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In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by S, so S(n) = n + 1. For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession.
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https://en.wikipedia.org/wiki/Successor_function
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In mathematics, the sum of two cubes is a cubed number added to another cubed number.
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https://en.wikipedia.org/wiki/Sum_of_two_cubes
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In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions, roots and logarithms, tetration has two inverse functions, super-roots and super-logarithms. There are several ways of interpreting super-logarithms: As the Abel function of exponential functions, As the inverse function of tetration with respect to the height, As a generalization of Robert Munafo's large number class system,For positive integer values, the super-logarithm with base-e is equivalent to the number of times a logarithm must be iterated to get to 1 (the Iterated logarithm).
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https://en.wikipedia.org/wiki/Super-logarithm
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However, this is not true for negative values and so cannot be considered a full definition. The precise definition of the super-logarithm depends on a precise definition of non-integer tetration (that is, y x {\displaystyle {^{y}x}} for y not an integer). There is no clear consensus on the definition of non-integer tetration and so there is likewise no clear consensus on the super-logarithm for non-integer inputs.
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https://en.wikipedia.org/wiki/Super-logarithm
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In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz: 249–251 in 1910 as a part of his work on field theory. A supernatural number ω {\displaystyle \omega } is a formal product: ω = ∏ p p n p , {\displaystyle \omega =\prod _{p}p^{n_{p}},} where p {\displaystyle p} runs over all prime numbers, and each n p {\displaystyle n_{p}} is zero, a natural number or infinity. Sometimes v p ( ω ) {\displaystyle v_{p}(\omega )} is used instead of n p {\displaystyle n_{p}} .
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https://en.wikipedia.org/wiki/Supernatural_number
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If no n p = ∞ {\displaystyle n_{p}=\infty } and there are only a finite number of non-zero n p {\displaystyle n_{p}} then we recover the positive integers. Slightly less intuitively, if all n p {\displaystyle n_{p}} are ∞ {\displaystyle \infty } , we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide ω {\displaystyle \omega } "infinitely often," by taking that prime's corresponding exponent to be the symbol ∞ {\displaystyle \infty } .
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https://en.wikipedia.org/wiki/Supernatural_number
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There is no natural way to add supernatural numbers, but they can be multiplied, with ∏ p p n p ⋅ ∏ p p m p = ∏ p p n p + m p {\displaystyle \prod _{p}p^{n_{p}}\cdot \prod _{p}p^{m_{p}}=\prod _{p}p^{n_{p}+m_{p}}} . Similarly, the notion of divisibility extends to the supernaturals with ω 1 ∣ ω 2 {\displaystyle \omega _{1}\mid \omega _{2}} if v p ( ω 1 ) ≤ v p ( ω 2 ) {\displaystyle v_{p}(\omega _{1})\leq v_{p}(\omega _{2})} for all p {\displaystyle p} . The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining lcm ( { ω i } ) = ∏ p p sup ( v p ( ω i ) ) {\displaystyle \displaystyle \operatorname {lcm} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\sup(v_{p}(\omega _{i}))}} and gcd ( { ω i } ) = ∏ p p inf ( v p ( ω i ) ) {\displaystyle \displaystyle \operatorname {gcd} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\inf(v_{p}(\omega _{i}))}} .With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number.
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https://en.wikipedia.org/wiki/Supernatural_number
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We can also extend the usual p {\displaystyle p} -adic order functions to supernatural numbers by defining v p ( ω ) = n p {\displaystyle v_{p}(\omega )=n_{p}} for each p {\displaystyle p} . Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of a finite field.Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.
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https://en.wikipedia.org/wiki/Supernatural_number
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In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the superellipses. The term may refer to the solid object or to its surface, depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs.
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https://en.wikipedia.org/wiki/Superquadrics
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The superquadrics include many shapes that resemble cubes, octahedra, cylinders, lozenges and spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation.Some authors, such as Alan Barr, define "superquadrics" as including both the superellipsoids and the supertoroids.
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https://en.wikipedia.org/wiki/Superquadrics
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In modern computer vision literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics. Comprehensive coverage of geometrical properties of superquadrics and methods of their recovery from range images and point clouds are covered in several computer vision literatures. Useful tools and algorithms for superquadrics visualization, sampling, and recovery are open-sourced here.
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https://en.wikipedia.org/wiki/Superquadrics
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In mathematics, the supersingular isogeny graphs are a class of expander graphs that arise in computational number theory and have been applied in elliptic-curve cryptography. Their vertices represent supersingular elliptic curves over finite fields and their edges represent isogenies between curves.
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https://en.wikipedia.org/wiki/Supersingular_isogeny_graph
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In mathematics, the support (sometimes topological support or spectrum) of a measure μ {\displaystyle \mu } on a measurable topological space ( X , Borel ( X ) ) {\displaystyle (X,\operatorname {Borel} (X))} is a precise notion of where in the space X {\displaystyle X} the measure "lives". It is defined to be the largest (closed) subset of X {\displaystyle X} for which every open neighbourhood of every point of the set has positive measure.
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https://en.wikipedia.org/wiki/Support_(measure_theory)
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In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on R n {\displaystyle \mathbb {R} ^{n}} . Any non-empty closed convex set A is uniquely determined by hA. Furthermore, the support function, as a function of the set A, is compatible with many natural geometric operations, like scaling, translation, rotation and Minkowski addition. Due to these properties, the support function is one of the most central basic concepts in convex geometry.
