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c_kq89pf087lgv | In mathematics, the braid group on n strands (denoted B n {\displaystyle B_{n}} ), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid gro... | Braid length |
c_ic9fznc1y5mc | In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrabilit... | Real Analysis |
c_8jv9sepxvoex | In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial. | Branching theorem |
c_hztygm9l2rrl | In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice. Lemma. Suppose G {\displaystyle G} is a group with subgroups A {\displaystyle A} and C {... | Butterfly lemma |
c_jfrtty49kab2 | Then there is an isomorphism of quotient groups: ( A ∩ C ) B ( A ∩ D ) B ≅ ( A ∩ C ) D ( B ∩ C ) D . {\displaystyle {\frac {(A\cap C)B}{(A\cap D)B}}\cong {\frac {(A\cap C)D}{(B\cap C)D}}.} This can be generalized to the case of a group with operators ( G , Ω ) {\displaystyle (G,\Omega )} with stable subgroups A {\displ... | Butterfly lemma |
c_jruwiwp4x2nd | Zassenhaus proved this lemma specifically to give the most direct proof of the Schreier refinement theorem. The 'butterfly' becomes apparent when trying to draw the Hasse diagram of the various groups involved. Zassenhaus' lemma for groups can be derived from a more general result known as Goursat's theorem stated in a... | Butterfly lemma |
c_80o0y02f11gq | In mathematics, the cake number, denoted by Cn, is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D a... | Cake number |
c_q0mpm27ublag | In mathematics, the caliber or calibre of a topological space X is a cardinal κ such that for every set of κ nonempty open subsets of X there is some point of X contained in κ of these subsets. This concept was introduced by Shanin (1948). There is a similar concept for posets. A pre-caliber of a poset P is a cardinal ... | Caliber (mathematics) |
c_ekdq2zm04z5c | In mathematics, the canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle \,\!\Omega ^{n}=\omega } , which is the nth exterior power of the cotangent bundle Ω {\displaystyle \Omega } on V {\displaystyle V} . Over ... | Canonical class |
c_eams46s1wzry | It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K {\displaystyle K} on V {\displaystyle V} giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V {\displaystyle V} , and any divisor in it may be called a canoni... | Canonical class |
c_g8z11afgttax | In mathematics, the capacitated arc routing problem (CARP) is that of finding the shortest tour with a minimum graph/travel distance of a mixed graph with undirected edges and directed arcs given capacity constraints for objects that move along the graph that represent snow-plowers, street sweeping machines, or winter ... | Capacitated arc routing problem |
c_i7dwwqjy6jpg | The CARP can be solved with combinatorial optimization including convex hulls. The large-scale capacitated arc routing problem (LSCARP) is a variant of the capacitated arc routing problem that applies to hundreds of edges and nodes to realistically simulate and model large complex environments. == References == | Capacitated arc routing problem |
c_rvh9vbeyw0vi | In mathematics, the capacity of a set in Euclidean space is a measure of the "size" of that set. Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge. More precisely, it is the capacitance of the set: the total ... | Harmonic capacity |
c_rxpe47di8alc | In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = { 2 , 4 , 6 } {\displaystyle A=\{2,4,6\}} contains 3 elements, and therefore A {\displaystyle A} has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets,... | Cardinality |
c_lr63ymzua0sx | The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A {\displaystyle A} is usually denoted | A | {\displaystyle |A|} , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The c... | Cardinality |
c_f7m5fnd8khqf | In mathematics, the caret can signify exponentiation (e.g. 3^5 for 35) where the usual superscript is not readily usable (as on some graphing calculators). It is also used to indicate a superscript in TeX typesetting. As Isaac Asimov described it in his 1974 "Skewered!" essay (on Skewes's number), "I make the exponent ... | Caret (punctuation) |
c_orrek7owgzsb | In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. | Category of abelian groups |
c_hn5zpb466hs5 | In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms. | Category of finite dimensional Hilbert spaces |
c_1vnlzc5r43id | In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. | Grp (category theory) |
c_9144w1som2up | In mathematics, the category Ord has preordered sets as objects and order-preserving functions as morphisms. This is a category because the composition of two order-preserving functions is order preserving and the identity map is order preserving. The monomorphisms in Ord are the injective order-preserving functions. T... | Category of preordered sets |
c_cjnvpconrhtd | There are thus no zero objects in Ord. The categorical product in Ord is given by the product order on the cartesian product. | Category of preordered sets |
c_6cgy8yh8p0qr | We have a forgetful functor Ord → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore Ord is a concrete category. This functor has a left adjoint (sending every set to that set equipped with the equality relat... | Category of preordered sets |
