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Pro-p group Summary P-adic_analytic_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pro-simplicial set Summary Pro-simplicial_set In mathematics, a pro-simplicial set is an inverse system of simplicial sets. A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets has finite homotopy groups. Pro-simplicial sets show up in shape theory, in the study of localizatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Probability measure Summary 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 inc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical product Summary Mathematical_product In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and x ⋅ ( 2 + x ) {\displaystyle x\cdot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical product Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Direct product of rings Summary Product_ring 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 dire... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Profinite groups Summary Profinite_groups In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Profinite groups Summary 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 ho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Profinite groups Summary Profinite_groups Every profinite group is compact and totally disconnected. A non-compact generalization of the concept is that of locally profinite groups. Even more general are the totally disconnected groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Profinite integer Summary Profinite_integers In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) Z ^ = lim ← Z / n Z = ∏ p Z p {\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} =\prod _{p}\mathbb {Z} _{p}} where lim ← Z / n Z {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Progressive function Summary 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Progressive function Summary 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 fun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Progressive function Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projection (mathematics) Summary Projection_(mathematics) In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projection (mathematics) Summary 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 proje... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projection (mathematics) Summary 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projection (mathematics) Summary 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 cart... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projectionless C*-algebra Summary 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 |
Projectionless C*-algebra Summary 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 equivalen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projectivized vector bundle Summary Projective_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} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective line Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Desarguesian plane Summary Projective_Plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective pl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Desarguesian plane Summary Projective_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 impo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Desarguesian plane Summary Projective_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 embe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic range Summary Projective_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic range Summary Projective_range A projectivity is said to act from one range to another, though the two ranges may coincide as sets. A projective range expresses projective invariance of the relation of projective harmonic conjugates. Indeed, three points on a projective line determine a fourth by this relation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic range Summary Projective_range Application of a projectivity to this quadruple results in four points likewise in the harmonic relation. Such a quadruple of points is termed a harmonic range. In 1940 Julian Coolidge described this structure and identified its originator: Two fundamental one-dimensional forms s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proof by infinite descent Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proof by infinite descent Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proof by infinite descent Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proof by infinite descent Summary 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 desce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Impossibility proof Summary 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 impossi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Impossibility proof Summary 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 |
Impossibility proof Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Impossibility proof Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Impossibility proof Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proof without words Summary Visual_proof 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 mathema... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Irreducible ideal Summary Irreducible_ideal In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Property (mathematics) Summary Mathematical_property 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Protorus Summary 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 soleno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prototile Summary Prototile In mathematics, a prototile is one of the shapes of a tile in a tessellation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo-canonical variety Summary Pseudo-canonical_variety In mathematics, a pseudo-canonical variety is an algebraic variety of "general type". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudofinite field Summary Pseudofinite_field In mathematics, a pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo-monotone operator Summary Pseudo-monotone_operator In mathematics, a pseudo-monotone operator from a reflexive Banach space into its continuous dual space is one that is, in some sense, almost as well-behaved as a monotone operator. Many problems in the calculus of variations can be expressed using operators tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo-reductive algebraic group Summary Pseudo-reductive_group In mathematics, a pseudo-reductive group over a field k (sometimes called a k-reductive group) is a smooth connected affine algebraic group defined over k whose k-unipotent radical (i.e., largest smooth connected unipotent normal k-subgroup) is trivial. Ov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo-reductive algebraic group Summary Pseudo-reductive_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 pse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudo-functor Summary Pseudo-functor In mathematics, a pseudofunctor F is a mapping between 2-categories, or from a category to a 2-category, that is just like a functor except that F ( f ∘ g ) = F ( f ) ∘ F ( g ) {\displaystyle F(f\circ g)=F(f)\circ F(g)} and F ( 1 ) = 1 {\displaystyle F(1)=1} do not hold as exact eq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudogamma function Summary 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudogamma function Summary 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 ) {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Local Lie group Summary Pseudogroup 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 differ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudomanifold Summary Pseudomanifold In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of z 2 = x 2 + y 2 {\displaystyle z^{2}=x^{2}+y^{2}} forms a pseudomanifold. A pseudomanifol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudometrizable space Summary Pseudometrisable_space In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric space, every semin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudoreflection Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Induced bundle Summary Pullback_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pullback Summary Pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Square pyramidal number Summary 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 figur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Square pyramidal number Summary Square_pyramidal_number They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-analog Summary 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 arbitraril... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-analog Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic algebra Summary 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 quad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic differential Summary 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 quadrati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Solving quadratic equations with continued fractions Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Solving quadratic equations with continued fractions Summary 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 evaluat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic space Summary Signature_of_a_quadratic_form 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 co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperbolic plane (quadratic forms) Summary Isotropic_subspace 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperbolic plane (quadratic forms) Summary Isotropic_subspace 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic integral Summary 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 |
Quadratic irrationalities Summary Quadratic_irrational_numbers 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 f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic irrationalities Summary Quadratic_irrational_numbers This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic irrationalities Summary Quadratic_irrational_numbers {\displaystyle {d \over a+b{\sqrt {c}}}={ad-bd{\sqrt {c}} \over a^{2}-b^{2}c}.} Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that all real quadratic irrationals, and only real quadrati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic irrationalities Summary Quadratic_irrational_numbers 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 eventua... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic functions Summary Second_degree_polynomial 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 associa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic functions Summary Second_degree_polynomial 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 ax... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic functions Summary Second_degree_polynomial 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic functions Summary Second_degree_polynomial 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+ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic set Summary Quadratic_set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic-linear algebra Summary 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 unive... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratically closed field Summary Quadratically_closed_field In mathematics, a quadratically closed field is a field in which every element has a square root. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadric (algebraic geometry) Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadric (algebraic geometry) Summary 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 considere... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadric Summary Quadric_hypersurface 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 irredu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadric Summary Quadric_hypersurface The values Q, P and R are often taken to be over real numbers or complex numbers, but a quadric may be defined over any field. A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see § Normal f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantaloid Summary Quantaloid In mathematics, a quantaloid is a category enriched over the category Sup of suplattices. In other words, for any objects a and b the morphism object between them is not just a set but a complete lattice, in such a way that composition of morphisms preserves all joins: ( ⋁ i f i ) ∘ ( ⋁ j ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantum affine algebra Summary 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 the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantum groupoid Summary 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)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantized enveloping algebra Summary Quantum_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}({\mathf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quartic equation Summary 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 polyno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-Frobenius Lie algebra Summary 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:... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-Lie algebra Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Denjoy–Carleman theorem Summary Quasi-analytic_function 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 iden... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-finite field Summary Quasi-finite_field In mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite (i.e. non-archimedean local fields), but the theory applies equally well when the residue fi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-invariant measure Summary 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-invariant measure Summary Quasi-invariant_measure Considering the whole equivalence class of measures ν, equivalent to μ, it is also the same to say that T preserves the class as a whole, mapping any such measure to another such. Therefore, the concept of quasi-invariant measure is the same as invariant measure c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-polynomial Summary 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 muc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-polynomial Summary 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 −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-projective variety Summary Quasiprojective_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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-split group Summary Quasi-split_group In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasitopological space Summary 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 introdu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasicircle Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasiconcave function Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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