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In mathematics, the braid group on n strands (denoted B n {\displaystyle B_{n}} ), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry.
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https://en.wikipedia.org/wiki/Braid_length
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In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.
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https://en.wikipedia.org/wiki/Real_Analysis
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In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.
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https://en.wikipedia.org/wiki/Branching_theorem
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In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice. Lemma. Suppose G {\displaystyle G} is a group with subgroups A {\displaystyle A} and C {\displaystyle C} . Suppose B ◃ A {\displaystyle B\triangleleft A} and D ◃ C {\displaystyle D\triangleleft C} are normal subgroups.
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https://en.wikipedia.org/wiki/Butterfly_lemma
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Then there is an isomorphism of quotient groups: ( A ∩ C ) B ( A ∩ D ) B ≅ ( A ∩ C ) D ( B ∩ C ) D . {\displaystyle {\frac {(A\cap C)B}{(A\cap D)B}}\cong {\frac {(A\cap C)D}{(B\cap C)D}}.} This can be generalized to the case of a group with operators ( G , Ω ) {\displaystyle (G,\Omega )} with stable subgroups A {\displaystyle A} and C {\displaystyle C} , the above statement being the case of Ω = G {\displaystyle \Omega =G} acting on itself by conjugation.
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https://en.wikipedia.org/wiki/Butterfly_lemma
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Zassenhaus proved this lemma specifically to give the most direct proof of the Schreier refinement theorem. The 'butterfly' becomes apparent when trying to draw the Hasse diagram of the various groups involved. Zassenhaus' lemma for groups can be derived from a more general result known as Goursat's theorem stated in a Goursat variety (of which groups are an instance); however the group-specific modular law also needs to be used in the derivation.
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https://en.wikipedia.org/wiki/Butterfly_lemma
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In mathematics, the cake number, denoted by Cn, is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence. The values of Cn for n = 0, 1, 2, ... are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, ... (sequence A000125 in the OEIS).
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https://en.wikipedia.org/wiki/Cake_number
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In mathematics, the caliber or calibre of a topological space X is a cardinal κ such that for every set of κ nonempty open subsets of X there is some point of X contained in κ of these subsets. This concept was introduced by Shanin (1948). There is a similar concept for posets. A pre-caliber of a poset P is a cardinal κ such that for any collection of elements of P indexed by κ, there is a subcollection of cardinality κ that is centered. Here a subset of a poset is called centered if for any finite subset there is an element of the poset less than or equal to all of them.
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https://en.wikipedia.org/wiki/Caliber_(mathematics)
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In mathematics, the canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle \,\!\Omega ^{n}=\omega } , which is the nth exterior power of the cotangent bundle Ω {\displaystyle \Omega } on V {\displaystyle V} . Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T ∗ V {\displaystyle T^{*}V} . Equivalently, it is the line bundle of holomorphic n-forms on V {\displaystyle V} . This is the dualising object for Serre duality on V {\displaystyle V} .
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https://en.wikipedia.org/wiki/Canonical_class
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It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K {\displaystyle K} on V {\displaystyle V} giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V {\displaystyle V} , and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor − K {\displaystyle K} with K {\displaystyle K} canonical. The anticanonical bundle is the corresponding inverse bundle ω − 1 {\displaystyle \omega ^{-1}} . When the anticanonical bundle of V {\displaystyle V} is ample, V {\displaystyle V} is called a Fano variety.
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https://en.wikipedia.org/wiki/Canonical_class
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In mathematics, the capacitated arc routing problem (CARP) is that of finding the shortest tour with a minimum graph/travel distance of a mixed graph with undirected edges and directed arcs given capacity constraints for objects that move along the graph that represent snow-plowers, street sweeping machines, or winter gritters, or other real-world objects with capacity constraints. The constraint can be imposed for the length of time the vehicle is away from the central depot, or a total distance traveled, or a combination of the two with different weighting factors. There are many different variations of the CARP described in the book Arc Routing:Problems, Methods, and Applications by Ángel Corberán and Gilbert Laporte.Solving the CARP involves the study of graph theory, arc routing, operations research, and geographical routing algorithms to find the shortest path efficiently. The CARP is NP-hard arc routing problem.
