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wiki_3100_chunk_0 | Polynomial matrix | An example 3×3 polynomial matrix, degree 2: P = ( 1 x 2 x 0 2 x 2 3 x + 2 x 2 − 1 0 ) = ( 1 0 0 0 0 2 2 − 1 0 ) + ( 0 0 1 0 2 0 3 0 0 ) x + ( 0 1 0 0 0 0 0 1 0 ) x 2 . {\displaystyle P={\begin{pmatrix}1&x^{2}&x\\0&2x&2\\3x+2&x^{2}-1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&2\\2&-1&0\end{pmatrix}}+{\begin{pmatrix}0&0&... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3101_chunk_0 | Polynomial sequence | In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3102_chunk_0 | Generalized Appell polynomials | In mathematics, a polynomial sequence { p n ( z ) } {\displaystyle \{p_{n}(z)\}} has a generalized Appell representation if the generating function for the polynomials takes on a certain form: K ( z , w ) = A ( w ) Ψ ( z g ( w ) ) = ∑ n = 0 ∞ p n ( z ) w n {\displaystyle K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\infty }p_{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3103_chunk_0 | Polynomial transformation | In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3104_chunk_0 | Sigma finite measure | In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞). A set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3105_chunk_0 | Herglotz representation theorem | In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911. It can be used to give ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3106_chunk_0 | Positive-semidefinite function | In mathematics, a positive-definite function is, depending on the context, either of two types of function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3107_chunk_0 | Power of three | In mathematics, a power of three is a number of the form 3n where n is an integer, that is, the result of exponentiation with number three as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to non-negative values, so there are 1, 3, and 3 multiplied by itself a c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3108_chunk_0 | Power series | In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that ev... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3109_chunk_0 | Power series | In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3110_chunk_0 | Pre-Lie algebra | In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space. The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras. Pre-Lie algebras have been con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3111_chunk_0 | Prehomogeneous vector space | In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space V together with a subgroup G of the general linear group GL(V) such that G has an open dense orbit in V. Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3112_chunk_0 | Primary cyclic group | In mathematics, a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p. That is, it is a cyclic group of order pm, Cpm, for some prime number p, and natural number m. Every finite abelian group G may be written as a finite direct sum of primary cyclic groups, as stat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3113_chunk_0 | Prime geodesic | In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3114_chunk_0 | Chen prime | In mathematics, a prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2p + 2 therefore satisfies Chen's theorem. The Chen primes are named after Chen Jingrun, who proved in 1966 that there are infinitely many such primes. This result wou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3115_chunk_0 | Primefree sequence | In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be composite numbers that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3116_chunk_0 | Primorial Prime | In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes).Primality tests show that pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... (sequence A057704 in the OEIS) pn# + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11, ... (sequence A01454... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3117_chunk_0 | Principal bundles | Likewise, there is not generally a projection onto G {\displaystyle G} generalizing the projection onto the second factor, X × G → G {\displaystyle X\times G\to G} that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3118_chunk_0 | Principal bundles | The group G , {\displaystyle G,} in this case, is the general linear group, which acts on the right in the usual way: by changes of basis. Since there is no natural way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section. Principal bundles have important appli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3119_chunk_0 | Principal ideal domain | In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3120_chunk_0 | Principal ideal domain | Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (altho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3121_chunk_0 | Principal ideal ring | Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains. A commutative principal ideal ring which is also an integral domain is said to be a principal ideal domain (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessaril... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3122_chunk_0 | Principal subalgebra | In mathematics, a principal subalgebra of a complex simple Lie algebra is a 3-dimensional simple subalgebra whose non-zero elements are regular. A finite-dimensional complex simple Lie algebra has a unique conjugacy class of principal subalgebras, each of which is the span of an sl2-triple. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3123_chunk_0 | Pro-p group | In mathematics, a pro-p group (for some prime number p) is a profinite group G {\displaystyle G} such that for any open normal subgroup N ◃ G {\displaystyle N\triangleleft G} the quotient group G / N {\displaystyle G/N} is a p-group. Note that, as profinite groups are compact, the open subgroups are exactly the closed ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3124_chunk_0 | Pro-p group | The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite rank, i.e. there exists a positive integer r {\displaystyle r} such that any closed subgroup has a topological generating set... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3125_chunk_0 | Probability measure | In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3126_chunk_0 | Mathematical product | When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well. There are many different kinds of products in mathematics: beside... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3127_chunk_0 | Direct product of rings | In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the category of rings. Since direct products are defined up to an isomorphism,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3128_chunk_0 | Profinite groups | Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems.To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjectiv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3129_chunk_0 | Progressive function | In mathematics, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform is supported by positive frequencies only: s u p p f ^ ⊆ R + . {\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{+}.} It is called super regressive if and only if the time reversed function f(−t) is progressive,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3130_chunk_0 | Progressive function | The complex conjugate of a progressive function is regressive, and vice versa. The space of progressive functions is sometimes denoted H + 2 ( R ) {\displaystyle H_{+}^{2}(R)} , which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula f ( t ) = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3131_chunk_0 | Progressive function | {\displaystyle f(t+iu)=\int _{0}^{\infty }e^{2\pi is(t+iu)}{\hat {f}}(s)\,ds=\int _{0}^{\infty }e^{2\pi ist}e^{-2\pi su}{\hat {f}}(s)\,ds.} Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal line will arise in this manner. Regressive functions are sim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3132_chunk_0 | Projection (mathematics) | The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: The projection from a point onto ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3133_chunk_0 | Projection (mathematics) | The points P such that the line CP is parallel to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see Projective geometry for a formalization of this terminology). The projection of the point C itself is not defined. The projection parallel ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3134_chunk_0 | Projection (mathematics) | This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.In cartography, a map projection is a map of a part of the surfac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3135_chunk_0 | Projectionless C*-algebra | In mathematics, a projectionless C*-algebra is a C*-algebra with no nontrivial projections. For a unital C*-algebra, the projections 0 and 1 are trivial. While for a non-unital C*-algebra, only 0 is considered trivial. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3136_chunk_0 | Projectionless C*-algebra | The problem of whether simple infinite-dimensional C*-algebras with this property exist was posed in 1958 by Irving Kaplansky, and the first example of one was published in 1981 by Bruce Blackadar. For commutative C*-algebras, being projectionless is equivalent to its spectrum being connected. Due to this, being projec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3137_chunk_0 | Projectivized vector bundle | In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle if it is locally a projective n-space; i.e., X × S U ≃ P U n {\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}} and transition automorphisms are linear. Ove... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3138_chunk_0 | Projective line | In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3139_chunk_0 | Desarguesian plane | Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3140_chunk_0 | Desarguesian plane | There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces. Such embeddability is a consequence of a property kno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3141_chunk_0 | Harmonic range | In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instance, a correlation interchanges the points of a projective range with the l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3142_chunk_0 | Proof by infinite descent | In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3143_chunk_0 | Proof by infinite descent | An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example—a minimal counterexample—can then be inferred. Once there, one would try to prove that if a smallest solution exists, then it must imply the existence of a smaller solution (in some sense)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3144_chunk_0 | Proof by infinite descent | The earliest uses of the method of infinite descent appear in Euclid's Elements. A typical example is Proposition 31 of Book 7, in which Euclid proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number.The method was much later developed by Fermat, who coined the term and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3145_chunk_0 | Proof by infinite descent | In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in arithmetic progression). In some cases, to the modern eye, his "method of infinite descent" is an exploitation of the inversion of the doub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3146_chunk_0 | Impossibility proof | In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Proofs of im... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3147_chunk_0 | Impossibility proof | Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic. The irrationality of the square root of 2 is one of the oldest proofs of impossibility. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3148_chunk_0 | Impossibility proof | It shows that it is impossible to express the square root of 2 as a ratio of two integers. Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic), and tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3149_chunk_0 | Impossibility proof | A problem that arose in the 16th century was creating a general formula using radicals to express the solution of any polynomial equation of fixed degree k, where k ≥ 5. In the 1820s, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) showed this to be impossible, using concepts such as solvable grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3150_chunk_0 | Impossibility proof | Gödel's incompleteness theorems were other examples that uncovered fundamental limitations in the provability of formal systems.In computational complexity theory, techniques like relativization (the addition of an oracle) allow for "weak" proofs of impossibility, in that proofs techniques that are not affected by rela... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3151_chunk_0 | Proof without words | In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3152_chunk_0 | Property (mathematics) | In mathematics, a property is any characteristic that applies to a given set. Rigorously, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or equivalently, as the subset of X for which p holds; i.e. the set {x | p(x) = true... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3153_chunk_0 | Protorus | In mathematics, a protorus is a compact connected topological abelian group. Equivalently, it is a projective limit of tori (products of a finite number of copies of the circle group), or the Pontryagin dual of a discrete torsion-free abelian group. Some examples of protori are given by solenoid groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3154_chunk_0 | Pseudo-reductive algebraic group | Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic (even over a perfect field of constants). Springer (1998) gives an exposition of Tits' results on pseudo-reductive groups, while Conrad, Gabber & Prasad (2010) build... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3155_chunk_0 | Pseudogamma function | In mathematics, a pseudogamma function is a function that interpolates the factorial. The gamma function is the most famous solution to the problem of extending the notion of the factorial beyond the positive integers only. However, it is clearly not the only solution, as, for any set of points, an infinite number of c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3156_chunk_0 | Pseudogamma function | Such a curve, namely one which interpolates the factorial but is not equal to the gamma function, is known as a pseudogamma function. The two most famous pseudogamma functions are Hadamard's gamma function: H ( x ) = ψ ( 1 − x 2 ) − ψ ( 1 2 − x 2 ) 2 Γ ( 1 − x ) {\displaystyle H(x)={\frac {\psi \left(1-{\frac {x}{2}}\r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3157_chunk_0 | Local Lie group | In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie to investigate symmetries of differential equations, rather than out of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3158_chunk_0 | Pseudoreflection | In mathematics, a pseudoreflection is an invertible linear transformation of a finite-dimensional vector space such that it is not the identity transformation, has a finite (multiplicative) order, and fixes a hyperplane. The concept of pseudoreflection generalizes the concepts of reflection and complex reflection and i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3159_chunk_0 | Induced bundle | In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π: E → B and a continuous map f: B′ → B one can define a "pullback" of E by f as a bundle f*E over B′. The fiber of f*E over a point b′ in B′ is just the fiber of E over f(b′). Thus f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3160_chunk_0 | Square pyramidal number | In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3161_chunk_0 | Q-analog | In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3162_chunk_0 | Q-analog | q-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic dynamical systems. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian groups in genera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3163_chunk_0 | Quadratic algebra | In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3164_chunk_0 | Quadratic differential | In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface has a natural int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3165_chunk_0 | Solving quadratic equations with continued fractions | In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is a x 2 + b x + c = 0 , {\displaystyle ax^{2}+bx+c=0,} where a ≠ 0. The quadratic equation on a number x {\displaystyle x} can be solved using the well-known quadratic formula, which can be derived by completing the sq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3166_chunk_0 | Solving quadratic equations with continued fractions | That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm. If the roots are real, there is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3167_chunk_0 | Quadratic space | In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3168_chunk_0 | Hyperbolic plane (quadratic forms) | In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3169_chunk_0 | Hyperbolic plane (quadratic forms) | A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that (V, q) is quadratic space and W is a subspace of V. Then W is called an isotropic subspace of V if some vector in it is isotropic, a totally isotropic subspace if all vectors in i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3170_chunk_0 | Quadratic integral | In mathematics, a quadratic integral is an integral of the form It can be evaluated by completing the square in the denominator. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3171_chunk_0 | Quadratic irrationalities | In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3172_chunk_0 | Quadratic irrationalities | The correspondence is explicitly provided by Minkowski's question mark function, and an explicit construction is given in that article. It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3173_chunk_0 | Quadratic functions | In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3174_chunk_0 | Quadratic functions | For example, a univariate (single-variable) quadratic function has the form f ( x ) = a x 2 + b x + c , a ≠ 0 , {\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0,} where x is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the y-axis. If a quadratic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3175_chunk_0 | Quadratic functions | The solutions of a quadratic equation are the zeros of the corresponding quadratic function. The bivariate case in terms of variables x and y has the form f ( x , y ) = a x 2 + b x y + c y 2 + d x + e y + f , {\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,} with at least one of a, b, c not equal to zero. The zeros of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3176_chunk_0 | Quadratic functions | A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: f ( x , y , z ) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z + g x + h y + i z + j , {\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,} where at least one of the coeffici... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3177_chunk_0 | Quadratic-linear algebra | In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by Polishchuk and Positselski (2005, p.101). An example is the universal enveloping algebra of a Lie algebra, with generators ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3178_chunk_0 | Quadric (algebraic geometry) | In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3179_chunk_0 | Quadric (algebraic geometry) | An example is the quadric surface x y = z w {\displaystyle xy=zw} in projective space P 3 {\displaystyle {\mathbf {P} }^{3}} over the complex numbers C. A quadric has a natural action of the orthogonal group, and so the study of quadrics can be considered as a descendant of Euclidean geometry. Many properties of quadri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3180_chunk_0 | Quadric | In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D +... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3181_chunk_0 | Quantum affine algebra | In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cart... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3182_chunk_0 | Quantum groupoid | In mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the information of a groupoid can be contained in its monoidal category of representations (by a version of Tannaka–Krein duality), in its groupoid algebra or in the commut... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3183_chunk_0 | Quantized enveloping algebra | In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the quantum enveloping algebra is typically denoted as U q ( g ) {\displaystyle U_{q}({\mathfrak {g}})} . The notation was introduced by Drinfeld and indepen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3184_chunk_0 | Quartic equation | In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is a x 4 + b x 3 + c x 2 + d x + e = 0 {\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0\,} where a ≠ 0. The quartic is the highest order polynomial equation that can be solved by radica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3185_chunk_0 | Quasi-Frobenius Lie algebra | In mathematics, a quasi-Frobenius Lie algebra ( g , , β ) {\displaystyle ({\mathfrak {g}},,\beta )} over a field k {\displaystyle k} is a Lie algebra ( g , ) {\displaystyle ({\mathfrak {g}},)} equipped with a nondegenerate skew-symmetric bilinear form β: g × g → k {\displaystyle \beta :{\mathfrak {g}}\times {\mathfra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3186_chunk_0 | Quasi-Lie algebra | In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom = 0 {\displaystyle =0} replaced by = − {\displaystyle =-} (anti-symmetry).In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when consider... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3187_chunk_0 | Denjoy–Carleman theorem | In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on an interval ⊂ R, and at some point f and all of its derivatives are zero, then f is identically zero on all of . Quasi-analytic classes are broa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3188_chunk_0 | Quasi-invariant measure | In mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T. An important class of examples occurs when X is a smooth manifold M, T is a diffeomorphism of M, and μ is any measure that l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3189_chunk_0 | Quasi-polynomial | In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3190_chunk_0 | Quasi-polynomial | If c d ( k ) {\displaystyle c_{d}(k)} is not identically zero, then the degree of q {\displaystyle q} is d {\displaystyle d} . Equivalently, a function f: N → N {\displaystyle f\colon \mathbb {N} \to \mathbb {N} } is a quasi-polynomial if there exist polynomials p 0 , … , p s − 1 {\displaystyle p_{0},\dots ,p_{s-1}} su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3191_chunk_0 | Quasi-projective variety | In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in scheme theory, where a quasi-projective scheme is a locally closed subsc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3192_chunk_0 | Quasitopological space | In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space. They were introduced by Spanier, who showed that there is a natural qua... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3193_chunk_0 | Quasicircle | In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by Pfluger (1961) and Tienari (1962), in the older literature (in German) they were referred to as quasiconformal curves, a term... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3194_chunk_0 | Quasiconcave function | In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form ( − ∞ , a ) {\displaystyle (-\infty ,a)} is a convex set. For a function of a single variable, along any stretch of the curve the highes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3195_chunk_0 | Quasiperfect number | In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far. The quasiperfect numbers are... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3196_chunk_0 | Quasiperiodic function | In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function f {\displaystyle f} is quasiperiodic with quasiperiod ω {\displaystyle \omega } if f ( z + ω ) = g ( z , f ( z ) ) {\displaystyle f(z+\omega )=g(z,f(z))} , where g {\displaystyle g} is a "simpler" fun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3197_chunk_0 | Quasiperiodic function | Another example is provided by the Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function. Functions with an additive functional equation f ( z + ω ) = f ( z ) + a z + b {\displaystyle f(z+\omega )=f(z)+az+b\ } are also called quasiper... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3198_chunk_0 | Quasirandom group | In mathematics, a quasirandom group is a group that does not contain a large product-free subset. Such groups are precisely those without a small non-trivial irreducible representation. The namesake of these groups stems from their connection to graph theory: bipartite Cayley graphs over any subset of a quasirandom gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3199_chunk_0 | Quasitoric manifold | In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth 2 n {\displaystyle 2n} -dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an n {\displaystyle n} -dimensional torus, with orbit space ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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