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wiki_2700_chunk_0 | Proper map | In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2701_chunk_0 | Bounded type (mathematics) | In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region Ω {\displaystyle \Omega } if and only if f {\displaystyle f} is analytic on Ω {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2702_chunk_0 | Rotational invariance | In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2703_chunk_0 | Bounded sequences | In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that | f ( x ) | ≤ M {\displaystyle |f(x)|\leq M} for all x in X. A function that is not bounded is said to be unbounded.If f is real-va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2704_chunk_0 | Cofunction | In mathematics, a function f is cofunction of a function g if f(A) = g(B) whenever A and B are complementary angles. This definition typically applies to trigonometric functions. The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).For example, sine (Latin: sinus) and cosine (Latin: cosinu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2705_chunk_0 | Logarithmically convex function | In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with f, is itself a convex function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2706_chunk_0 | Chisini mean | In mathematics, a function f of n variables x1, ..., xn leads to a Chisini mean M if, for every vector ⟨x1, ..., xn⟩, there exists a unique M such that f(M,M, ..., M) = f(x1,x2, ..., xn).The arithmetic, harmonic, geometric, generalised, Heronian and quadratic means are all Chisini means, as are their weighted variants.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2707_chunk_0 | Luzin N property | In mathematics, a function f on the interval has the Luzin N property, named after Nikolai Luzin (also called Luzin property or N property) if for all N ⊂ {\displaystyle N\subset } such that λ ( N ) = 0 {\displaystyle \lambda (N)=0} , there holds: λ ( f ( N ) ) = 0 {\displaystyle \lambda (f(N))=0} , where λ {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2708_chunk_0 | Superadditivity | In mathematics, a function f {\displaystyle f} is superadditive if for all x {\displaystyle x} and y {\displaystyle y} in the domain of f . {\displaystyle f.} Similarly, a sequence a 1 , a 2 , … {\displaystyle a_{1},a_{2},\ldots } is called superadditive if it satisfies the inequality for all m {\displaystyle m} and n ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2709_chunk_0 | Weakly harmonic | In mathematics, a function f {\displaystyle f} is weakly harmonic in a domain D {\displaystyle D} if ∫ D f Δ g = 0 {\displaystyle \int _{D}f\,\Delta g=0} for all g {\displaystyle g} with compact support in D {\displaystyle D} and continuous second derivatives, where Δ is the Laplacian. This is the same notion as a weak... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2710_chunk_0 | Supermodular function | In mathematics, a function f: R k → R {\displaystyle f\colon \mathbb {R} ^{k}\to \mathbb {R} } is supermodular if f ( x ↑ y ) + f ( x ↓ y ) ≥ f ( x ) + f ( y ) {\displaystyle f(x\uparrow y)+f(x\downarrow y)\geq f(x)+f(y)} for all x {\displaystyle x} , y ∈ R k {\displaystyle y\in \mathbb {R} ^{k}} , where x ↑ y {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2711_chunk_0 | Closed convex function | In mathematics, a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is said to be closed if for each α ∈ R {\displaystyle \alpha \in \mathbb {R} } , the sublevel set { x ∈ dom f | f ( x ) ≤ α } {\displaystyle \{x\in {\mbox{dom}}f\vert f(x)\leq \alpha \}} is a closed set. Equivalently, if th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2712_chunk_0 | Symmetrically continuous function | In mathematics, a function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is symmetrically continuous at a point x if lim h → 0 f ( x + h ) − f ( x − h ) = 0. {\displaystyle \lim _{h\to 0}f(x+h)-f(x-h)=0.} The usual definition of continuity implies symmetric continuity, but the converse is not true. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2713_chunk_0 | Symmetrically continuous function | For example, the function x − 2 {\displaystyle x^{-2}} is symmetrically continuous at x = 0 {\displaystyle x=0} , but not continuous. Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability. The set of the symmetrically con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2714_chunk_0 | Antidual space | In mathematics, a function f: V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors x , y ∈ V {\displaystyle x,y\in V} and every complex number s , {\displaystyle s,} where s ¯ {\displaystyle {\overline {s}}} denotes the complex conjugate o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2715_chunk_0 | Antidual space | If the vector spaces are real then antilinearity is the same as linearity. Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-value... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2716_chunk_0 | Function evaluation | In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2717_chunk_0 | Function evaluation | A function is most often denoted by letters such as f, g and h, and the value of a function f at an element x of its domain is denoted by f(x); the numerical value resulting from the function evaluation at a particular input value is denoted by replacing x with this value; for example, the value of f at x = 4 is denote... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2718_chunk_0 | Function evaluation | (x+1)^{2}\right\vert _{x=4}} (which results in 25).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2719_chunk_0 | Response variable | In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) and providing an output (which may also be a number). A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2720_chunk_0 | Locally bounded | In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2721_chunk_0 | Rapidly decreasing function | In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces. Aside from thi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2722_chunk_0 | Motor variable | In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873).... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2723_chunk_0 | Motor variable | For example, f ( z ) = u ( z ) + j v ( z ) , z = x + j y , x , y ∈ R , j 2 = + 1 , u ( z ) , v ( z ) ∈ R . {\displaystyle f(z)=u(z)+j\ v(z),\ z=x+jy,\ x,y\in R,\quad j^{2}=+1,\quad u(z),v(z)\in R.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2724_chunk_0 | Motor variable | Functions of a motor variable provide a context to extend real analysis and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2725_chunk_0 | Bounded deformation | In mathematics, a function of bounded deformation is a function whose distributional derivatives are not quite well-behaved-enough to qualify as functions of bounded variation, although the symmetric part of the derivative matrix does meet that condition. Thought of as deformations of elasto-plastic bodies, functions o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2726_chunk_0 | Bounded deformation | One can show that if u is of bounded deformation then the measure ε(u) can be decomposed into three parts: one absolutely continuous with respect to Lebesgue measure, denoted e(u) dx; a jump part, supported on a rectifiable (n − 1)-dimensional set Ju of points where u has two different approximate limits u+ and u−, tog... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2727_chunk_0 | Complete symmetric function | In mathematics, a function of n {\displaystyle n} variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f ( x 1 , x 2 ) {\displaystyle f\left(x_{1},x_{2}\right)} of two arguments is a symmetric function if and only if f ( x 1 , x 2 ) = f ( x 2 , x 1 ) {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2728_chunk_0 | Complete symmetric function | A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle k} -tensors on a vector space V {\displaystyle V} is isomorphic to the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2729_chunk_0 | Step function | In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2730_chunk_0 | Quadratic growth | In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position. "Quadratic growth" often means more generally "quadratic growth in the limit", as the argument or sequence position goes to infinity – in big Theta not... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2731_chunk_0 | Function space | In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2732_chunk_0 | Functional (mathematics) | In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). In linear algebra, it is synonymous with linear forms, which are linear mappings from a vector space V {\displaystyle V} into its field of scalars (t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2733_chunk_0 | Functional (mathematics) | The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name linear form. The third concept is detailed in the computer science article on higher-order functions. In the case where the space X {\displaystyle X} is a space of functions, the functional is a "function ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2734_chunk_0 | Functional calculus | In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2735_chunk_0 | Functional calculus | If f {\displaystyle f} is a function, say a numerical function of a real number, and M {\displaystyle M} is an operator, there is no particular reason why the expression f ( M ) {\displaystyle f(M)} should make sense. If it does, then we are no longer using f {\displaystyle f} on its original function domain. In the tr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2736_chunk_0 | Functional calculus | This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f ( x ) = x 2 {\displaystyle f(x)=x^{2}} and M {\displaystyle M} an n × n {\displaystyle n\times n} matrix. The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2737_chunk_0 | Functional calculus | Furthermore, it is a nontrivial ideal: let n {\displaystyle n} be the finite dimension of the algebra of matrices, then { I , T , T 2 , … , T n } {\displaystyle \{I,T,T^{2},\ldots ,T^{n}\}} is linearly dependent. So ∑ i = 0 n α i T i = 0 {\displaystyle \sum _{i=0}^{n}\alpha _{i}T^{i}=0} for some scalars α i {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2738_chunk_0 | Functional calculus | Consider the unilateral shift with the polynomials calculus; the ideal defined above is now trivial. Thus one is interested in functional calculi more general than polynomials. The subject is closely linked to spectral theory, since for a diagonal matrix or multiplication operator, it is rather clear what the definitio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2739_chunk_0 | Abel's functional equation | In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates sever... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2740_chunk_0 | Abel's functional equation | {\displaystyle \log(xy)=\log(x)+\log(y).} If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2741_chunk_0 | Abel's functional equation | Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2742_chunk_0 | Functional square root | In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function g is a function f satisfying f(f(x)) = g(x) for all x. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2743_chunk_0 | Fundamental discriminant | In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax2 + bxy + cy2 is a quadratic form with integer coefficients, then D = b2 − 4ac is the discriminant of Q(x, y). Conversely, every integer D with D ≡ 0, 1 (mod 4) is the discriminant of so... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2744_chunk_0 | Fundamental matrix (linear differential equation) | In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations is a matrix-valued function Ψ ( t ) {\displaystyle \Psi (t)} whose columns are linearly independent solutions of the system. Then every solution to the system can be written as x ( t ) = Ψ ( t ) c {\displaystyle \ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2745_chunk_0 | Fundamental polygon | In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2746_chunk_0 | Fundamental polygon | Using the theory of quasiconformal mappings and the Beltrami equation, it can be shown there is a canonical convex Dirichlet polygon with 4g sides, first defined by Fricke, which corresponds to the standard presentation of Γ as the group with 2g generators a1, b1, a2, b2, ..., ag, bg and the single relation ⋅⋅⋅ = 1, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2747_chunk_0 | Fundamental solution | In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta "function" δ(x), ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2748_chunk_0 | Fundamental lemma | In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundament... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2749_chunk_0 | Fundamental lemma | For instance, the fundamental theorem of curves describe classification of regular curves in space up to translation and rotation. Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2750_chunk_0 | Fusion frame | In mathematics, a fusion frame of a vector space is a natural extension of a frame. It is an additive construct of several, potentially "overlapping" frames. The motivation for this concept comes from the event that a signal can not be acquired by a single sensor alone (a constraint found by limitations of hardware or ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2751_chunk_0 | General hypergeometric function | In mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced by Gelfand (1986). The general hypergeometric function is a function that is (more or less) defined on a Grassmannian, and depends on a choice of some comple... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2752_chunk_0 | Borcherds algebra | In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known exam... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2753_chunk_0 | Generalized Korteweg–De Vries equation | In mathematics, a generalized Korteweg–De Vries equation (Masayoshi Tsutsumi, Toshio Mukasa & Riichi Iino 1970) is the nonlinear partial differential equation ∂ t u + ∂ x 3 u + ∂ x f ( u ) = 0. {\displaystyle \partial _{t}u+\partial _{x}^{3}u+\partial _{x}f(u)=0.\,} The function f is sometimes taken to be f(u) = uk+1/(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2754_chunk_0 | Generalized conic | In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of some defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2755_chunk_0 | Generalized conic | In rectangular Cartesian coordinates, the equation y = x2 represents a parabola. The generalized equation y = x r, for r ≠ 0 and r ≠ 1, can be treated as defining a generalized parabola. The idea of generalized conic has found applications in approximation theory and optimization theory.Among the several possible ways ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2756_chunk_0 | Generalized conic | In the infinite case, the weighted arithmetic mean has to be replaced by an appropriate integral. Generalized conics in this sense are also called polyellipses, egglipses, or, generalized ellipses. Since such curves were considered by the German mathematician Ehrenfried Walther von Tschirnhaus (1651 – 1708) they are al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2757_chunk_0 | Generalized flag manifold | In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2758_chunk_0 | Generalized flag manifold | For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth project... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2759_chunk_0 | Generalized flag manifold | If X has an F-rational point, then it is isomorphic to G/P for some parabolic subgroup P of G. A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2760_chunk_0 | Generalized hypergeometric series | In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2761_chunk_0 | Generalized maps | In mathematics, a generalized map is a topological model which allows one to represent and to handle subdivided objects. This model was defined starting from combinatorial maps in order to represent non-orientable and open subdivisions, which is not possible with combinatorial maps. The main advantage of generalized ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2762_chunk_0 | Monomial matrix | In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero en... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2763_chunk_0 | Exponential generating series | In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the gener... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2764_chunk_0 | Exponential generating series | One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet serie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2765_chunk_0 | Exponential generating series | The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal ser... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2766_chunk_0 | Exponential generating series | However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2767_chunk_0 | Generic polynomial | However in Galois theory, a branch of algebra, and in this article, the term generic polynomial has a different, although related, meaning: a generic polynomial for a finite group G and a field F is a monic polynomial P with coefficients in the field of rational functions L = F(t1, ..., tn) in n indeterminates over F, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2768_chunk_0 | Double torus | In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g many tori: the interior of a disk is removed from each of g many tori and the boundaries of the g many disks are identified (glued together), forming a g-torus. The genus of such a surface is g. A ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2769_chunk_0 | Hirzebruch polynomial | In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a seq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2770_chunk_0 | Geometric algebra | In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional ob... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2771_chunk_0 | Geometric algebra | The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2772_chunk_0 | Geometric algebra | Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra (CGA) providing a framework for classical geometries. In practice, these and several derive... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2773_chunk_0 | Geometric algebra | The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2774_chunk_0 | Geometric algebra | A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of V {\displaystyle V} and orthogonal projections onto that subspace. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2775_chunk_0 | Geometric algebra | Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2776_chunk_0 | Geometric algebra | Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2777_chunk_0 | Geometrical progression | Examples of a geometric sequence are powers rk of a fixed non-zero number r, such as 2k and 3k. The general form of a geometric sequence is a , a r , a r 2 , a r 3 , a r 4 , … {\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots } where r ≠ 0 is the common ratio and a ≠ 0 is a scale factor, equal to the sequence's... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2778_chunk_0 | Geometric series | In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series 1 2 + 1 4 + 1 8 + 1 16 + ⋯ {\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots } is geometric, because each successive ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2779_chunk_0 | Geometric series | . {\displaystyle a+ar+ar^{2}+ar^{3}+...} , where a {\displaystyle a} is the coefficient of each term and r {\displaystyle r} is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to fr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2780_chunk_0 | Transformation (geometry) | In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both R 2 {\displaystyle \mathbb {R} ^{2}} or both R 3 {\displaystyle \mathbb {R} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2781_chunk_0 | Gerbe | In mathematics, a gerbe (; French: ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2782_chunk_0 | Gerbe | Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2783_chunk_0 | Global field | In mathematics, a global field is one of two types of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: Algebraic number field: A finite extension of Q {\displaystyle \mathbb {Q} } Global function field: The function field of an algebraic curve over a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2784_chunk_0 | Graded Lie algebra | In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2785_chunk_0 | Graded Lie algebra | A graded Lie superalgebra extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily anticommutative. These arise in the study of derivations on graded algebras, in the deformation theory of Murray Gerstenhaber, Kunihiko Kodaira, and Donald C. Spencer, and in th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2786_chunk_0 | Graded Lie algebra | A supergraded Lie superalgebra is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } -gradation. These arise when one forms a graded Lie superalgebra in a classical (non-supers... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2787_chunk_0 | Graded dimension | In mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers. For "pure" vector spaces, the concept has been introduced in homological algebra, and it ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2788_chunk_0 | Gradually varied surface | In mathematics, a gradually varied surface is a special type of digital surfaces. It is a function from a 2D digital space (see digital geometry) to an ordered set or a chain. A gradually varied function is a function from a digital space Σ {\displaystyle \Sigma } to { A 1 , … , A m } {\displaystyle \{A_{1},\dots ,A_{m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2789_chunk_0 | Graph C*-algebra | In mathematics, a graph C*-algebra is a universal C*-algebra constructed from a directed graph. Graph C*-algebras are direct generalizations of the Cuntz algebras and Cuntz-Krieger algebras, but the class of graph C*-algebras has been shown to also include several other widely studied classes of C*-algebras. As a resul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2790_chunk_0 | Graph C*-algebra | Although graph C*-algebras include numerous examples, they provide a class of C*-algebras that are surprisingly amenable to study and much more manageable than general C*-algebras. The graph not only determines the associated C*-algebra by specifying relations for generators, it also provides a useful tool for describi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2791_chunk_0 | Graph partitioning | In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. If the number of resulting edges is small compared to the original grap... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2792_chunk_0 | Graph partitioning | Recently, the graph partition problem has gained importance due to its application for clustering and detection of cliques in social, pathological and biological networks. For a survey on recent trends in computational methods and applications see Buluc et al. (2013). Two common examples of graph partitioning are minim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2793_chunk_0 | Graph polynomial | In mathematics, a graph polynomial is a graph invariant whose values are polynomials. Invariants of this type are studied in algebraic graph theory. Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2794_chunk_0 | Graph polynomial | The chromatic polynomial, a polynomial whose values at integer arguments give the number of colorings of the graph with that many colors. The dichromatic polynomial, a 2-variable generalization of the chromatic polynomial The flow polynomial, a polynomial whose values at integer arguments give the number of nowhere-zer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2795_chunk_0 | Great circle | In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of dist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2796_chunk_0 | Great circle | Any other circle of the sphere is called a small circle, and is the intersection of the sphere with a plane not passing through its center. Small circles are the spherical-geometry analog of circles in Euclidean space. Every circle in Euclidean 3-space is a great circle of exactly one sphere. The disk bounded by a grea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2797_chunk_0 | GCD matrix | In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2798_chunk_0 | Direct sum of groups | In mathematics, a group G is called the direct sum of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2799_chunk_0 | Group action | A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of GL(n, K), the group of the invertible matrices of dimension n over a field K. The symmetric group Sn acts on any set with n elements by perm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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