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Quadratic bottleneck assignment problem Summary Quadratic_bottleneck_assignment_problem 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 prob... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic eigenvalue problem Summary Quadratic_eigenvalue_problem 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 , {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadruple product Summary Quadruple_product 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 produc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pointwise Summary Pointwise 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Qualitative theory of differential equations Summary Qualitative_theory_of_differential_equations 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 Lyap... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantum Markov chain Summary Quantum_Markov_chain 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantum dilogarithm Summary Quantum_dilogarithm 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantum dilogarithm Summary Quantum_dilogarithm 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 ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantum q-Krawtchouk polynomials Summary Quantum_q-Krawtchouk_polynomials 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-commutative property Summary Quasi-commutative_property 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-derivative Summary Quasi-derivative 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 continuo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-derivative Summary Quasi-derivative 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-dihedral group Summary Semidihedral_group 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 h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-dihedral group Summary Semidihedral_group 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-dihedral group Summary Semidihedral_group 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}\ran... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-dihedral group Summary Semidihedral_group 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-dihedral group Summary Semidihedral_group .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 {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-dihedral group Summary Semidihedral_group 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 gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Norm of a quaternion Summary Scalar_quaternion 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 di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Norm of a quaternion Summary Scalar_quaternion 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Norm of a quaternion Summary Scalar_quaternion 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Norm of a quaternion Summary Scalar_quaternion 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 di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Norm of a quaternion Summary Scalar_quaternion 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 algeb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Convergence of Fourier series Summary Classic_harmonic_analysis 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quotient of an abelian category Summary Quotient_of_an_abelian_category 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}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quotient of subspace theorem Summary Quotient_of_subspace_theorem 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quotient of subspace theorem Summary Quotient_of_subspace_theorem 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Radical symbol Summary Root_symbol 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 {\s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Disc of convergence Summary Convergence_radius 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inertia group Ramification theory of valuations Higher_ramification_group > Ramification theory of valuations 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.Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Embree–Trefethen constant Summary Embree–Trefethen_constant 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 probab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Range of a function Summary Range_of_a_function 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rank (differential topology) Summary Rank_(differential_topology) 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 deri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Average rank of elliptic curves Summary Rank_of_an_elliptic_curve 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 bas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Average rank of elliptic curves Summary Rank_of_an_elliptic_curve 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Average rank of elliptic curves Summary Rank_of_an_elliptic_curve 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 tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Torsion-free rank Summary Rank_of_an_abelian_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ratio test Summary Ratio_test 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rational normal curve Summary Rational_normal_curve 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 Z... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Group of rational points on the unit circle Summary Group_of_rational_points_on_the_unit_circle 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 prim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rational sieve Summary Rational_sieve 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 under... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real coordinate plane Summary Standard_topology 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real coordinate plane Summary Standard_topology 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.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real projective plane Summary Real_projective_plane 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 geome... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real projective plane Summary Real_projective_plane (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 gen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real rank (C*-algebras) Summary Real_rank_(C*-algebras) 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rearrangement inequality Summary Rearrangement_inequality 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reciprocal Gamma function Summary Reciprocal_Gamma_function 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 re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reduced derivative Summary Reduced_derivative 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 deri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reflexive closure Summary Reflexive_closure 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 {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
False position Summary Illinois_algorithm 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
False position Summary Illinois_algorithm 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
False position Summary Illinois_algorithm 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regular part Summary Regular_part 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Regulated integral Summary Regulated_integral 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 Dieudon... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relative interior Summary Relative_interior 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relative interior Summary Relative_interior 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linear programming relaxation Summary Linear_programming_relaxation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Remainder Summary Remainder 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 polynomi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Remainder Summary Remainder 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Replicator equation Summary Replicator_equation 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of semisimple Lie algebras Summary Representation_theory_of_semisimple_Lie_algebras 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 b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of semisimple Lie algebras Summary Representation_theory_of_semisimple_Lie_algebras 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of the Poincaré group Summary Representation_theory_of_the_Poincaré_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of the Poincaré group Summary Representation_theory_of_the_Poincaré_group 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of the Poincaré group Summary Representation_theory_of_the_Poincaré_group 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of the Poincaré group Summary Representation_theory_of_the_Poincaré_group 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 n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of the Poincaré group Summary Representation_theory_of_the_Poincaré_group 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of symmetric groups Summary Representation_theory_of_the_symmetric_group 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of symmetric groups Summary Representation_theory_of_the_symmetric_group 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 clas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of symmetric groups Summary Representation_theory_of_the_symmetric_group 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of symmetric groups Summary Representation_theory_of_the_symmetric_group 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theory of symmetric groups Summary Representation_theory_of_the_symmetric_group 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 exampl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Residue field Summary Residue_field 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 con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sum of residues formula Summary Sum_of_residues_formula In mathematics, the residue formula says that the sum of the residues of a meromorphic differential form on a smooth proper algebraic curve vanishes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Resolvent formalism Summary Resolvent_operator 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 fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Resolvent formalism Summary Resolvent_operator 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Restricted product Summary Restricted_product 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 {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Restricted product Summary Restricted_product 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 {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Restricted function Summary Restriction_of_a_function 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 .... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Modulo operation Variants of the definition Modulo_operation > Variants of the definition 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 in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Modulo operation Variants of the definition Modulo_operation > Variants of the definition 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate resultant Summary Resultant 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 som... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate resultant Summary Resultant 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 res... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Number ring Summary Number_ring 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^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Number ring Summary Number_ring 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. N... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Number ring Summary Number_ring 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 imagin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Number ring Summary Number_ring 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ring of modular forms Summary Ring_of_modular_forms 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomials on vector spaces Summary Polynomials_on_vector_spaces 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 coordin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomials on vector spaces Summary Polynomials_on_vector_spaces 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 wri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomials on vector spaces Summary Polynomials_on_vector_spaces 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 infini... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hexagram Group theory Hexagram > Group theory 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cauchy's radical test Summary Root_test 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Map winding number Summary Rotation_number In mathematics, the rotation number is an invariant of homeomorphisms of the circle. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scalar projection Summary Scalar_projection 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 = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Logarithmic convolution Summary Logarithmic_convolution 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 {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Second moment method Summary Second_moment_method 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 flu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Second partial derivative test Summary Second_partial_derivative_test 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Secondary measure Summary Secondary_measure 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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