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https://en.wikipedia.org/wiki/Support_function
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In mathematics, the support of a real-valued function f {\displaystyle f} is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f {\displaystyle f} is a topological space, then the support of f {\displaystyle f} is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis.
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https://en.wikipedia.org/wiki/Singular_support
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In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in Robion Kirby's problem list.Assuming the geometrization conjecture, the only open case was that of closed hyperbolic 3-manifolds.
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https://en.wikipedia.org/wiki/Surface_subgroup_conjecture
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A proof of this case was announced in the summer of 2009 by Jeremy Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared in the arxiv.org server in October 2009. Their paper was published in the Annals of Mathematics in 2012. In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.
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https://en.wikipedia.org/wiki/Surface_subgroup_conjecture
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In mathematics, the surgery structure set S ( X ) {\displaystyle {\mathcal {S}}(X)} is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or homeomorphic). There are different versions of the structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsion is taken into account or not.
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https://en.wikipedia.org/wiki/Surgery_structure_set
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In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.
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https://en.wikipedia.org/wiki/Surreal_form
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If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field.
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https://en.wikipedia.org/wiki/Surreal_form
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In mathematics, the symbol × has a number of uses, including Multiplication of two numbers, where it is read as "times" or "multiplied by" Cross product of two vectors, where it is usually read as "cross" Cartesian product of two sets, where it is usually read as "cross" Geometric dimension of an object, such as noting that a room is 10 feet × 12 feet in area, where it is usually read as "by" (e.g., "10 feet by 12 feet") Screen resolution in pixels, such as 1920 pixels across × 1080 pixels down. Read as "by". Dimensions of a matrix, where it is usually read as "by" A statistical interaction between two explanatory variables, where it is usually read as "by"In biology, the multiplication sign is used in a botanical hybrid name, for instance Ceanothus papillosus × impressus (a hybrid between C. papillosus and C. impressus) or Crocosmia × crocosmiiflora (a hybrid between two other species of Crocosmia).
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https://en.wikipedia.org/wiki/Multiplication_sign
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However, the communication of these hybrid names with a Latin letter "x" is common, when the actual "×" symbol is not readily available. The multiplication sign is also used by historians for an event between two dates. When employed between two dates – for example 1225 and 1232 – the expression "1225×1232" means "no earlier than 1225 and no later than 1232".A monadic × symbol is used by the APL programming language to denote the sign function.
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https://en.wikipedia.org/wiki/Multiplication_sign
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In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, Siegfried Heinrich Aronhold, Alfred Clebsch, and Paul Gordan in the 19th century for computing invariants of algebraic forms. It is based on treating the form as if it were a power of a degree one form, which corresponds to embedding a symmetric power of a vector space into the symmetric elements of a tensor product of copies of it.
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https://en.wikipedia.org/wiki/Symbolic_method_of_invariant_theory
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In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. Here, "minimal" means that S(V) satisfies the following universal property: for every linear map f from V to a commutative algebra A, there is a unique algebra homomorphism g: S(V) → A such that f = g ∘ i, where i is the inclusion map of V in S(V). If B is a basis of V, the symmetric algebra S(V) can be identified, through a canonical isomorphism, to the polynomial ring K, where the elements of B are considered as indeterminates. Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V. The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring.
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https://en.wikipedia.org/wiki/Symmetric_square
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In mathematics, the symmetric closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest symmetric relation on X {\displaystyle X} that contains R . {\displaystyle R.} For example, if X {\displaystyle X} is a set of airports and x R y {\displaystyle xRy} means "there is a direct flight from airport x {\displaystyle x} to airport y {\displaystyle y} ", then the symmetric closure of R {\displaystyle R} is the relation "there is a direct flight either from x {\displaystyle x} to y {\displaystyle y} or from y {\displaystyle y} to x {\displaystyle x} ". Or, if X {\displaystyle X} is the set of humans and R {\displaystyle R} is the relation 'parent of', then the symmetric closure of R {\displaystyle R} is the relation " x {\displaystyle x} is a parent or a child of y {\displaystyle y} ".
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https://en.wikipedia.org/wiki/Symmetric_closure
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In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.
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https://en.wikipedia.org/wiki/Symmetric_decreasing_rearrangement
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In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as The expression under the limit is sometimes called the symmetric difference quotient. A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point. If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true.
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https://en.wikipedia.org/wiki/Second_symmetric_derivative
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A well-known counterexample is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient.The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. : 6 Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.
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https://en.wikipedia.org/wiki/Second_symmetric_derivative
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In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 3 , 4 } {\displaystyle \{3,4\}} is { 1 , 2 , 4 } {\displaystyle \{1,2,4\}} . The symmetric difference of the sets A and B is commonly denoted by A △ B {\displaystyle A\operatorname {\vartriangle } B} (traditionally, A Δ B {\displaystyle A\ \Delta \ B} ), A ⊕ B {\displaystyle A\oplus B} , or A ⊖ B {\displaystyle A\ominus B} . It can be viewed as a form of addition modulo 2. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
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https://en.wikipedia.org/wiki/Symmetric_difference
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In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function f ( x 1 , x 2 , … , x n ) {\displaystyle f\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)} of n variables without changing the result under certain conditions (see below). The symmetry is the assertion that the second-order partial derivatives satisfy the identity ∂ ∂ x i ( ∂ f ∂ x j ) = ∂ ∂ x j ( ∂ f ∂ x i ) {\displaystyle {\frac {\partial }{\partial x_{i}}}\left({\frac {\partial f}{\partial x_{j}}}\right)\ =\ {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial f}{\partial x_{i}}}\right)} so that they form an n × n symmetric matrix, known as the function's Hessian matrix. Sufficient conditions for the above symmetry to hold are established by a result known as Schwarz's theorem, Clairaut's theorem, or Young's theorem.In the context of partial differential equations it is called the Schwarz integrability condition.