c_qukxvkhevpx1 | In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) R: A → B in this category is a relation between the sets A and B, so R ⊆ A × B. The composition of two relations R: A → B and S: B → C is given by (a, c) ∈ S o R ⇔ for some b ∈ B, (a, b) ∈ R and (b... | Category of relations |
c_tanptrmcq5es | In mathematics, the category number of a mathematician is a humorous construct invented by Dan Freed, intended to measure the capacity of that mathematician to stomach the use of higher categories. It is defined as the largest number n such that they can think about n-categories for a half hour without getting a splitt... | N-category number |
c_v1z5wtyxwxgy | In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the latter lacks some of the pleasant properties one would desire. There is also such a category for based spaces, defined b... | Category of compactly generated weak Hausdorff spaces |
c_pg4g370qu1aj | In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp. One is often interested only ... | Category of manifolds |
c_g8azs85elwly | In mathematics, the category of medial magmas, also known as the medial category, and denoted Med, is the category whose objects are medial magmas (that is, sets with a medial binary operation), and whose morphisms are magma homomorphisms (which are equivalent to homomorphisms in the sense of universal algebra). The ca... | Category of medial magmas |
c_su5f4182n9i4 | In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper. | Category of commutative algebras |
c_ohnq791b5c2k | In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of pro... | Categorical topology |
c_dfju0gtp38z0 | In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category because the composition of two continuous linear maps is again a continuous linear map. The category is often denoted TVec... | Category of topological vector spaces |
c_asv1scwztc4b | In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneou... | Characteristic equation (calculus) |
c_hctftwerh91w | The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is... | Characteristic equation (calculus) |
c_bepwknoazatv | For both types of equation, persistent fluctuations occur if there is at least one pair of complex roots. The method of integrating linear ordinary differential equations with constant coefficients was discovered by Leonhard Euler, who found that the solutions depended on an algebraic 'characteristic' equation. The qua... | Characteristic equation (calculus) |
c_xddilfb5sbvn | In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, char(... | Characteristic subring |
c_9w056ysuj8eo | In mathematics, the chromatic spectral sequence is a spectral sequence, introduced by Ravenel (1978), used for calculating the initial term of the Adams spectral sequence for Brown–Peterson cohomology, which is in turn used for calculating the stable homotopy groups of spheres. | Chromatic spectral sequence |
c_j2c80vq9yzca | In mathematics, the circle group, denoted by T {\displaystyle \mathbb {T} } or S 1 {\displaystyle \mathbb {S} ^{1}} , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C × {\disp... | Circle action |
c_cvymbmbvtuep | The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation T {\displaystyle \mathbb {T} } for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, T n {\displaystyle \mathbb {T} ^{n}} (the direct ... | Circle action |
c_f3pr7sdtbt70 | In mathematics, the circumflex is used to modify variable names; it is usually read "hat", e.g., î is "i hat". The Fourier transform of a function ƒ is often denoted by f ^ {\displaystyle {\hat {f}}} . In the notation of sets, a hat above an element signifies that the element was removed from the set, such as in { x 0 ... | Circumflex accent |
c_bmi1jx9vyxq6 | In geometry, a hat is sometimes used for an angle. For instance, the angles A ^ {\displaystyle {\hat {A}}} or A B ^ C {\displaystyle A{\hat {B}}C} . In vector notation, a hat above a letter indicates a unit vector (a dimensionless vector with a magnitude of 1). | Circumflex accent |
c_u9l408qptmyf | For instance, ı ^ {\displaystyle {\hat {\mathbf {\imath } }}} , x ^ {\displaystyle {\hat {\mathbf {x} }}} , or e ^ 1 {\displaystyle {\hat {\mathbf {e} }}_{1}} stands for a unit vector in the direction of the x-axis of a Cartesian coordinate system. In statistics, the hat is used to denote an estimator or an estimated v... | Circumflex accent |
c_mhdtulzujiuy | In mathematics, the class of L-matrices are those matrices whose off-diagonal entries are less than or equal to zero and whose diagonal entries are positive; that is, an L-matrix L satisfies L = ( ℓ i j ) ; ℓ i i > 0 ; ℓ i j ≤ 0 , i ≠ j . {\displaystyle L=(\ell _{ij});\quad \ell _{ii}>0;\quad \ell _{ij}\leq 0,\quad i\n... | L-matrix |
c_z9q54jn3gr6i | In mathematics, the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: Z = ( z i j ) ; z i j ≤ 0 , i ≠ j . {\displaystyle Z=(z_{ij});\quad z_{ij}\leq 0,\quad i\neq j.} Note that this definition coincides precisely with that of a negated M... | Z-matrix (mathematics) |
c_dvqyscyvca0o | The Jacobian of a competitive dynamical system is a Z-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is J, then (−J) is a Z-matrix. Related classes are L-matrices, M-matrices, P-matrices, Hurwitz matrices and Metzler matrices. | Z-matrix (mathematics) |
c_g1ljhqvoix6q | L-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices and P-matrices are nonsingular M-matri... | Z-matrix (mathematics) |
