedia

Metrization theorem - Wikipedia: "metrizable"



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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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Symmetry group - Wikipedia

Symmetry group - Wikipedia: "Not to be confused with Symmetric group.
"



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Symmetric group - Wikipedia

Symmetric group - Wikipedia:



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Group action - Wikipedia

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



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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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Online Etymology Dictionary

Online Etymology Dictionary: "bungled scribal transliterations of Arabic samt"



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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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年・月・日の干支(No.0020)

A good result from googling '干支日付'

年・月・日の干支(No.0020):



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Watch 101

Watch 101:



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Pitch class - Wikipedia

Pitch class - Wikipedia: "(thus, middle C is 60)"



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Liner notes - Wikipedia

Liner notes - Wikipedia:



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definite article - Can we use the preposition "on" before "the night"? - English Language Learners Stack Exchange

definite article - Can we use the preposition "on" before "the night"? - English Language Learners Stack Exchange: "on night"



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Number theory - Wikipedia

Number theory - Wikipedia:



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Morphism - Wikipedia

Morphism - Wikipedia: "the structures (called "objects") "



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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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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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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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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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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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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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Materials Used In Construction And Repair Of Watches

Materials Used In Construction And Repair Of Watches:



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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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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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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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Drum kit - Wikipedia

Drum kit - Wikipedia: "Carpets"



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Drum kit - Wikipedia

Drum kit - Wikipedia: "Terminology"



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Exterior derivative - Wikipedia

Exterior derivative - Wikipedia: "If  f  is a smooth function (a 0-form), then the exterior derivative of  f  is the differential of  f . That is, df  is the unique 1-form such that for every smooth vector field X, df (X) = dX f , where dX f  is the directional derivative of  f  in the direction of X.
"



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Omission of the indefinite article to eliminate ambiguity - English Language & Usage Stack Exchange

Omission of the indefinite article to eliminate ambiguity - English Language & Usage Stack Exchange: "He was Managing Director at Boots.
Who's going to be Best Man?
We elected her treasurer."



These sentences when spoken let the listener hear the capitalized wording.

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articles - Is it necessary to use "the" multiple times? - English Language & Usage Stack Exchange

articles - Is it necessary to use "the" multiple times? - English Language & Usage Stack Exchange: "I would like to buy the red, blue and yellow cloth.
I would like to buy the red, the blue, and the yellow cloth."



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Zero-marking in English - Wikipedia

Zero-marking in English - Wikipedia: "The definite article is sometimes omitted before some words for specific institutions, such as prison, school, and (in standard non-American dialects) hospital.[5]
She is in hospital.
The criminal went to prison.
I'm going to school.
The article may also be omitted between a preposition and the word bed when describing activities typically associated with beds.[5]
He is lying in bed.
They went to bed."



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pycckuu: Crash Course in Russian | The Alphabet ... | language o'clock

pycckuu: Crash Course in Russian | The Alphabet ... | language o'clock:



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Greek handwriting | Handwritten Greek letters | How to handwrite in Greek

Greek handwriting | Handwritten Greek letters | How to handwrite in Greek:



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Toque Girl — kuronchu: hangul crash course

Toque Girl — kuronchu: hangul crash course:



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Vintage Citizen Diamond Flake – World’s thinnest watch – Vintage Citizen Watches

Vintage Citizen Diamond Flake – World’s thinnest watch – Vintage Citizen Watches:



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Citizen history and data – Vintage Citizen Watches

Citizen history and data – Vintage Citizen Watches: "The “Center Second” came in many versions, and even a solid gold one, and a total of 6-7 generations for about 10 years. The first one was produced in 1948. All of them were running at 18,000 bph and came  equipped with 7, 8, 9, 10, 11, 15, 16, 17 and even 19 jewels. The Center Seconds are non-hacking and don’t have a date complication. During this time they became water protected and starting from 1956 they had the Citizen Parashock system installed.

The first watch presented here is ParaShock and Water Protected and is made in about 1961. The case back is SS and screws in even though most of them had snap on case backs.  I love the simplicity of the pale white dial with gold accents! The Citizen “C” logo at 12, the hour markers and the hands are all golden while all the rest of the marking are printed in black.

