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wiki_7200_chunk_0 | 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 ( z − c ) n . {\displaystyle \sum ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7201_chunk_0 | 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 Dieudonné. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7202_chunk_0 | 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 full vector space (always the case when the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7203_chunk_0 | 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 x_{i}\in \{0,1\}} .The relaxation of the original integer program i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7204_chunk_0 | 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 polynomial "left over" after dividin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7205_chunk_0 | 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 textbooks; colloquially it ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7206_chunk_0 | 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 allows the fitness function to incorporate the di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7207_chunk_0 | 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 because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural descri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7208_chunk_0 | 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 g {\displaystyle {\mathfrak {g}}} that is the complexification of the Lie algebra of K (this fact is esse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7209_chunk_0 | 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 the projection to Minkowski space is an equivariant map. Therefore, the Poincaré group also acts o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7210_chunk_0 | 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 mechanics, the state of the system is determined by the Schrödinger equation, which is invariant... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7211_chunk_0 | 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 no finite unitary representations of the full Lorentz (and thus Poincaré) transformations due to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7212_chunk_0 | Representation theory of symmetric groups | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7213_chunk_0 | Representation theory of symmetric groups | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7214_chunk_0 | 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 construction is applied in algebraic ge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7215_chunk_0 | 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 |
wiki_7216_chunk_0 | Resolvent formalism | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7217_chunk_0 | Restricted 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 . {\displaystyle f.} The function f {\displaystyle f} i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7218_chunk_0 | Modulo operation | 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7219_chunk_0 | Modulo operation | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7220_chunk_0 | Multivariate 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 some older texts, the resultant is also call... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7221_chunk_0 | Multivariate 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 resultant for distinguishing it from the usu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7222_chunk_0 | 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. Namely, Z = O Q {\displaystyle \m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7223_chunk_0 | 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 imaginary parts are integers. It is th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7224_chunk_0 | 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 coordinate ring of V. The explicit definition of the ring can be given as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7225_chunk_0 | 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 write t i {\displaystyle t_{i}} for its dual basis, then k consists o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7226_chunk_0 | 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 infinite. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7227_chunk_0 | Cauchy's radical 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 series, and states that the series conve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7228_chunk_0 | 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 = ‖ a ‖ cos θ = a ⋅ b ^ , {\displaystyle s=\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7229_chunk_0 | 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 {\displaystyle s*_{l}r(t)=r*_{l}s(t)=\int _{0}^{\infty }s\left({\fra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7230_chunk_0 | 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 fluctuates far from its mean, by using its moments.Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7231_chunk_0 | 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 |
