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c_9n5ta3ce2tgx | 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 t... | Classification theorem |
c_ptv7tf7yaac2 | 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 ... | Classification theorem |
c_shb3608j86ej | 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. | Classifying topos |
c_iua2afaom1cj | 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. | Clean ring |
c_2a7hwj8prnvb | Every clean ring is an exchange ring. A matrix ring over a clean ring is itself clean. == References == | Clean ring |
c_ipzvc8z0gpaj | 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. | Compact surface |
c_w2pfnfhwu5d2 | 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. | Boundary parallel |
c_x844b78lkhmw | 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 deter... | Closure space |
c_a7inu3icntpw | 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 fur... | Semi-s-cobordism |
c_of4l44ehcip4 | 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 compl... | Cocountability |
c_tpy6uxf0tkja | 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... | Leading coefficient |
c_du4kip19l6b9 | 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 R... | Leading coefficient |
c_pchft4jotcdo | 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 t... | Leading coefficient |
c_mcaf16bk3sgp | 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, se... | Leading coefficient |
c_e1aosxvprstw | 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,... | Leading coefficient |
c_weqix8cffytk | 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 a... | Leading coefficient |
c_2kfj3xdkq2lc | 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... | Leading coefficient |
c_9gbbgfiq3vkh | 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... | Leading coefficient |
c_863gxm8i157w | 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. | Coercive operator |
c_t37ducjix5y3 | 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.} | Cofinite set |
c_sz87ilu2frce | 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'... | Cofinite set |
c_cnf8wqgh4vid | 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... | Coframe |
c_21i28ijxj55f | In mathematics, a coherent topos is a topos generated by a collection of quasi-compact quasi-separated objects closed under finite products. | Coherent topos |
c_izrid7gqa4zg | 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. | Cohomological invariant |
c_gy5qbrgu6owf | 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 st... | Coincidence point |
c_jh2ksoyec718 | 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 point |
c_ermop115ksl9 | 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 generat... | Collapsing algebra |
c_nr3ukecrg9hu | 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 ... | Rational dependence |
c_xwb29pzdof4c | 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 eve... | Point finite |
c_iomn41iy2b5o | 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. | Point finite |
c_65i6bn2zuxm9 | 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... | Collocation point |
c_99m3m1kbh1sq | 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 polynomia... | Colored matroid |
c_o3kniffxw2dq | 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 int... | Colossally abundant number |
c_otfyfbxxqon5 | 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 pea... | Combination |
c_hanu8wvrw2i9 | 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 ) ! | Combination |
c_98ll4sjyr9ev | {\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 ! | Combination |
c_rhm8zbxcg3or | {\displaystyle k!} permutations so P k n = C k n × k ! {\displaystyle P_{k}^{n}=C_{k}^{n}\times k!} | Combination |
c_xdswwev3vb06 | or C k n = P k n / k ! {\displaystyle C_{k}^{n}=P_{k}^{n}/k!} . | Combination |
c_77gdlcrmk0jg | 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,... | Combination |
c_g85qtri32y3j | 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 s... | Combination |
c_poinyrc13197 | 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. | Combinatorial class |
c_1t48kye38v1e | 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 problem... | Combinatorial explosion (communication) |
c_zcb0crz7eoog | 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. Lawver... | Comma category |
c_v3ni28auk0ed | 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 ... | Comma category |
c_2482f2goyjrr | 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 d... | Commutation theory |
c_9mp27ipozx4z | 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 algebrai... | Universally catenary |
c_szld8065b0zb | 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 | Universally catenary |
c_ugsf7yblxda2 | 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 numb... | Commutative rings |
c_nqjnjylrpwh9 | 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 ... | Braided monoidal category |
c_a499kopbt53w | 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 modi... | Braided monoidal category |
c_im7v61hm031u | 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. | Comodule |
c_7jso8heh0dty | 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 t... | Compact group |
c_oroqe65bpqvr | 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 compl... | Compact quantum group |
c_1szus72kon3e | 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 o... | Compact quantum group |
c_b3pu6bo0vpqc | 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. | Compact semigroup |
c_pyzl66fbgr3q | 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 solut... | Compact semigroup |
c_ep0ds9wc2dqv | 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 se... | Compactly generated group |
c_nt5iiou17vrg | 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 Boolea... | Complete Boolean algebra |
c_73hcjhi4vy28 | 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 cocom... | Finitely complete category |
c_ldsd8gf240nv | 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 categ... | Finitely complete category |
c_j7debfbinxm6 | 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). | Complete field |
c_ctwi3mfw8sqo | 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. | Complete lattices |
c_u1fax9iiykhc | 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 or... | Complete lattices |
c_3di353g9ph6g | 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. Equivalentl... | Geodesically complete |
c_imll1o7amjid | 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 ... | Complete measure |
c_9635usu7wbqf | 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 s... | Complete set of invariants |
c_y85h4f7d6lu8 | 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... | Completely metrizable |
c_l8cxeasd07ft | 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 fi... | Completely regular semigroup |
c_5wg58ihra48l | 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. | Completely regular semigroup |
c_6lbrnqa5tedv | 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. | Completely regular semigroup |
c_390f56ko1xfe | 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.} | Butson-type Hadamard matrix |
c_27csys9n9gc1 | 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}... | Complex Lie algebra |
c_ljuvodlr52gg | 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... | K3 surface |
c_nmzicduild8v | 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 negativel... | K3 surface |
c_yb326d7oih2k | 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 applie... | K3 surface |
c_zzro6k5j60fh | 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,... | D-bar operator |
c_eol7wm8z5cqh | 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 c... | D-bar operator |
c_poqd7iv0988m | In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces. | Complex geodesic |
c_36qlu8fts43q | 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 ... | Complex line |
c_0tr7mtvokmgc | 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 {... | Complex logarithm |
c_wy1dggwhcfq1 | 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 {\displayst... | Complex logarithm |
c_ifcpten6mddb | 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... | Complex Numbers |
c_w3nnrtxynuat | 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 ne... | Complex Numbers |
c_jdolnje4hm6h | 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. | Complex Numbers |
c_22wuucwx94x1 | 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 li... | Complex Numbers |
c_s8edadfe0pzt | 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 absolut... | Complex Numbers |
c_rx68ecqci1ih | 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-20... | Complex reflection group |
c_9htphk3jvce9 | 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, t... | Complex representation |
c_bp8pu9yrqox6 | 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 ex... | Normal matrix |
c_lcwpbauk1268 | 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 vec... | Normal matrix |
c_tg5rprwldqsy | 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 ... | Linear complex structure |
c_7tkd3xgv8dy4 | 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 follow... | Complex torus |
c_j9p85shxt8x2 | 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 >... | Complex torus |
c_9bxdk55zw49e | 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 ; {\displayst... | Conjugate bundle |
c_dqxauv6a65rm | 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. | Conjugate bundle |
c_bnnqiyr2dt1a | 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 ∗... | Multiplicative quadratic form |
c_koi0se9jplvw | 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 finitel... | Composition (combinatorics) |
c_ytapq6tdd4pt | 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 comp... | Composition (combinatorics) |
c_qzkavxj0n3mu | 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} on... | Composition ring |
c_w5g2epah9afr | 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. | Concave function |
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