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c_eyttpqdyg9v5 | In mathematics, the quadratic bottleneck assignment problem (QBAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research, from the category of the facilities location problems.It is related to the quadratic assignment problem in the same way as the linear bot... | Quadratic bottleneck assignment problem |
c_eautqxyqgs0o | In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues λ {\displaystyle \lambda } , left eigenvectors y {\displaystyle y} and right eigenvectors x {\displaystyle x} such that Q ( λ ) x = 0 and y ∗ Q ( λ ) = 0 , {\displaystyle Q(\lambda )x=0~{\text{ and }}~y^{\ast }Q(\lambda )=0,} where Q ... | Quadratic eigenvalue problem |
c_hpzw84096qji | In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors. | Quadruple product |
c_zbbvd85qin2v | In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f ( x ) {\displaystyle f(x)} of some function f . {\displaystyle f.} An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the op... | Pointwise |
c_28v8agz0rn9e | In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relatively few differential equations that can be solved explicitly, but using too... | Qualitative theory of differential equations |
c_tzga1ygam2it | In mathematics, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain, replacing the classical definitions of probability with quantum probability. | Quantum Markov chain |
c_tf0xzaovktoc | In mathematics, the quantum dilogarithm is a special function defined by the formula ϕ ( x ) ≡ ( x ; q ) ∞ = ∏ n = 0 ∞ ( 1 − x q n ) , | q | < 1 {\displaystyle \phi (x)\equiv (x;q)_{\infty }=\prod _{n=0}^{\infty }(1-xq^{n}),\quad |q|<1} It is the same as the q-exponential function E q ( x ) {\displaystyle E_{q}(x)} . L... | Quantum dilogarithm |
c_a0l80oi4s0g1 | The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity. Faddeev's quantum dilogarithm Φ b ( w ) {\displaystyle \Phi _{b}(w)} is defined by the following formula: Φ b ( z ) = exp ( 1 4 ∫ C e − 2 i z w sinh ( w b ) sinh ( w / b ) d w w ) , {\displaystyle \Phi _{b}(z)=\exp \left... | Quantum dilogarithm |
c_msrqsri5tylk | In mathematics, the quantum q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | Quantum q-Krawtchouk polynomials |
c_l44wpdp2zypp | In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in specific applications with various definitions. | Quasi-commutative property |
c_4vigm3in55mo | In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative. Let f: A → F be a continuous function from an open set A in a Banach... | Quasi-derivative |
c_m1uewr2ni23j | If such a linear map u exists, then f is said to be quasi-differentiable at x0. Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its F... | Quasi-derivative |
c_ukcsb46i8l3e | In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of non-abelian groups of order 2n which have a cyclic subgroup of index 2. Two are well k... | Quasi-dihedral group |
c_p24wub59of2c | One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class. In Bertram Huppert's text Endliche Gruppen, this group is called a "Quasidiedergruppe". In Daniel Gorenstein's text, Finite Groups, this group is called the "semidihedral group". | Quasi-dihedral group |
c_hkavrouphnut | Dummit and Foote refer to it as the "quasidihedral group"; we adopt that name in this article. All give the same presentation for this group: ⟨ r , s ∣ r 2 n − 1 = s 2 = 1 , s r s = r 2 n − 2 − 1 ⟩ {\displaystyle \langle r,s\mid r^{2^{n-1}}=s^{2}=1,\ srs=r^{2^{n-2}-1}\rangle \,\!} .The other non-abelian 2-group with cy... | Quasi-dihedral group |
c_vnkg8szvfekf | When this group has order 16, Dummit and Foote refer to this group as the "modular group of order 16", as its lattice of subgroups is modular. In this article this group will be called the modular maximal-cyclic group of order 2 n {\displaystyle 2^{n}} . Its presentation is: ⟨ r , s ∣ r 2 n − 1 = s 2 = 1 , s r s = r 2 ... | Quasi-dihedral group |
c_nxiifrxgy9ad | .Both these two groups and the dihedral group are semidirect products of a cyclic group of order 2n−1 with a cyclic group of order 2. Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the group of units of the ring Z / 2 n − 1 Z {\displaystyle \mathbb {Z} /2^{n-1}\mathbb {Z} } and... | Quasi-dihedral group |