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https://en.wikipedia.org/wiki/Capacitated_arc_routing_problem
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The CARP can be solved with combinatorial optimization including convex hulls. The large-scale capacitated arc routing problem (LSCARP) is a variant of the capacitated arc routing problem that applies to hundreds of edges and nodes to realistically simulate and model large complex environments. == References ==
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https://en.wikipedia.org/wiki/Capacitated_arc_routing_problem
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In mathematics, the capacity of a set in Euclidean space is a measure of the "size" of that set. Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge. More precisely, it is the capacitance of the set: the total charge a set can hold while maintaining a given potential energy. The potential energy is computed with respect to an idealized ground at infinity for the harmonic or Newtonian capacity, and with respect to a surface for the condenser capacity.
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https://en.wikipedia.org/wiki/Harmonic_capacity
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In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = { 2 , 4 , 6 } {\displaystyle A=\{2,4,6\}} contains 3 elements, and therefore A {\displaystyle A} has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers.
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https://en.wikipedia.org/wiki/Cardinality
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The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A {\displaystyle A} is usually denoted | A | {\displaystyle |A|} , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A {\displaystyle A} may alternatively be denoted by n ( A ) {\displaystyle n(A)} , A {\displaystyle A} , card ( A ) {\displaystyle \operatorname {card} (A)} , or # A {\displaystyle \#A} .
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https://en.wikipedia.org/wiki/Cardinality
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In mathematics, the caret can signify exponentiation (e.g. 3^5 for 35) where the usual superscript is not readily usable (as on some graphing calculators). It is also used to indicate a superscript in TeX typesetting. As Isaac Asimov described it in his 1974 "Skewered!" essay (on Skewes's number), "I make the exponent a figure of normal size and it is as though it is being held up by a lever, and its added weight when its size grows bends the lever down. "The use of the caret for exponentiation can be traced back to ALGOL 60, which expressed the exponentiation operator as an upward-pointing arrow, intended to evoke the superscript notation common in mathematics. The upward-pointing arrow is now used to signify hyperoperations in Knuth's up-arrow notation.
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https://en.wikipedia.org/wiki/Caret_(punctuation)
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In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
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https://en.wikipedia.org/wiki/Category_of_abelian_groups
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In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms.
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https://en.wikipedia.org/wiki/Category_of_finite_dimensional_Hilbert_spaces
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In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
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https://en.wikipedia.org/wiki/Grp_(category_theory)
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In mathematics, the category Ord has preordered sets as objects and order-preserving functions as morphisms. This is a category because the composition of two order-preserving functions is order preserving and the identity map is order preserving. The monomorphisms in Ord are the injective order-preserving functions. The empty set (considered as a preordered set) is the initial object of Ord, and the terminal objects are precisely the singleton preordered sets.
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https://en.wikipedia.org/wiki/Category_of_preordered_sets
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There are thus no zero objects in Ord. The categorical product in Ord is given by the product order on the cartesian product.
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https://en.wikipedia.org/wiki/Category_of_preordered_sets
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We have a forgetful functor Ord → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore Ord is a concrete category. This functor has a left adjoint (sending every set to that set equipped with the equality relation) and a right adjoint (sending every set to that set equipped with the total relation).
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https://en.wikipedia.org/wiki/Category_of_preordered_sets
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In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) R: A → B in this category is a relation between the sets A and B, so R ⊆ A × B. The composition of two relations R: A → B and S: B → C is given by (a, c) ∈ S o R ⇔ for some b ∈ B, (a, b) ∈ R and (b, c) ∈ S.Rel has also been called the "category of correspondences of sets".
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https://en.wikipedia.org/wiki/Category_of_relations
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In mathematics, the category number of a mathematician is a humorous construct invented by Dan Freed, intended to measure the capacity of that mathematician to stomach the use of higher categories. It is defined as the largest number n such that they can think about n-categories for a half hour without getting a splitting headache.
