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wiki_6500_chunk_0 | Annulus theorem | In mathematics, the annulus theorem (formerly called the annulus conjecture) states roughly that the region between two well-behaved spheres is an annulus. It is closely related to the stable homeomorphism conjecture (now proved) which states that every orientation-preserving homeomorphism of Euclidean space is stable. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6501_chunk_0 | Antilimit | In mathematics, the antilimit is the equivalent of a limit for a divergent series. The concept not necessarily unique or well-defined, but the general idea is to find a formula for a series and then evaluate it outside its radius of convergence. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6502_chunk_0 | Arg max | In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized. In contrast to global maxima, which refers to the largest outputs of a function, arg max refers to the inputs, or arguments, at which the fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6503_chunk_0 | Manin–Mumford conjecture | In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of res... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6504_chunk_0 | Arithmetic zeta function | In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most-fundamental objects of number theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6505_chunk_0 | Arithmetic-geometric mean | In mathematics, the arithmetic–geometric mean of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means: Begin the sequences with x and y: Then define the two interdependent sequences (an) and (gn) as These two sequences converge to the same number, the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6506_chunk_0 | Associated Legendre function | In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or equivalently where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsing... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6507_chunk_0 | Associated Legendre function | In general, when ℓ and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. The fully general class of functions with arbitrary real or complex values of ℓ and m are Legendre functions. In that case the parameters are usually l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6508_chunk_0 | Associated Legendre function | The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical ha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6509_chunk_0 | Associative property | Associativity is not the same as commutativity, which addresses whether the order of two operands affects the result. For example, the order does not matter in the multiplication of real numbers, that is, a × b = b × a, so we say that the multiplication of real numbers is a commutative operation. However, operations su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6510_chunk_0 | Associative property | Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; some examples include subtraction, exponentiation, and the vect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6511_chunk_0 | Academic authorship | In mathematics, the authors are usually listed in alphabetical order (the so-called Hardy-Littlewood Rule). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6512_chunk_0 | Automorphism group | In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X).... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6513_chunk_0 | Axiom of Choice | In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6514_chunk_0 | Axiom of Choice | An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6515_chunk_0 | Axiom of Choice | Even if infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set. That is, the choice function provides the set of chosen elements. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6516_chunk_0 | Axiom of Choice | However, no definite choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropria... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6517_chunk_0 | Axiom of Choice | For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that forms a set out of selecting one sock from each pair, without invoking the axiom of choice.Although originally controversial, the axiom of choice is now used without reservation by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6518_chunk_0 | Axiom of determinacy | In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6519_chunk_0 | Axiom of determinacy | Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. Steel and W. Hugh Woodin concluded a long line of research. Assumi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6520_chunk_0 | Power set axiom | In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: ∀ x ∃ y ∀ z {\displaystyle \forall x\,\exists y\,\forall z\,} where y is the power set of x, P ( x ) {\displaystyle {\mathcal {P}}(x)} . In Engli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6521_chunk_0 | Power set axiom | By the axiom of extensionality, the set P ( x ) {\displaystyle {\mathcal {P}}(x)} is unique. The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6522_chunk_0 | Axiom of real determinacy | In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states the following: The axiom of real determinacy is a stronger version of the axiom of determinacy (AD), which makes the same statement about games where both players choose integers; ADR is inconsistent with the axiom o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6523_chunk_0 | Axiom of Regularity | In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads: ∀ x ( x ≠ ∅ → ∃ y ( y ∈ x ∧ y ∩ x = ∅ ) ) . {\displaystyle \forall x\,(x\n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6524_chunk_0 | Axiom of Regularity | The axiom was introduced by von Neumann (1925); it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo (1930). Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). However, regularity makes s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6525_chunk_0 | Axiom of Regularity | {\displaystyle \{(n,\alpha )\mid n\in \omega \land \alpha {\text{ is an ordinal }}\}\,.} Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6526_chunk_0 | Axis angle | In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction (geometry) of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the rotation about the axis. Onl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6527_chunk_0 | Axis angle | By Rodrigues' rotation formula, the angle and axis determine a transformation that rotates three-dimensional vectors. The rotation occurs in the sense prescribed by the right-hand rule. The rotation axis is sometimes called the Euler axis. The axis–angle representation is predicated on Euler's rotation theorem, which d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6528_chunk_0 | Azimuth angle | In mathematics, the azimuth angle of a point in cylindrical coordinates or spherical coordinates is the anticlockwise angle between the positive x-axis and the projection of the vector onto the xy-plane. A special case of an azimuth angle is the angle in polar coordinates of the component of the vector in the xy-plane,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6529_chunk_0 | Ca space | In mathematics, the ba space b a ( Σ ) {\displaystyle ba(\Sigma )} of an algebra of sets Σ {\displaystyle \Sigma } is the Banach space consisting of all bounded and finitely additive signed measures on Σ {\displaystyle \Sigma } . The norm is defined as the variation, that is ‖ ν ‖ = | ν | ( X ) . {\displaystyle \|\nu \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6530_chunk_0 | Ca space | If Σ is a sigma-algebra, then the space c a ( Σ ) {\displaystyle ca(\Sigma )} is defined as the subset of b a ( Σ ) {\displaystyle ba(\Sigma )} consisting of countably additive measures. The notation ba is a mnemonic for bounded additive and ca is short for countably additive. If X is a topological space, and Σ is the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6531_chunk_0 | Bagpipe theorem | In mathematics, the bagpipe theorem of Peter Nyikos (1984) describes the structure of the connected (but possibly non-paracompact) ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact "bag" with several "long pipes". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6532_chunk_0 | Base change theorems | In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: g ∗ ( R r f ∗ F ) → R r f ∗ ′ ( g ′ ∗ F ) {\displaystyle g^{*}(R^{r}f_{*}{\mathcal {F}})\to R^{r}f'_{*}(g'^{*}{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6533_chunk_0 | Base flow (random dynamical systems) | In mathematics, the base flow of a random dynamical system is the dynamical system defined on the "noise" probability space that describes how to "fast forward" or "rewind" the noise when one wishes to change the time at which one "starts" the random dynamical system. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6534_chunk_0 | Euler beta function | In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral B ( z 1 , z 2 ) = ∫ 0 1 t z 1 − 1 ( 1 − t ) z 2 − 1 d t {\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6535_chunk_0 | Bicyclic semigroup | In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. It is perhaps most easily understood as the syntactic monoid describing the Dyck language of balanced pairs of parentheses. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6536_chunk_0 | Big q-Legendre polynomials | In mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as P n ( x ; c ; q ) = 3 ϕ 2 ( q − n , q n + 1 , x ; q , c q ; q , q ) {\displaystyle \displaystyle P_{n}(x;c;q)={}_{3}\phi _{2}(q^{-n},q^{n+1},x;q,cq;q,q)} .They obey the ortho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6537_chunk_0 | Biharmonic equation | In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6538_chunk_0 | Base-2 logarithm | In mathematics, the binary logarithm (log2 n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x, x = log 2 n ⟺ 2 x = n . {\displaystyle x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.} For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6539_chunk_0 | Base-2 logarithm | As well as log2, an alternative notation for the binary logarithm is lb (the notation preferred by ISO 31-11 and ISO 80000-2). Historically, the first application of binary logarithms was in music theory, by Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves by wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6540_chunk_0 | Base-2 logarithm | In computer science, they count the number of steps needed for binary search and related algorithms. Other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6541_chunk_0 | Base-2 logarithm | Binary logarithms are included in the standard C mathematical functions and other mathematical software packages. The integer part of a binary logarithm can be found using the find first set operation on an integer value, or by looking up the exponent of a floating point value. The fractional part of the logarithm can ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6542_chunk_0 | Dyadic logarithm | In mathematics, the binary logarithm of a number n is often written as log2 n. However, several other notations for this function have been used or proposed, especially in application areas. Some authors write the binary logarithm as lg n, the notation listed in The Chicago Manual of Style. Donald Knuth credits this no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6543_chunk_0 | Dyadic logarithm | The binary logarithm has also been written as log n with a prior statement that the default base for the logarithm is 2. Another notation that is often used for the same function (especially in the German scientific literature) is ld n, from Latin logarithmus dualis or logarithmus dyadis. The DIN 1302, ISO 31-11 and IS... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6544_chunk_0 | Binomial coefficient | In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ( n k ) . {\displaystyle {\tbinom {n}{k}}.} It is the coefficient of the xk term in the polynomial expansion ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6545_chunk_0 | Binomial coefficient | Arranging the numbers ( n 0 ) , ( n 1 ) , … , ( n n ) {\displaystyle {\tbinom {n}{0}},{\tbinom {n}{1}},\ldots ,{\tbinom {n}{n}}} in successive rows for n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\ldots } gives a triangular array called Pascal's triangle, satisfying the recurrence relation ( n k ) = ( n − 1 k − 1 ) + ( n ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6546_chunk_0 | Binomial differential equation | In mathematics, the binomial differential equation is an ordinary differential equation containing one or more functions of one independent variable and the derivatives of those functions. For example: ( y ′ ) m = f ( x , y ) , {\displaystyle \left(y'\right)^{m}=f(x,y),} when m {\displaystyle m} is a natural number (i.