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In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues related to classification are the following. The equivalence problem is "given two objects, determine if they are equivalent".
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https://en.wikipedia.org/wiki/Classification_theorem
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A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it. A computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the equivalence problem. A canonical form solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class.There exist many classification theorems in mathematics, as described below.
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https://en.wikipedia.org/wiki/Classification_theorem
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In mathematics, a classifying topos for some sort of structure is a topos T such that there is a natural equivalence between geometric morphisms from a cocomplete topos E to T and the category of models for the structure in E.
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https://en.wikipedia.org/wiki/Classifying_topos
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In mathematics, a clean ring is a ring in which every element can be written as the sum of a unit and an idempotent. A ring is a local ring if and only if it is clean and has no idempotents other than 0 and 1. The endomorphism ring of a continuous module is a clean ring.
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https://en.wikipedia.org/wiki/Clean_ring
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Every clean ring is an exchange ring. A matrix ring over a clean ring is itself clean. == References ==
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https://en.wikipedia.org/wiki/Clean_ring
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In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components.
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https://en.wikipedia.org/wiki/Compact_surface
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In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a boundary component of M.
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https://en.wikipedia.org/wiki/Boundary_parallel
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In mathematics, a closure operator on a set S is a function cl: P ( S ) → P ( S ) {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal {P}}(S)} from the power set of S to itself that satisfies the following conditions for all sets X , Y ⊆ S {\displaystyle X,Y\subseteq S} Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families". A set together with a closure operator on it is sometimes called a closure space. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in topology.
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https://en.wikipedia.org/wiki/Closure_space
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In mathematics, a cobordism (W, M, M−) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M−, is called a semi-s-cobordism if (and only if) the inclusion M ↪ W {\displaystyle M\hookrightarrow W} is a simple homotopy equivalence (as in an s-cobordism), with no further requirement on the inclusion M − ↪ W {\displaystyle M^{-}\hookrightarrow W} (not even being a homotopy equivalence).
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https://en.wikipedia.org/wiki/Semi-s-cobordism
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In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. Since the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then one says Y is cofinite.
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https://en.wikipedia.org/wiki/Cocountability
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In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. For example, in the polynomial with variables x {\displaystyle x} and y {\displaystyle y} , the first two terms have the coefficients 7 and −3. The third term 1.5 is the constant coefficient. In the final term, the coefficient is 1 and is not explicitly written.
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https://en.wikipedia.org/wiki/Leading_coefficient
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In many scenarios, coefficients are numbers (as is the case for each term of the previous example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish between symbols representing variables and symbols representing parameters. Following René Descartes, the variables are often denoted by x, y, ..., and the parameters by a, b, c, ..., but this is not always the case.
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https://en.wikipedia.org/wiki/Leading_coefficient
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For example, if y is considered a parameter in the above expression, then the coefficient of x would be −3y, and the constant coefficient (with respect to x) would be 1.5 + y. When one writes it is generally assumed that x is the only variable, and that a, b and c are parameters; thus the constant coefficient is c in this case. Any polynomial in a single variable x can be written as for some nonnegative integer k {\displaystyle k} , where a k , … , a 1 , a 0 {\displaystyle a_{k},\dotsc ,a_{1},a_{0}} are the coefficients. This includes the possibility that some terms have coefficient 0; for example, in x 3 − 2 x + 1 {\displaystyle x^{3}-2x+1} , the coefficient of x 2 {\displaystyle x^{2}} is 0, and the term 0 x 2 {\displaystyle 0x^{2}} does not appear explicitly.
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https://en.wikipedia.org/wiki/Leading_coefficient
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For the largest i {\displaystyle i} such that a i ≠ 0 {\displaystyle a_{i}\neq 0} (if any), a i {\displaystyle a_{i}} is called the leading coefficient of the polynomial. For example, the leading coefficient of the polynomial is 4. This can be generalised to multivariate polynomials with respect to a monomial order, see Gröbner basis § Leading term, coefficient and monomial.
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https://en.wikipedia.org/wiki/Leading_coefficient
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In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression. It may be a number (dimensionless), in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression (including variables such as a, b and c).