c_09ntglk1jop4 | In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.A large generalization of this formula applies to summation over an arbitrary locally finit... | Moebius inversion |
c_kxze4e199cpp | In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker. | Kronecker limit formula |
c_vmaklet6lx7z | In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory and representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture,... | Geometric Langlands correspondence |
c_qxlmd3p9u8ni | In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory to the branch of mathematics known as representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such a... | S-duality |
c_zqfblyhzhe4g | Starting with two Yang–Mills theories related by S-duality, Kapustin and Witten showed that one can construct a pair of quantum field theories in two-dimensional spacetime. By analyzing what this dimensional reduction does to certain physical objects called D-branes, they showed that one can recover the mathematical in... | S-duality |
c_8ihu91m8mh31 | In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any generalized circle, and thus a natural setting for planar inversive geometry.... | Moebius plane |
c_o87x5ezt6j8z | In inversive geometry a straight line is considered to be a generalized circle containing the point at infinity; inversion of the plane with respect to a line is a Euclidean reflection. More generally, a Möbius plane is an incidence structure with the same incidence relationships as the classical Möbius plane. It is on... | Moebius plane |
c_f4nbu7i8s0z5 | In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic... | Classical groups |
c_vx72mfl3pe0z | The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. | Classical groups |
c_oy7ppc3jzem1 | A few examples are the following. The rotation group SO(3) is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp(m)... | Classical groups |
c_yqafunscjb1d | In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such... | Classical orthogonal polynomials |
c_n760swr93ocp | Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials. For given polynomials Q , L: R → R {\displaystyle Q,L:\mathbb {R} \to \mathbb {R} } and ∀ n ∈ N 0 {\displaystyle \forall \,n\in \mathbb {N} _{0}} the classical orthogonal polynomials f n: R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R... | Classical orthogonal polynomials |
c_wukm2201mvgw | In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof co... | Classification of finite simple groups |
c_wa5a0joj6qvd | The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in a... | Classification of finite simple groups |
c_feaa1qy5vewz | In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size o... | List of finite simple groups |
c_so04vrlh5g2f | In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space R ∞ {\displaystyle \mathbb {R} ^{\infty }} . It is analogous to the classifying space for U(n). | Classifying space for O(n) |
c_mose32pfcpzv | In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. This space with its universal fibration may be constructed as either the Gr... | Classifying space for U(n) |
c_afirssp2gzaa | In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. | Closed graph theorem |
c_75e9vvkfd6be | In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of a Lie group G, then H is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding. One of severa... | Closed subgroup theorem |
c_qxk8pt3si3yq | In mathematics, the coadjoint representation K {\displaystyle K} of a Lie group G {\displaystyle G} is the dual of the adjoint representation. If g {\displaystyle {\mathfrak {g}}} denotes the Lie algebra of G {\displaystyle G} , the corresponding action of G {\displaystyle G} on g ∗ {\displaystyle {\mathfrak {g}}^{*}} ... | Coadjoint representation |
c_qseq0k0pwveh | The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups G {\displaystyle G} a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of G {\displaystyle G} are constructed ge... | Coadjoint representation |
c_4yhh7dygohdp | In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism hypothesis, though the details of his approach have yet to appear in the literature as of 2022. In 202... | Tangle hypothesis |
c_8lcc5nb99plb | In mathematics, the coclass of a finite p-group of order pn is n − c, where c is the class. | Coclass conjectures |
c_lwygktllc3uw | In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or image of a function. A codomain is part of a function... | Codomain |
c_8kuz2k67b4iv | Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution. A codomain is not part of a function f if f is defined as just a graph. For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case th... | Codomain |
c_1negkfqg1g17 | In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Through... | Cohomology operation |
c_qjd4kwfoqdkq | In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level. The convergence is to groups in stable homotopy theory, about which information is ... | Cohomology operation |
c_e8hdixd3yzl6 | In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given set. It is typically ... | Collage theorem |
c_akasb1qnt87i | In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm. Historically, it was known as logar... | Base-10 logarithm |
c_5s28kncsqz0c | On calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity, the ISO 80000 specification recommends that log10 (x) should be written lg(x), and loge (x) should be ln(x). Before ... | Base-10 logarithm |
