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wiki_2400_chunk_0 | Binary functions | In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs. Precisely stated, a function f {\displaystyle f} is binary if there exists sets X , Y , Z {\displaystyle X,Y,Z} such that f: X × Y → Z {\displaystyle \,f\colon X\times Y\rightarrow Z} wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2401_chunk_0 | Binary operation | Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a binary operation. For example, scalar multiplication of vector spaces takes a scalar and a vector... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2402_chunk_0 | Binary quadratic form | In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables q ( x , y ) = a x 2 + b x y + c y 2 , {\displaystyle q(x,y)=ax^{2}+bxy+cy^{2},\,} where a, b, c are the coefficients. When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2403_chunk_0 | Binary quadratic form | This choice is motivated by their status as the driving force behind the development of algebraic number theory. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to quadratic and more general number fields, but advances specific to binary quadratic for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2404_chunk_0 | Binary quadratic form | x 2 + y 2 , x 2 + 2 y 2 , x 2 − 3 y 2 {\displaystyle x^{2}+y^{2},x^{2}+2y^{2},x^{2}-3y^{2}} and so on are quadratic forms, and the theory of quadratic forms gives a unified way of looking at and proving these theorems. Another instance of quadratic forms is Pell's equation x 2 − n y 2 = 1 {\displaystyle x^{2}-ny^{2}=1}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2405_chunk_0 | Well-founded order | In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded. A relation R is converse well-founded, upwards well-founded or No... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2406_chunk_0 | Left-total relation | In mathematics, a binary relation R ⊆ X×Y between two sets X and Y is total (or left total) if the source set X equals the domain {x: there is a y with xRy }. Conversely, R is called right total if Y equals the range {y: there is an x with xRy }. When f: X → Y is a function, the domain of f is all of X, hence f is a to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2407_chunk_0 | Left-unique relation | In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y. It is a generalization of the more widely understood idea of a unary ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2408_chunk_0 | Left-unique relation | These include, among others: the "is greater than", "is equal to", and "divides" relations in arithmetic; the "is congruent to" relation in geometry; the "is adjacent to" relation in graph theory; the "is orthogonal to" relation in linear algebra.A function may be defined as a special kind of binary relation. Binary re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2409_chunk_0 | Left-unique relation | {\displaystyle X\times Y.} A binary relation is called a homogeneous relation when X = Y. A binary relation is also called a heterogeneous relation when it is not necessary that X = Y. Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2410_chunk_0 | Left-unique relation | Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called conce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2411_chunk_0 | Relation (mathematics) | For most common relations in mathematics, special symbols are introduced, like "<" for "is less than", and "|" for "is a nontrivial divisor of", and, most popular "=" for "is equal to". For example, "1<3", "1 is less than 3", and "(1,3) ∈ Rless" mean all the same; some authors also write "(1,3) ∈ (<)". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2412_chunk_0 | Relation (mathematics) | Mathematical theorems are known about combinations of relation properties, such as "A transitive relation is irreflexive if, and only if, it is asymmetric". Of particular importance are relations that satisfy certain combinations of properties. A partial order is a relation that is reflexive, antisymmetric, and transit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2413_chunk_0 | Biorthogonal system | In mathematics, a biorthogonal system is a pair of indexed families of vectors such that where E {\displaystyle E} and F {\displaystyle F} form a pair of topological vector spaces that are in duality, ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } is a bilinear mapping and δ i , j {\displaystyle \delta _{i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2414_chunk_0 | Biquaternion algebra | In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2415_chunk_0 | Bisymmetric matrix | In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = AT and AJ = JA where J is the n × n exchange matrix. For example, any matrix of the form = {\displaystyle {\begin{bmatrix}a&b&c&d&e\\b&f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2416_chunk_0 | Bivector (complex) | In mathematics, a bivector is the vector part of a biquaternion. For biquaternion q = w + xi + yj + zk, w is called the biscalar and xi + yj + zk is its bivector part. The coordinates w, x, y, z are complex numbers with imaginary unit h: x = x 1 + h x 2 , y = y 1 + h y 2 , z = z 1 + h z 2 , h 2 = − 1 = i 2 = j 2 = k 2 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2417_chunk_0 | Bivector (complex) | The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r1 and r2 are right versors so that r 1 2 = − 1 = r 2 2 {\displaystyle r_{1}^{2}=-1=r_{2}^{2}} , then the biquaternion curve {exp θr1: θ ∈ R} traces over and over the unit circle in the plane {x + yr1: x, y ∈ R}. Such a circle corresponds... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2418_chunk_0 | Bivector (complex) | "The commutator product of this Lie algebra is just twice the cross product on R3, for instance, = ij − ji = 2k, which is twice i × j. As Shaw wrote in 1970: Now it is well known that the Lie algebra of the homogeneous Lorentz group can be considered to be that of bivectors under commutation. The Lie algebra of bivect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2419_chunk_0 | Bivector (complex) | : 665 The popular text Vector Analysis (1901) used the term. : 249 Given a bivector r = r1 + hr2, the ellipse for which r1 and r2 are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r.: 436 In the standard linear representation of biquaternions as 2 × 2 complex matrices acting on th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2420_chunk_0 | Bivector (complex) | Ludwik Silberstein studied a complexified electromagnetic field E + hB, where there are three components, each a complex number, known as the Riemann–Silberstein vector. "Bivectors help describe elliptically polarized homogeneous and inhomogeneous plane waves – one vector for direction of propagation, one for amplitud... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2421_chunk_0 | Bivector | In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector can be thought of as being of degree two. Bivectors have applications in ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2422_chunk_0 | Partitioned matrix | In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2423_chunk_0 | Partitioned matrix | This notion can be made more precise for an n {\displaystyle n} by m {\displaystyle m} matrix M {\displaystyle M} by partitioning n {\displaystyle n} into a collection rowgroups {\displaystyle {\text{rowgroups}}} , and then partitioning m {\displaystyle m} into a collection colgroups {\displaystyle {\text{colgroups}}} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2424_chunk_0 | Block matrix pseudoinverse | In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing, which are based on the least squares method. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2425_chunk_0 | Bouquet graph | In mathematics, a bouquet graph B m {\displaystyle B_{m}} , for an integer parameter m {\displaystyle m} , is an undirected graph with one vertex and m {\displaystyle m} edges, all of which are self-loops. It is the graph-theoretic analogue of the topological bouquet, a space of m {\displaystyle m} circles joined at a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2426_chunk_0 | Boxcar function | In mathematics, a boxcar function is any function which is zero over the entire real line except for a single interval where it is equal to a constant, A. The function is named after its graph's resemblance to a boxcar, a type of railroad car. The boxcar function can be expressed in terms of the uniform distribution as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2427_chunk_0 | Bracket algebra | In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants. Given that L is a proper signed alphabet and Super is the supersymmetric algebra, the bracket algebra Bracket of dimension n over the field K is the quoti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2428_chunk_0 | Braided Hopf algebra | In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra H, particularly the Nichols algebra of a braided vector space in that category. The notion should not be confused with quasitriangula... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2429_chunk_0 | Braided vector space | In mathematics, a braided vector space V {\displaystyle \;V} is a vector space together with an additional structure map τ {\displaystyle \tau } symbolizing interchanging of two vector tensor copies: τ: V ⊗ V ⟶ V ⊗ V {\displaystyle \tau :\;V\otimes V\longrightarrow V\otimes V} such that the Yang–Baxter equation is fulf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2430_chunk_0 | Braided vector space | A superspace has a braiding with negative sign in braiding two odd vectors. More generally, a diagonal braiding means that for a V {\displaystyle \;V} -base x i {\displaystyle x_{i}} we have τ ( x i ⊗ x j ) = q i j ( x j ⊗ x i ) {\displaystyle \tau (x_{i}\otimes x_{j})=q_{ij}(x_{j}\otimes x_{i})} A good source for brai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2431_chunk_0 | Branched manifold | In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restricted type and admits a well-defined tangent space at each point. A branched n-manifold is covered by n-dimensional "coordinate charts", each of which involves one or several "branches" homeomo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2432_chunk_0 | Bullet-nose curve | In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation a 2 y 2 − b 2 x 2 = x 2 y 2 {\displaystyle a^{2}y^{2}-b^{2}x^{2}=x^{2}y^{2}\,} The bullet curve has three double points in the real projective plane, at x = 0 and y = 0, x = 0 and z = 0, and y = 0 and z ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2433_chunk_0 | Test function | In mathematics, a bump function (also called a test function) is a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } on a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2434_chunk_0 | Bundle (mathematics) | In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E→ B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2435_chunk_0 | Bundle homomorphism | In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2436_chunk_0 | Canonical basis | In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta. In a polynomial ring, it refers to its standard basis given... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2437_chunk_0 | Character (topology) | In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2438_chunk_0 | Cardinal addition | A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2439_chunk_0 | Cardinal addition | If the axiom of choice is not true (see Axiom of choice § Independence), there are infinite cardinals that are not aleph numbers. Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra and mathematical anal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2440_chunk_0 | Category (mathematics) | Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2441_chunk_0 | Category (mathematics) | In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the semantics of programming languages. Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2442_chunk_0 | Cochain complexes | In algebraic topology, the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2443_chunk_0 | Change of variables | In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. Change of variables is an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2444_chunk_0 | Change of variables | A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial: x 6 − 9 x 3 + 8 = 0. {\displaystyle x^{6}-9x^{3}+8=0.} Sixth-degree polynomial equations are generally impossible to solve in terms of radicals (see Abel–Ruffini theorem). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2445_chunk_0 | Change of variables | This particular equation, however, may be written ( x 3 ) 2 − 9 ( x 3 ) + 8 = 0 {\displaystyle (x^{3})^{2}-9(x^{3})+8=0} (this is a simple case of a polynomial decomposition). Thus the equation may be simplified by defining a new variable u = x 3 {\displaystyle u=x^{3}} . Substituting x by u 3 {\displaystyle {\sqrt{u}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2446_chunk_0 | Change of variables | {\displaystyle u=1\quad {\text{and}}\quad u=8.} The solutions in terms of the original variable are obtained by substituting x3 back in for u, which gives x 3 = 1 and x 3 = 8. {\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.} Then, assuming that one is interested only in real solutions, the solutions of the origi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2447_chunk_0 | Chaos machine | In mathematics, a chaos machine is a class of algorithms constructed on the base of chaos theory (mainly deterministic chaos) to produce pseudo-random oracle. It represents the idea of creating a universal scheme with modular design and customizable parameters, which can be applied wherever randomness and sensitiveness... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2448_chunk_0 | Character group | In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that arise in the related context of character theory. Whenever a group is represented by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2449_chunk_0 | Character group | The characters of irreducible representations are orthogonal.The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the discrete Fourier transform. For locally ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2450_chunk_0 | Character (mathematics) | In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2451_chunk_0 | Pólya-Vinogradov inequality | In mathematics, a character sum is a sum ∑ χ ( n ) {\textstyle \sum \chi (n)} of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of findin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2452_chunk_0 | Characteristic number | In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2453_chunk_0 | Characterization theorem | In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property P characterizes object X" is to say that not only does X have property P, but that X is the only thing that has property P (i.e., P is a de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2454_chunk_0 | Characterization theorem | Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P". It is also common to find statements such as "Property Q characterizes Y up to isomorphism". The first type of statement says in different words that the extension of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2455_chunk_0 | Characterization theorem | A reference on mathematical terminology notes that characteristic originates from the Greek term kharax, "a pointed stake":From Greek kharax came kharakhter, an instrument used to mark or engrave an object. Once an object was marked, it became distinctive, so the character of something came to mean its distinctive natu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2456_chunk_0 | Characterization theorem | Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in Mathematical Reviews, as of 2018, more than 24,000 articles contain the word in the article title, and 93,600 somewhere in the review. In an arbitrary context o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2457_chunk_0 | Chiral algebra | In mathematics, a chiral algebra is an algebraic structure introduced by Beilinson & Drinfeld (2004) as a rigorous version of the rather vague concept of a chiral algebra in physics. In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-module... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2458_chunk_0 | Chord diagram (mathematics) | The crossing pattern of chords in a chord diagram may be described by a circle graph, the intersection graph of the chords: it has a vertex for each chord and an edge for each two chords that cross.In knot theory, a chord diagram can be used to describe the sequence of crossings along the planar projection of a knot, w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2459_chunk_0 | Principal circle bundle | In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S 1 {\displaystyle S^{1}} . Oriented circle bundles are also known as principal U(1)-bundles, or equivalently, as principal SO(2)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2460_chunk_0 | Classification theorem | In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues related to classification are the following. The equivalence problem is "given t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2461_chunk_0 | Classification theorem | A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it. A computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the equivalence problem. A canonical form ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2462_chunk_0 | Clean ring | Every clean ring is an exchange ring. A matrix ring over a clean ring is itself clean. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2463_chunk_0 | Closure space | In mathematics, a closure operator on a set S is a function cl: P ( S ) → P ( S ) {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal {P}}(S)} from the power set of S to itself that satisfies the following conditions for all sets X , Y ⊆ S {\displaystyle X,Y\subseteq S} Closure operators are deter... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2464_chunk_0 | Leading coefficient | In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. For example, in the polynomial with variables x {\displaystyle x} and y {\displaystyle y} , the first two terms have the coefficients 7 and −3. The third term 1.5 is the constant coefficient. In the final... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2465_chunk_0 | Leading coefficient | For example, if y is considered a parameter in the above expression, then the coefficient of x would be −3y, and the constant coefficient (with respect to x) would be 1.5 + y. When one writes it is generally assumed that x is the only variable, and that a, b and c are parameters; thus the constant coefficient is c in t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2466_chunk_0 | Leading coefficient | In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression. It may be a number (dimensionless), in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2467_chunk_0 | Leading coefficient | When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter.For example, the polynomial 2 x 2 − x + 3 {\displaystyle 2x^{2}-x+3} has coefficients 2, −1, and 3, and the powers of the variable x {\displaystyle x} in the polynomial a x 2 + b x + c {\displaystyle a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2468_chunk_0 | Leading coefficient | The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and a, respectively. In the context of differential equations, an equation can often be written as equating to zero a polynomial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2469_chunk_0 | Leading coefficient | In this case, the coefficients of the differential equation are the coefficients of this polynomial, and are generally non-constant functions. A coefficient is a constant coefficient when it is a constant function. For avoiding confusion, the coefficient that is not attached to unknown functions and their derivative is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2470_chunk_0 | Coercive operator | In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context different exact definitions of this idea are in use. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2471_chunk_0 | Coframe | In mathematics, a coframe or coframe field on a smooth manifold M {\displaystyle M} is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M {\displaystyle M} , one has a natural map from v k: ⨁ k T ∗ M → ⋀ k T ∗ M {\displaystyle v_{k}:\bigoplus ^{k}T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2472_chunk_0 | Coincidence point | In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain having the same image. Formally, given two functions f , g: X → Y {\displaystyle f,g\colon X\rightarrow Y} we say that a point x in X is a coincidence point of f and g if f(x) = g(x).Coincidence theory (the st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2473_chunk_0 | Coincidence point | Notable among them, in the setting of manifolds, is the Lefschetz coincidence theorem, which is typically known only in its special case formulation for fixed points.Coincidence points, like fixed points, are today studied using many tools from mathematical analysis and topology. An equaliser is a generalization of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2474_chunk_0 | Collapsing algebra | In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963.The collapsing algebra of λω is a complete Boolean algebra with at least λ elements but generat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2475_chunk_0 | Rational dependence | In mathematics, a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients. A collection of numbers which is not rationally independent is called rationally dependent. For instance we have the following ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2476_chunk_0 | Collocation point | In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2477_chunk_0 | Colored matroid | In mathematics, a colored matroid is a matroid whose elements are labeled from a set of colors, which can be any set that suits the purpose, for instance the set of the first n positive integers, or the sign set {+, −}. The interest in colored matroids is through their invariants, especially the colored Tutte polynomia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2478_chunk_0 | Colossally abundant number | In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it's defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2479_chunk_0 | Combination | {\displaystyle \textstyle {\frac {n!