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https://en.wikipedia.org/wiki/N-category_number
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In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the latter lacks some of the pleasant properties one would desire. There is also such a category for based spaces, defined by requiring maps to preserve the base points.The articles compactly generated space and weak Hausdorff space define the respective topological properties. For the historical motivation behind these conditions on spaces, see Compactly generated space#Motivation. This article focuses on the properties of the category.
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https://en.wikipedia.org/wiki/Category_of_compactly_generated_weak_Hausdorff_spaces
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In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp. One is often interested only in Cp-manifolds modeled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modeled on a fixed space E is denoted Manp(E). One may also speak of the category of smooth manifolds, Man∞, or the category of analytic manifolds, Manω.
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https://en.wikipedia.org/wiki/Category_of_manifolds
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In mathematics, the category of medial magmas, also known as the medial category, and denoted Med, is the category whose objects are medial magmas (that is, sets with a medial binary operation), and whose morphisms are magma homomorphisms (which are equivalent to homomorphisms in the sense of universal algebra). The category Med has direct products, so the concept of a medial magma object (internal binary operation) makes sense. As a result, Med has all its objects as medial objects, and this characterizes it. There is an inclusion functor from Set to Med as trivial magmas, with operations being the right projections (x, y) → y.An injective endomorphism can be extended to an automorphism of a magma extension—the colimit of the constant sequence of the endomorphism.
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https://en.wikipedia.org/wiki/Category_of_medial_magmas
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In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.
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https://en.wikipedia.org/wiki/Category_of_commutative_algebras
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In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces.
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https://en.wikipedia.org/wiki/Categorical_topology
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In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category because the composition of two continuous linear maps is again a continuous linear map. The category is often denoted TVect or TVS. Fixing a topological field K, one can also consider the subcategory TVectK of topological vector spaces over K with continuous K-linear maps as the morphisms.
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https://en.wikipedia.org/wiki/Category_of_topological_vector_spaces
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In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with y as the dependent variable, superscript (n) denoting nth-derivative, and an, an − 1, ..., a1, a0 as constants, a n y ( n ) + a n − 1 y ( n − 1 ) + ⋯ + a 1 y ′ + a 0 y = 0 , {\displaystyle a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+\cdots +a_{1}y'+a_{0}y=0,} will have a characteristic equation of the form a n r n + a n − 1 r n − 1 + ⋯ + a 1 r + a 0 = 0 {\displaystyle a_{n}r^{n}+a_{n-1}r^{n-1}+\cdots +a_{1}r+a_{0}=0} whose solutions r1, r2, ..., rn are the roots from which the general solution can be formed. Analogously, a linear difference equation of the form y t + n = b 1 y t + n − 1 + ⋯ + b n y t {\displaystyle y_{t+n}=b_{1}y_{t+n-1}+\cdots +b_{n}y_{t}} has characteristic equation r n − b 1 r n − 1 − ⋯ − b n = 0 , {\displaystyle r^{n}-b_{1}r^{n-1}-\cdots -b_{n}=0,} discussed in more detail at Linear recurrence with constant coefficients#Solution to homogeneous case.
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https://en.wikipedia.org/wiki/Characteristic_equation_(calculus)
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The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative. For difference equations, there is stability if and only if the modulus of each root is less than 1.
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https://en.wikipedia.org/wiki/Characteristic_equation_(calculus)
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For both types of equation, persistent fluctuations occur if there is at least one pair of complex roots. The method of integrating linear ordinary differential equations with constant coefficients was discovered by Leonhard Euler, who found that the solutions depended on an algebraic 'characteristic' equation. The qualities of the Euler's characteristic equation were later considered in greater detail by French mathematicians Augustin-Louis Cauchy and Gaspard Monge.
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https://en.wikipedia.org/wiki/Characteristic_equation_(calculus)
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In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, char(R) is the smallest positive number n such that:(p 198, Thm. 23.14) 1 + ⋯ + 1 ⏟ n summands = 0 {\displaystyle \underbrace {1+\cdots +1} _{n{\text{ summands}}}=0} if such a number n exists, and 0 otherwise.
