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wiki_2900_chunk_0 | Nonlinear science | The conditions of additivity and homogeneity are often combined in the superposition principle f ( α x + β y ) = α f ( x ) + β f ( y ) {\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)} An equation written as f ( x ) = C {\displaystyle f(x)=C} is called linear if f ( x ) {\displaystyle f(x)} is a linear map (as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2901_chunk_0 | Linearity | In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties: Additivity: f(x + y) = f(x) + f(y). Homogeneity of degree 1: f(αx) = α f(x) for all α.These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in genera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2902_chunk_0 | Linearity | Additivity alone implies homogeneity for rational α, since f ( x + x ) = f ( x ) + f ( x ) {\displaystyle f(x+x)=f(x)+f(x)} implies f ( n x ) = n f ( x ) {\displaystyle f(nx)=nf(x)} for any natural number n by mathematical induction, and then n f ( x ) = f ( n x ) = f ( m n m x ) = m f ( n m x ) {\displaystyle nf(x)=f(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2903_chunk_0 | Semi-simple operator | In mathematics, a linear operator T on a vector space is semisimple if every T-invariant subspace has a complementary T-invariant subspace; in other words, the vector space is a semisimple representation of the operator T. Equivalently, a linear operator is semisimple if the minimal polynomial of it is a product of dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2904_chunk_0 | Local martingale | In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2905_chunk_0 | Locally compact group | For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral. In the general locally compact setting, such techniques need not hold. The resulting theory is a central part of harmonic analysis. The representation theory for locally compact abelian gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2906_chunk_0 | Kazhdan's property (T) | In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero invariant vector. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2907_chunk_0 | Locally constant function | In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2908_chunk_0 | Locally finite partially ordered set | In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval consists of finitely many elements. Given a locally finite poset P we can define its incidence algebra. Elements of the incidence algebra are functions ƒ that assign to each interval of P a real number ƒ(x, y)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2909_chunk_0 | Locally integrable function | In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but its ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2910_chunk_0 | Logarithm of a matrix | In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do ha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2911_chunk_0 | Disjunct matrix | In mathematics, a logical matrix may be described as d-disjunct and/or d-separable. These concepts play a pivotal role in the mathematical area of non-adaptive group testing. In the mathematical literature, d-disjunct matrices may also be called super-imposed codes or d-cover-free families.According to Chen and Hwang (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2912_chunk_0 | Disjunct matrix | A matrix is said to be d-disjunct if no set of d columns has a boolean sum which is a superset of any other single column.The following relationships are "well-known": Every d + 1 ¯ {\displaystyle {\overline {d+1}}} -separable matrix is also d {\displaystyle d} -disjunct. Every d {\displaystyle d} -disjunct matrix is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2913_chunk_0 | Loop (topology) | In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point.A loop may also be seen as a continuous map f from the pointed unit circle S1 into X, because S1 may be regar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2914_chunk_0 | Nasik magic hypercube | Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks. Marian Trenkler proved the following theorem: A p-dimensional magic hypercube of order n exists if and only if p > 1 and n is different from 2 or p = 1. A construction of a magic hypercube follows ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2915_chunk_0 | Manifold theory | Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simple... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2916_chunk_0 | Manifold theory | A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. The study of manifolds requires working knowledge of calcul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2917_chunk_0 | Map (mathematics) | In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2918_chunk_0 | Map (mathematics) | In category theory, a map may refer to a morphism. The term transformation can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2919_chunk_0 | Principal submatrix | In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2920_chunk_0 | Principal submatrix | Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents the composition of linear maps. Not all matrices are related to linear algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2921_chunk_0 | Principal submatrix | This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such. Square matrices, matrices with the same number of rows and columns, p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2922_chunk_0 | Principal submatrix | Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. The determinant of a square matrix is a number associated to the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2923_chunk_0 | Principal submatrix | In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimension. Matrices are used in most areas of mathematics and most scientific fields, either directly, or through their use in geometry and numerical analysis. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2924_chunk_0 | Matrix coefficient | In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2925_chunk_0 | Matrix coefficient | They arise naturally from finite-dimensional representations of G as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on G are dense in the Hilbert space of square-integrable functions on G. Matrix coefficients of representations of Lie gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2926_chunk_0 | Matrix factorization of a polynomial | In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p without linear terms can be written as a AB = pI, where A and B are square matrices and I is the identity matrix. Given the po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2927_chunk_0 | Matrix group | In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over K). Any finite group is linear, becau... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2928_chunk_0 | M-theory | In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mech... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2929_chunk_0 | M-theory | In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2930_chunk_0 | Gauge–gravity duality | In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mech... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2931_chunk_0 | Gauge–gravity duality | In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2932_chunk_0 | Gauge–gravity duality | This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra. In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2933_chunk_0 | Conformable matrix | In mathematics, a matrix is conformable if its dimensions are suitable for defining some operation (e.g. addition, multiplication, etc.). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2934_chunk_0 | Spectral norm | In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2935_chunk_0 | Matrix of ones | In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below: J 2 = ( 1 1 1 1 ) ; J 3 = ( 1 1 1 1 1 1 1 1 1 ) ; J 2 , 5 = ( 1 1 1 1 1 1 1 1 1 1 ) ; J 1 , 2 = ( 1 1 ) . {\displaystyle J_{2}={\begin{pmatrix}1&1\\1&1\end{pmatrix}};\quad J_... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2936_chunk_0 | Matrix geometrical series | In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial P ( x ) = ∑ i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + ⋯ + a n x n , {\displaystyle P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},} this polynomial evaluated at... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2937_chunk_0 | Measurable space | In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2938_chunk_0 | Measure algebra | In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2939_chunk_0 | Transverse measure | In mathematics, a measure on a real vector space is said to be transverse to a given set if it assigns measure zero to every translate of that set, while assigning finite and positive (i.e. non-zero) measure to some compact set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2940_chunk_0 | Measure preserving dynamical system | In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2941_chunk_0 | Riemannian connection | In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to: A connection for which the covariant derivativ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2942_chunk_0 | Doubling dimension | In mathematics, a metric space X with metric d is said to be doubling if there is some doubling constant M > 0 such that for any x ∈ X and r > 0, it is possible to cover the ball B(x, r) = {y | d(x, y) < r} with the union of at most M balls of radius r/2. The base-2 logarithm of M is called the doubling dimension of X.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2943_chunk_0 | Homogeneous metric | In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The mos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2944_chunk_0 | Homogeneous metric | Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs. In abstract algebra, the p-adi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2945_chunk_0 | Hard analysis | In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2946_chunk_0 | Microbundle | In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situations where they would not ordinarily be thought to exist. For example, the tangent bundle is defined for a smooth manifo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2947_chunk_0 | Minimal counterexample | In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2948_chunk_0 | Minimal counterexample | In which case, there may be multiple and more complex ways to structure the argument of the proof. The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2949_chunk_0 | Minimal surfaces | In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2950_chunk_0 | Mixed boundary condition | In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2951_chunk_0 | Mock theta functions | In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2952_chunk_0 | Modular Lie algebra | In mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic. The theory of modular Lie algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2953_chunk_0 | Modular equation | In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli. The most frequent use of the term modular equation i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2954_chunk_0 | Modular equation | That implies that any two rational functions F and G, in the function field of the modular curve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial of two variables over the complex numbers. For suitable non-degenerate choice of F and G, the equation P(X,Y) = 0 will actually define the modular cur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2955_chunk_0 | Elliptic modular form | In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition.The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2956_chunk_0 | Modular invariant of a group | In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic (usually dividing the order of the group). The study of modular invariants was originated in about 1914 by Dickson (2004). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2957_chunk_0 | Right module | In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2958_chunk_0 | Countably generated module | In mathematics, a module over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance of the notion comes from Kaplansky's theorem (Kaplansky 1958), which states that a projective module is a direct sum of countably generated modules. More general... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2959_chunk_0 | Fine moduli scheme | In mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept (algebrai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2960_chunk_0 | Moment matrix | In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.) Moment matrices play an important role in polynomial fitting, polynomial optimization (since positiv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2961_chunk_0 | Monogenic field | In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers OK is the subring Z of K generated by a. Then OK is a quotient of the polynomial ring Z and the powers of a constitute a power integral basis. In a monogenic field K, the field discriminan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2962_chunk_0 | Monoidal category of endofunctors | In mathematics, a monoidal category (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗: C × C → C {\displaystyle \otimes :\mathbf {C} \times \mathbf {C} \to \mathbf {C} } that is associative up to a natural isomorphism, and an object I that is both a left and right identity fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2963_chunk_0 | Monoidal category of endofunctors | Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category'... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2964_chunk_0 | Simple expression | In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. F... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2965_chunk_0 | Simple expression | If only a single variable x {\displaystyle x} is considered, this means that a monomial is either 1 {\displaystyle 1} or a power x n {\displaystyle x^{n}} of x {\displaystyle x} , with n {\displaystyle n} a positive integer. If several variables are considered, say, x , y , z , {\displaystyle x,y,z,} then each can be g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2966_chunk_0 | Simple expression | A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is 1 {\displaystyle 1} . For example, in this interpretation − 7 x 5 {\displaystyle -7x^{5}} and ( 3 − 4 i ) x 4 y z 13 {\displaystyle (3-4i)x^{4}yz^{13}} are monomials (in the second example, the variables are x , ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2967_chunk_0 | Monotonically non-decreasing | In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2968_chunk_0 | Moving frame | In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2969_chunk_0 | Multiplication tables | In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2970_chunk_0 | Multiplicative cascade | In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2971_chunk_0 | Reciprocal value | The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution). Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2972_chunk_0 | Reciprocal value | In these cases it can happen that ab ≠ ba; then "inverse" typically implies that an element is both a left and right inverse. The notation f −1 is sometimes also used for the inverse function of the function f, which is for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2973_chunk_0 | Multiperfect number | In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k, a number n is called k-perfect (or k-fold perfect) if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2974_chunk_0 | Series multisection | In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series ∑ n = − ∞ ∞ a n ⋅ z n {\displaystyle \sum _{n=-\infty }^{\infty }a_{n}\cdot z^{n}} then its multisection is a power series of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2975_chunk_0 | Multisymplectic integrator | In mathematics, a multisymplectic integrator is a numerical method for the solution of a certain class of partial differential equations, that are said to be multisymplectic. Multisymplectic integrators are geometric integrators, meaning that they preserve the geometry of the problems; in particular, the numerical meth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2976_chunk_0 | Many-valued function | In mathematics, a multivalued function, also called multifunction and many-valued function, is a set-valued function with continuity properties that allow considering it locally as an ordinary function. Multivalued functions arise commonly in applications of the implicit function theorem, since this theorem can be view... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2977_chunk_0 | Many-valued function | It cannot be considered as an ordinary function, since, when one follows one value of the logarithm along a circle centered at 0, one gets another value than the starting one after a complete turn. This phenomenon is called monodromy. Another common way for defining a multivalued function is analytic continuation, whic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2978_chunk_0 | Isotope (Jordan algebra) | In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first intr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2979_chunk_0 | Isotope (Jordan algebra) | Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2980_chunk_0 | Natural bundle | In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle F s ( M ) {\displaystyle F^{s}(M)} for some s ≥ 1 {\displaystyle s\geq 1} . It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold M {\displaystyle M} together with their pa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2981_chunk_0 | Bi-unitary divisor | Equivalently, a divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b. The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of the k-th powers of the unitary divisors is denoted by σ*k(n): σ k ∗ ( n ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2982_chunk_0 | Kaprekar number | In mathematics, a natural number in a given number base is a p {\displaystyle p} -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p {\displaystyle p} digits, that add up to the original number. The numbers are named after D. R. Kaprekar. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2983_chunk_0 | Near polygon | In mathematics, a near polygon is an incidence geometry introduced by Ernest E. Shult and Arthur Yanushka in 1980. Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. These structures g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2984_chunk_0 | Nearly Kähler manifold | In mathematics, a nearly Kähler manifold is an almost Hermitian manifold M {\displaystyle M} , with almost complex structure J {\displaystyle J} , such that the (2,1)-tensor ∇ J {\displaystyle \nabla J} is skew-symmetric. So, ( ∇ X J ) X = 0 {\displaystyle (\nabla _{X}J)X=0} for every vector field X {\displaystyle X} o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2985_chunk_0 | Nearly Kähler manifold | Each of these admits such a unique nearly Kähler metric that is also homogeneous, and these examples are in fact the only compact homogeneous strictly nearly Kähler 6-manifolds. However, Foscolo and Haskins recently showed that S 6 {\displaystyle S^{6}} and S 3 × S 3 {\displaystyle S^{3}\times S^{3}} also admit strict ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2986_chunk_0 | Negligible set | In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function. Negligible sets define several useful concepts t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2987_chunk_0 | Nil manifold | A Riemannian manifold is called a homogeneous nilmanifold if there exist a nilpotent group of isometries acting transitively on it. The requirement that the transitive nilpotent group acts by isometries leads to the following rigid characterization: every homogeneous nilmanifold is isometric to a nilpotent Lie group wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2988_chunk_0 | Non-Desarguesian plane | In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective spaces of dimension not 2; in other words, the only projective spaces of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2989_chunk_0 | Non-Euclidean crystallographic group | In mathematics, a non-Euclidean crystallographic group, NEC group or N.E.C. group is a discrete group of isometries of the hyperbolic plane. These symmetry groups correspond to the wallpaper groups in euclidean geometry. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2990_chunk_0 | Relativistic system (mathematics) | In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } over R {\displaystyle \mathbb {R} } . For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2991_chunk_0 | Relativistic system (mathematics) | Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space Q = R 4 {\displaystyle Q=\mathbb {R} ^{4}} is of this type. Since a configuration space Q {\displaystyle Q} of a relativistic system has no preferable fibration over R {\displaystyle \mathbb {R} } , a velocity spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2992_chunk_0 | Relativistic system (mathematics) | The notion of jets of submanifolds generalizes that of jets of sections of fiber bundles which are utilized in covariant classical field theory and non-autonomous mechanics. A first order jet bundle J 1 1 Q → Q {\displaystyle J_{1}^{1}Q\to Q} is projective and, following the terminology of Special Relativity, one can t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2993_chunk_0 | Relativistic system (mathematics) | Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads ( ∂ λ G μ α 2 … α 2 N 2 N − ∂ μ G λ α 2 … α 2 N ) q τ μ q τ α 2 ⋯ q τ α 2 N − ( 2 N − 1 ) G λ μ α 3 … α 2 N q τ τ μ q τ α 3 ⋯ q τ α 2 N + F λ μ q τ μ = 0 , {\displaystyle \left({\frac {\partial _{\lambda }G_{\mu \alpha ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2994_chunk_0 | Non-measurable set | In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zermelo–Fraenkel set theory, the axiom of choice entails that non-measurable sub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2995_chunk_0 | Non-measurable set | These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable. In 1970, Robert M. Solovay constructed the Solovay model, which shows that it is consistent with standard set theory without uncountable... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2996_chunk_0 | Nonabelian sheaf cohomology | In mathematics, a nonabelian cohomology is any cohomology with coefficients in a nonabelian group, a sheaf of nonabelian groups or even in a topological space. If homology is thought of as the abelianization of homotopy (cf. Hurewicz theorem), then the nonabelian cohomology may be thought of as a dual of homotopy group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2997_chunk_0 | Non-commutative ring | In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of propert... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2998_chunk_0 | Noncototient | In mathematics, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m. The cototient of n is defined as n − φ(n), so a noncototi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2999_chunk_0 | Nonelementary Integral | In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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