c_jg5d7zx0lkjs | Instead, tables of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a slide rule. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplica... | Base-10 logarithm |
c_svpc87ejbb6r | In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. | Commutator (ring theory) |
c_5709wl9mmr75 | In mathematics, the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of operators in the ideal with bounded operators. Modern characterisation of the commutator subspace is through the Calkin correspondence and it involves th... | Commutator subspace |
c_7ykukcs8u1d2 | In mathematics, the compact complement topology is a topology defined on the set R {\displaystyle \scriptstyle \mathbb {R} } of real numbers, defined by declaring a subset X ⊆ R {\displaystyle \scriptstyle X\subseteq \mathbb {R} } open if and only if it is either empty or its complement R ∖ X {\displaystyle \scriptstyl... | Compact complement topology |
c_r2fag0z78wj6 | In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945.If the codo... | Compact open topology |
c_6n5e0vrjwzdh | In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the giv... | Direct comparison test |
c_iqw610rwephf | In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by F j ( x ) = 1 Γ ( j + 1 ) ∫ 0 ∞ t j e t − x + 1 d t , ( j > − 1 ) {\displaystyle F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt,\qquad (j>-1)} This equals − Li j ... | Complete Fermi–Dirac integral |
c_7vzlce4rx2hk | In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations ... | Witt algebra |
c_wbssbd7opfqu | In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a... | Conjugate pair |
c_vdcg2ye0kwh6 | In polar form, if r {\displaystyle r} and φ {\displaystyle \varphi } are real numbers then the conjugate of r e i φ {\displaystyle re^{i\varphi }} is r e − i φ . {\displaystyle re^{-i\varphi }.} | Conjugate pair |
c_5bakd6yxa7gn | This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a 2 + b 2 {\displaystyle a^{2}+b^{2}} (or r 2 {\displaystyle r^{2}} in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. | Conjugate pair |
c_17jz1mns2ajm | In mathematics, the complex conjugate of a complex vector space V {\displaystyle V\,} is a complex vector space V ¯ {\displaystyle {\overline {V}}} , which has the same elements and additive group structure as V , {\displaystyle V,} but whose scalar multiplication involves conjugation of the scalars. In other words, th... | Complex conjugate of a vector space |
c_6wc4aw12x1h8 | The letter v {\displaystyle v} stands for a vector in V , {\displaystyle V,} α {\displaystyle \alpha } is a complex number, and α ¯ {\displaystyle {\overline {\alpha }}} denotes the complex conjugate of α . {\displaystyle \alpha .} More concretely, the complex conjugate vector space is the same underlying real vector s... | Complex conjugate of a vector space |
c_eact71g8z3fj | In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P.It follows from this (and the fundamental theorem of algebra) that, if the degree of a re... | Complex conjugate root theorem |
c_1j162dlqfszu | In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation o... | Gauss plane |
c_vpgerc8idf0s | The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number... | Gauss plane |
c_gaq63ov2l3dq | In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates ( Z 1 , Z 2 , Z 3 ) ∈ C 3 , ( Z 1 , Z 2 , Z 3 ) ≠ ( 0 , 0 , 0 ) {\displaystyle (Z_{1},Z_{2},Z_{3})\in \mathbf {C} ... | Complex projective plane |
c_w7j7onpsrzqa | In mathematics, the complex squaring map, a polynomial mapping of degree two, is a simple and accessible demonstration of chaos in dynamical systems. It can be constructed by performing the following steps: Choose any complex number on the unit circle whose argument (angle) is not a rational multiple of π, Repeatedly s... | Complex squaring map |
c_vcd4g7176ubo | In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for V (a... | Complexification |
c_w8nsg8bdkw7m | In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic ... | Complexification (Lie group) |
c_e2q0wrr0vvqd | They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear. For compact Lie groups, the complexification, sometimes called the Chevalley complexification after Claude Chevalley, can be defined as the group of complex characters of the Hopf algebra of representative functions,... | Complexification (Lie group) |
c_2fixyh3kgaoi | In mathematics, the composition operator C ϕ {\displaystyle C_{\phi }} with symbol ϕ {\displaystyle \phi } is a linear operator defined by the rule where f ∘ ϕ {\displaystyle f\circ \phi } denotes function composition. The study of composition operators is covered by AMS category 47B33. | Composition operator |
c_qfx8ymtswiht | In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who u... | Compound of three octahedra |
c_aeqrmyq3zah4 | In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field. In general, when we define metric space the distance function is taken to be a real-valued function. The real numbers form an ordered field w... | Generalised metric |
c_fjmoio9p7t8k | In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical ... | Positive measure |
c_pryvc7dyxsbg | Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and ea... | Positive measure |
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