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https://en.wikipedia.org/wiki/Characteristic_subring
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In mathematics, the chromatic spectral sequence is a spectral sequence, introduced by Ravenel (1978), used for calculating the initial term of the Adams spectral sequence for Brown–Peterson cohomology, which is in turn used for calculating the stable homotopy groups of spheres.
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https://en.wikipedia.org/wiki/Chromatic_spectral_sequence
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In mathematics, the circle group, denoted by T {\displaystyle \mathbb {T} } or S 1 {\displaystyle \mathbb {S} ^{1}} , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C × {\displaystyle \mathbb {C} ^{\times }} , the multiplicative group of all nonzero complex numbers. Since C × {\displaystyle \mathbb {C} ^{\times }} is abelian, it follows that T {\displaystyle \mathbb {T} } is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure θ {\displaystyle \theta }: This is the exponential map for the circle group.
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https://en.wikipedia.org/wiki/Circle_action
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The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation T {\displaystyle \mathbb {T} } for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, T n {\displaystyle \mathbb {T} ^{n}} (the direct product of T {\displaystyle \mathbb {T} } with itself n {\displaystyle n} times) is geometrically an n {\displaystyle n} -torus. The circle group is isomorphic to the special orthogonal group S O ( 2 ) {\displaystyle \mathrm {SO} (2)} .
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https://en.wikipedia.org/wiki/Circle_action
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In mathematics, the circumflex is used to modify variable names; it is usually read "hat", e.g., î is "i hat". The Fourier transform of a function ƒ is often denoted by f ^ {\displaystyle {\hat {f}}} . In the notation of sets, a hat above an element signifies that the element was removed from the set, such as in { x 0 , … , x ^ i , … , x n } {\displaystyle \{x_{0},\dotsc ,{\hat {x}}_{i},\dotsc ,x_{n}\}} , the set containing all elements x 0 , … , x n {\displaystyle x_{0},\dotsc ,x_{n}} except x i {\displaystyle x_{i}} .
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https://en.wikipedia.org/wiki/Circumflex_accent
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In geometry, a hat is sometimes used for an angle. For instance, the angles A ^ {\displaystyle {\hat {A}}} or A B ^ C {\displaystyle A{\hat {B}}C} . In vector notation, a hat above a letter indicates a unit vector (a dimensionless vector with a magnitude of 1).
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https://en.wikipedia.org/wiki/Circumflex_accent
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For instance, ı ^ {\displaystyle {\hat {\mathbf {\imath } }}} , x ^ {\displaystyle {\hat {\mathbf {x} }}} , or e ^ 1 {\displaystyle {\hat {\mathbf {e} }}_{1}} stands for a unit vector in the direction of the x-axis of a Cartesian coordinate system. In statistics, the hat is used to denote an estimator or an estimated value, as opposed to its theoretical counterpart. For example, in errors and residuals, the hat in ε ^ {\displaystyle {\hat {\varepsilon }}} indicates an observable estimate (the residual) of an unobservable quantity called ε {\displaystyle \varepsilon } (the statistical error). It is read x-hat or x-roof, where x represents the character under the hat.
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https://en.wikipedia.org/wiki/Circumflex_accent
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In mathematics, the class of L-matrices are those matrices whose off-diagonal entries are less than or equal to zero and whose diagonal entries are positive; that is, an L-matrix L satisfies L = ( ℓ i j ) ; ℓ i i > 0 ; ℓ i j ≤ 0 , i ≠ j . {\displaystyle L=(\ell _{ij});\quad \ell _{ii}>0;\quad \ell _{ij}\leq 0,\quad i\neq j.}
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https://en.wikipedia.org/wiki/L-matrix
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In mathematics, the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: Z = ( z i j ) ; z i j ≤ 0 , i ≠ j . {\displaystyle Z=(z_{ij});\quad z_{ij}\leq 0,\quad i\neq j.} Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term quasinegative matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made.