"



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Hmt kohinoor Vintage Watch Nos 17J hand winding Men's Watch | eBay

Hmt kohinoor Vintage Watch Nos 17J hand winding Men&#039;s Watch | eBay:



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Best Free OCR API & Online OCR Service - Fresh 2017 OCR Software

Best Free OCR API & Online OCR Service - Fresh 2017 OCR Software: "하드웨어 세트"



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Finitely generated module - Wikipedia

Finitely generated module - Wikipedia: "Finitely generated module
"



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Differential form - Wikipedia

Differential form - Wikipedia: "In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to defining integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression f(x) dx from one-variable calculus is an example of a 1-form, and can be integrated over an interval [a, b] in the domain of f:
{\displaystyle \int _{a}^{b}f(x)\,dx}"



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SKKとは (エスケイケイとは) [単語記事] - ニコニコ大百科

SKKとは (エスケイケイとは) [単語記事] - ニコニコ大百科: "Ctrlキー+"y" 単語登録時、ペーストする"



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Section (fiber bundle) - Wikipedia

Section (fiber bundle) - Wikipedia: "The space of continuous sections of a fiber bundle E over U is sometimes denoted C(U,E), while the space of global sections of E is often denoted Γ(E) or Γ(B,E).
"



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Jewelry for him - Etsy

Jewelry for him - Etsy: "for him"



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レースカーテン | ニトリ公式通販 家具・インテリア・生活雑貨通販のニトリネット

レースカーテン | ニトリ公式通販　家具・インテリア・生活雑貨通販のニトリネット: "
遮熱・遮像・ミラーレースカーテン(アラン)
834  ～  3,695円(税別）
"



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シチズン - アンティーク時計ルチクル

シチズン - アンティーク時計ルチクル: "シチズン　エース
新入荷！
６０年代、手巻き
SSケース、ケースサイズ３６ミリ
※若者に向けて発売されていた手巻き時計「エース」です。
初期の防水ケースで、シンプルでスタイルを選ばない実用性の高い１本です。"



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シチズン・ホーマーデート・昭和３０年代紳士手巻・小ぶり 美品 - アンティーク時計専門店 時計屋なかの

シチズン・ホーマーデート・昭和３０年代紳士手巻・小ぶり 美品 - アンティーク時計専門店　時計屋なかの:



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How does versioning work in a list or library? - DefaultTitleSuffixPlaceholder

How does versioning work in a list or library? - DefaultTitleSuffixPlaceholder: "Version numbers "



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n-sphere - Wikipedia

n-sphere - Wikipedia: "the pair of points at the ends of a (one-dimensional) line segment is 0-sphere,
"



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Furstenberg's proof of the infinitude of primes - Wikipedia

Furstenberg's proof of the infinitude of primes - Wikipedia:



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↯ - Wikipedia

↯ - Wikipedia: "It is used for indicating a contradiction (the relationship between incompatible propositions) in mathematical logic"



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Set-builder notation - Wikipedia

Set-builder notation - Wikipedia: "Defined sets by properties is also known as set comprehension, set abstraction or as defining a set's intension.
"



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Why do we call them headphone “jacks”? – mashed radish

Why do we call them headphone “jacks”? – mashed radish:



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What is the difference between a bolt, a screw and a stud? - Quora

What is the difference between a bolt, a screw and a stud? - Quora:

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Morphism - Wikipedia

Morphism - Wikipedia: "Examples[edit]
In the concrete categories studied in universal algebra (groups, rings, modules, etc.), morphisms are usually homomorphisms. Likewise, the notions of automorphism, endomorphism, epimorphism, homeomorphism, isomorphism, and monomorphism all find use in universal algebra.
In the category of topological spaces, morphisms are continuous functions and isomorphisms are called homeomorphisms.
In the category of smooth manifolds, morphisms are smooth functions and isomorphisms are called diffeomorphisms."

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symbols - Typeset an = with an ! above - TeX - LaTeX Stack Exchange

symbols - Typeset an = with an ! above - TeX - LaTeX Stack Exchange: "equals because of data that was given in the problem"

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