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https://en.wikipedia.org/wiki/Schwarz's_theorem
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In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
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https://en.wikipedia.org/wiki/Symplectization
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In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + ⋯ + 1 {\displaystyle 1+1+\cdots +1} (for any finite number of terms).Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since sin ( π n ) = 0 {\displaystyle \sin({\pi n})=0} for all integers n, one also has sin ( π H ) = 0 {\displaystyle \sin({\pi H})=0} for all hyperintegers H {\displaystyle H} .
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https://en.wikipedia.org/wiki/Hyperreal_number
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The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955. Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were.
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https://en.wikipedia.org/wiki/Hyperreal_number
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This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes f ′ ( x ) = st ( f ( x + Δ x ) − f ( x ) Δ x ) {\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+\Delta x)-f(x)}{\Delta x}}\right)} for an infinitesimal Δ x {\displaystyle \Delta x} , where st(·) denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.
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https://en.wikipedia.org/wiki/Hyperreal_number
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In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation λ ( x 3 + y 3 + z 3 ) + μ x y z = 0. {\displaystyle \lambda (x^{3}+y^{3}+z^{3})+\mu xyz=0.} Each curve in the family is determined by a pair of parameter values ( λ , μ {\displaystyle \lambda ,\mu } ) (not both zero) and consists of the points in the plane whose homogeneous coordinates ( x , y , z ) {\displaystyle (x,y,z)} satisfy the equation for those parameters. Multiplying both λ {\displaystyle \lambda } and μ {\displaystyle \mu } by the same scalar does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either λ {\displaystyle \lambda } or μ {\displaystyle \mu } (but not both) to be set to zero.
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https://en.wikipedia.org/wiki/Syzygetic_pencil
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Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are equivalent in pairs giving only nine points. The family of cubics through these nine points forms the Hesse pencil.
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https://en.wikipedia.org/wiki/Syzygetic_pencil
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More generally, one can replace the complex numbers by any field containing a cube root of unity and define the Hesse pencil over this field to be the family of cubics through these nine points. The nine common points of the Hesse pencil are the inflection points of each of the cubics in the pencil. Any line that passes through at least two of these nine points passes through exactly three of them; the nine points and twelve lines through triples of points form the Hesse configuration. Every elliptic curve is birationally equivalent to a curve of the Hesse pencil; this is the Hessian form of an elliptic curve. However, the parameters ( λ , μ {\displaystyle \lambda ,\mu } ) of the Hessian form may belong to an extension field of the field of definition of the original curve.
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https://en.wikipedia.org/wiki/Syzygetic_pencil
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In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by Marden (1974). It was proved by Agol (2004) and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem. It also implies the Ahlfors measure conjecture.
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https://en.wikipedia.org/wiki/Marden_conjecture
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In mathematics, the tanc function is defined for z ≠ 0 {\displaystyle z\neq 0} as
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https://en.wikipedia.org/wiki/Tanc_function
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In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.
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https://en.wikipedia.org/wiki/Tangent_spaces
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In mathematics, the tanhc function is defined for z ≠ 0 {\displaystyle z\neq 0} asThe tanhc function is the hyperbolic analogue of the tanc function.
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https://en.wikipedia.org/wiki/Tanhc_function
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In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle k} -dimensional subspaces of V {\displaystyle V} , given a point in the Grassmannian corresponding to a k {\displaystyle k} -dimensional vector subspace W ⊆ V {\displaystyle W\subseteq V} , the fiber over W {\displaystyle W} is the subspace W {\displaystyle W} itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes.
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https://en.wikipedia.org/wiki/Universal_vector_bundle
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Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is O P n ( − 1 ) , {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1),} the dual of the hyperplane bundle or Serre's twisting sheaf O P n ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(1)} . The hyperplane bundle is the line bundle corresponding to the hyperplane (divisor) P n − 1 {\displaystyle \mathbb {P} ^{n-1}} in P n {\displaystyle \mathbb {P} ^{n}} .
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https://en.wikipedia.org/wiki/Universal_vector_bundle
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The tautological line bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space.In Michael Atiyah's "K-theory", the tautological line bundle over a complex projective space is called the standard line bundle. The sphere bundle of the standard bundle is usually called the Hopf bundle.
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https://en.wikipedia.org/wiki/Universal_vector_bundle
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(cf. Bott generator.) More generally, there are also tautological bundles on a projective bundle of a vector bundle as well as a Grassmann bundle. The older term canonical bundle has dropped out of favour, on the grounds that canonical is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the canonical class in algebraic geometry could scarcely be avoided.