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https://en.wikipedia.org/wiki/Z-matrix_(mathematics)
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The Jacobian of a competitive dynamical system is a Z-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is J, then (−J) is a Z-matrix. Related classes are L-matrices, M-matrices, P-matrices, Hurwitz matrices and Metzler matrices.
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https://en.wikipedia.org/wiki/Z-matrix_(mathematics)
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L-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices and P-matrices are nonsingular M-matrices. In the context of quantum complexity theory, these are referred to as stoquastic operators.
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https://en.wikipedia.org/wiki/Z-matrix_(mathematics)
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In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra.
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https://en.wikipedia.org/wiki/Moebius_inversion
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In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker.
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https://en.wikipedia.org/wiki/Kronecker_limit_formula
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In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory and representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture, which includes Fermat's Last Theorem as a special case. Establishing the Langlands correspondence in the number theoretic context has proven extremely difficult. As a result, some mathematicians have posed the geometric Langlands correspondence.
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https://en.wikipedia.org/wiki/Geometric_Langlands_correspondence
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In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory to the branch of mathematics known as representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture, which includes Fermat's Last Theorem as a special case.In spite of its importance in number theory, establishing the Langlands correspondence in the number theoretic context has proved extremely difficult. As a result, some mathematicians have worked on a related conjecture known as the geometric Langlands correspondence. This is a geometric reformulation of the classical Langlands correspondence which is obtained by replacing the number fields appearing in the original version by function fields and applying techniques from algebraic geometry.In a paper from 2007, Anton Kapustin and Edward Witten suggested that the geometric Langlands correspondence can be viewed as a mathematical statement of Montonen–Olive duality.
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https://en.wikipedia.org/wiki/S-duality
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Starting with two Yang–Mills theories related by S-duality, Kapustin and Witten showed that one can construct a pair of quantum field theories in two-dimensional spacetime. By analyzing what this dimensional reduction does to certain physical objects called D-branes, they showed that one can recover the mathematical ingredients of the geometric Langlands correspondence. Their work shows that the Langlands correspondence is closely related to S-duality in quantum field theory, with possible applications in both subjects.
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https://en.wikipedia.org/wiki/S-duality
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In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any generalized circle, and thus a natural setting for planar inversive geometry. An inversion of the Möbius plane with respect to any circle is an involution which fixes the points on the circle and exchanges the points in the interior and exterior, the center of the circle exchanged with the point at infinity.
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https://en.wikipedia.org/wiki/Moebius_plane
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In inversive geometry a straight line is considered to be a generalized circle containing the point at infinity; inversion of the plane with respect to a line is a Euclidean reflection. More generally, a Möbius plane is an incidence structure with the same incidence relationships as the classical Möbius plane. It is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane.
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https://en.wikipedia.org/wiki/Moebius_plane
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In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type.
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https://en.wikipedia.org/wiki/Classical_groups
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The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics.
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https://en.wikipedia.org/wiki/Classical_groups
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A few examples are the following. The rotation group SO(3) is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp(m) finds application in Hamiltonian mechanics and quantum mechanical versions of it.
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https://en.wikipedia.org/wiki/Classical_groups
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In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others. Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A.
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https://en.wikipedia.org/wiki/Classical_orthogonal_polynomials
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Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials. For given polynomials Q , L: R → R {\displaystyle Q,L:\mathbb {R} \to \mathbb {R} } and ∀ n ∈ N 0 {\displaystyle \forall \,n\in \mathbb {N} _{0}} the classical orthogonal polynomials f n: R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} } are characterized by being solutions of the differential equation Q ( x ) f n ′ ′ + L ( x ) f n ′ + λ n f n = 0 {\displaystyle Q(x)\,f_{n}^{\prime \prime }+L(x)\,f_{n}^{\prime }+\lambda _{n}f_{n}=0} with to be determined constants λ n ∈ R {\displaystyle \lambda _{n}\in \mathbb {R} } . There are several more general definitions of orthogonal classical polynomials; for example, Andrews & Askey (1985) use the term for all polynomials in the Askey scheme.
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https://en.wikipedia.org/wiki/Classical_orthogonal_polynomials
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In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers.