}{k!(n-k)!}}} whenever k ≤ n {\displaystyle k\leq n} , and which is zero when k > n {\displaystyle k>n} . This formula can be derived from the fact that each k-combination of a set S of n members has k ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2480_chunk_0 | Combinatorial class | In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2481_chunk_0 | Combinatorial explosion (communication) | In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the problem. Combinatorial explosion is sometimes used to justify the intractability of certain problems. Examples of such problem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2482_chunk_0 | Commutation theory | In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2483_chunk_0 | Commutative rings | In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2484_chunk_0 | Braided monoidal category | Partly for this reason, braided monoidal categories and other topics are related in the theory of knot invariants. Alternatively, a braided monoidal category can be seen as a tricategory with one 0-cell and one 1-cell. Braided monoidal categories were introduced by André Joyal and Ross Street in a 1986 preprint. A modi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2485_chunk_0 | Comodule | In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2486_chunk_0 | Compact quantum group | In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of "continuous complex-valued functions" on a compact quantum group. The basic motivation for this theory comes from the following analogy. The space of compl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2487_chunk_0 | Compact quantum group | On the other hand, by the Gelfand Theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism. S. L. Woronowicz introduced the important concept o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2488_chunk_0 | Complete Boolean algebra | In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2489_chunk_0 | Finitely complete category | In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F: J → C (where J is small) has a limit in C. Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2490_chunk_0 | Complete lattices | Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. Complete lattices must not be confused with complete partial orders (cpos), which constitute a strictly more general class of partially or... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2491_chunk_0 | Butson-type Hadamard matrix | In mathematics, a complex Hadamard matrix H of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-type H(q, N) if all its elements are powers of q-th root of unity, ( H j k ) q = 1 f o r j , k = 1 , 2 , … , N . {\displaystyle (H_{jk})^{q}=1{\quad {\rm {for\quad }}}j,k=1,2,\dots ,N.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2492_chunk_0 | Complex Lie algebra | In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle {\overline {\mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector space but with i = − 1 {\displaystyle i={\sqrt {-1}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2493_chunk_0 | D-bar operator | In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the basis for much of algebraic geometry,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2494_chunk_0 | Complex line | In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has dimension one over C (hence the term "line"), it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2495_chunk_0 | Complex logarithm | In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number z {\displaystyle z} , defined to be any complex number w {\displaystyle w} for which e w = z {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2496_chunk_0 | Complex logarithm | These logarithms are equally spaced along a vertical line in the complex plane. A complex-valued function log: U → C {\displaystyle \log \colon U\to \mathbb {C} } , defined on some subset U {\displaystyle U} of the set C ∗ {\displaystyle \mathbb {C} ^{*}} of nonzero complex numbers, satisfying e log z = z {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2497_chunk_0 | Complex Numbers | In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation i 2 = − 1 {\displaystyle i^{2}=-1} ; every complex number can be expressed in the form a + b i {\displaystyle a+bi} , where a and b are... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2498_chunk_0 | Complex Numbers | More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation ( x + 1 ) 2 = − 9 {\displaystyle (x+1)^{2}=-9} has no real solution, since the square of a real number cannot be ne... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2499_chunk_0 | Complex Numbers | Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with {1, i} as a standard basis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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