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https://en.wikipedia.org/wiki/Universal_vector_bundle
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In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T ∗ Q {\displaystyle T^{*}Q} of a manifold Q . {\displaystyle Q.} In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics and Hamiltonian mechanics (on the manifold Q {\displaystyle Q} ). The exterior derivative of this form defines a symplectic form giving T ∗ Q {\displaystyle T^{*}Q} the structure of a symplectic manifold.
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https://en.wikipedia.org/wiki/Canonical_symplectic_form
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The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle.
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https://en.wikipedia.org/wiki/Canonical_symplectic_form
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To define the tautological one-form, select a coordinate chart U {\displaystyle U} on T ∗ Q {\displaystyle T^{*}Q} and a canonical coordinate system on U . {\displaystyle U.} Pick an arbitrary point m ∈ T ∗ Q .
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https://en.wikipedia.org/wiki/Canonical_symplectic_form
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{\displaystyle m\in T^{*}Q.} By definition of cotangent bundle, m = ( q , p ) , {\displaystyle m=(q,p),} where q ∈ Q {\displaystyle q\in Q} and p ∈ T q ∗ Q . {\displaystyle p\in T_{q}^{*}Q.}
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https://en.wikipedia.org/wiki/Canonical_symplectic_form
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The tautological one-form θ m: T m T ∗ Q → R {\displaystyle \theta _{m}:T_{m}T^{*}Q\to \mathbb {R} } is given by with n = dim Q {\displaystyle n=\mathop {\text{dim}} Q} and ( p 1 , … , p n ) ∈ U ⊆ R n {\displaystyle (p_{1},\ldots ,p_{n})\in U\subseteq \mathbb {R} ^{n}} being the coordinate representation of p . {\displaystyle p.} Any coordinates on T ∗ Q {\displaystyle T^{*}Q} that preserve this definition, up to a total differential (exact form), may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations.
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https://en.wikipedia.org/wiki/Canonical_symplectic_form
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The canonical symplectic form, also known as the Poincaré two-form, is given by The extension of this concept to general fibre bundles is known as the solder form. By convention, one uses the phrase "canonical form" whenever the form has a unique, canonical definition, and one uses the term "solder form", whenever an arbitrary choice has to be made. In algebraic geometry and complex geometry the term "canonical" is discouraged, due to confusion with the canonical class, and the term "tautological" is preferred, as in tautological bundle.
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https://en.wikipedia.org/wiki/Canonical_symplectic_form
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In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways n people can be connected by person-to-person telephone calls. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on n vertices, the number of permutations on n elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with n cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values (starting from n = 0)
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https://en.wikipedia.org/wiki/Telephone_number_(mathematics)
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In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
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https://en.wikipedia.org/wiki/Tensor_coalgebra
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The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.
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https://en.wikipedia.org/wiki/Tensor_coalgebra
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In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors.
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https://en.wikipedia.org/wiki/Tensor_bundle
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In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam as key concept for higher-order singular value decomposition of functions. It transforms a function (which can be given via closed formulas or neural networks, fuzzy logic, etc.) into TP function form if such a transformation is possible. If an exact transformation is not possible, then the method determines a TP function that is an approximation of the given function. Hence, the TP model transformation can provide a trade-off between approximation accuracy and complexity.A free MATLAB implementation of the TP model transformation can be downloaded at or an old version of the toolbox is available at MATLAB Central .
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https://en.wikipedia.org/wiki/TP_model_transformation
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A key underpinning of the transformation is the higher-order singular value decomposition.Besides being a transformation of functions, the TP model transformation is also a new concept in qLPV based control which plays a central role in the providing a valuable means of bridging between identification and polytopic systems theories. The TP model transformation is uniquely effective in manipulating the convex hull of polytopic forms, and, as a result has revealed and proved the fact that convex hull manipulation is a necessary and crucial step in achieving optimal solutions and decreasing conservativeness in modern LMI based control theory. Thus, although it is a transformation in a mathematical sense, it has established a conceptually new direction in control theory and has laid the ground for further new approaches towards optimality.
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https://en.wikipedia.org/wiki/TP_model_transformation
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Further details on the control theoretical aspects of the TP model transformation can be found here: TP model transformation in control theory. The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions", on which further information can be found here. It has been proved that the TP model transformation is capable of numerically reconstructing this HOSVD based canonical form.
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https://en.wikipedia.org/wiki/TP_model_transformation
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Thus, the TP model transformation can be viewed as a numerical method to compute the HOSVD of functions, which provides exact results if the given function has a TP function structure and approximative results otherwise. The TP model transformation has recently been extended in order to derive various types of convex TP functions and to manipulate them. This feature has led to new optimization approaches in qLPV system analysis and design, as described at TP model transformation in control theory.
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https://en.wikipedia.org/wiki/TP_model_transformation
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In mathematics, the tensor product V ⊗ W {\displaystyle V\otimes W} of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V × W → V ⊗ W {\displaystyle V\times W\to V\otimes W} that maps a pair ( v , w ) , v ∈ V , w ∈ W {\displaystyle (v,w),\ v\in V,w\in W} to an element of V ⊗ W {\displaystyle V\otimes W} denoted v ⊗ w . {\displaystyle v\otimes w.} An element of the form v ⊗ w {\displaystyle v\otimes w} is called the tensor product of v and w. An element of V ⊗ W {\displaystyle V\otimes W} is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor.