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https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
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The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution. Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.
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https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
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In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates.
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https://en.wikipedia.org/wiki/List_of_finite_simple_groups
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In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space R ∞ {\displaystyle \mathbb {R} ^{\infty }} . It is analogous to the classifying space for U(n).
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https://en.wikipedia.org/wiki/Classifying_space_for_O(n)
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In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. This space with its universal fibration may be constructed as either the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or, the direct limit, with the induced topology, of Grassmannians of n planes.Both constructions are detailed here.
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https://en.wikipedia.org/wiki/Classifying_space_for_U(n)
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In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.
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https://en.wikipedia.org/wiki/Closed_graph_theorem
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In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of a Lie group G, then H is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding. One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan, who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.
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https://en.wikipedia.org/wiki/Closed_subgroup_theorem
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In mathematics, the coadjoint representation K {\displaystyle K} of a Lie group G {\displaystyle G} is the dual of the adjoint representation. If g {\displaystyle {\mathfrak {g}}} denotes the Lie algebra of G {\displaystyle G} , the corresponding action of G {\displaystyle G} on g ∗ {\displaystyle {\mathfrak {g}}^{*}} , the dual space to g {\displaystyle {\mathfrak {g}}} , is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on G {\displaystyle G} .
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https://en.wikipedia.org/wiki/Coadjoint_representation
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The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups G {\displaystyle G} a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of G {\displaystyle G} are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of G {\displaystyle G} , which again may be complicated, while the orbits are relatively tractable.
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https://en.wikipedia.org/wiki/Coadjoint_representation
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In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism hypothesis, though the details of his approach have yet to appear in the literature as of 2022. In 2021, Daniel Grady and Dmitri Pavlov claimed a complete proof of the cobordism hypothesis, as well as a generalization to bordisms with arbitrary geometric structures.
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https://en.wikipedia.org/wiki/Tangle_hypothesis
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In mathematics, the coclass of a finite p-group of order pn is n − c, where c is the class.
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https://en.wikipedia.org/wiki/Coclass_conjectures
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In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or image of a function. A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it.
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https://en.wikipedia.org/wiki/Codomain
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Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution. A codomain is not part of a function f if f is defined as just a graph. For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: X → Y.
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https://en.wikipedia.org/wiki/Codomain
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In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Throughout there have been two basic points: the operations can be studied by combinatorial means; and the effect of the operations is to yield an interesting bicommutant theory.The origin of these studies was the work of Pontryagin, Postnikov, and Norman Steenrod, who first defined the Pontryagin square, Postnikov square, and Steenrod square operations for singular cohomology, in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a natural diagonal map, at cochain level. The general theory of the Steenrod algebra of operations has been brought into close relation with that of the symmetric group.
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https://en.wikipedia.org/wiki/Cohomology_operation
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In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level. The convergence is to groups in stable homotopy theory, about which information is hard to come by. This connection established the deep interest of the cohomology operations for homotopy theory, and has been a research topic ever since. An extraordinary cohomology theory has its own cohomology operations, and these may exhibit a richer set on constraints.
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https://en.wikipedia.org/wiki/Cohomology_operation
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In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given set. It is typically used in fractal compression.
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https://en.wikipedia.org/wiki/Collage_theorem
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In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm. Historically, it was known as logarithmus decimalis or logarithmus decadis. It is indicated by log(x), log10 (x), or sometimes Log(x) with a capital L (however, this notation is ambiguous, since it can also mean the complex natural logarithmic multi-valued function).
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https://en.wikipedia.org/wiki/Base-10_logarithm
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On calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity, the ISO 80000 specification recommends that log10 (x) should be written lg(x), and loge (x) should be ln(x). Before the early 1970s, handheld electronic calculators were not available, and mechanical calculators capable of multiplication were bulky, expensive and not widely available.
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https://en.wikipedia.org/wiki/Base-10_logarithm
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Instead, tables of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a slide rule. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions. Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well. For the history of such tables, see log table.