c_p4561f3nwtx0 | The groups of order pn and nilpotency class n − 1 were the beginning of the classification of all p-groups via coclass. The modular maximal-cyclic group of order 2n always has nilpotency class 2. This makes the modular maximal-cyclic group less interesting, since most groups of order pn for large n have nilpotency clas... | Quasi-dihedral group |
c_zwd6wo5z6duj | In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, ... | Norm of a quaternion |
c_xplovmwsvdic | Quaternions are generally represented in the form where a, b, c, and d are real numbers; and 1, i, j, and k are the basis vectors or basis elements.Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as ... | Norm of a quaternion |
c_av2tj7pgs90f | In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, being both a division ring and a domain. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by H . {\displaystyle \mathbb {H} .} | Norm of a quaternion |
c_hxap1bljxv3d | It can also be given by the Clifford algebra classifications Cl 0 , 2 ( R ) ≅ Cl 3 , 0 + ( R ) . {\displaystyle \operatorname {Cl} _{0,2}(\mathbb {R} )\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).} In fact, it was the first noncommutative division algebra to be discovered. | Norm of a quaternion |
c_eyoy4vsgqdw4 | According to the Frobenius theorem, the algebra H {\displaystyle \mathbb {H} } is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest a... | Norm of a quaternion |
c_gm1fiioq71vc | In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur... | Convergence of Fourier series |
c_cwwtk6jmf3h6 | In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category A {\displaystyle {\mathcal {A}}} by a Serre subcategory B {\displaystyle {\mathcal {B}}} is the abelian category A / B {\displaystyle {\mathcal {A}}/{\mathcal {B}}} which, intuitively, is obtained from A {\displaystyle ... | Quotient of an abelian category |
c_cn2z8r0hfkm4 | In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman.Let (X, ||·||) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds: The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0... | Quotient of subspace theorem |
c_1s3ode8qlapw | The statement is relative easy to prove by induction on the dimension of Z (even for Y=Z, X=0, c=1) with a K that depends only on N; the point of the theorem is that K is independent of N. In fact, the constant c can be made arbitrarily close to 1, at the expense of the constant K becoming large. The original proof all... | Quotient of subspace theorem |
c_o09ksg38khqn | In mathematics, the radical symbol, radical sign, root symbol, radix, or surd is a symbol for the square root or higher-order root of a number. The square root of a number x is written as 11 , {\displaystyle {\sqrt {11}},} while the nth root of x is written as x n . {\displaystyle {\sqrt{x}}.} It is also used for other... | Radical symbol |
c_316h5pbv3b39 | In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or ∞ {\displaystyle \infty } . When it is positive, the power series converges absolutely and uniformly on compact sets inside ... | Disc of convergence |
c_taztrsow4vi7 | In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.The structure of the set of extensions is known better when L/K is Galois. | Inertia group |
c_1sbh4u6qrqas | In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation f n = f n − 1 ± f n − 2 {\displaystyle f_{n}=f_{n-1}\pm f_{n-2}} , where the signs + or − are chosen at random with equal probability 1 2 {\displaystyle {\tfrac {1}{2}}} , independently fo... | Embree–Trefethen constant |
c_1sz4zfly34mw | In mathematics, the range of a function may refer to either of two closely related concepts: The codomain of the function The image of the functionGiven two sets X and Y, a binary relation f between X and Y is a (total) function (from X to Y) if for every x in X there is exactly one y in Y such that f relates x to y. T... | Range of a function |
c_hc7uhmmac0lm | In mathematics, the rank of a differentiable map f: M → N {\displaystyle f:M\to N} between differentiable manifolds at a point p ∈ M {\displaystyle p\in M} is the rank of the derivative of f {\displaystyle f} at p {\displaystyle p} . Recall that the derivative of f {\displaystyle f} at p {\displaystyle p} is a linear m... | Rank (differential topology) |