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https://en.wikipedia.org/wiki/Tensor_product_representation
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The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense that every element of V ⊗ W {\displaystyle V\otimes W} is a sum of elementary tensors. If bases are given for V and W, a basis of V ⊗ W {\displaystyle V\otimes W} is formed by all tensor products of a basis element of V and a basis element of W. The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V × W {\displaystyle V\times W} into another vector space Z factors uniquely through a linear map V ⊗ W → Z {\displaystyle V\otimes W\to Z} (see Universal property). Tensor products are used in many application areas, including physics and engineering. For example, in general relativity, the gravitational field is described through the metric tensor, which is a vector field of tensors, one at each point of the space-time manifold, and each belonging to the tensor product with itself of the cotangent space at the point.
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https://en.wikipedia.org/wiki/Tensor_product_representation
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In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.
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https://en.wikipedia.org/wiki/Tensor_product_of_abelian_groups
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In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative ring where 2 is invertible (that is, R has characteristic char ( R ) ≠ 2 {\displaystyle {\text{char}}(R)\neq 2} ), and if ( V 1 , q 1 ) {\displaystyle (V_{1},q_{1})} and ( V 2 , q 2 ) {\displaystyle (V_{2},q_{2})} are two quadratic spaces over R, then their tensor product ( V 1 ⊗ V 2 , q 1 ⊗ q 2 ) {\displaystyle (V_{1}\otimes V_{2},q_{1}\otimes q_{2})} is the quadratic space whose underlying R-module is the tensor product V 1 ⊗ V 2 {\displaystyle V_{1}\otimes V_{2}} of R-modules and whose quadratic form is the quadratic form associated to the tensor product of the bilinear forms associated to q 1 {\displaystyle q_{1}} and q 2 {\displaystyle q_{2}} . In particular, the form q 1 ⊗ q 2 {\displaystyle q_{1}\otimes q_{2}} satisfies ( q 1 ⊗ q 2 ) ( v 1 ⊗ v 2 ) = q 1 ( v 1 ) q 2 ( v 2 ) ∀ v 1 ∈ V 1 , v 2 ∈ V 2 {\displaystyle (q_{1}\otimes q_{2})(v_{1}\otimes v_{2})=q_{1}(v_{1})q_{2}(v_{2})\quad \forall v_{1}\in V_{1},\ v_{2}\in V_{2}} (which does uniquely characterize it however). It follows from this that if the quadratic forms are diagonalizable (which is always possible if 2 is invertible in R), i.e., q 1 ≅ ⟨ a 1 , .
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https://en.wikipedia.org/wiki/Tensor_product_of_quadratic_forms
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. . , a n ⟩ {\displaystyle q_{1}\cong \langle a_{1},...,a_{n}\rangle } q 2 ≅ ⟨ b 1 , .
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https://en.wikipedia.org/wiki/Tensor_product_of_quadratic_forms
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. . , b m ⟩ {\displaystyle q_{2}\cong \langle b_{1},...,b_{m}\rangle } then the tensor product has diagonalization q 1 ⊗ q 2 ≅ ⟨ a 1 b 1 , a 1 b 2 , .
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https://en.wikipedia.org/wiki/Tensor_product_of_quadratic_forms
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. . a 1 b m , a 2 b 1 , .
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https://en.wikipedia.org/wiki/Tensor_product_of_quadratic_forms
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. . , a 2 b m , .
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https://en.wikipedia.org/wiki/Tensor_product_of_quadratic_forms
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. . , a n b 1 , .
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https://en.wikipedia.org/wiki/Tensor_product_of_quadratic_forms
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. . a n b m ⟩ . {\displaystyle q_{1}\otimes q_{2}\cong \langle a_{1}b_{1},a_{1}b_{2},...a_{1}b_{m},a_{2}b_{1},...,a_{2}b_{m},...,a_{n}b_{1},...a_{n}b_{m}\rangle .}
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https://en.wikipedia.org/wiki/Tensor_product_of_quadratic_forms
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In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few.
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https://en.wikipedia.org/wiki/Hom_representation
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In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations.
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https://en.wikipedia.org/wiki/Tensor_product_algebra
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In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield. The tensor product of two fields is sometimes a field, and often a direct product of fields; In some cases, it can contain non-zero nilpotent elements. The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field.
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https://en.wikipedia.org/wiki/Complex_embedding
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In mathematics, the tensor representations of the general linear group are those that are obtained by taking finitely many tensor products of the fundamental representation and its dual. The irreducible factors of such a representation are also called tensor representations, and can be obtained by applying Schur functors (associated to Young tableaux). These coincide with the rational representations of the general linear group.
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https://en.wikipedia.org/wiki/Tensor_representation
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More generally, a matrix group is any subgroup of the general linear group. A tensor representation of a matrix group is any representation that is contained in a tensor representation of the general linear group. For example, the orthogonal group O(n) admits a tensor representation on the space of all trace-free symmetric tensors of order two. For orthogonal groups, the tensor representations are contrasted with the spin representations. The classical groups, like the symplectic group, have the property that all finite-dimensional representations are tensor representations (by Weyl's construction), while other representations (like the metaplectic representation) exist in infinite dimensions.
