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wiki_3300_chunk_0 | Logarithmically concave sequence | In mathematics, a sequence a = (a0, a1, ..., an) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if ai2 ≥ ai−1ai+1 holds for 0 < i < n . Remark: some authors (explicitly or not) add two further conditions in the definition of log-concave sequences: a is non... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3301_chunk_0 | Finite sequence | Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set. For example, (M, A, R, Y) is a sequence of letters wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3302_chunk_0 | Seven-dimensional space | In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3303_chunk_0 | Seven-dimensional space | An especially distinctive property is that a cross product can be defined only in three or seven dimensions. This is related to Hurwitz's theorem, which prohibits the existence of algebraic structures like the quaternions and octonions in dimensions other than 2, 4, and 8. The first exotic spheres ever discovered were ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3304_chunk_0 | Eight-dimensional space | In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Eight-dimensional Euclidean space is eight-dimensional space equip... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3305_chunk_0 | Nested sequences of intervals | Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3306_chunk_0 | Irrationality sequence | In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers, the sum of the series ∑ n = 1 ∞ 1 a n x n {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{a_{n}x_{n}}}} exists (that is, it converges) and is an irrational number... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3307_chunk_0 | Linear sequence transformation | In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improvi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3308_chunk_0 | Series expansion | In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).The resulting so-called series often can b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3309_chunk_0 | Convergent series | In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence ( a 0 , a 1 , a 2 , … ) {\displaystyle (a_{0},a_{1},a_{2},\ldots )} defines a series S that is denoted S = a 0 + a 1 + a 2 + ⋯ = ∑ k = 0 ∞ a k . {\displaystyle S=a_{0}+a_{1}+a_{2}+\cdots =\sum _{k=0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3310_chunk_0 | Convergent series | {\displaystyle S_{n}=\sum _{k=1}^{n}a_{k}.} A series is convergent (or converges) if the sequence ( S 1 , S 2 , S 3 , … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} of its partial sums tends to a limit; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices, one get... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3311_chunk_0 | Convergent series | {\displaystyle \left|S_{n}-\ell \right|<\varepsilon .} If the series is convergent, the (necessarily unique) number ℓ {\displaystyle \ell } is called the sum of the series. The same notation ∑ k = 1 ∞ a k {\displaystyle \sum _{k=1}^{\infty }a_{k}} is used for the series, and, if it is convergent, to its sum. This conve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3312_chunk_0 | Partial sums | In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite stru... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3313_chunk_0 | Partial sums | In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3314_chunk_0 | Partial sums | This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3315_chunk_0 | Partial sums | Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3316_chunk_0 | Partial sums | To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like or, using the summation sign, The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time).... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3317_chunk_0 | Partial sums | This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is, When this limit exists, one says that the series is convergent or summable, or that the sequence ( a 1 , a 2 , a 3 , … ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3318_chunk_0 | Partial sums | Otherwise, the series is said to be divergent.The notation ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}} denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generali... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3319_chunk_0 | Conditionally convergent | In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3320_chunk_0 | Hermitian product | In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the nam... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3321_chunk_0 | Hermitian product | This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any field and the twist is provided by a field automorphism. An application in projective geometry requires that the scalars come from a division ring (skew field), K, and this means that the "vectors... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3322_chunk_0 | Directly finite ring | In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3323_chunk_0 | Directly finite ring | Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth century, Zermelo–Fraenkel set theory, today the most commonly used form of axiomatic set theory, was ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3324_chunk_0 | Directly finite ring | However, there exists a model of Zermelo–Fraenkel set theory without the axiom of choice (ZF) in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF are not strong enough to prove that every set that is Dedekind-finite is finite. There are definitions of finiteness and infiniteness of set... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3325_chunk_0 | Inhabited set | In mathematics, a set A {\displaystyle A} is inhabited if there exists an element a ∈ A {\displaystyle a\in A} . In classical mathematics, the property of being inhabited is equivalent to being non-empty. However, this equivalence is not valid in constructive or intuitionistic logic, and so this separate terminology is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3326_chunk_0 | Vector decomposition | In mathematics, a set B of vectors in a vector space V is called a basis (PL: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The element... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3327_chunk_0 | Separating set | In mathematics, a set S {\displaystyle S} of functions with domain D {\displaystyle D} is called a separating set for D {\displaystyle D} and is said to separate the points of D {\displaystyle D} (or just to separate points) if for any two distinct elements x {\displaystyle x} and y {\displaystyle y} of D , {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3328_chunk_0 | Denumerable set | In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3329_chunk_0 | Unisolvent functions | In mathematics, a set of n functions f1, f2, ..., fn is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors , , … , {\displaystyle {\begin{bmatrix}f_{1}(x_{1})\\f_{1}(x_{2})\\\vdots \\f_{1}(x_{n})\end{bmatrix}},{\begin{bmatrix}f_{2}(x_{1})\\f_{2}(x_{2})\\\vdots \\f_{2}(x_{n})\end{bmatrix}},\dots ,{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3330_chunk_0 | K-trivial set | In mathematics, a set of natural numbers is called a K-trivial set if its initial segments viewed as binary strings are easy to describe: the prefix-free Kolmogorov complexity is as low as possible, close to that of a computable set. Solovay proved in 1975 that a set can be K-trivial without being computable. The Schno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3331_chunk_0 | K-trivial set | Thus the K-trivials are far from random. This is why these sets are studied in the field of algorithmic randomness, which is a subfield of Computability theory and related to algorithmic information theory in computer science. At the same time, K-trivial sets are close to computable. For instance, they are all superlow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3332_chunk_0 | Simultaneous equation | In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: System of linear equations, System of nonlinear eq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3333_chunk_0 | Set of uniqueness | In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3334_chunk_0 | Extensional set | In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set.Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics. Often in mathematics, when one defines an equivalence relation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3335_chunk_0 | Sheaf of sets | On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of she... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3336_chunk_0 | Sheaf of sets | First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3337_chunk_0 | Sheaf of sets | Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3338_chunk_0 | Shift matrix | In mathematics, a shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere. A shift matrix U with ones on the superdiagonal is an upper shift matrix. The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix. The (i,j):th component of U and L ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3339_chunk_0 | Shift matrix | For example, the 5×5 shift matrices are U 5 = ( 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 ) L 5 = ( 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ) . {\displaystyle U_{5}={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\quad L_{5}={\begin{pmatrix}0&0&0&0&0\\1&0&0&0&0\\0&1&... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3340_chunk_0 | Shift matrix | As a linear transformation, a lower shift matrix shifts the components of a column vector one position down, with a zero appearing in the first position. An upper shift matrix shifts the components of a column vector one position up, with a zero appearing in the last position.Premultiplying a matrix A by a lower shift ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3341_chunk_0 | Shift matrix | Similar operations involving an upper shift matrix result in the opposite shift. Clearly all finite-dimensional shift matrices are nilpotent; an n by n shift matrix S becomes the null matrix when raised to the power of its dimension n. Shift matrices act on shift spaces. The infinite-dimensional shift matrices are part... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3342_chunk_0 | Shuffle algebra | In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of two words X, Y: the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation. The shuffle algebra on a finite set is the grad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3343_chunk_0 | Shuffle algebra | Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words. The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3344_chunk_0 | Signalizer functor | In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup. The idea is to try to construct a p ′ {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3345_chunk_0 | Signalizer functor | {\displaystyle G.