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wiki_3000_chunk_0 | Polynomial eigenvalue problem | In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form M ( λ ) x = 0 , {\displaystyle M(\lambda )x=0,} where x ≠ 0 {\displaystyle x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3001_chunk_0 | Dynamic systems theory | In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous system, which is linear... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3002_chunk_0 | Nonlocal operator | In mathematics, a nonlocal operator is a mapping which maps functions on a topological space to functions, in such a way that the value of the output function at a given point cannot be determined solely from the values of the input function in any neighbourhood of any point. An example of a nonlocal operator is the Fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3003_chunk_0 | Nonnegative matrices | In mathematics, a nonnegative matrix, written X ≥ 0 , {\displaystyle \mathbf {X} \geq 0,} is a matrix in which all the elements are equal to or greater than zero, that is, x i j ≥ 0 ∀ i , j . {\displaystyle x_{ij}\geq 0\qquad \forall {i,j}.} A positive matrix is a matrix in which all the elements are strictly greater t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3004_chunk_0 | Nonnegative matrices | The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A matrix which is both non-negative and is positive semidefinite is called a doubly non-ne... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3005_chunk_0 | Nonrecursive filter | In mathematics, a nonrecursive filter only uses input values like x, unlike recursive filter where it uses previous output values like y. In signal processing, non-recursive digital filters are often known as Finite Impulse Response (FIR) filters, as a non-recursive digital filter has a finite number of coefficients in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3006_chunk_0 | Norm form | In mathematics, a norm form is a homogeneous form in n variables constructed from the field norm of a field extension L/K of degree n. That is, writing N for the norm mapping to K, and selecting a basis e1, ..., en for L as a vector space over K, the form is given by N(x1e1 + ... + xnen)in variables x1, ..., xn. In num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3007_chunk_0 | Vector length | In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3008_chunk_0 | Vector length | A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3009_chunk_0 | Vector length | It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality " ≤ {\displaystyle \,\leq \,} " in the homogeneity axiom. It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3010_chunk_0 | Norm variety | In mathematics, a norm variety is a particular type of algebraic variety V over a field F, introduced for the purposes of algebraic K-theory by Voevodsky. The idea is to relate Milnor K-theory of F to geometric objects V, having function fields F(V) that 'split' given 'symbols' (elements of Milnor K-groups).The formula... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3011_chunk_0 | Normal surface | The concept of normal surface is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds. Later Wolfgang Haken extended and refined the notion to create normal surface theory, which is at the basis of many of the algorithms in 3-manifold theory. The notion of almost norma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3012_chunk_0 | Normed algebra | In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: ∀ x , y ∈ A ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ . {\displaystyle \forall x,y\in A\qquad \|xy\|\leq \|x\|\|y\|.} Some authors require it to have a multiplicative identity 1A such that ║1A║ = 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3013_chunk_0 | Normed spaces | In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If V {\displaystyle V} is a vector space over K {\displaystyle K} , where K {\displaystyle K} is a fi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3014_chunk_0 | Normed spaces | Absolute homogeneity: for every λ ∈ K {\displaystyle \lambda \in K} and x ∈ V {\displaystyle x\in V} , Triangle inequality: for every x ∈ V {\displaystyle x\in V} and y ∈ V {\displaystyle y\in V} ,If V {\displaystyle V} is a real or complex vector space as above, and ‖ ⋅ ‖ {\displaystyle \lVert \cdot \rVert } is a norm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3015_chunk_0 | Normed spaces | If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3016_chunk_0 | Normed spaces | For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm, but it is not complete for this norm. An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3017_chunk_0 | Nowhere continuous function | In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f {\displaystyle f} is a function from real numbers to real numbers, then f {\displaystyle f} is nowhere continuous if for each point x {\displaystyle x} t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3018_chunk_0 | Nullform | In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of the group vanish. Nullforms were introduced by Hilbert (1893). (Dieudonné & Carrell 1970, 1971, p.57). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3019_chunk_0 | Harmonic (mathematics) | In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3020_chunk_0 | Fixed-point theorems in infinite-dimensional spaces | In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Ju... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3021_chunk_0 | Fixed-point theorems in infinite-dimensional spaces | One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray who founded sh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3022_chunk_0 | Fixed-point theorems in infinite-dimensional spaces | Tikhonov (Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f: X → X has a fixed point. Browder fixed-point theorem: Let K be a nonempty closed bounded convex set in a uniformly convex Banach space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3023_chunk_0 | Fixed-point theorems in infinite-dimensional spaces | Then any non-expansive function f: K → K has a fixed point. (A function f {\displaystyle f} is called non-expansive if ‖ f ( x ) − f ( y ) ‖ ≤ ‖ x − y ‖ {\displaystyle \|f(x)-f(y)\|\leq \|x-y\|} for each x {\displaystyle x} and y {\displaystyle y} .) Other results include the Markov–Kakutani fixed-point theorem (1936-1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3024_chunk_0 | Numerical monoid | Numerical semigroups are commutative monoids and are also known as numerical monoids.The definition of numerical semigroup is intimately related to the problem of determining nonnegative integers that can be expressed in the form x1n1 + x2 n2 + ... + xr nr for a given set {n1, n2, ..., nr} of positive integers and for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3025_chunk_0 | Data cube | In mathematics, a one-dimensional array corresponds to a vector, a two-dimensional array resembles a matrix; more generally, a tensor may be represented as an n-dimensional data cube. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3026_chunk_0 | 1-parameter group | It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension. The action of a one-parameter group on a set is known as a flow. A smooth vector field on a manifold, at a point, induces a local flow - a one parameter group of local diffeomorphisms, sending po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3027_chunk_0 | P-adic distribution | In mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of p-adic numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3028_chunk_0 | P-adic zeta function | In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3029_chunk_0 | P-adic zeta function | For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a p-adic L-function, the p-adic Riemann zeta function ζp(s), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). p-adic L-functions arising i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3030_chunk_0 | P-adic zeta function | A p-adic L-function arising in this way is typically called an arithmetic p-adic L-function as it encodes arithmetic data of the Galois module involved. The main conjecture of Iwasawa theory (now a theorem due to Barry Mazur and Andrew Wiles) is the statement that the Kubota–Leopoldt p-adic L-function and an arithmetic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3031_chunk_0 | Packing in a hypergraph | In mathematics, a packing in a hypergraph is a partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in k-uniform hypergraphs. One of them is a random greedy a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3032_chunk_0 | Pair of pants (mathematics) | In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants. Pairs of pants are used as building blocks for compact surfaces in various theories. Two important appli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3033_chunk_0 | Cantor's pairing function | In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3034_chunk_0 | Palindromic prime | In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are: 2, 3, 5, 7, 11, 10... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3035_chunk_0 | Pandigital number | In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 (one billion two hundred thirty four million five hundred sixty seven thousand eight hundred ninety) is a pandigital number in base 10. The first few... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3036_chunk_0 | Pandigital number | In a trivial sense, all positive integers are pandigital in unary (or tallying). In binary, all integers are pandigital except for 0 and numbers of the form 2 n − 1 {\displaystyle 2^{n}-1} (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of b x {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3037_chunk_0 | Pandigital number | Conversely, the smaller the base, the fewer pandigital numbers without redundant digits there are. 2 is the only such pandigital number in base 2, while there are more of these in base 10. Sometimes, the term is used to refer only to pandigital numbers with no redundant digits. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3038_chunk_0 | Pandigital number | In some cases, a number might be called pandigital even if it doesn't have a zero as a significant digit, for example, 923456781 (these are sometimes referred to as "zeroless pandigital numbers"). No base 10 pandigital number can be a prime number if it doesn't have redundant digits. The sum of the digits 0 to 9 is 45,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3039_chunk_0 | Pandigital number | The first base 10 pandigital prime is 10123457689; OEIS: A050288 lists more. For different reasons, redundant digits are also required for a pandigital number (in any base except unary) to also be a palindromic number in that base. The smallest pandigital palindromic number in base 10 is 1023456789876543201. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3040_chunk_0 | Pantachy | In mathematics, a pantachy or pantachie (from the Greek word πανταχη meaning everywhere) is a maximal totally ordered subset of a partially ordered set, especially a set of equivalence classes of sequences of real numbers. The term was introduced by du Bois-Reymond (1879, 1882) to mean a dense subset of an ordered set,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3041_chunk_0 | Parabola | Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3042_chunk_0 | Paracompact manifold | For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracompact spaces may not be paracompact. Compare this to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. However, the product of a paracompact sp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3043_chunk_0 | Parallelization (mathematics) | In mathematics, a parallelization of a manifold M {\displaystyle M\,} of dimension n is a set of n global smooth linearly independent vector fields. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3044_chunk_0 | Parametric formula | In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, called parametric curve and parametric surf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3045_chunk_0 | Parametric formula | {\displaystyle (x,y)=(\cos t,\sin t).} Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.In addition to curves and surfaces, parametric equations can describe manifolds and algebraic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3046_chunk_0 | Parametric formula | Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled t; however, parameters can represent other physical quantities (such as geometric variables) or can be ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3047_chunk_0 | Paramodular group | In mathematics, a paramodular group is a special sort of arithmetic subgroup of the symplectic group. It is a generalization of the Siegel modular group, and has the same relation to polarized abelian varieties that the Siegel modular group has to principally polarized abelian varieties. It is the group of automorphism... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3048_chunk_0 | Paraproduct | {\displaystyle fg=\Lambda (f,g)+\Lambda (g,f).} For any appropriate functions f {\displaystyle f} and h {\displaystyle h} with h ( 0 ) = 0 {\displaystyle h(0)=0} , it is the case that h ( f ) = Λ ( f , h ′ ( f ) ) {\displaystyle h(f)=\Lambda (f,h'(f))} . It should satisfy some form of the Leibniz rule.A paraproduct may... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3049_chunk_0 | Paratopological group | In mathematics, a paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group G with a topology such that the group's product operation is a continuous function from G × G to G. This differs from the definition of a topological group in that the group inverse is not req... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3050_chunk_0 | Parent function | In mathematics, a parent function is the core representation of a function type without manipulations such as translation and dilation. For example, for the family of quadratic functions having the general form y = a x 2 + b x + c , {\displaystyle y=ax^{2}+bx+c\,,} the simplest function is y = x 2 {\displaystyle y=x^{2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3051_chunk_0 | Parent function | For example, the graph of y = x2 − 4x + 7 can be obtained from the graph of y = x2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2)2. For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3052_chunk_0 | Parent function | For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄B), then stretching it parallel to the Y axis using a stretch factor R, where R2 = A2 + B2. This is because A sin(x) + B cos(x) can be written a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3053_chunk_0 | Partial derivatives | In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3054_chunk_0 | Partial derivatives | {\displaystyle {\tfrac {\partial z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3055_chunk_0 | Linear partial differential equation | In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3056_chunk_0 | Linear partial differential equation | Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3057_chunk_0 | Linear partial differential equation | For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics (Schrödinger equation, Pauli equation, etc.). They also arise from many purely mathematical conside... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3058_chunk_0 | Linear partial differential equation | As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a sing... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3059_chunk_0 | Partial functions | In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3060_chunk_0 | Partial functions | This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3061_chunk_0 | Partial functions | In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function f {\displaystyle f} from X {\displaystyle X} to Y {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3062_chunk_0 | Partial group algebra | In mathematics, a partial group algebra is an associative algebra related to the partial representations of a group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3063_chunk_0 | Partition matroid | In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a capacity constraint - a maximum number of allowed elements from this category. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3064_chunk_0 | Set partition | In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence rela... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3065_chunk_0 | Partition of unity | In mathematics, a partition of unity of a topological space X {\displaystyle X} is a set R {\displaystyle R} of continuous functions from X {\displaystyle X} to the unit interval such that for every point x ∈ X {\displaystyle x\in X}: there is a neighbourhood of x {\displaystyle x} where all but a finite number of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3066_chunk_0 | Arc (topology) | In mathematics, a path in a topological space X {\displaystyle X} is a continuous function from the closed unit interval {\displaystyle } into X . {\displaystyle X.} Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3067_chunk_0 | Pedal curve | In mathematics, a pedal curve of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve C and a given fixed pedal point P, the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3068_chunk_0 | Perfect lattice | In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by Korkine & Zolotareff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3069_chunk_0 | Perfect matrix | In mathematics, a perfect matrix is an m-by-n binary matrix that has no possible k-by-k submatrix K that satisfies the following conditions: k > 3 the row and column sums of K are each equal to b, where b ≥ 2 there exists no row of the (m − k)-by-k submatrix formed by the rows not included in K with a row sum greater t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3070_chunk_0 | Periodic travelling wave | In mathematics, a periodic travelling wave (or wavetrain) is a periodic function of one-dimensional space that moves with constant speed. Consequently, it is a special type of spatiotemporal oscillation that is a periodic function of both space and time. Periodic travelling waves play a fundamental role in many mathema... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3071_chunk_0 | Primitive permutation group | The simplest example is the Klein four-group acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the trivial group acting on a 2-element set. This is because for a non-trans... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3072_chunk_0 | Permutation group | By Cayley's theorem, every group is isomorphic to some permutation group. The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3073_chunk_0 | Circular notation | Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3074_chunk_0 | Circular notation | In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3075_chunk_0 | Circular notation | This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). For example, the permutation (3, 1, 2) mentioned above is described by the function α {\displaystyle \alpha } defined as α ( 1 ) = 3 , α ( 2 ) = 1 , α ( 3 ) = 2 {\displaystyle \alpha (1)=3,\quad \alp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3076_chunk_0 | Circular notation | As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} that are considered for studying permutations. In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3077_chunk_0 | Permutation polynomial | In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x ↦ g ( x ) {\displaystyle x\mapsto g(x)} is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3078_chunk_0 | Phase line (mathematics) | In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, d y d x = f ( y ) {\displaystyle {\tfrac {dy}{dx}}=f(y)} . The phase line is the 1-dimensional form of the general n {\displaystyle n} -dimensional phase space, and can be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3079_chunk_0 | Phase portrait | This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An att... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3080_chunk_0 | Piecewise algebraic space | In mathematics, a piecewise algebraic space is a generalization of a semialgebraic set, introduced by Maxim Kontsevich and Yan Soibelman. The motivation was for the proof of Deligne's conjecture on Hochschild cohomology. Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić later developed the theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3081_chunk_0 | Piecewise smooth | In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3082_chunk_0 | Planar Riemann surface | In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3083_chunk_0 | Two-dimensional space | In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclid... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3084_chunk_0 | Point process | In mathematics, a point process is a random element whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3085_chunk_0 | Discrete subset | In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S and there exists a neighborhood of x that does not contain any other points of S. This is equivalent to saying that the singleton {x} is an open set in the topological space S (considered as a subspace... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3086_chunk_0 | Pointed set | In mathematics, a pointed set (also based set or rooted set) is an ordered pair ( X , x 0 ) {\displaystyle (X,x_{0})} where X {\displaystyle X} is a set and x 0 {\displaystyle x_{0}} is an element of X {\displaystyle X} called the base point, also spelled basepoint. : 10–11 Maps between pointed sets ( X , x 0 ) {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3087_chunk_0 | Pointed set | : 226 There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent. : 44 In particular, the empty set is not a pointed set because it has no element that can be chosen as the basepoint.The category of pointed sets and based maps is equivalent to the category of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3088_chunk_0 | Pointed set | One textbook notes that "This formal completion of sets and partial maps by adding 'improper', 'infinite' elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science. "The category of pointed sets and pointed maps is isomorphic to the coslice category ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3089_chunk_0 | Pointed set | The category of pointed sets and pointed maps has both products and coproducts, but it is not a distributive category. It is also an example of a category where 0 × A {\displaystyle 0\times A} is not isomorphic to 0 {\displaystyle 0} .Many algebraic structures are pointed sets in a rather trivial way. For example, grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3090_chunk_0 | Polyadic space | In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3091_chunk_0 | Polyhedral complex | In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3092_chunk_0 | Polylogarithmic function | In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, a k ( log n ) k + a k − 1 ( log n ) k − 1 + ⋯ + a 1 ( log n ) + a 0 . {\displaystyle a_{k}(\log n)^{k}+a_{k-1}(\log n)^{k-1}+\cdots +a_{1}(\log n)+a_{0}.} The notation logkn is often used as a shorthand for (log n)k, analogous ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3093_chunk_0 | Polylogarithmic function | In computer science, polylogarithmic functions occur as the order of time or memory used by some algorithms (e.g., "it has polylogarithmic order"), such as in the definition of QPTAS (see PTAS). All polylogarithmic functions of n are o(nε) for every exponent ε > 0 (for the meaning of this symbol, see small o notation),... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3094_chunk_0 | Polymatroid | In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also described as the multiset analogue of the matroid. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3095_chunk_0 | Polynomial Diophantine equation | In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation for which one seeks solutions restricted to be polynomials in the indeterminate. A Diophantine equation, in general, is one where the solutions are restricted to some algebraic system, typically integers. (In another usage ) Dioph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3096_chunk_0 | Polynomial Diophantine equation | {\displaystyle t=x^{2}+x.} A necessary and sufficient condition for a polynomial Diophantine equation to have a solution is for c to be a multiple of the GCD of a and b. In the example above, the GCD of a and b was 1, so solutions would exist for any value of c. Solutions to polynomial Diophantine equations are not uni... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3097_chunk_0 | Separable polynomial | In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial.This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3098_chunk_0 | Polynomial multiplication | Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3099_chunk_0 | Polynomial matrix | In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices. A univariate polynomial matrix P of degree p is defined as: P = ∑ n = 0 p A ( n ) x n = A ( 0 ) + A ( 1 )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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