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https://en.wikipedia.org/wiki/Tensor_representation
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In mathematics, the tensor-hom adjunction is that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle \operatorname {Hom} (X,-)} form an adjoint pair: Hom ( Y ⊗ X , Z ) ≅ Hom ( Y , Hom ( X , Z ) ) . {\displaystyle \operatorname {Hom} (Y\otimes X,Z)\cong \operatorname {Hom} (Y,\operatorname {Hom} (X,Z)).} This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint.
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https://en.wikipedia.org/wiki/Tensor-hom_adjunction
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In mathematics, the tent map with parameter μ is the real-valued function fμ defined by f μ ( x ) := μ min { x , 1 − x } , {\displaystyle f_{\mu }(x):=\mu \min\{x,\,1-x\},} the name being due to the tent-like shape of the graph of fμ. For the values of the parameter μ within 0 and 2, fμ maps the unit interval into itself, thus defining a discrete-time dynamical system on it (equivalently, a recurrence relation). In particular, iterating a point x0 in gives rise to a sequence x n {\displaystyle x_{n}}: x n + 1 = f μ ( x n ) = { μ x n f o r x n < 1 2 μ ( 1 − x n ) f o r 1 2 ≤ x n {\displaystyle x_{n+1}=f_{\mu }(x_{n})={\begin{cases}\mu x_{n}&\mathrm {for} ~~x_{n}<{\frac {1}{2}}\\\mu (1-x_{n})&\mathrm {for} ~~{\frac {1}{2}}\leq x_{n}\end{cases}}} where μ is a positive real constant.
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https://en.wikipedia.org/wiki/Tent_map
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Choosing for instance the parameter μ = 2, the effect of the function fμ may be viewed as the result of the operation of folding the unit interval in two, then stretching the resulting interval to get again the interval . Iterating the procedure, any point x0 of the interval assumes new subsequent positions as described above, generating a sequence xn in . The μ = 2 {\displaystyle \mu =2} case of the tent map is a non-linear transformation of both the bit shift map and the r = 4 case of the logistic map.
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https://en.wikipedia.org/wiki/Tent_map
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In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X {\displaystyle X} is a set, "almost all elements of X {\displaystyle X} " means "all elements of X {\displaystyle X} but those in a negligible subset of X {\displaystyle X} ". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of X {\displaystyle X} " means "a negligible quantity of elements of X {\displaystyle X} ".
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https://en.wikipedia.org/wiki/Almost_all
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In mathematics, the term "characteristic function" can refer to any of several distinct concepts: The indicator function of a subset, that is the function 1 A: X → { 0 , 1 } , {\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},} which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.There is an indicator function for affine varieties over a finite field: given a finite set of functions f α ∈ F q {\displaystyle f_{\alpha }\in \mathbb {F} _{q}} let V = { x ∈ F q n: f α ( x ) = 0 } {\displaystyle V=\left\{x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\right\}} be their vanishing locus. Then, the function P ( x ) = ∏ ( 1 − f α ( x ) q − 1 ) {\textstyle P(x)=\prod \left(1-f_{\alpha }(x)^{q-1}\right)} acts as an indicator function for V {\displaystyle V} . If x ∈ V {\displaystyle x\in V} then P ( x ) = 1 {\displaystyle P(x)=1} , otherwise, for some f α {\displaystyle f_{\alpha }} , we have f α ( x ) ≠ 0 {\displaystyle f_{\alpha }(x)\neq 0} , which implies that f α ( x ) q − 1 = 1 {\displaystyle f_{\alpha }(x)^{q-1}=1} , hence P ( x ) = 0 {\displaystyle P(x)=0} . The characteristic function in convex analysis, closely related to the indicator function of a set: χ A ( x ) := { 0 , x ∈ A ; + ∞ , x ∉ A .
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https://en.wikipedia.org/wiki/Characteristic_function
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{\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}} In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: φ X ( t ) = E ( e i t X ) , {\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),} where E {\displaystyle \operatorname {E} } denotes expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors. The characteristic function of a cooperative game in game theory.
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https://en.wikipedia.org/wiki/Characteristic_function
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The characteristic polynomial in linear algebra. The characteristic state function in statistical mechanics. The Euler characteristic, a topological invariant.
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https://en.wikipedia.org/wiki/Characteristic_function
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The receiver operating characteristic in statistical decision theory. The point characteristic function in statistics. == References ==
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https://en.wikipedia.org/wiki/Characteristic_function
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In mathematics, the term "graded" has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: An algebraic structure X {\displaystyle X} is said to be I {\displaystyle I} -graded for an index set I {\displaystyle I} if it has a gradation or grading, i.e. a decomposition into a direct sum X = ⨁ i ∈ I X i {\textstyle X=\bigoplus _{i\in I}X_{i}} of structures; the elements of X i {\displaystyle X_{i}} are said to be "homogeneous of degree i ". The index set I {\displaystyle I} is most commonly N {\displaystyle \mathbb {N} } or Z {\displaystyle \mathbb {Z} } , and may be required to have extra structure depending on the type of X {\displaystyle X} . Grading by Z 2 {\displaystyle \mathbb {Z} _{2}} (i.e. Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ) is also important; see e.g. signed set (the Z 2 {\displaystyle \mathbb {Z} _{2}} -graded sets). The trivial ( Z {\displaystyle \mathbb {Z} } - or N {\displaystyle \mathbb {N} } -) gradation has X 0 = X , X i = 0 {\displaystyle X_{0}=X,X_{i}=0} for i ≠ 0 {\displaystyle i\neq 0} and a suitable trivial structure 0 {\displaystyle 0} .