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https://en.wikipedia.org/wiki/Leading_coefficient
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When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter.For example, the polynomial 2 x 2 − x + 3 {\displaystyle 2x^{2}-x+3} has coefficients 2, −1, and 3, and the powers of the variable x {\displaystyle x} in the polynomial a x 2 + b x + c {\displaystyle ax^{2}+bx+c} have coefficient parameters a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} . The constant coefficient, also known as constant term or simply constant is the quantity not attached to variables in an expression. For example, the constant coefficients of the expressions above are the number 3 and the parameter c, respectively.
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https://en.wikipedia.org/wiki/Leading_coefficient
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The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and a, respectively. In the context of differential equations, an equation can often be written as equating to zero a polynomial in the unknown functions and their derivatives.
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https://en.wikipedia.org/wiki/Leading_coefficient
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In this case, the coefficients of the differential equation are the coefficients of this polynomial, and are generally non-constant functions. A coefficient is a constant coefficient when it is a constant function. For avoiding confusion, the coefficient that is not attached to unknown functions and their derivative is generally called the constant term rather the constant coefficient. In particular, in a linear differential equation with constant coefficient, the constant term is generally not supposed to be a constant function.
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https://en.wikipedia.org/wiki/Leading_coefficient
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In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context different exact definitions of this idea are in use.
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https://en.wikipedia.org/wiki/Coercive_operator
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In mathematics, a cofinite subset of a set X {\displaystyle X} is a subset A {\displaystyle A} whose complement in X {\displaystyle X} is a finite set. In other words, A {\displaystyle A} contains all but finitely many elements of X . {\displaystyle X.}
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https://en.wikipedia.org/wiki/Cofinite_set
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If the complement is not finite, but is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum. This use of the prefix "co" to describe a property possessed by a set's complement is consistent with its use in other terms such as "comeagre set".
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https://en.wikipedia.org/wiki/Cofinite_set
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In mathematics, a coframe or coframe field on a smooth manifold M {\displaystyle M} is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M {\displaystyle M} , one has a natural map from v k: ⨁ k T ∗ M → ⋀ k T ∗ M {\displaystyle v_{k}:\bigoplus ^{k}T^{*}M\to \bigwedge ^{k}T^{*}M} , given by v k: ( ρ 1 , … , ρ k ) ↦ ρ 1 ∧ … ∧ ρ k {\displaystyle v_{k}:(\rho _{1},\ldots ,\rho _{k})\mapsto \rho _{1}\wedge \ldots \wedge \rho _{k}} . If M {\displaystyle M} is n {\displaystyle n} dimensional a coframe is given by a section σ {\displaystyle \sigma } of ⨁ n T ∗ M {\displaystyle \bigoplus ^{n}T^{*}M} such that v n ∘ σ ≠ 0 {\displaystyle v_{n}\circ \sigma \neq 0} . The inverse image under v n {\displaystyle v_{n}} of the complement of the zero section of ⋀ n T ∗ M {\displaystyle \bigwedge ^{n}T^{*}M} forms a G L ( n ) {\displaystyle GL(n)} principal bundle over M {\displaystyle M} , which is called the coframe bundle.
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https://en.wikipedia.org/wiki/Coframe
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In mathematics, a coherent topos is a topos generated by a collection of quasi-compact quasi-separated objects closed under finite products.
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https://en.wikipedia.org/wiki/Coherent_topos
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In mathematics, a cohomological invariant of an algebraic group G over a field is an invariant of forms of G taking values in a Galois cohomology group.
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https://en.wikipedia.org/wiki/Cohomological_invariant
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In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain having the same image. Formally, given two functions f , g: X → Y {\displaystyle f,g\colon X\rightarrow Y} we say that a point x in X is a coincidence point of f and g if f(x) = g(x).Coincidence theory (the study of coincidence points) is, in most settings, a generalization of fixed point theory, the study of points x with f(x) = x. Fixed point theory is the special case obtained from the above by letting X = Y and taking g to be the identity function. Just as fixed point theory has its fixed-point theorems, there are theorems that guarantee the existence of coincidence points for pairs of functions.