wiki_7232_chunk_0 | Semi-implicit Euler method | In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7233_chunk_0 | Logarithmic measure | In mathematics, the set of positive real numbers, R > 0 = { x ∈ R ∣ x > 0 } , {\displaystyle \mathbb {R} _{>0}=\left\{x\in \mathbb {R} \mid x>0\right\},} is the subset of those real numbers that are greater than zero. The non-negative real numbers, R ≥ 0 = { x ∈ R ∣ x ≥ 0 } , {\displaystyle \mathbb {R} _{\geq 0}=\left\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7234_chunk_0 | Seven-dimensional cross product | In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in R 7 {\displaystyle \mathbb {R} ^{7}} a vector a × b also in R 7 {\displaystyle \mathbb {R} ^{7}} . Like the cross product in three dimensions, the seven-dime... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7235_chunk_0 | Seven-dimensional cross product | The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions. The seven-dimensional cross product is one way of generalizing the cross product to other than three dimensions, and it is the only other bilinear product of two vectors that is vecto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7236_chunk_0 | Sieve of Eratosthenes | In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7237_chunk_0 | Sieve of Pritchard | In mathematics, the sieve of Pritchard is an algorithm for finding all prime numbers up to a specified bound. Like the ancient sieve of Eratosthenes, it has a simple conceptual basis in number theory. It is especially suited to quick hand computation for small bounds. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7238_chunk_0 | Sieve of Pritchard | Whereas the sieve of Eratosthenes marks off each non-prime for each of its prime factors, the sieve of Pritchard avoids considering almost all non-prime numbers by building progressively larger wheels, which represent the pattern of numbers not divisible by any of the primes processed thus far. It thereby achieves a be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7239_chunk_0 | Sign function | In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that returns the sign of a real number. In mathematical notation the sign function is often represented as sgn ( x ) {\displaystyle \operatorname {sgn}(x)} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7240_chunk_0 | Metric signature | In mathematics, the signature (v, p, r) of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix gab of the metric tens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7241_chunk_0 | Metric signature | A Lorentzian metric is a metric with signature (p, 1), or (1, p). There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as (v − p), where v and p are as above, which is equivalent to the above definition when the dimension n = v + p is given or implicit. For example, s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7242_chunk_0 | Signature defect | In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. Hirzebruch (1973) introduced the signature defect for the cusp singularities of Hilbert modular surfaces. Michael Francis Atiyah, H. Donnelly, and I. M. Singer (1983) defined the signat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7243_chunk_0 | Negative area | In mathematics, the signed area or oriented area of a region of an affine plane is its area with orientation specified by the ("plus" or "minus") sign. More generally, the signed area of an arbitrary surface region is its surface area with specified orientation. When the boundary of the region is a simple curve, the si... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7244_chunk_0 | Negative area | Negative area arises in the study of natural logarithm as signed area under the curve y = 1/x for x in the positive real numbers: Definition: ln x = ∫ 1 x d t t , x > 0. {\displaystyle \ln x=\int _{1}^{x}{\frac {dt}{t}},\quad x>0.} "For 0 < x < 1, ln x = ∫ 1 x d t t = − ∫ x 1 d t t < 0 {\displaystyle \ln x=\int _{1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7245_chunk_0 | Negative area | In the case of the natural logarithm, obtained by integrating area under the hyperbola xy=1, the density dx ∧ dy is positive for x>1, but since the logarithm is anchored to 1, the orientation of the x-axis is reversed in the unit interval. For this integration the (− dx) orientation yields the opposite density to the o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7246_chunk_0 | Parallelogram equality | In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7247_chunk_0 | Real analytic Eisenstein series | In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function. There are many generalizations associated to more complicated groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7248_chunk_0 | Simplicial approximation theorem | In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices—that is, finit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7249_chunk_0 | Simplicial approximation theorem | This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness). It served to put the homology theory of the time—the first decade of the twentieth century—on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mapping... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7250_chunk_0 | Simplicial approximation theorem | This must be seen against the background of a realisation at the time that continuity was in general compatible with the pathological, in some other areas. This initiated, one could say, the era of combinatorial topology. There is a further simplicial approximation theorem for homotopies, stating that a homotopy betwee... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7251_chunk_0 | Bers's theorem | In mathematics, the simultaneous uniformization theorem, proved by Bers (1960), states that it is possible to simultaneously uniformize two different Riemann surfaces of the same genus using a quasi-Fuchsian group of the first kind. The quasi-Fuchsian group is essentially uniquely determined by the two Riemann surfaces... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7252_chunk_0 | Sinhc function | In mathematics, the sinhc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For z ≠ 0 {\displaystyle z\neq 0} , it is defined as The sinhc function is the hyperbolic analogue of the sinc function, defined by sin x / x {\displaystyle \sin x/x} . It is a solut... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7253_chunk_0 | Slice genus | In mathematics, the slice genus of a smooth knot K in S3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that K is the boundary of a connected, orientable 2-manifold S of genus g properly embedded in the 4-ball D4 bounded by S3. More precisely, if S is required to be smoothly embedded,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7254_chunk_0 | Slope of a line | In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7255_chunk_0 | Slope of a line | This is a constant function. If a line is vertical the slope is undefined (see below).The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances, where the Earth's curvature ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7256_chunk_0 | Slope of a line | The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of inclination θ by the tangent function m = tan ( θ ) {\displaystyle m=\tan(\theta )} Thus, a 45° rising line has a slope of +1 and a 45° falling line h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7257_chunk_0 | Slope of a line | When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic expression, then the differential calculus provides rules giving a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7258_chunk_0 | Small Veblen ordinal | The small Veblen ordinal θ Ω ω ( 0 ) {\displaystyle \theta _{\Omega ^{\omega }}(0)} or ψ ( Ω Ω ω ) {\displaystyle \psi (\Omega ^{\Omega ^{\omega }})} is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7259_chunk_0 | Soul conjecture | In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7260_chunk_0 | Special linear Lie algebra | In mathematics, the special linear Lie algebra of order n (denoted s l n ( F ) {\displaystyle {\mathfrak {sl}}_{n}(F)} or s l ( n , F ) {\displaystyle {\mathfrak {sl}}(n,F)} ) is the Lie algebra of n × n {\displaystyle n\times n} matrices with trace zero and with the Lie bracket := X Y − Y X {\displaystyle :=XY-YX} . ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7261_chunk_0 | SL2(R) | In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: SL ( 2 , R ) = { ( a b c d ): a , b , c , d ∈ R and a d − b c = 1 } . {\displaystyle {\mbox{SL}}(2,\mathbf {R} )=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\colon a,b,c,d\in \mathbf {R} {\mbox{ and }}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7262_chunk_0 | SL2(R) | SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL(2, R) (the 2 × 2 projective special linear group over R). More specifically, PSL(2, R) = SL(2, R) / {±I},where I denotes the 2 × 2 identity matrix. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7263_chunk_0 | Special linear group | In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant det: GL ( n , F... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7264_chunk_0 | Special linear group | where F× is the multiplicative group of F (that is, F excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When F is a finite field of order q, the notation SL(n, q) is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7265_chunk_0 | Special unitary group | In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7266_chunk_0 | Special unitary group | As a compact classical group, U(n) is the group that preserves the standard inner product on C n {\displaystyle \mathbb {C} ^{n}} . It is itself a subgroup of the general linear group, SU ( n ) ⊂ U ( n ) ⊂ GL ( n , C ) {\displaystyle \operatorname {SU} (n)\subset \operatorname {U} (n)\subset \operatorname {GL} (n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7267_chunk_0 | Special unitary group | The groups SU(2n) are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n {\displaystyle n} qubits and thus 2 n {\displaystyle 2^{n}} basis states. (Alternatively, the more general unitary group U ( 2 n ) {\displaystyle U(2^{n})} can be used, since mu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7268_chunk_0 | Spectral abscissa | In mathematics, the spectral abscissa of a matrix or a bounded linear operator is the greatest real part of the matrix's spectrum (its set of eigenvalues). It is sometimes denoted α ( A ) {\displaystyle \alpha (A)} . As a transformation α: M n → R {\displaystyle \alpha :\mathrm {M} ^{n}\rightarrow \mathbb {R} } , the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7269_chunk_0 | Spectral gap | In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7270_chunk_0 | Spectral radius | In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by ρ(·). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7271_chunk_0 | Spectral measure | In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7272_chunk_0 | Spectral measure | In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7273_chunk_0 | Hull-kernel topology | In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant und... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7274_chunk_0 | Energy spectrum | In mathematics, the spectrum of a matrix is the multiset of the eigenvalues of the matrix. In functional analysis, the concept of the spectrum of a bounded operator is a generalization of the eigenvalue concept for matrices. In algebraic topology, a spectrum is an object representing a generalized cohomology theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7275_chunk_0 | Spectrum of a matrix | In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if T: V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7276_chunk_0 | Spectrum of a matrix | Similarly, the trace of the matrix equals the sum of its eigenvalues. From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this quantity). In many applications, such as PageRank, one is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7277_chunk_0 | Spherical mean | In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7278_chunk_0 | Spheroidal wave equation | In mathematics, the spheroidal wave equation is given by ( 1 − t 2 ) d 2 y d t 2 − 2 ( b + 1 ) t d y d t + ( c − 4 q t 2 ) y = 0 {\displaystyle (1-t^{2}){\frac {d^{2}y}{dt^{2}}}-2(b+1)t\,{\frac {dy}{dt}}+(c-4qt^{2})\,y=0} It is a generalization of the Mathieu differential equation. If y ( t ) {\displaystyle y(t)} is a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7279_chunk_0 | Spin representation | Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron. The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7280_chunk_0 | Spin representation | Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures. The properties of the spin representations depend, in a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7281_chunk_0 | Spiral optimization algorithm | In mathematics, the spiral optimization (SPO) algorithm is a metaheuristic inspired by spiral phenomena in nature. The first SPO algorithm was proposed for two-dimensional unconstrained optimization based on two-dimensional spiral models. This was extended to n-dimensional problems by generalizing the two-dimensional s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7282_chunk_0 | Splitting circle method | In mathematics, the splitting circle method is a numerical algorithm for the numerical factorization of a polynomial and, ultimately, for finding its complex roots. It was introduced by Arnold Schönhage in his 1982 paper The fundamental theorem of algebra in terms of computational complexity (Technical report, Mathemat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7283_chunk_0 | Splitting principle | In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct sums of line bundles.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7284_chunk_0 | Splitting principle | The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with Z 2 {\displaystyle \mathbb {Z} _{2}} coefficients. In the complex case, the line bundles L i {\displaystyle L_{i}} or their first characteristic classes are called Chern roots. The fact that p ∗: H ∗ ( X ) → H ∗ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7285_chunk_0 | Splitting principle | The point is that these equations are easier to understand for direct sums of line bundles than for arbitrary vector bundles, so equations should be understood in Y {\displaystyle Y} and then pushed down to X {\displaystyle X} . Since vector bundles on X {\displaystyle X} are used to define the K-theory group K ( X ) {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7286_chunk_0 | Square root of a matrix | In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is equal to A.Some authors use the name square root or the notation A1/2 only for the specific case when A is positive semidefinite, to denote the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7287_chunk_0 | Stability radius | In mathematics, the stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest ball, centered at the nominal point, all of whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this: where p ^ {\displaystyle {\h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7288_chunk_0 | Standard basis | In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as R n {\displaystyle \mathbb {R} ^{n}} or C n {\displaystyle \mathbb {C} ^{n}} ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the E... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7289_chunk_0 | Standard basis | {\displaystyle \mathbf {e} _{x}=(1,0,0),\quad \mathbf {e} _{y}=(0,1,0),\quad \mathbf {e} _{z}=(0,0,1).} Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7290_chunk_0 | Standard basis | There are several common notations for standard-basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7291_chunk_0 | Standard basis | For example, every vector v in three-dimensional space can be written uniquely as v x e x + v y e y + v z e z , {\displaystyle v_{x}\,\mathbf {e} _{x}+v_{y}\,\mathbf {e} _{y}+v_{z}\,\mathbf {e} _{z},} the scalars v x {\displaystyle v_{x}} , v y {\displaystyle v_{y}} , v z {\displaystyle v_{z}} being the scalar componen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7292_chunk_0 | Bar resolution | In mathematics, the standard complex, also called standard resolution, bar resolution, bar complex, bar construction, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by Samuel Eilenberg and Saunders Mac Lane (1953) and Henri C... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7293_chunk_0 | Standard conjectures | In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his construction of pure motives gave an abe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7294_chunk_0 | Structure constant | In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors. Because the product operation in the algebra is bilinear, by linearity knowing the product of basis vector... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7295_chunk_0 | Structure constant | Given the structure constants, the resulting product is obtained by bilinearity and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra. Structure constants are used whenever an explicit form for the algebra must be given. Thus, they are frequently used whe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7296_chunk_0 | Structure Tensor | In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant respect the observing coordinates. The structure tensor is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7297_chunk_0 | Structure theorem for Gaussian measures | In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space. It was proved in the 1970s by Kallianpur–Sato–Stefan and Dudley–Feldman–le Cam. There is the earlier resul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7298_chunk_0 | Interchange of limiting operations | In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7299_chunk_0 | Bloch-Kato conjecture on Tamagawa numbers | In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely by the recognition that expression on the left-hand side is also L ( 1 ) {\displaystyle L(1)} where L ( s ) {\displaystyle L(s)} is the Dirichlet L-functi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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