c_t0qcagjosfw0 | In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve E {\displaystyle E} defined over the field of rational numbers. Mordell's theorem says the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite subse... | Average rank of elliptic curves |
c_re26osulb6qd | The number of independent basis points with infinite order is the rank of the curve. The rank is related to several outstanding problems in number theory, most notably the Birch–Swinnerton-Dyer conjecture. | Average rank of elliptic curves |
c_1um5hgb3mwse | It is widely believed that there is no maximum rank for an elliptic curve, and it has been shown that there exist curves with rank as large as 28, but it is widely believed that such curves are rare. Indeed, Goldfeld and later Katz–Sarnak conjectured that in a suitable asymptotic sense (see below), the rank of elliptic... | Average rank of elliptic curves |
c_ad610s9xsst0 | In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimensio... | Torsion-free rank |
c_rb4oft30yb75 | In mathematics, the ratio test is a test (or "criterion") for the convergence of a series ∑ n = 1 ∞ a n , {\displaystyle \sum _{n=1}^{\infty }a_{n},} where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alember... | Ratio test |
c_ao0y9aq5f01p | In mathematics, the rational normal curve is a smooth, rational curve C of degree n in projective n-space Pn. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For n = 2 it is the plane conic Z0Z2 = Z21, and for n = 3 it is the twisted cubic. Th... | Rational normal curve |
c_mry7g1u48ruj | In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1. The set of such points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integer side lengt... | Group of rational points on the unit circle |
c_q6udveoe9i24 | In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field ... | Rational sieve |
c_vkj5ha5qfs1g | In mathematics, the real coordinate space of dimension n, denoted Rn or R n {\displaystyle \mathbb {R} ^{n}} , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. Special cases are called the real line R1 and the real coordinate plane R2. With component-wise addition and scal... | Real coordinate plane |
c_60nvra0bdqxx | Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n form a real coordinate space of dimension n. These one to one correspondences between vectors, points and coordinate vectors explain the names of coordinate space and coordinate vector. It allows using geometric terms and methods for... | Real coordinate plane |
c_dlzvwn2g0ws9 | In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real proje... | Real projective plane |
c_2vcj68szbt52 | (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. Since the Möbius strip, in tu... | Real projective plane |
c_j80rql79uyon | In mathematics, the real rank of a C*-algebra is a noncommutative analogue of Lebesgue covering dimension. The notion was first introduced by Lawrence G. Brown and Gert K. Pedersen. | Real rank (C*-algebras) |
c_ffiajkxinz9m | In mathematics, the rearrangement inequality states that for every choice of real numbers and every permutation y σ ( 1 ) , … , y σ ( n ) {\displaystyle y_{\sigma (1)},\ldots ,y_{\sigma (n)}} of y 1 , … , y n . {\displaystyle y_{1},\ldots ,y_{n}.} If the numbers x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are dif... | Rearrangement inequality |
c_of3gwx2kh02e | In mathematics, the reciprocal gamma function is the function f ( z ) = 1 Γ ( z ) , {\displaystyle f(z)={\frac {1}{\Gamma (z)}},} where Γ(z) denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is... | Reciprocal Gamma function |
c_jdxb4o40k7ub | In mathematics, the reduced derivative is a generalization of the notion of derivative that is well-suited to the study of functions of bounded variation. Although functions of bounded variation have derivatives in the sense of Radon measures, it is desirable to have a derivative that takes values in the same space as ... | Reduced derivative |
c_zv8wl8i4hjbd | In mathematics, the reflexive closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest reflexive relation on X {\displaystyle X} that contains R . {\displaystyle R.} A relation is called reflexive if it relates every element of X {\displaystyle X} to itself. For example, if X {\disp... | Reflexive closure |
c_681k7bc5a42q | In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and error technique of using test ("false") values for the variable and then adjust... | False position |