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https://en.wikipedia.org/wiki/Base-10_logarithm
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In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
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https://en.wikipedia.org/wiki/Commutator_(ring_theory)
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In mathematics, the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of operators in the ideal with bounded operators. Modern characterisation of the commutator subspace is through the Calkin correspondence and it involves the invariance of the Calkin sequence space of an operator ideal to taking Cesàro means. This explicit spectral characterisation reduces problems and questions about commutators and traces on two-sided ideals to (more resolvable) problems and conditions on sequence spaces.
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https://en.wikipedia.org/wiki/Commutator_subspace
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In mathematics, the compact complement topology is a topology defined on the set R {\displaystyle \scriptstyle \mathbb {R} } of real numbers, defined by declaring a subset X ⊆ R {\displaystyle \scriptstyle X\subseteq \mathbb {R} } open if and only if it is either empty or its complement R ∖ X {\displaystyle \scriptstyle \mathbb {R} \setminus X} is compact in the standard Euclidean topology on R {\displaystyle \scriptstyle \mathbb {R} } .
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https://en.wikipedia.org/wiki/Compact_complement_topology
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In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945.If the codomain of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain.
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https://en.wikipedia.org/wiki/Compact_open_topology
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In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.
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https://en.wikipedia.org/wiki/Direct_comparison_test
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In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by F j ( x ) = 1 Γ ( j + 1 ) ∫ 0 ∞ t j e t − x + 1 d t , ( j > − 1 ) {\displaystyle F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt,\qquad (j>-1)} This equals − Li j + 1 ( − e x ) , {\displaystyle -\operatorname {Li} _{j+1}(-e^{x}),} where Li s ( z ) {\displaystyle \operatorname {Li} _{s}(z)} is the polylogarithm. Its derivative is d F j ( x ) d x = F j − 1 ( x ) , {\displaystyle {\frac {dF_{j}(x)}{dx}}=F_{j-1}(x),} and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for F j {\displaystyle F_{j}} appears in the literature, for instance some authors omit the factor 1 / Γ ( j + 1 ) {\displaystyle 1/\Gamma (j+1)} . The definition used here matches that in the NIST DLMF.
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https://en.wikipedia.org/wiki/Complete_Fermi–Dirac_integral
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In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations of the ring C. There are some related Lie algebras defined over finite fields, that are also called Witt algebras. The complex Witt algebra was first defined by Élie Cartan (1909), and its analogues over finite fields were studied by Witt in the 1930s.
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https://en.wikipedia.org/wiki/Witt_algebra
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In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.} The complex conjugate of z {\displaystyle z} is often denoted as z ¯ {\displaystyle {\overline {z}}} or z ∗ {\displaystyle z^{*}} .
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https://en.wikipedia.org/wiki/Conjugate_pair
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In polar form, if r {\displaystyle r} and φ {\displaystyle \varphi } are real numbers then the conjugate of r e i φ {\displaystyle re^{i\varphi }} is r e − i φ . {\displaystyle re^{-i\varphi }.}
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https://en.wikipedia.org/wiki/Conjugate_pair
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This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a 2 + b 2 {\displaystyle a^{2}+b^{2}} (or r 2 {\displaystyle r^{2}} in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.
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https://en.wikipedia.org/wiki/Conjugate_pair
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In mathematics, the complex conjugate of a complex vector space V {\displaystyle V\,} is a complex vector space V ¯ {\displaystyle {\overline {V}}} , which has the same elements and additive group structure as V , {\displaystyle V,} but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of V ¯ {\displaystyle {\overline {V}}} satisfies where ∗ {\displaystyle *} is the scalar multiplication of V ¯ {\displaystyle {\overline {V}}} and ⋅ {\displaystyle \cdot } is the scalar multiplication of V . {\displaystyle V.}
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https://en.wikipedia.org/wiki/Complex_conjugate_of_a_vector_space
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The letter v {\displaystyle v} stands for a vector in V , {\displaystyle V,} α {\displaystyle \alpha } is a complex number, and α ¯ {\displaystyle {\overline {\alpha }}} denotes the complex conjugate of α . {\displaystyle \alpha .} More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure J {\displaystyle J} (different multiplication by i {\displaystyle i} ).