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https://en.wikipedia.org/wiki/Coincidence_point
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Notable among them, in the setting of manifolds, is the Lefschetz coincidence theorem, which is typically known only in its special case formulation for fixed points.Coincidence points, like fixed points, are today studied using many tools from mathematical analysis and topology. An equaliser is a generalization of the coincidence set. == References ==
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https://en.wikipedia.org/wiki/Coincidence_point
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In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963.The collapsing algebra of λω is a complete Boolean algebra with at least λ elements but generated by a countable number of elements. As the size of countably generated complete Boolean algebras is unbounded, this shows that there is no free complete Boolean algebra on a countable number of elements.
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https://en.wikipedia.org/wiki/Collapsing_algebra
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In mathematics, a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients. A collection of numbers which is not rationally independent is called rationally dependent. For instance we have the following example. independent 3 , 8 ⏞ , 1 + 2 ⏟ dependent {\displaystyle {\begin{matrix}{\mbox{independent}}\qquad \\\underbrace {\overbrace {3,\quad {\sqrt {8}}\quad } ,1+{\sqrt {2}}} \\{\mbox{dependent}}\\\end{matrix}}} Because if we let x = 3 , y = 8 {\displaystyle x=3,y={\sqrt {8}}} , then 1 + 2 = 1 3 x + 1 2 y {\displaystyle 1+{\sqrt {2}}={\frac {1}{3}}x+{\frac {1}{2}}y} .
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https://en.wikipedia.org/wiki/Rational_dependence
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In mathematics, a collection or family U {\displaystyle {\mathcal {U}}} of subsets of a topological space X {\displaystyle X} is said to be point-finite if every point of X {\displaystyle X} lies in only finitely many members of U . {\displaystyle {\mathcal {U}}.} A metacompact space is a topological space in which every open cover admits a point-finite open refinement.
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https://en.wikipedia.org/wiki/Point_finite
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Every locally finite collection of subsets of a topological space is also point-finite. A topological space in which every open cover admits a locally finite open refinement is called a paracompact space. Every paracompact space is therefore metacompact.
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https://en.wikipedia.org/wiki/Point_finite
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In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the given equation at the collocation points.
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https://en.wikipedia.org/wiki/Collocation_point
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In mathematics, a colored matroid is a matroid whose elements are labeled from a set of colors, which can be any set that suits the purpose, for instance the set of the first n positive integers, or the sign set {+, −}. The interest in colored matroids is through their invariants, especially the colored Tutte polynomial, which generalizes the Tutte polynomial of a signed graph of Kauffman (1989).There has also been study of optimization problems on matroids where the objective function of the optimization depends on the set of colors chosen as part of a matroid basis.
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https://en.wikipedia.org/wiki/Colored_matroid
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In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it's defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number. Formally, a number n is said to be colossally abundant if there is an ε > 0 such that for all k > 1, σ ( n ) n 1 + ε ≥ σ ( k ) k 1 + ε {\displaystyle {\frac {\sigma (n)}{n^{1+\varepsilon }}}\geq {\frac {\sigma (k)}{k^{1+\varepsilon }}}} where σ denotes the sum-of-divisors function.The first 15 colossally abundant numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A004490 in the OEIS) are also the first 15 superior highly composite numbers, but neither set is a subset of the other.
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https://en.wikipedia.org/wiki/Colossally_abundant_number
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In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.)
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https://en.wikipedia.org/wiki/Combination
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If the set has n elements, the number of k-combinations, denoted by C ( n , k ) {\displaystyle C(n,k)} or C k n {\displaystyle C_{k}^{n}} , is equal to the binomial coefficient which can be written using factorials as n ! k ! ( n − k ) !
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https://en.wikipedia.org/wiki/Combination
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{\displaystyle \textstyle {\frac {n!}{k!(n-k)!}}} whenever k ≤ n {\displaystyle k\leq n} , and which is zero when k > n {\displaystyle k>n} . This formula can be derived from the fact that each k-combination of a set S of n members has k !
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https://en.wikipedia.org/wiki/Combination
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{\displaystyle k!} permutations so P k n = C k n × k ! {\displaystyle P_{k}^{n}=C_{k}^{n}\times k!}
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https://en.wikipedia.org/wiki/Combination
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or C k n = P k n / k ! {\displaystyle C_{k}^{n}=P_{k}^{n}/k!} .