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6547_chunk_0 | Binomial series | In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like ( 1 + x ) n {\displaystyle (1+x)^{n}} for a nonnegative integer n {\displaystyle n} . Specifically, the binomial series is the Taylor series for the function f ( x ) = ( 1 + x ) α {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6548_chunk_0 | Bipolar theorem | In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6549_chunk_0 | Method of bisection | In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6550_chunk_0 | Method of bisection | Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method, the binary search method, or the dichotomy method.For polynomials, more elaborate methods exist for testing t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6551_chunk_0 | Bounded inverse theorem | In mathematics, the bounded inverse theorem ( also called inverse mapping theorem or Banach isomorphism theorem) is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T−1. It is equivalent to both... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6552_chunk_0 | Braid length | In mathematics, the braid group on n strands (denoted B n {\displaystyle B_{n}} ), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6553_chunk_0 | Real Analysis | In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrabilit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6554_chunk_0 | Branching theorem | In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6555_chunk_0 | Butterfly lemma | Zassenhaus proved this lemma specifically to give the most direct proof of the Schreier refinement theorem. The 'butterfly' becomes apparent when trying to draw the Hasse diagram of the various groups involved. Zassenhaus' lemma for groups can be derived from a more general result known as Goursat's theorem stated in a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6556_chunk_0 | Capacitated arc routing problem | In mathematics, the capacitated arc routing problem (CARP) is that of finding the shortest tour with a minimum graph/travel distance of a mixed graph with undirected edges and directed arcs given capacity constraints for objects that move along the graph that represent snow-plowers, street sweeping machines, or winter ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6557_chunk_0 | Harmonic capacity | In mathematics, the capacity of a set in Euclidean space is a measure of the "size" of that set. Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge. More precisely, it is the capacitance of the set: the total ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6558_chunk_0 | Category of preordered sets | We have a forgetful functor Ord → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore Ord is a concrete category. This functor has a left adjoint (sending every set to that set equipped with the equality relat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6559_chunk_0 | Category of compactly generated weak Hausdorff spaces | In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the latter lacks some of the pleasant properties one would desire. There is also such a category for based spaces, defined b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6560_chunk_0 | Category of medial magmas | In mathematics, the category of medial magmas, also known as the medial category, and denoted Med, is the category whose objects are medial magmas (that is, sets with a medial binary operation), and whose morphisms are magma homomorphisms (which are equivalent to homomorphisms in the sense of universal algebra). The ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6561_chunk_0 | Categorical topology | In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6562_chunk_0 | Category of topological vector spaces | In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category because the composition of two continuous linear maps is again a continuous linear map. The category is often denoted TVec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6563_chunk_0 | Characteristic equation (calculus) | In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6564_chunk_0 | Characteristic equation (calculus) | The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6565_chunk_0 | Characteristic equation (calculus) | For both types of equation, persistent fluctuations occur if there is at least one pair of complex roots. The method of integrating linear ordinary differential equations with constant coefficients was discovered by Leonhard Euler, who found that the solutions depended on an algebraic 'characteristic' equation. The qua... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6566_chunk_0 | Circumflex accent | In mathematics, the circumflex is used to modify variable names; it is usually read "hat", e.g., î is "i hat". The Fourier transform of a function ƒ is often denoted by f ^ {\displaystyle {\hat {f}}} . In the notation of sets, a hat above an element signifies that the element was removed from the set, such as in { x 0 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6567_chunk_0 | Circumflex accent | In geometry, a hat is sometimes used for an angle. For instance, the angles A ^ {\displaystyle {\hat {A}}} or A B ^ C {\displaystyle A{\hat {B}}C} . In vector notation, a hat above a letter indicates a unit vector (a dimensionless vector with a magnitude of 1). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6568_chunk_0 | Circumflex accent | For instance, ı ^ {\displaystyle {\hat {\mathbf {\imath } }}} , x ^ {\displaystyle {\hat {\mathbf {x} }}} , or e ^ 1 {\displaystyle {\hat {\mathbf {e} }}_{1}} stands for a unit vector in the direction of the x-axis of a Cartesian coordinate system. In statistics, the hat is used to denote an estimator or an estimated v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6569_chunk_0 | L-matrix | In mathematics, the class of L-matrices are those matrices whose off-diagonal entries are less than or equal to zero and whose diagonal entries are positive; that is, an L-matrix L satisfies L = ( ℓ i j ) ; ℓ i i > 0 ; ℓ i j ≤ 0 , i ≠ j . {\displaystyle L=(\ell _{ij});\quad \ell _{ii}>0;\quad \ell _{ij}\leq 0,\quad i\n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6570_chunk_0 | Z-matrix (mathematics) | In mathematics, the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: Z = ( z i j ) ; z i j ≤ 0 , i ≠ j . {\displaystyle Z=(z_{ij});\quad z_{ij}\leq 0,\quad i\neq j.