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wiki_2800_chunk_0 | Extension (algebra) | In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q {\displaystyle Q} and N {\displaystyle N} are two groups, then G {\displaystyle G} is an extension of Q {\displaystyle Q} by N {\displaystyle N} if there is a short exact sequence... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2801_chunk_0 | Extension (algebra) | Group extensions arise in the context of the extension problem, where the groups Q {\displaystyle Q} and N {\displaystyle N} are known and the properties of G {\displaystyle G} are to be determined. Note that the phrasing " G {\displaystyle G} is an extension of N {\displaystyle N} by Q {\displaystyle Q} " is also used... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2802_chunk_0 | Elementary group theory | Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2803_chunk_0 | Elementary group theory | Point groups describe symmetry in molecular chemistry. The concept of a group arose in the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term group (French: groupe) for the symmetry group of the roots of an equation, now called a Galois group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2804_chunk_0 | Elementary group theory | After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2805_chunk_0 | Iwasawa group | Kenkichi Iwasawa (1941) proved that a p-group G is an Iwasawa group if and only if one of the following cases happens: G is a Dedekind group, or G contains an abelian normal subgroup N such that the quotient group G/N is a cyclic group and if q denotes a generator of G/N, then for all n ∈ N, q−1nq = n1+ps where s ≥ 1 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2806_chunk_0 | Iwasawa group | As part of Schmidt's proof, he proves that a finite p-group is a modular group if and only if every subgroup is permutable, by (Schmidt 1994, Lemma 2.3.2, p. 55). Every subgroup of a finite p-group is subnormal, and those finite groups in which subnormality and permutability coincide are called PT-groups. In other word... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2807_chunk_0 | Infinite conjugacy class property | In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite.The von Neumann group algebra of a group is a factor if and only if the group has the infinite conjugacy class property. It will then be, provi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2808_chunk_0 | Supersoluble group | In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2809_chunk_0 | Multiplicative group scheme | In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2810_chunk_0 | Half range Fourier series | In mathematics, a half range Fourier series is a Fourier series defined on an interval {\displaystyle } instead of the more common {\displaystyle } , with the implication that the analyzed function f ( x ) , x ∈ {\displaystyle f(x),x\in } should be extended to {\displaystyle } as either an even (f(-x)=f(x)) or odd ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2811_chunk_0 | Half-exponential function | In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function f {\displaystyle f} such that f {\displaystyle f} composed with itself results in an exponential function: for some constants a {\displaystyle a} and b {\displaystyle b} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2812_chunk_0 | Half-integer | Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient. Note that halving an integer does not always produce a half-integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2813_chunk_0 | Handle decomposition | In mathematics, a handle decomposition of an m-manifold M is a union where each M i {\displaystyle M_{i}} is obtained from M i − 1 {\displaystyle M_{i-1}} by the attaching of i {\displaystyle i} -handles. A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2814_chunk_0 | Ore's harmonic number | In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are: 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2815_chunk_0 | Harmonic progression (mathematics) | In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequenc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2816_chunk_0 | Harshad numbers | In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2817_chunk_0 | Helix | {\displaystyle h(t)=t.\,} A circular helix of radius a and slope a/b (or pitch 2πb) is described by the following parametrisation: x ( t ) = a cos ( t ) , {\displaystyle x(t)=a\cos(t),\,} y ( t ) = a sin ( t ) , {\displaystyle y(t)=a\sin(t),\,} z ( t ) = b t . {\displaystyle z(t)=bt.\,} Another way of mathematicall... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2818_chunk_0 | Induced-hereditary property | In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of subobject depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2819_chunk_0 | Heteroclinic network | In mathematics, a heteroclinic network is an invariant set in the phase space of a dynamical system. It can be thought of loosely as the union of more than one heteroclinic cycle. Heteroclinic networks arise naturally in a number of different types of applications, including fluid dynamics and populations dynamics. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2820_chunk_0 | Hierarchy (mathematics) | In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2821_chunk_0 | Hierarchy (mathematics) | This idea can be applied to any commutative monoid. On the other hand, the set of integers Z requires a more sophisticated argument for its hierarchical structure, since we can always solve the equation n + m = n ′ {\displaystyle n+m=n'} by writing m = ( n ′ − n ) {\displaystyle m=(n'-n)} .A mathematical hierarchy (a p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2822_chunk_0 | Hierarchy (mathematics) | This is not just a pedantic claim; there are also mathematical hierarchies, in the general sense, that are not describable using set theory.Other natural hierarchies arise in computer science, where the word refers to partially ordered sets whose elements are classes of objects of increasing complexity. In that case, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2823_chunk_0 | Higher local field | In contrast to one-dimensional local fields, higher local fields have a sequence of residue fields. There are different integral structures on higher local fields, depending how many residue fields information one wants to take into account.Geometrically, higher local fields appear via a process of localization and com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2824_chunk_0 | Higher spin alternating sign matrix | In mathematics, a higher spin alternating sign matrix is a generalisation of the alternating sign matrix (ASM), where the columns and rows sum to an integer r (the spin) rather than simply summing to 1 as in the usual alternating sign matrix definition. HSASMs are square matrices whose elements may be integers in the r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2825_chunk_0 | Higher spin alternating sign matrix | {\displaystyle {\begin{pmatrix}0&0&2&0\\0&2&-1&1\\2&-1&2&-1\\0&1&-1&2\end{pmatrix}};\quad {\begin{pmatrix}0&0&2&0&0\\0&1&-1&2&0\\2&-1&-1&0&2\\0&0&2&0&0\\0&2&0&0&0\end{pmatrix}};\quad {\begin{pmatrix}0&0&0&2\\0&2&0&0\\2&-2&2&0\\0&2&0&0\end{pmatrix}};\quad {\begin{pmatrix}0&2&0&0\\0&0&0&2\\2&0&0&0\\0&0&2&0\end{pmatrix}}.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2826_chunk_0 | Hollow matrix | In mathematics, a hollow matrix may refer to one of several related classes of matrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal entries all zero. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2827_chunk_0 | Holomorphic discrete series representation | In mathematics, a holomorphic discrete series representation is a discrete series representation of a semisimple Lie group that can be represented in a natural way as a Hilbert space of holomorphic functions. The simple Lie groups with holomorphic discrete series are those whose symmetric space is Hermitian. Holomorphi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2828_chunk_0 | Holomorphic map | In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2829_chunk_0 | Holomorphic map | That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.Holomorphic functions are also sometimes referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2830_chunk_0 | Holomorphic vector bundle | In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π: E → X is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2831_chunk_0 | Homogeneous distribution | In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or Rn \ {0} that is homogeneous in the sense that, roughly speaking, S ( t x ) = t m S ( x ) {\displaystyle S(tx)=t^{m}S(x)\,} for all t > 0. More precisely, let μ t: x ↦ x / t {\displaystyle \mu _{t}:x\mapsto x/t} be the scalar divisi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2832_chunk_0 | Homogeneous distribution | The number m can be real or complex. It can be a non-trivial problem to extend a given homogeneous distribution from Rn \ {0} to a distribution on Rn, although this is necessary for many of the techniques of Fourier analysis, in particular the Fourier transform, to be brought to bear. Such an extension exists in most c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2833_chunk_0 | Homogenous function | In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the degree; that is, if k is an integer, a function f of n variables is homogeneous... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2834_chunk_0 | Homogenous function | This definition is often further generalized to functions whose domain is not V, but a cone in V, that is, a subset C of V such that v ∈ C {\displaystyle \mathbf {v} \in C} implies s v ∈ C {\displaystyle s\mathbf {v} \in C} for every nonzero scalar s. In the case of functions of several real variables and real vector s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2835_chunk_0 | Homogenous function | A norm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the absolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2836_chunk_0 | Inhomogeneous polynomial | In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each te... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2837_chunk_0 | Inhomogeneous polynomial | An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2838_chunk_0 | Inhomogeneous polynomial | In geometry, the Euclidean distance is the square root of a quadratic form. Homogeneous polynomials are ubiquitous in mathematics and physics. They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2839_chunk_0 | Identity relation | In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, wher... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2840_chunk_0 | Homogeneous space | The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2841_chunk_0 | Compactly-supported homology | In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group Hn(X, A) of every pair of spaces (X, A)is naturally isomorphic to the direct limit of the nth relative homology groups of pairs (Y, B), where Y varies over compact subspaces of X and B varie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2842_chunk_0 | Homothety | In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number k {\displaystyle k} called its ratio, which sends point X {\displaystyle X} to a point X ′ {\displaystyle X'} by the rule S X ′ → = k S X → {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2843_chunk_0 | Hyper-finite field | In mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field F is called hyper-finite if it is uncountable and quasi-finite, and for every subfield E, every absolutely entire E-algebra (regular field extension of E) of smaller cardinality than F can be embed... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2844_chunk_0 | Rectangular hyperbola | In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others. Each br... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2845_chunk_0 | Rectangular hyperbola | So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve y ( x ) = 1 / x {\displaystyle y(x)=1/x} the asymptotes are the two coordinate axes.Hyperbolas sh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2846_chunk_0 | Hyperbolic knot | In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component. As a consequence of the work of William Thurston, it is known that every knot is pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2847_chunk_0 | Hyperbolic manifolds | In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2848_chunk_0 | Gromov hyperbolic space | In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a larg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2849_chunk_0 | Hyperbolic PDE | In mathematics, a hyperbolic partial differential equation of order n {\displaystyle n} is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n − 1 {\displaystyle n-1} derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary init... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2850_chunk_0 | Hyperbolic PDE | In one spatial dimension, this is The equation has the property that, if u and its first time derivative are arbitrarily specified initial data on the line t = 0 (with sufficient smoothness properties), then there exists a solution for all time t. The solutions of hyperbolic equations are "wave-like". If a disturbance ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2851_chunk_0 | Hyperbolic PDE | They travel along the characteristics of the equation. This feature qualitatively distinguishes hyperbolic equations from elliptic partial differential equations and parabolic partial differential equations. A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essenti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2852_chunk_0 | Hyperbolic PDE | Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration. There is a well-developed theory for linear differential operators, due to Lars Gårding, in the context of microlocal analysis. Nonline... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2853_chunk_0 | Dual hypergraph | In computational geometry, an undirected hypergraph may sometimes be called a range space and then the hyperedges are called ranges. In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory. In some literature edges are referred to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2854_chunk_0 | Signed symmetric group | As a wreath product it is S 2 ≀ S n {\displaystyle S_{2}\wr S_{n}} where Sn is the symmetric group of degree n. As a permutation group, the group is the signed symmetric group of permutations π either of the set { − n , − n + 1 , ⋯ , − 1 , 1 , 2 , ⋯ , n } {\displaystyle \{-n,-n+1,\cdots ,-1,1,2,\cdots ,n\}} or of the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2855_chunk_0 | Signed symmetric group | 2). In three dimensions, the hyperoctahedral group is known as O × S2 where O ≅ S4 is the octahedral group, and S2 is a symmetric group (here a cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry, named after the regular octahedron, or 3-orthople... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2856_chunk_0 | Hyperplane section | In algebraic geometry, assuming therefore that X is V, a subvariety not lying completely in any H, the hyperplane sections are algebraic sets with irreducible components all of dimension dim(V) − 1. What more can be said is addressed by a collection of results known collectively as Bertini's theorem. The topology of hy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2857_chunk_0 | Hypertoric variety | In mathematics, a hypertoric variety or toric hyperkähler variety is a quaternionic analog of a toric variety constructed by applying the hyper-Kähler quotient construction of N. J. Hitchin, A. Karlhede, and U. Lindström et al. (1987) to a torus acting on a quaternionic vector space. Roger Bielawski and Andrew S. Dance... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2858_chunk_0 | Hypocontinuous bilinear map | In mathematics, a hypocontinuous is a condition on bilinear maps of topological vector spaces that is weaker than continuity but stronger than separate continuity. Many important bilinear maps that are not continuous are, in fact, hypocontinuous. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2859_chunk_0 | Jacket matrix | In mathematics, a jacket matrix is a square symmetric matrix A = ( a i j ) {\displaystyle A=(a_{ij})} of order n if its entries are non-zero and real, complex, or from a finite field, and A B = B A = I n {\displaystyle \ AB=BA=I_{n}} where In is the identity matrix, and B = 1 n ( a i j − 1 ) T . {\displaystyle \ B={1 \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2860_chunk_0 | Jumping line | In mathematics, a jumping line or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behavior, in other words the structure of its restriction to the line "jumps". Jumping lines were introduced by R. L. E. Schwarzenberger (1961). Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2861_chunk_0 | Jumping line | The Birkhoff–Grothendieck theorem classifies the n-dimensional vector bundles over a projective line as corresponding to unordered n-tuples of integers. This phenomenon cannot be generalized to higher dimensional projective spaces, namely, one cannot decompose an arbitrary bundle in terms of a Whitney sum of powers of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2862_chunk_0 | Jumping line | Given a bundle on C P n {\displaystyle \mathbb {CP} ^{n}} , E {\displaystyle {\mathcal {E}}} , we may take a line L {\displaystyle L} in C P n {\displaystyle \mathbb {CP} ^{n}} , or equivalently, a 2-dimensional subspace of C n + 1 {\displaystyle \mathbb {C} ^{n+1}} . This forms a variety equivalent to C P 1 {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2863_chunk_0 | Hyperperfect number | In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, whic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2864_chunk_0 | Knee of a curve | In mathematics, a knee of a curve (or elbow of a curve) is a point where the curve visibly bends, specifically from high slope to low slope (flat or close to flat), or in the other direction. This is particularly used in optimization, where a knee point is the optimum point for some decision, for example when there is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2865_chunk_0 | Framed knot | Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of S j in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2866_chunk_0 | Leaky integrator | In mathematics, a leaky integrator equation is a specific differential equation, used to describe a component or system that takes the integral of an input, but gradually leaks a small amount of input over time. It appears commonly in hydraulics, electronics, and neuroscience where it can represent either a single neur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2867_chunk_0 | Quaternionic vector space | In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H is the (non-commutative) division ring of quaternions. The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication: q ( q 1 , q 2 , … q n ) = ( q q 1 , q ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2868_chunk_0 | Semi-differentiability | In mathematics, a left derivative and a right derivative are derivatives (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2869_chunk_0 | Lethargy theorem | In mathematics, a lethargy theorem is a statement about the distance of points in a metric space from members of a sequence of subspaces; one application in numerical analysis is to approximation theory, where such theorems quantify the difficulty of approximating general functions by functions of special form, such as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2870_chunk_0 | Level sets | In mathematics, a level set of a real-valued function f of n real variables is a set where the function takes on a given constant value c, that is: L c ( f ) = { ( x 1 , … , x n ) ∣ f ( x 1 , … , x n ) = c } , {\displaystyle L_{c}(f)=\left\{(x_{1},\ldots ,x_{n})\mid f(x_{1},\ldots ,x_{n})=c\right\}~,} When the number o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2871_chunk_0 | Mathematical limit | In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the conce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2872_chunk_0 | Limit points | In mathematics, a limit point, accumulation point, or cluster point of a set S {\displaystyle S} in a topological space X {\displaystyle X} is a point x {\displaystyle x} that can be "approximated" by points of S {\displaystyle S} in the sense that every neighbourhood of x {\displaystyle x} with respect to the topology... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2873_chunk_0 | Limit points | A cluster point or accumulation point of a sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} in a topological space X {\displaystyle X} is a point x {\displaystyle x} such that, for every neighbourhood V {\displaystyle V} of x , {\displaystyle x,} there are infinitely many natural numbers n {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2874_chunk_0 | Limit points | The similarly named notion of a limit point of a sequence (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is synonymous with "clust... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2875_chunk_0 | Limit points | The limit points of a set should not be confused with adherent points (also called points of closure) for which every neighbourhood of x {\displaystyle x} contains another point of S {\displaystyle S} . Unlike for limit points, an adherent point x {\displaystyle x} of S {\displaystyle S} may have a neighbourhood not co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2876_chunk_0 | Limit points | Limit points of a set should also not be confused with boundary points. For example, 0 {\displaystyle 0} is a boundary point (but not a limit point) of the set { 0 } {\displaystyle \{0\}} in R {\displaystyle \mathbb {R} } with standard topology. However, 0.5 {\displaystyle 0.5} is a limit point (though not a boundary p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2877_chunk_0 | Limiting case (mathematics) | In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example: In statistics, the limiting case of the binomial distribution is the Poisson distribution. As the number of events tends to infinity i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2878_chunk_0 | Limiting case (mathematics) | Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached. Archimedes calculated an approximate value of π by treating the circle as the limiting case of a regular polygon with 3 × 2n sides, as n gets large. In ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2879_chunk_0 | Limiting case (mathematics) | In economics, two limiting cases of a demand curve or supply curve are those in which the elasticity is zero (the totally inelastic case) or infinity (the infinitely elastic case). In finance, continuous compounding is the limiting case of compound interest in which the compounding period becomes infinitesimally small,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2880_chunk_0 | Complex line bundle | In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2881_chunk_0 | Complex line bundle | The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2882_chunk_0 | Complex line bundle | In algebraic geometry, an invertible sheaf (i.e., locally free sheaf of rank one) is often called a line bundle. Every line bundle arises from a divisor with the following conditions (I) If X is reduced and irreducible scheme, then every line bundle comes from a divisor. (II) If X is projective scheme then the same sta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2883_chunk_0 | Curve integral | In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane. The function to b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2884_chunk_0 | Curve integral | The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from sim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2885_chunk_0 | Linear algebraic group action | In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M T M = I n {\displaystyle M^{T}M=I_{n}} where M T {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2886_chunk_0 | Tangent line approximation | In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2887_chunk_0 | Linear combination | In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2888_chunk_0 | First-order linear differential equation | In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2889_chunk_0 | First-order linear differential equation | An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2890_chunk_0 | First-order linear differential equation | The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2891_chunk_0 | First degree equation | In mathematics, a linear equation is an equation that may be put in the form a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,} where x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are the variables (or unknowns), and b , a 1 , … , a n {\displaystyle b,a_{1},\ldots ,a_{n}} are the coe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2892_chunk_0 | First degree equation | Alternatively, a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken. The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true. In the case of just one variable, there is exactly one soluti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2893_chunk_0 | First degree equation | Often, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown. In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2894_chunk_0 | First degree equation | This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n. Linear equations occur frequently in all mathematics and their applications in physics a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2895_chunk_0 | Dual vector | In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2896_chunk_0 | Dual vector | It is often denoted Hom(V, k), or, when the field k is understood, V ∗ {\displaystyle V^{*}} ; other notations are also used, such as V ′ {\displaystyle V'} , V # {\displaystyle V^{\#}} or V ∨ . {\displaystyle V^{\vee }.} When vectors are represented by column vectors (as is common when a basis is fixed), then linear f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2897_chunk_0 | Linear fractional transformations | The invertibility condition is then ad – bc ≠ 0. Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. When a, b, c, d are integer (or, more generally, belong to an integral domain), z is supposed to be a rational number (or... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2898_chunk_0 | Linear fractional transformations | In this case, the invertibility condition is that ad – bc must be a unit of the domain (that is 1 or −1 in the case of integers).In the most general setting, the a, b, c, d and z are elements of a ring, such as square matrices. An example of such linear fractional transformation is the Cayley transform, which was origi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2899_chunk_0 | Nonlinear science | In mathematics, a linear map (or linear function) f ( x ) {\displaystyle f(x)} is one which satisfies both of the following properties: Additivity or superposition principle: f ( x + y ) = f ( x ) + f ( y ) ; {\displaystyle \textstyle f(x+y)=f(x)+f(y);} Homogeneity: f ( α x ) = α f ( x ) . {\displaystyle \textstyle f(\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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