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https://en.wikipedia.org/wiki/Combination
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The set of all k-combinations of a set S is often denoted by ( S k ) {\displaystyle \textstyle {\binom {S}{k}}} . A combination is a combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-combination with repetition, k-multiset, or k-selection, are often used.
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https://en.wikipedia.org/wiki/Combination
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If, in the above example, it were possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears. Although the set of three fruits was small enough to write a complete list of combinations, this becomes impractical as the size of the set increases. For example, a poker hand can be described as a 5-combination (k = 5) of cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.
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https://en.wikipedia.org/wiki/Combination
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In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size.
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https://en.wikipedia.org/wiki/Combinatorial_class
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In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the problem. Combinatorial explosion is sometimes used to justify the intractability of certain problems. Examples of such problems include certain mathematical functions, the analysis of some puzzles and games, and some pathological examples which can be modelled as the Ackermann function.
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https://en.wikipedia.org/wiki/Combinatorial_explosion_(communication)
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In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later.
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https://en.wikipedia.org/wiki/Comma_category
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Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13).
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https://en.wikipedia.org/wiki/Comma_category
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In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a measurable transformation preserving a probability measure. Another important application is in the theory of unitary representations of unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s by Jacques Dixmier as part of the theory of Hilbert algebras. It was not until the late 1960s, prompted partly by results in algebraic quantum field theory and quantum statistical mechanics due to the school of Rudolf Haag, that the more general non-tracial Tomita–Takesaki theory was developed, heralding a new era in the theory of von Neumann algebras.
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https://en.wikipedia.org/wiki/Commutation_theory
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In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains p = p0 ⊂ p1 ⊂ ... ⊂ pn = qof prime ideals are contained in maximal strictly increasing chains from p to q of the same (finite) length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain n is usually the difference in dimensions. A ring is called universally catenary if all finitely generated algebras over it are catenary rings.
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https://en.wikipedia.org/wiki/Universally_catenary
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The word 'catenary' is derived from the Latin word catena, which means "chain". There is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings
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https://en.wikipedia.org/wiki/Universally_catenary
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In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
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https://en.wikipedia.org/wiki/Commutative_rings
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In mathematics, a commutativity constraint γ {\displaystyle \gamma } on a monoidal category C {\displaystyle {\mathcal {C}}} is a choice of isomorphism γ A , B: A ⊗ B → B ⊗ A {\displaystyle \gamma _{A,B}:A\otimes B\rightarrow B\otimes A} for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have A ⊗ B ≅ B ⊗ A {\displaystyle A\otimes B\cong B\otimes A} for all pairs of objects A , B ∈ C {\displaystyle A,B\in {\mathcal {C}}} . A braided monoidal category is a monoidal category C {\displaystyle {\mathcal {C}}} equipped with a braiding—that is, a commutativity constraint γ {\displaystyle \gamma } that satisfies axioms including the hexagon identities defined below. The term braided references the fact that the braid group plays an important role in the theory of braided monoidal categories.
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https://en.wikipedia.org/wiki/Braided_monoidal_category
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Partly for this reason, braided monoidal categories and other topics are related in the theory of knot invariants. Alternatively, a braided monoidal category can be seen as a tricategory with one 0-cell and one 1-cell. Braided monoidal categories were introduced by André Joyal and Ross Street in a 1986 preprint. A modified version of this paper was published in 1993.
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https://en.wikipedia.org/wiki/Braided_monoidal_category
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In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.
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https://en.wikipedia.org/wiki/Comodule
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In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces.
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https://en.wikipedia.org/wiki/Compact_group
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In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of "continuous complex-valued functions" on a compact quantum group. The basic motivation for this theory comes from the following analogy. The space of complex-valued functions on a compact Hausdorff topological space forms a commutative C*-algebra.
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https://en.wikipedia.org/wiki/Compact_quantum_group
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On the other hand, by the Gelfand Theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism. S. L. Woronowicz introduced the important concept of compact matrix quantum groups, which he initially called compact pseudogroups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry.
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https://en.wikipedia.org/wiki/Compact_quantum_group
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In mathematics, a compact semigroup is a semigroup in which the sets of solutions to equations can be described by finite sets of equations. The term "compact" here does not refer to any topology on the semigroup. Let S be a semigroup and X a finite set of letters.