c_3vnv410ng5z7 | As an example, consider problem 26 in the Rhind papyrus, which asks for a solution of (written in modern notation) the equation x + x/4 = 15. This is solved by false position. First, guess that x = 4 to obtain, on the left, 4 + 4/4 = 5. | False position |
c_pqfvcybs906g | This guess is a good choice since it produces an integer value. However, 4 is not the solution of the original equation, as it gives a value which is three times too small. To compensate, multiply x (currently set to 4) by 3 and substitute again to get 12 + 12/4 = 15, verifying that the solution is x = 12. Modern versi... | False position |
c_mx6yy0wpx5tw | In mathematics, the regular part of a Laurent series consists of the series of terms with positive powers. That is, if f ( z ) = ∑ n = − ∞ ∞ a n ( z − c ) n , {\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n},} then the regular part of this Laurent series is ∑ n = 0 ∞ a n ( z − c ) n . {\displaystyle \sum ... | Regular part |
c_61rzn3642a8g | In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The use of the regulated integral instead of the Riemann integral has been advocated by Nicolas Bourbaki and Jean Dieudonné. | Regulated integral |
c_aebx1n9ubd4h | In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces. Formally, the relative interior of a set S {\displaystyle S} (denoted relint ( S ) {\displaystyle \operatorname {relint}... | Relative interior |
c_xnq7fl52ygt8 | Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior. A set is relatively open iff it is equal to its relative interior. Note that when aff ( S ) {\displaystyle \operatorname {aff} (S)} is a closed subspace of the full vector space (always the case when the... | Relative interior |
c_dysfn1mgd0ft | In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form x i ∈ { 0 , 1 } {\displaystyle x_{i}\in \{0,1\}} .The relaxation of the original integer program i... | Linear programming relaxation |
c_o1cdsxgfn7kc | In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividin... | Remainder |
c_5bv0j8fuik1s | The modulo operation is the operation that produces such a remainder when given a dividend and divisor. Alternatively, a remainder is also what is left after subtracting one number from another, although this is more precisely called the difference. This usage can be found in some elementary textbooks; colloquially it ... | Remainder |
c_iyf5qp1m6b04 | In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory. The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the fitness function to incorporate the di... | Replicator equation |
c_8bvd9xupijbk | In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural descri... | Representation theory of semisimple Lie algebras |
c_md1pay42qbq4 | There is a natural one-to-one correspondence between the finite-dimensional representations of a simply connected compact Lie group K and the finite-dimensional representations of the complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} that is the complexification of the Lie algebra of K (this fact is esse... | Representation theory of semisimple Lie algebras |
c_ied5te6c3h47 | In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics. In a physical theory having Minkowski space as the underlying spacetime, the space of physical state... | Representation theory of the Poincaré group |
c_8ek8ze32gv8o | In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space. The equivariance condition means that the group acts on the total space of the vector bundle, and the projection to Minkowski space is an equivariant map. Therefore, the Poincaré group also acts o... | Representation theory of the Poincaré group |
c_ttife27h1bka | Representations arising in this way (and their subquotients) are called covariant field representations, and are not usually unitary. For a discussion of such unitary representations, see Wigner's classification. In quantum mechanics, the state of the system is determined by the Schrödinger equation, which is invariant... | Representation theory of the Poincaré group |
c_1wmixk0ys87s | Quantum field theory is the relativistic extension of quantum mechanics, where relativistic (Lorentz/Poincaré invariant) wave equations are solved, "quantized", and act on a Hilbert space composed of Fock states. There are no finite unitary representations of the full Lorentz (and thus Poincaré) transformations due to ... | Representation theory of the Poincaré group |