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https://en.wikipedia.org/wiki/Complex_conjugate_of_a_vector_space
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In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P.It follows from this (and the fundamental theorem of algebra) that, if the degree of a real polynomial is odd, it must have at least one real root. That fact can also be proved by using the intermediate value theorem.
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https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem
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In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors.
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https://en.wikipedia.org/wiki/Gauss_plane
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The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes known as the Argand plane or Gauss plane.
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https://en.wikipedia.org/wiki/Gauss_plane
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In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates ( Z 1 , Z 2 , Z 3 ) ∈ C 3 , ( Z 1 , Z 2 , Z 3 ) ≠ ( 0 , 0 , 0 ) {\displaystyle (Z_{1},Z_{2},Z_{3})\in \mathbf {C} ^{3},\qquad (Z_{1},Z_{2},Z_{3})\neq (0,0,0)} where, however, the triples differing by an overall rescaling are identified: ( Z 1 , Z 2 , Z 3 ) ≡ ( λ Z 1 , λ Z 2 , λ Z 3 ) ; λ ∈ C , λ ≠ 0. {\displaystyle (Z_{1},Z_{2},Z_{3})\equiv (\lambda Z_{1},\lambda Z_{2},\lambda Z_{3});\quad \lambda \in \mathbf {C} ,\qquad \lambda \neq 0.} That is, these are homogeneous coordinates in the traditional sense of projective geometry.
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https://en.wikipedia.org/wiki/Complex_projective_plane
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In mathematics, the complex squaring map, a polynomial mapping of degree two, is a simple and accessible demonstration of chaos in dynamical systems. It can be constructed by performing the following steps: Choose any complex number on the unit circle whose argument (angle) is not a rational multiple of π, Repeatedly square that number.This repetition (iteration) produces a sequence of complex numbers that can be described alone by their arguments. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. It can be shown that the sequence will be chaotic, i.e. it is sensitive to the detailed choice of starting angle.
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https://en.wikipedia.org/wiki/Complex_squaring_map
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In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for V (a space over the real numbers) may also serve as a basis for VC over the complex numbers.
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https://en.wikipedia.org/wiki/Complexification
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In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group.
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https://en.wikipedia.org/wiki/Complexification_(Lie_group)
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They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear. For compact Lie groups, the complexification, sometimes called the Chevalley complexification after Claude Chevalley, can be defined as the group of complex characters of the Hopf algebra of representative functions, i.e. the matrix coefficients of finite-dimensional representations of the group. In any finite-dimensional faithful unitary representation of the compact group it can be realized concretely as a closed subgroup of the complex general linear group. It consists of operators with polar decomposition g = u • exp iX, where u is a unitary operator in the compact group and X is a skew-adjoint operator in its Lie algebra. In this case the complexification is a complex algebraic group and its Lie algebra is the complexification of the Lie algebra of the compact Lie group.
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https://en.wikipedia.org/wiki/Complexification_(Lie_group)
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In mathematics, the composition operator C ϕ {\displaystyle C_{\phi }} with symbol ϕ {\displaystyle \phi } is a linear operator defined by the rule where f ∘ ϕ {\displaystyle f\circ \phi } denotes function composition. The study of composition operators is covered by AMS category 47B33.
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https://en.wikipedia.org/wiki/Composition_operator
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In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut Stars.
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https://en.wikipedia.org/wiki/Compound_of_three_octahedra
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In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field. In general, when we define metric space the distance function is taken to be a real-valued function. The real numbers form an ordered field which is Archimedean and order complete. These metric spaces have some nice properties like: in a metric space compactness, sequential compactness and countable compactness are equivalent etc. These properties may not, however, hold so easily if the distance function is taken in an arbitrary ordered field, instead of in R . {\displaystyle \scriptstyle \mathbb {R} .}
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https://en.wikipedia.org/wiki/Generalised_metric
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In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
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https://en.wikipedia.org/wiki/Positive_measure
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Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.
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https://en.wikipedia.org/wiki/Positive_measure
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