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https://en.wikipedia.org/wiki/Compact_semigroup
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A system of equations is a subset E of the Cartesian product X∗ × X∗ of the free monoid (finite strings) over X with itself. The system E is satisfiable in S if there is a map f from X to S, which extends to a semigroup morphism f from X+ to S, such that for all (u,v) in E we have f(u) = f(v) in S. Such an f is a solution, or satisfying assignment, for the system E.Two systems of equations are equivalent if they have the same set of satisfying assignments. A system of equations if independent if it is not equivalent to a proper subset of itself. A semigroup is compact if every independent system of equations is finite.
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https://en.wikipedia.org/wiki/Compact_semigroup
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In mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. This should not be confused with the unrelated notion (widely used in algebraic topology) of a compactly generated space -- one whose topology is generated (in a suitable sense) by its compact subspaces.
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https://en.wikipedia.org/wiki/Compactly_generated_group
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In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebra containing A such that every element is the supremum of some subset of A. As a partially ordered set, this completion of A is the Dedekind–MacNeille completion. More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum.
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https://en.wikipedia.org/wiki/Complete_Boolean_algebra
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In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F: J → C (where J is small) has a limit in C. Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete. The existence of all limits (even when J is a proper class) is too strong to be practically relevant.
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https://en.wikipedia.org/wiki/Finitely_complete_category
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Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category J). Dually, a category is finitely cocomplete if all finite colimits exist.
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https://en.wikipedia.org/wiki/Finitely_complete_category
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In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers).
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https://en.wikipedia.org/wiki/Complete_field
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In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a conditionally complete lattice. Specifically, every non-empty finite lattice is complete.
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https://en.wikipedia.org/wiki/Complete_lattices
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Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. Complete lattices must not be confused with complete partial orders (cpos), which constitute a strictly more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales).
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https://en.wikipedia.org/wiki/Complete_lattices
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In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, you can follow a "straight" line indefinitely along any direction. More formally, the exponential map at point p, is defined on TpM, the entire tangent space at p. Equivalently, consider a maximal geodesic ℓ: I → M {\displaystyle \ell \colon I\to M} . Here I {\displaystyle I} is an open interval of R {\displaystyle \mathbb {R} } , and, because geodesics are parameterized with "constant speed", it is uniquely defined up to transversality. Because I {\displaystyle I} is maximal, ℓ {\displaystyle \ell } maps the ends of I {\displaystyle I} to points of ∂M, and the length of I {\displaystyle I} measures the distance between those points. A manifold is geodesically complete if for any such geodesic ℓ {\displaystyle \ell } , we have that I = ( − ∞ , ∞ ) {\displaystyle I=(-\infty ,\infty )} .
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https://en.wikipedia.org/wiki/Geodesically_complete
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In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if S ⊆ N ∈ Σ and μ ( N ) = 0 ⇒ S ∈ Σ . {\displaystyle S\subseteq N\in \Sigma {\mbox{ and }}\mu (N)=0\ \Rightarrow \ S\in \Sigma .}
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https://en.wikipedia.org/wiki/Complete_measure
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In mathematics, a complete set of invariants for a classification problem is a collection of maps f i: X → Y i {\displaystyle f_{i}:X\to Y_{i}} (where X {\displaystyle X} is the collection of objects being classified, up to some equivalence relation ∼ {\displaystyle \sim } , and the Y i {\displaystyle Y_{i}} are some sets), such that x ∼ x ′ {\displaystyle x\sim x'} if and only if f i ( x ) = f i ( x ′ ) {\displaystyle f_{i}(x)=f_{i}(x')} for all i {\displaystyle i} . In words, such that two objects are equivalent if and only if all invariants are equal.Symbolically, a complete set of invariants is a collection of maps such that ( ∏ f i ): ( X / ∼ ) → ( ∏ Y i ) {\displaystyle \left(\prod f_{i}\right):(X/\sim )\to \left(\prod Y_{i}\right)} is injective. As invariants are, by definition, equal on equivalent objects, equality of invariants is a necessary condition for equivalence; a complete set of invariants is a set such that equality of these is also sufficient for equivalence. In the context of a group action, this may be stated as: invariants are functions of coinvariants (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants).