c_eac9j15lciyv | These spinors transform under Lorentz transformations generated by the gamma matrices ( γ μ {\displaystyle \gamma _{\mu }} ). It can be shown that the scalar product ⟨ ψ | ϕ ⟩ = ψ ¯ ϕ = ψ † γ 0 ϕ {\displaystyle \langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi } is preserved. It is not, ho... | Representation theory of the Poincaré group |
c_emwk6unvo9tc | In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules ... | Representation theory of symmetric groups |
c_mo0jfb8y9844 | To each irreducible representation ρ we can associate an irreducible character, χρ. To compute χρ(π) where π is a permutation, one can use the combinatorial Murnaghan–Nakayama rule . Note that χρ is constant on conjugacy classes, that is, χρ(π) = χρ(σ−1πσ) for all permutations σ. Over other fields the situation can bec... | Representation theory of symmetric groups |
c_m88u0dcti5u7 | If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KSn is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary). However... | Representation theory of symmetric groups |
c_ksvlyptrgklg | In this context it is more usual to use the language of modules rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. | Representation theory of symmetric groups |
c_crnyd1mylpwd | The modules so constructed are called Specht modules, and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimensions are not known in general. The determination of the irreducible modules... | Representation theory of symmetric groups |
c_9wfbuuf04ued | In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal. This construction is applied in algebraic ge... | Residue field |
c_y8hyqdkt57ua | In mathematics, the residue formula says that the sum of the residues of a meromorphic differential form on a smooth proper algebraic curve vanishes. | Sum of residues formula |
c_n4gpxfg12aeg | In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus. The resolvent captures the s... | Resolvent formalism |
c_uznp2ntd45h3 | Given an operator A, the resolvent may be defined as R ( z ; A ) = ( A − z I ) − 1 . {\displaystyle R(z;A)=(A-zI)^{-1}~.} Among other uses, the resolvent may be used to solve the inhomogeneous Fredholm integral equations; a commonly used approach is a series solution, the Liouville–Neumann series. The resolvent of A ca... | Resolvent formalism |
c_b36xsm1mqlvu | In mathematics, the restricted product is a construction in the theory of topological groups. Let I {\displaystyle I} be an index set; S {\displaystyle S} a finite subset of I {\displaystyle I} . If G i {\displaystyle G_{i}} is a locally compact group for each i ∈ I {\displaystyle i\in I} , and K i ⊂ G i {\displaystyle... | Restricted product |
c_o208isa8f8wa | This group is given the topology whose basis of open sets are those of the form ∏ i A i , {\displaystyle \prod _{i}A_{i}\,,} where A i {\displaystyle A_{i}} is open in G i {\displaystyle G_{i}} and A i = K i {\displaystyle A_{i}=K_{i}} for all but finitely many i {\displaystyle i} . One can easily prove that the restri... | Restricted product |
c_i0dgvzecbz0c | In mathematics, the restriction of a function f {\displaystyle f} is a new function, denoted f | A {\displaystyle f\vert _{A}} or f ↾ A , {\displaystyle f{\upharpoonright _{A}},} obtained by choosing a smaller domain A {\displaystyle A} for the original function f . {\displaystyle f.} The function f {\displaystyle f} i... | Restricted function |
c_equc61fthbpc | In mathematics, the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division). However,... | Modulo operation |
c_hmpksie56ss7 | In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of a or n. Standard Pascal and ALGOL 68, for example, give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it to th... | Modulo operation |
c_1lm54xtf13vq | In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also call... | Multivariate resultant |
c_tgfmbwd3dxzp | It is used, among others, for cylindrical algebraic decomposition, integration of rational functions and drawing of curves defined by a bivariate polynomial equation. The resultant of n homogeneous polynomials in n variables (also called multivariate resultant, or Macaulay's resultant for distinguishing it from the usu... | Multivariate resultant |