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called "bidegrees" (e.g. see Spectral sequence). A I {\displaystyle I} -graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = ⨁ i ∈ I V i {\textstyle V=\bigoplus _{i\in I}V_{i}} of spaces. A graded linear map is a map between graded vector spaces respecting their gradations.
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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A graded ring is a ring that is a direct sum of additive abelian groups R i {\displaystyle R_{i}} such that R i R j ⊆ R i + j {\displaystyle R_{i}R_{j}\subseteq R_{i+j}} , with i {\displaystyle i} taken from some monoid, usually N {\displaystyle \mathbb {N} } or Z {\displaystyle \mathbb {Z} } , or semigroup (for a ring without identity). The associated graded ring of a commutative ring R {\displaystyle R} with respect to a proper ideal I {\displaystyle I} is gr I R = ⨁ n ∈ N I n / I n + 1 {\textstyle \operatorname {gr} _{I}R=\bigoplus _{n\in \mathbb {N} }I^{n}/I^{n+1}} . A graded module is left module M {\displaystyle M} over a graded ring that is a direct sum ⨁ i ∈ I M i {\textstyle \bigoplus _{i\in I}M_{i}} of modules satisfying R i M j ⊆ M i + j {\displaystyle R_{i}M_{j}\subseteq M_{i+j}} .
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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The associated graded module of an R {\displaystyle R} -module M {\displaystyle M} with respect to a proper ideal I {\displaystyle I} is gr I M = ⨁ n ∈ N I n M / I n + 1 M {\textstyle \operatorname {gr} _{I}M=\bigoplus _{n\in \mathbb {N} }I^{n}M/I^{n+1}M} . A differential graded module, differential graded Z {\displaystyle \mathbb {Z} } -module or DG-module is a graded module M {\displaystyle M} with a differential d: M → M: M i → M i + 1 {\displaystyle d\colon M\to M\colon M_{i}\to M_{i+1}} making M {\displaystyle M} a chain complex, i.e. d ∘ d = 0 {\displaystyle d\circ d=0} . A graded algebra is an algebra A {\displaystyle A} over a ring R {\displaystyle R} that is graded as a ring; if R {\displaystyle R} is graded we also require A i R j ⊆ A i + j ⊇ R i A j {\displaystyle A_{i}R_{j}\subseteq A_{i+j}\supseteq R_{i}A_{j}} .
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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The graded Leibniz rule for a map d: A → A {\displaystyle d\colon A\to A} on a graded algebra A {\displaystyle A} specifies that d ( a ⋅ b ) = ( d a ) ⋅ b + ( − 1 ) | a | a ⋅ ( d b ) {\displaystyle d(a\cdot b)=(da)\cdot b+(-1)^{|a|}a\cdot (db)} . A differential graded algebra, DG-algebra or DGAlgebra is a graded algebra that is a differential graded module whose differential obeys the graded Leibniz rule. A homogeneous derivation on a graded algebra A is a homogeneous linear map of grade d = |D| on A such that D ( a b ) = D ( a ) b + ε | a | | D | a D ( b ) , ε = ± 1 {\displaystyle D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b),\varepsilon =\pm 1} acting on homogeneous elements of A. A graded derivation is a sum of homogeneous derivations with the same ε {\displaystyle \varepsilon } .
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see Differential graded algebra). A superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra.
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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A graded-commutative superalgebra satisfies the "supercommutative" law y x = ( − 1 ) | x | | y | x y . {\displaystyle yx=(-1)^{|x||y|}xy.}
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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for homogeneous x,y, where | a | {\displaystyle |a|} represents the "parity" of a {\displaystyle a} , i.e. 0 or 1 depending on the component in which it lies. CDGA may refer to the category of augmented differential graded commutative algebras. A graded Lie algebra is a Lie algebra that is graded as a vector space by a gradation compatible with its Lie bracket.
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed. A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super Z 2 {\displaystyle \mathbb {Z} _{2}} -gradation. A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map : L i ⊗ L j → L i + j {\displaystyle \colon L_{i}\otimes L_{j}\to L_{i+j}} and a differential d: L i → L i − 1 {\displaystyle d\colon L_{i}\to L_{i-1}} satisfying = ( − 1 ) | x | | y | + 1 , {\displaystyle =(-1)^{|x||y|+1},} for any homogeneous elements x, y in L, the "graded Jacobi identity" and the graded Leibniz rule.