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https://en.wikipedia.org/wiki/Complete_set_of_invariants
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In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric d on X such that (X, d) is a complete metric space and d induces the topology T. The term topologically complete space is employed by some authors as a synonym for completely metrizable space, but sometimes also used for other classes of topological spaces, like completely uniformizable spaces or Čech-complete spaces.
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https://en.wikipedia.org/wiki/Completely_metrizable
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In mathematics, a completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. The class of completely regular semigroups forms an important subclass of the class of regular semigroups, the class of inverse semigroups being another such subclass. Alfred H. Clifford was the first to publish a major paper on completely regular semigroups though he used the terminology "semigroups admitting relative inverses" to refer to such semigroups. The name "completely regular semigroup" stems from Lyapin's book on semigroups.
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https://en.wikipedia.org/wiki/Completely_regular_semigroup
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In the Russian literature, completely regular semigroups are often called "Clifford semigroups". In the English literature, the name "Clifford semigroup" is used synonymously to "inverse Clifford semigroup", and refers to a completely regular inverse semigroup.
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https://en.wikipedia.org/wiki/Completely_regular_semigroup
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In a completely regular semigroup, each Green H-class is a group and the semigroup is the union of these groups. Hence completely regular semigroups are also referred to as "unions of groups". Epigroups generalize this notion and their class includes all completely regular semigroups.
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https://en.wikipedia.org/wiki/Completely_regular_semigroup
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In mathematics, a complex Hadamard matrix H of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-type H(q, N) if all its elements are powers of q-th root of unity, ( H j k ) q = 1 f o r j , k = 1 , 2 , … , N . {\displaystyle (H_{jk})^{q}=1{\quad {\rm {for\quad }}}j,k=1,2,\dots ,N.}
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https://en.wikipedia.org/wiki/Butson-type_Hadamard_matrix
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In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle {\overline {\mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector space but with i = − 1 {\displaystyle i={\sqrt {-1}}} acting as − i {\displaystyle -i} instead. As a real Lie algebra, a complex Lie algebra g {\displaystyle {\mathfrak {g}}} is trivially isomorphic to its conjugate. A complex Lie algebra is isomorphic to its conjugate if and only if it admits a real form (and is said to be defined over the real numbers).
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https://en.wikipedia.org/wiki/Complex_Lie_algebra
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In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface x 4 + y 4 + z 4 + w 4 = 0 {\displaystyle x^{4}+y^{4}+z^{4}+w^{4}=0} in complex projective 3-space.
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https://en.wikipedia.org/wiki/K3_surface
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Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable).
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https://en.wikipedia.org/wiki/K3_surface
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K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties, and yet where a substantial understanding is possible. A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds. K3 surfaces have been applied to Kac–Moody algebras, mirror symmetry and string theory. It can be useful to think of complex algebraic K3 surfaces as part of the broader family of complex analytic K3 surfaces. Many other types of algebraic varieties do not have such non-algebraic deformations.
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https://en.wikipedia.org/wiki/K3_surface
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In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the basis for much of algebraic geometry, Kähler geometry, and Hodge theory. Over non-complex manifolds, they also play a role in the study of almost complex structures, the theory of spinors, and CR structures.
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https://en.wikipedia.org/wiki/D-bar_operator
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Typically, complex forms are considered because of some desirable decomposition that the forms admit. On a complex manifold, for instance, any complex k-form can be decomposed uniquely into a sum of so-called (p, q)-forms: roughly, wedges of p differentials of the holomorphic coordinates with q differentials of their complex conjugates. The ensemble of (p, q)-forms becomes the primitive object of study, and determines a finer geometrical structure on the manifold than the k-forms. Even finer structures exist, for example, in cases where Hodge theory applies.
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https://en.wikipedia.org/wiki/D-bar_operator
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In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces.
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https://en.wikipedia.org/wiki/Complex_geodesic
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In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has dimension one over C (hence the term "line"), it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line.The "complex plane" commonly refers to the graphical representation of the complex line on the real plane, and is thus generally synonymous with the complex line, and not a two-dimensional space over the complex numbers.