c_owiplqlhd36u | In mathematics, the ring of integers of an algebraic number field K {\displaystyle K} is the ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is a root of a monic polynomial with integer coefficients: x n + c n − 1 x n − 1 + ⋯ + c 0 {\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}... | Number ring |
c_69d0cqszb2g5 | Since any integer belongs to K {\displaystyle K} and is an integral element of K {\displaystyle K} , the ring Z {\displaystyle \mathbb {Z} } is always a subring of O K {\displaystyle O_{K}} . The ring of integers Z {\displaystyle \mathbb {Z} } is the simplest possible ring of integers. Namely, Z = O Q {\displaystyle \m... | Number ring |
c_zw30hv5yv6jo | And indeed, in algebraic number theory the elements of Z {\displaystyle \mathbb {Z} } are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers Z {\displaystyle \mathbb {Z} } , consisting of complex numbers whose real and imaginary parts are integers. It is th... | Number ring |
c_z6rq8tlan63n | Like the rational integers, Z {\displaystyle \mathbb {Z} } is a Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain. | Number ring |
c_efmipnna9uu2 | In mathematics, the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is the graded ring generated by the modular forms of Γ. The study of rings of modular forms describes the algebraic structure of the space of modular forms. | Ring of modular forms |
c_e5dv35yku3q5 | In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k. If V is finite dimensional and is viewed as an algebraic variety, then k is precisely the coordinate ring of V. The explicit definition of the ring can be given as... | Polynomials on vector spaces |
c_wevvo6i4ztel | This suggests the following: given a vector space V, let k be the commutative k-algebra generated by the dual space V ∗ {\displaystyle V^{*}} , which is a subring of the ring of all functions V → k {\displaystyle V\to k} . If we fix a basis for V and write t i {\displaystyle t_{i}} for its dual basis, then k consists o... | Polynomials on vector spaces |
c_e24tyrp1o4bx | In applications, one also defines k when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies. Throughout the article, for simplicity, the base field k is assumed to be infinite. | Polynomials on vector spaces |
c_302jc8j3csrd | In mathematics, the root system for the simple Lie group G2 is in the form of a hexagram, with six long roots and six short roots. | Hexagram |
c_obrd8h5d9rrf | In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity lim sup n → ∞ | a n | n , {\displaystyle \limsup _{n\rightarrow \infty }{\sqrt{|a_{n}|}},} where a n {\displaystyle a_{n}} are the terms of the series, and states that the series conve... | Cauchy's radical test |
c_oxxvgamxzvlo | In mathematics, the rotation number is an invariant of homeomorphisms of the circle. | Map winding number |
c_j6qu7shafwy3 | In mathematics, the scalar projection of a vector a {\displaystyle \mathbf {a} } on (or onto) a vector b , {\displaystyle \mathbf {b} ,} also known as the scalar resolute of a {\displaystyle \mathbf {a} } in the direction of b , {\displaystyle \mathbf {b} ,} is given by: s = ‖ a ‖ cos θ = a ⋅ b ^ , {\displaystyle s=\... | Scalar projection |
c_lt15j9r58qs7 | In mathematics, the scale convolution of two functions s ( t ) {\displaystyle s(t)} and r ( t ) {\displaystyle r(t)} , also known as their logarithmic convolution is defined as the function s ∗ l r ( t ) = r ∗ l s ( t ) = ∫ 0 ∞ s ( t a ) r ( a ) d a a {\displaystyle s*_{l}r(t)=r*_{l}s(t)=\int _{0}^{\infty }s\left({\fra... | Logarithmic convolution |
c_duzngynftid2 | In mathematics, the second moment method is a technique used in probability theory and analysis to show that a random variable has positive probability of being positive. More generally, the "moment method" consists of bounding the probability that a random variable fluctuates far from its mean, by using its moments.Th... | Second moment method |
c_aak3zfzrmsiu | In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point. | Second partial derivative test |
c_8akc20a7x5iz | In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system. | Secondary measure |
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