} The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including Gorenstein (1969) who defined signalizer functors, Glauberman (1976) who proved the Solvable Signalizer Functor Theorem for solvable groups, and McBride (1982a, 1982b) who proved it for all... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3346_chunk_0 | Signature matrix | In mathematics, a signature matrix is a diagonal matrix whose diagonal elements are plus or minus 1, that is, any matrix of the form: A = ( ± 1 0 ⋯ 0 0 0 ± 1 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ ± 1 0 0 0 ⋯ 0 ± 1 ) {\displaystyle A={\begin{pmatrix}\pm 1&0&\cdots &0&0\\0&\pm 1&\cdots &0&0\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3347_chunk_0 | Signature matrix | Noting that signature matrices are both symmetric and involutory, they are thus orthogonal. Consequently, any linear transformation corresponding to a signature matrix constitutes an isometry. Geometrically, signature matrices represent a reflection in each of the axes corresponding to the negated rows or columns. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3348_chunk_0 | Simple group | In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3349_chunk_0 | Friedman's SSCG function | In mathematics, a simple subcubic graph (SSCG) is a finite simple graph in which each vertex has a degree of at most three. Suppose we have a sequence of simple subcubic graphs G1, G2, ... such that each graph Gi has at most i + k vertices (for some integer k) and for no i < j is Gi homeomorphically embeddable into (i.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3350_chunk_0 | Friedman's SSCG function | The function SSCG(k) denotes that length for simple subcubic graphs. The function SCG(k) denotes that length for (general) subcubic graphs. The SCG sequence begins SCG(0) = 6, but then explodes to a value equivalent to fε2*2 in the fast-growing hierarchy. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3351_chunk_0 | Friedman's SSCG function | The SSCG sequence begins slower than SCG, SSCG(0) = 2, SSCG(1) = 5, but then grows rapidly. SSCG(2) = 3 × 2(3 × 295) − 8 ≈ 3.241704 × 1035775080127201286522908640065. Its first and last 20 digits are 32417042291246009846...34057047399148290040. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3352_chunk_0 | Friedman's SSCG function | SSCG(3) is much larger than both TREE(3) and TREETREE(3)(3), that is, the TREE function nested TREE(3) times with 3 at the bottom. Adam P. Goucher claims there is no qualitative difference between the asymptotic growth rates of SSCG and SCG. He writes "It's clear that SCG(n) ≥ SSCG(n), but I can also prove SSCG(4n + 3)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3353_chunk_0 | Singular trace | In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operato... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3354_chunk_0 | Singular trace | American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of trace class operators. Therefore, in distinction to the finite-dimensional case, in infinite dimensions the canonical operator trace is no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3355_chunk_0 | Singular trace | Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes. In heuristic terms, a singular trace corresponds to a way of summing numbers a1, a2, a3, ... that is completely orthogonal or 'singular' with respect to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3356_chunk_0 | Mathematical singularities | In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.For example, the function f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} has a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3357_chunk_0 | Skew gradient | In mathematics, a skew gradient of a harmonic function over a simply connected domain with two real dimensions is a vector field that is everywhere orthogonal to the gradient of the function and that has the same magnitude as the gradient. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3358_chunk_0 | Thom conjecture | In mathematics, a smooth algebraic curve C {\displaystyle C} in the complex projective plane, of degree d {\displaystyle d} , has genus given by the genus–degree formula g = ( d − 1 ) ( d − 2 ) / 2 {\displaystyle g=(d-1)(d-2)/2} .The Thom conjecture, named after French mathematician René Thom, states that if Σ {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3359_chunk_0 | Thom conjecture | There is at least one generalization of this conjecture, known as the symplectic Thom conjecture (which is now a theorem, as proved for example by Peter Ozsváth and Szabó in 2000). It states that a symplectic surface of a symplectic 4-manifold is genus minimizing within its homology class. This would imply the previous... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3360_chunk_0 | Gromov-Ruh theorem | In mathematics, a smooth compact manifold M is called almost flat if for any ε > 0 {\displaystyle \varepsilon >0} there is a Riemannian metric g ε {\displaystyle g_{\varepsilon }} on M such that diam ( M , g ε ) ≤ 1 {\displaystyle {\mbox{diam}}(M,g_{\varepsilon })\leq 1} and g ε {\displaystyle g_{\varepsilon }} is ε {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3361_chunk_0 | Smooth maximum | In mathematics, a smooth maximum of an indexed family x1, ..., xn of numbers is a smooth approximation to the maximum function max ( x 1 , … , x n ) , {\displaystyle \max(x_{1},\ldots ,x_{n}),} meaning a parametric family of functions m α ( x 1 , … , x n ) {\displaystyle m_{\alpha }(x_{1},\ldots ,x_{n})} such that for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3362_chunk_0 | Smooth structure | In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3363_chunk_0 | Sofic group | A finitely generated group is sofic if it is the limit of a sequence of sofic groups. The limit of a sequence of amenable groups (that is, an initially subamenable group) is necessarily sofic, but there exist sofic groups that are not initially subamenable groups.As Gromov proved, Sofic groups are surjunctive. That is,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3364_chunk_0 | Convolution quotient | In mathematics, a space of convolution quotients is a field of fractions of a convolution ring of functions: a convolution quotient is to the operation of convolution as a quotient of integers is to multiplication. The construction of convolution quotients allows easy algebraic representation of the Dirac delta functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3365_chunk_0 | Convolution quotient | The kind of convolution ( f , g ) ↦ f ∗ g {\textstyle (f,g)\mapsto f*g} with which this theory is concerned is defined by ( f ∗ g ) ( x ) = ∫ 0 x f ( u ) g ( x − u ) d u . {\displaystyle (f*g)(x)=\int _{0}^{x}f(u)g(x-u)\,du.} It follows from the Titchmarsh convolution theorem that if the convolution f ∗ g {\textstyle f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3366_chunk_0 | Convolution quotient | As with the construction of the rational numbers from the integers, the field of convolution quotients is a direct extension of the convolution ring from which it was built. Every "ordinary" function f {\displaystyle f} in the original space embeds canonically into the space of convolution quotients as the (equivalence... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3367_chunk_0 | Sparse polynomial | In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables would suggest. For example, x10 + 3x3 - 1 is a sparse polynomial as it is a trinomial with a degree of 10. The motivation for studying sparse polynomials is to con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3368_chunk_0 | Sparsely totient number | In mathematics, a sparsely totient number is a certain kind of natural number. A natural number, n, is sparsely totient if for all m > n, φ ( m ) > φ ( n ) {\displaystyle \varphi (m)>\varphi (n)} where φ {\displaystyle \varphi } is Euler's totient function. The first few sparsely totient numbers are: 2, 6, 12, 18, 30, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3369_chunk_0 | Spherical conic | Like planar conics, spherical conics also satisfy a "reflection property": the great-circle arcs from the two foci to any point on the conic have the tangent and normal to the conic at that point as their angle bisectors. Many theorems about conics in the plane extend to spherical conics. For example, Graves's theorem ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3370_chunk_0 | Spline curve | In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3371_chunk_0 | Split-biquaternion | In mathematics, a split-biquaternion is a hypercomplex number of the form q = w + x i + y j + z k {\displaystyle q=w+xi+yj+zk} where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an ele... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3372_chunk_0 | Split-biquaternion | This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the tensor product of algebras, and as an illustration of the direct sum of alge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3373_chunk_0 | Sporadic groups | In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3374_chunk_0 | Modulus squared | In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3375_chunk_0 | Modulus squared | The adjective which corresponds to squaring is quadratic. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3376_chunk_0 | Modulus squared | For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1)2 = x2 + 2x + 1. One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers x), the square of x is the same as the square of its additive inverse −x. That is, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3377_chunk_0 | Square matrices | In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n {\displaystyle n} . Any two square matrices of the same order can be added and multiplied. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3378_chunk_0 | Square matrices | Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R {\displaystyle R} is a square matrix representing a rotation (rotation matrix) and v {\displaystyle \mathbf {v} } is a column vector describing the position of a point in space, the product R v {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3379_chunk_0 | Diagonally dominant matrix | In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row. More precisely, the matrix A is diagonally dominant if | a i i | ≥ ∑ j... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3380_chunk_0 | Square root | The two roots can be written more concisely using the ± sign as ± x {\displaystyle \pm {\sqrt {x}}} . Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.Square roots of negative numbers can be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3381_chunk_0 | Square-difference-free set | In mathematics, a square-difference-free set is a set of natural numbers, no two of which differ by a square number. Hillel Furstenberg and András Sárközy proved in the late 1970s the Furstenberg–Sárközy theorem of additive number theory showing that, in a certain sense, these sets cannot be very large. In the game of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3382_chunk_0 | Square-free factorization | In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if it has no multiple root in an algebraically closed field containing its coe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3383_chunk_0 | Square-free factorization | In the case of univariate polynomials, the product rule implies that, if p2 divides f, then p divides the formal derivative f ' of f. The converse is also true and hence, f {\displaystyle f} is square-free if and only if 1 {\displaystyle 1} is a greatest common divisor of the polynomial and its derivative.A square-free... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3384_chunk_0 | Square-free factorization | Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra. Over a field of characteristic 0, the quotient of f {\displaystyle f} by its GCD with its derivativ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3385_chunk_0 | Square-free factorization | Over a perfect field of non-zero characteristic p, this quotient is the product of the a i {\displaystyle a_{i}} such that i is not a multiple of p. Further GCD computations and exact divisions allow computing the square-free factorization (see square-free factorization over a finite field). In characteristic zero, a b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3386_chunk_0 | Square integrable | In mathematics, a square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3387_chunk_0 | Square integrable | For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L p {\displaystyle L^{p}} space with p = 2. {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3388_chunk_0 | Square integrable | Among the L p {\displaystyle L^{p}} spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the L p {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3389_chunk_0 | Stable holomorphic vector bundle | In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in Mumford (1963) and later built u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3390_chunk_0 | Stacky curve | In mathematics, a stacky curve is an object in algebraic geometry that is roughly an algebraic curve with potentially "fractional points" called stacky points. A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms. Stacky curves are deeply related to 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3391_chunk_0 | Statistical manifold | In mathematics, a statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. Statistical manifolds provide a setting for the field of information geometry. The Fisher information metric provides a metric on these manifolds. Following this definition, the log-likelihood function i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3392_chunk_0 | Stiff equation | In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3393_chunk_0 | Stiff equation | In order for a numerical method to give a reliable solution to the differential system sometimes the step size is required to be at an unacceptably small level in a region where the solution curve is very smooth. The phenomenon is known as stiffness. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3394_chunk_0 | Stiff equation | In some cases there may be two different problems with the same solution, yet one is not stiff and the other is. The phenomenon cannot therefore be a property of the exact solution, since this is the same for both problems, and must be a property of the differential system itself. Such systems are thus known as stiff s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3395_chunk_0 | Right stochastic matrix | In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. : 9–11 It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. : 9–11 The stochastic matrix wa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3396_chunk_0 | Right stochastic matrix | : 1–8 There are several different definitions and types of stochastic matrices:: 9–11 A right stochastic matrix is a real square matrix, with each row summing to 1. A left stochastic matrix is a real square matrix, with each column summing to 1. A doubly stochastic matrix is a square matrix of nonnegative real numbers ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3397_chunk_0 | Right stochastic matrix | Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a stochastic vector. : 9–11 A common convention in English language mathematics literature is to use row vectors of probabilities and right stochastic matrices rather than column vectors of probabilities and left stochastic matrices;... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3398_chunk_0 | Strange nonchaotic attractor | In mathematics, a strange nonchaotic attractor (SNA) is a form of attractor which, while converging to a limit, is strange, because it is not piecewise differentiable, and also non-chaotic, in that its Lyapunov exponents are non-positive. SNAs were introduced as a topic of study by Grebogi et al. in 1984. SNAs can be d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3399_chunk_0 | Strange nonchaotic attractor | The first experiment to demonstrate a robust strange nonchaotic attractor involved the buckling of a magnetoelastic ribbon driven quasiperiodically by two incommensurate frequencies in the golden ratio. Strange nonchaotic attractors have been robustly observed in laboratory experiments involving magnetoelastic ribbons,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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