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https://en.wikipedia.org/wiki/Complex_line
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In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number z {\displaystyle z} , defined to be any complex number w {\displaystyle w} for which e w = z {\displaystyle e^{w}=z} . Such a number w {\displaystyle w} is denoted by log z {\displaystyle \log z} . If z {\displaystyle z} is given in polar form as z = r e i θ {\displaystyle z=re^{i\theta }} , where r {\displaystyle r} and θ {\displaystyle \theta } are real numbers with r > 0 {\displaystyle r>0} , then ln r + i θ {\displaystyle \ln r+i\theta } is one logarithm of z {\displaystyle z} , and all the complex logarithms of z {\displaystyle z} are exactly the numbers of the form ln r + i ( θ + 2 π k ) {\displaystyle \ln r+i\left(\theta +2\pi k\right)} for integers k {\displaystyle k} .
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https://en.wikipedia.org/wiki/Complex_logarithm
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These logarithms are equally spaced along a vertical line in the complex plane. A complex-valued function log: U → C {\displaystyle \log \colon U\to \mathbb {C} } , defined on some subset U {\displaystyle U} of the set C ∗ {\displaystyle \mathbb {C} ^{*}} of nonzero complex numbers, satisfying e log z = z {\displaystyle e^{\log z}=z} for all z {\displaystyle z} in U {\displaystyle U} . Such complex logarithm functions are analogous to the real logarithm function ln: R > 0 → R {\displaystyle \ln \colon \mathbb {R} _{>0}\to \mathbb {R} } , which is the inverse of the real exponential function and hence satisfies eln x = x for all positive real numbers x. Complex logarithm functions can be constructed by explicit formulas involving real-valued functions, by integration of 1 / z {\displaystyle 1/z} , or by the process of analytic continuation.There is no continuous complex logarithm function defined on all of C ∗ {\displaystyle \mathbb {C} ^{*}} . Ways of dealing with this include branches, the associated Riemann surface, and partial inverses of the complex exponential function. The principal value defines a particular complex logarithm function Log: C ∗ → C {\displaystyle \operatorname {Log} \colon \mathbb {C} ^{*}\to \mathbb {C} } that is continuous except along the negative real axis; on the complex plane with the negative real numbers and 0 removed, it is the analytic continuation of the (real) natural logarithm.
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https://en.wikipedia.org/wiki/Complex_logarithm
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In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation i 2 = − 1 {\displaystyle i^{2}=-1} ; every complex number can be expressed in the form a + b i {\displaystyle a+bi} , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number a + b i {\displaystyle a+bi} , a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols C {\displaystyle \mathbb {C} } or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers.
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https://en.wikipedia.org/wiki/Complex_Numbers
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More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation ( x + 1 ) 2 = − 9 {\displaystyle (x+1)^{2}=-9} has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions − 1 + 3 i {\displaystyle -1+3i} and − 1 − 3 i {\displaystyle -1-3i} . Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i 2 = − 1 {\displaystyle i^{2}=-1} combined with the associative, commutative, and distributive laws.
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https://en.wikipedia.org/wiki/Complex_Numbers
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Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with {1, i} as a standard basis.
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https://en.wikipedia.org/wiki/Complex_Numbers
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This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane.
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https://en.wikipedia.org/wiki/Complex_Numbers
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The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm. In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.
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https://en.wikipedia.org/wiki/Complex_Numbers
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In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra).
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https://en.wikipedia.org/wiki/Complex_reflection_group
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In mathematics, a complex representation is a representation of a group (or that of Lie algebra) on a complex vector space. Sometimes (for example in physics), the term complex representation is reserved for a representation on a complex vector space that is neither real nor pseudoreal (quaternionic). In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation. For compact groups, the Frobenius-Schur indicator can be used to tell whether a representation is real, complex, or pseudo-real. For example, the N-dimensional fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the antifundamental representation.
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https://en.wikipedia.org/wiki/Complex_representation
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In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*: The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.
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https://en.wikipedia.org/wiki/Normal_matrix
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The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A*A = AA* is diagonalizable. The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces. The left and right singular vectors in the singular value decomposition of a normal matrix A = U Σ V ∗ {\displaystyle \mathbf {A} =\mathbf {U} {\boldsymbol {\Sigma }}\mathbf {V} ^{*}} differ only in complex phase from each other and from the corresponding eigenvectors, since the phase must be factored out of the eigenvalues to form singular values.