} Note that this definition coincides precisely with that of a negated M... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6571_chunk_0 | Z-matrix (mathematics) | The Jacobian of a competitive dynamical system is a Z-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is J, then (−J) is a Z-matrix. Related classes are L-matrices, M-matrices, P-matrices, Hurwitz matrices and Metzler matrices. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6572_chunk_0 | Z-matrix (mathematics) | L-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices and P-matrices are nonsingular M-matri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6573_chunk_0 | Moebius inversion | In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.A large generalization of this formula applies to summation over an arbitrary locally finit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6574_chunk_0 | Kronecker limit formula | In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6575_chunk_0 | Geometric Langlands correspondence | In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory and representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6576_chunk_0 | S-duality | In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory to the branch of mathematics known as representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6577_chunk_0 | S-duality | Starting with two Yang–Mills theories related by S-duality, Kapustin and Witten showed that one can construct a pair of quantum field theories in two-dimensional spacetime. By analyzing what this dimensional reduction does to certain physical objects called D-branes, they showed that one can recover the mathematical in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6578_chunk_0 | Moebius plane | In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any generalized circle, and thus a natural setting for planar inversive geometry.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6579_chunk_0 | Moebius plane | In inversive geometry a straight line is considered to be a generalized circle containing the point at infinity; inversion of the plane with respect to a line is a Euclidean reflection. More generally, a Möbius plane is an incidence structure with the same incidence relationships as the classical Möbius plane. It is on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6580_chunk_0 | Classical groups | In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6581_chunk_0 | Classical groups | A few examples are the following. The rotation group SO(3) is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp(m)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6582_chunk_0 | Classical orthogonal polynomials | Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials. For given polynomials Q , L: R → R {\displaystyle Q,L:\mathbb {R} \to \mathbb {R} } and ∀ n ∈ N 0 {\displaystyle \forall \,n\in \mathbb {N} _{0}} the classical orthogonal polynomials f n: R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6583_chunk_0 | Classification of finite simple groups | In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6584_chunk_0 | Classification of finite simple groups | The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6585_chunk_0 | Classifying space for U(n) | In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. This space with its universal fibration may be constructed as either the Gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6586_chunk_0 | Closed graph theorem | In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6587_chunk_0 | Closed subgroup theorem | In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of a Lie group G, then H is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding. One of severa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6588_chunk_0 | Coadjoint representation | In mathematics, the coadjoint representation K {\displaystyle K} of a Lie group G {\displaystyle G} is the dual of the adjoint representation. If g {\displaystyle {\mathfrak {g}}} denotes the Lie algebra of G {\displaystyle G} , the corresponding action of G {\displaystyle G} on g ∗ {\displaystyle {\mathfrak {g}}^{*}} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6589_chunk_0 | Tangle hypothesis | In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism hypothesis, though the details of his approach have yet to appear in the literature as of 2022. In 202... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6590_chunk_0 | Codomain | In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or image of a function. A codomain is part of a function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6591_chunk_0 | Codomain | Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution. A codomain is not part of a function f if f is defined as just a graph. For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6592_chunk_0 | Cohomology operation | In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Through... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6593_chunk_0 | Cohomology operation | In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level. The convergence is to groups in stable homotopy theory, about which information is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6594_chunk_0 | Collage theorem | In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given set. It is typically ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6595_chunk_0 | Base-10 logarithm | In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm. Historically, it was known as logar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6596_chunk_0 | Base-10 logarithm | On calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity, the ISO 80000 specification recommends that log10 (x) should be written lg(x), and loge (x) should be ln(x). Before ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6597_chunk_0 | Base-10 logarithm | Instead, tables of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a slide rule. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6598_chunk_0 | Compact open topology | In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945.If the codo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6599_chunk_0 | Direct comparison test | In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the giv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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