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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The Graded Brauer group is a synonym for the Brauer–Wall group B W ( F ) {\displaystyle BW(F)} classifying finite-dimensional graded central division algebras over the field F. An A {\displaystyle {\mathcal {A}}} -graded category for a category A {\displaystyle {\mathcal {A}}} is a category C {\displaystyle {\mathcal {C}}} together with a functor F: C → A {\displaystyle F\colon {\mathcal {C}}\rightarrow {\mathcal {A}}} . A differential graded category or DG category is a category whose morphism sets form differential graded Z {\displaystyle \mathbb {Z} } -modules. Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on Graded function Graded vector fields Graded exterior forms Graded differential geometry Graded differential calculusIn other areas of mathematics: Functionally graded elements are used in finite element analysis. A graded poset is a poset P {\displaystyle P} with a rank function ρ: P → N {\displaystyle \rho \colon P\to \mathbb {N} } compatible with the ordering (i.e. ρ ( x ) < ρ ( y ) ⟹ x < y {\displaystyle \rho (x)<\rho (y)\implies x
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https://en.wikipedia.org/wiki/Graded_(mathematics)
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In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure. These include, among others: Empty set: the set containing no or null members Trivial group: the mathematical group containing only the identity element Trivial ring: a ring defined on a singleton set"Trivial" can also be used to describe solutions to an equation that have a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the trivial solutions. For example, consider the differential equation where y = y ( x ) {\displaystyle y=y(x)} is a function whose derivative is y ′ {\displaystyle y'} .
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https://en.wikipedia.org/wiki/Trivial_solution
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The trivial solution is the zero function while a nontrivial solution is the exponential function The differential equation f ″ ( x ) = − λ f ( x ) {\displaystyle f''(x)=-\lambda f(x)} with boundary conditions f ( 0 ) = f ( L ) = 0 {\displaystyle f(0)=f(L)=0} is important in mathematics and physics, as it could be used to describe a particle in a box in quantum mechanics, or a standing wave on a string. It always includes the solution f ( x ) = 0 {\displaystyle f(x)=0} , which is considered obvious and hence is called the "trivial" solution. In some cases, there may be other solutions (sinusoids), which are called "nontrivial" solutions.Similarly, mathematicians often describe Fermat's last theorem as asserting that there are no nontrivial integer solutions to the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} , where n is greater than 2. Clearly, there are some solutions to the equation. For example, a = b = c = 0 {\displaystyle a=b=c=0} is a solution for any n, but such solutions are obvious and obtainable with little effort, and hence "trivial".
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https://en.wikipedia.org/wiki/Trivial_solution
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In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like displacement or velocity, which have components that relate directly to the three dimensions of space, or relativistically, to the four of spacetime. Such vectors are typically denoted with over arrows ( r → {\displaystyle {\vec {r}}} ), boldface ( p {\displaystyle \mathbf {p} } ) or indices ( v μ {\displaystyle v^{\mu }} ). In quantum mechanics, a quantum state is typically represented as an element of a complex Hilbert space, for example, the infinite-dimensional vector space of all possible wavefunctions (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically.
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https://en.wikipedia.org/wiki/Braket_notation
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To distinguish this type of vector from those described above, it is common and useful in physics to denote an element ϕ {\displaystyle \phi } of an abstract complex vector space as a ket | ϕ ⟩ {\displaystyle |\phi \rangle } , to refer to it as a "ket" rather than as a vector, and to pronounce it "ket- ϕ {\displaystyle \phi } " or "ket-A" for |A⟩. Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the | ⟩ {\displaystyle |\ \rangle } making clear that the label indicates a vector in vector space.
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https://en.wikipedia.org/wiki/Braket_notation
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In other words, the symbol "|A⟩" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "A" by itself does not. For example, |1⟩ + |2⟩ is not necessarily equal to |3⟩. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as x ^ {\displaystyle {\hat {x}}} , p ^ {\displaystyle {\hat {p}}} , L ^ z {\displaystyle {\hat {L}}_{z}} , etc.
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https://en.wikipedia.org/wiki/Braket_notation
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In mathematics, the term Beurling algebra is used for different algebras introduced by Arne Beurling (1949), usually it is an algebra of periodic functions with Fourier series f ( x ) = ∑ a n e i n x {\displaystyle f(x)=\sum a_{n}e^{inx}} Example We may consider the algebra of those functions f where the majorants c k = sup | n | ≥ k | a n | {\displaystyle c_{k}=\sup _{|n|\geq k}|a_{n}|} of the Fourier coefficients an are summable. In other words ∑ k ≥ 0 c k < ∞ . {\displaystyle \sum _{k\geq 0}c_{k}<\infty .} Example We may consider a weight function w on Z {\displaystyle \mathbb {Z} } such that w ( m + n ) ≤ w ( m ) w ( n ) , w ( 0 ) = 1 {\displaystyle w(m+n)\leq w(m)w(n),\quad w(0)=1} in which case A w ( T ) = { f: f ( t ) = ∑ n a n e i n t , ‖ f ‖ w = ∑ n | a n | w ( n ) < ∞ } ( ∼ ℓ w 1 ( Z ) ) {\displaystyle A_{w}(\mathbb {T} )=\{f:f(t)=\sum _{n}a_{n}e^{int},\,\|f\|_{w}=\sum _{n}|a_{n}|w(n)<\infty \}\,(\sim \ell _{w}^{1}(\mathbb {Z} ))} is a unitary commutative Banach algebra. These algebras are closely related to the Wiener algebra.
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https://en.wikipedia.org/wiki/Beurling_algebra
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In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.
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https://en.wikipedia.org/wiki/Cartan_matrices
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In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generalizations of this. The original setting appearing in Hilbert's twenty-first problem was for the Riemann sphere, where it was about the existence of systems of linear regular differential equations with prescribed monodromy representations. First the Riemann sphere may be replaced by an arbitrary Riemann surface and then, in higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1.
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https://en.wikipedia.org/wiki/Riemann-Hilbert_correspondence
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