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https://en.wikipedia.org/wiki/Normal_matrix
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In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space. Every complex vector space can be equipped with a compatible complex structure, however, there is in general no canonical such structure. Complex structures have applications in representation theory as well as in complex geometry where they play an essential role in the definition of almost complex manifolds, by contrast to complex manifolds. The term "complex structure" often refers to this structure on manifolds; when it refers instead to a structure on vector spaces, it may be called a linear complex structure.
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https://en.wikipedia.org/wiki/Linear_complex_structure
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In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles). Here N must be the even number 2n, where n is the complex dimension of M. All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to Cn considered as real vector space; then the quotient group is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For n = 1 this is the classical period lattice construction of elliptic curves.
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https://en.wikipedia.org/wiki/Complex_torus
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For n > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when n > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing as simple as the cubic curve description for n = 1. Computer algebra can handle cases for small n reasonably well. By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space.
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https://en.wikipedia.org/wiki/Complex_torus
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In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle E can be promoted to a complex vector bundle, the complexification E ⊗ C ; {\displaystyle E\otimes \mathbb {C} ;} whose fibers are Ex ⊗R C. Any complex vector bundle over a paracompact space admits a hermitian metric.
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https://en.wikipedia.org/wiki/Conjugate_bundle
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The basic invariant of a complex vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X is a complex manifold and if the local trivializations are biholomorphic.
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https://en.wikipedia.org/wiki/Conjugate_bundle
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In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N(y)} for all x and y in A. A composition algebra includes an involution called a conjugation: x ↦ x ∗ . {\displaystyle x\mapsto x^{*}.} The quadratic form N ( x ) = x x ∗ {\displaystyle N(x)=xx^{*}} is called the norm of the algebra. A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such that N(v) = 0, called a null vector. When x is not a null vector, the multiplicative inverse of x is x ∗ N ( x ) {\textstyle {\frac {x^{*}}{N(x)}}} . When there is a non-zero null vector, N is an isotropic quadratic form, and "the algebra splits".
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https://en.wikipedia.org/wiki/Multiplicative_quadratic_form
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In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence.
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https://en.wikipedia.org/wiki/Composition_(combinatorics)
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Each positive integer n has 2n−1 distinct compositions. A weak composition of an integer n is similar to a composition of n, but allowing terms of the sequence to be zero: it is a way of writing n as the sum of a sequence of non-negative integers. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the end of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions are assumed to be implicitly extended indefinitely by terms 0. To further generalize, an A-restricted composition of an integer n, for a subset A of the (nonnegative or positive) integers, is an ordered collection of one or more elements in A whose sum is n.
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https://en.wikipedia.org/wiki/Composition_(combinatorics)
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In mathematics, a composition ring, introduced in (Adler 1962), is a commutative ring (R, 0, +, −, ·), possibly without an identity 1 (see non-unital ring), together with an operation ∘: R × R → R {\displaystyle \circ :R\times R\rightarrow R} such that, for any three elements f , g , h ∈ R {\displaystyle f,g,h\in R} one has ( f + g ) ∘ h = ( f ∘ h ) + ( g ∘ h ) {\displaystyle (f+g)\circ h=(f\circ h)+(g\circ h)} ( f ⋅ g ) ∘ h = ( f ∘ h ) ⋅ ( g ∘ h ) {\displaystyle (f\cdot g)\circ h=(f\circ h)\cdot (g\circ h)} ( f ∘ g ) ∘ h = f ∘ ( g ∘ h ) . {\displaystyle (f\circ g)\circ h=f\circ (g\circ h).} It is not generally the case that f ∘ g = g ∘ f {\displaystyle f\circ g=g\circ f} , nor is it generally the case that f ∘ ( g + h ) {\displaystyle f\circ (g+h)} (or f ∘ ( g ⋅ h ) {\displaystyle f\circ (g\cdot h)} ) has any algebraic relationship to f ∘ g {\displaystyle f\circ g} and f ∘ h {\displaystyle f\circ h} .
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https://en.wikipedia.org/wiki/Composition_ring
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In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
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https://en.wikipedia.org/wiki/Concave_function
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