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wiki_7400_chunk_0 | Optimal Stopping | In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7401_chunk_0 | Theta correspondence | In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7402_chunk_0 | Riemann–Kempf singularity theorem | In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta-function. It is therefore an algebraic subvariety of A of dimension dim A − 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7403_chunk_0 | Theta function of a lattice | In mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7404_chunk_0 | Theta operator | {\displaystyle \theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.} As in one variable, the eigenspaces of θ are the spaces of homogeneous functions. (Euler's homogeneous function theorem) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7405_chunk_0 | Theta representation | In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. The representation was popularized by David Mumford. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7406_chunk_0 | Pythagorean mean | In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7407_chunk_0 | Three spheres inequality | In mathematics, the three spheres inequality bounds the L 2 {\displaystyle L^{2}} norm of a harmonic function on a given sphere in terms of the L 2 {\displaystyle L^{2}} norm of this function on two spheres, one with bigger radius and one with smaller radius. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7408_chunk_0 | Three-gap theorem | In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ, ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7409_chunk_0 | ~ | The symbol "≈" is also used for this purpose. In physics and astronomy, a tilde can be used between two expressions (e.g. h ~ 10−34 J s) to state that the two are of the same order of magnitude.In statistics and probability theory, the tilde means "is distributed as"; see random variable(e.g. X ~ B(n,p) for a binomial ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7410_chunk_0 | ~ | A triple tilde (≋) is often used to show congruence, an equivalence relation in geometry. In graph theory, the tilde can be used to represent adjacency between vertices. The edge ( x , y ) {\displaystyle (x,y)} connects vertices x {\displaystyle x} and y {\displaystyle y} which can be said to be adjacent, and this adja... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7411_chunk_0 | Halmos box | In mathematics, the tombstone, halmos, end-of-proof, or Q.E.D. symbol "∎" (or "□") is a symbol used to denote the end of a proof, in place of the traditional abbreviation "Q.E.D." for the Latin phrase "quod erat demonstrandum". It is inspired by the typographic practice of end marks, an element that marks the end of an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7412_chunk_0 | Halmos box | Its graphic form varies, as it may be a hollow or filled rectangle or square. In AMS-LaTeX, the symbol is automatically appended at the end of a proof environment \begin{proof} ... \end{proof}. It can also be obtained from the commands \qedsymbol, \qedhere or \qed (the latter causes the symbol to be right aligned).It i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7413_chunk_0 | Halmos box | He got the idea of using it from seeing end marks in magazines, that is, typographic signs that indicate the end of an article. In his memoir I Want to Be a Mathematician, he wrote the following: The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, bu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7414_chunk_0 | Topological entropy | In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov–Sinai, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7415_chunk_0 | Topological entropy | Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension. The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7416_chunk_0 | Total derivative | In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7417_chunk_0 | Total variation norm | In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval ⊂ R, its total variation on the interval of definition is a measure of the one-dimen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7418_chunk_0 | Trace operator | In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions (boundary value problems), where weak sol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7419_chunk_0 | Transcendental law of homogeneity | In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali. Henk J. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7420_chunk_0 | Bernoulli operator | In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the invariant measure of the system.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7421_chunk_0 | Transitive closure logic | 337). We have R+ = R if, and only if, R itself is transitive. Conversely, transitive reduction adduces a minimal relation S from a given relation R such that they have the same closure, that is, S+ = R+; however, many different S with this property may exist. Both transitive closure and transitive reduction are also us... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7422_chunk_0 | Segment addition postulate | In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclud... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7423_chunk_0 | Segment addition postulate | In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proved without these theorems. The inequality can be viewed intuitively in either R2 or R3. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7424_chunk_0 | Segment addition postulate | The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7425_chunk_0 | Tricorn (mathematics) | In mathematics, the tricorn, sometimes called the Mandelbar set, is a fractal defined in a similar way to the Mandelbrot set, but using the mapping z ↦ z ¯ 2 + c {\displaystyle z\mapsto {\bar {z}}^{2}+c} instead of z ↦ z 2 + c {\displaystyle z\mapsto z^{2}+c} used for the Mandelbrot set. It was introduced by W. D. Crow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7426_chunk_0 | Trigamma function | In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by ψ 1 ( z ) = d 2 d z 2 ln Γ ( z ) {\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)} .It follows from this definition that ψ 1 ( z ) = d d z ψ ( z ) {\displaystyle \psi _{1}(z)={... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7427_chunk_0 | Cotangent (trigonometric function) | In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7428_chunk_0 | Cotangent (trigonometric function) | Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7429_chunk_0 | Cotangent (trigonometric function) | The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7430_chunk_0 | Truncated power function | In mathematics, the truncated power function with exponent n {\displaystyle n} is defined as x + n = { x n: x > 0 0: x ≤ 0. {\displaystyle x_{+}^{n}={\begin{cases}x^{n}&:\ x>0\\0&:\ x\leq 0.\end{cases}}} In particular, x + = { x: x > 0 0: x ≤ 0. {\displaystyle x_{+}={\begin{cases}x&:\ x>0\\0&:\ x\leq 0.\end{cases}}} an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7431_chunk_0 | Twisted Poincaré duality | In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7432_chunk_0 | Bilateral Laplace transform | In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7433_chunk_0 | Bilateral Laplace transform | The integral is most commonly understood as an improper integral, which converges if and only if both integrals ∫ 0 ∞ e − s t f ( t ) d t , ∫ − ∞ 0 e − s t f ( t ) d t {\displaystyle \int _{0}^{\infty }e^{-st}f(t)\,dt,\quad \int _{-\infty }^{0}e^{-st}f(t)\,dt} exist. There seems to be no generally accepted notation for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7434_chunk_0 | Bilateral Laplace transform | {\displaystyle {\mathcal {T}}\{f\}(s)=s{\mathcal {B}}\{f\}(s)=sF(s)=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.} In pure mathematics the argument t can be any variable, and Laplace transforms are used to study how differential operators transform the function. In science and engineering applications, the argument t oft... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7435_chunk_0 | Bilateral Laplace transform | In these cases, the signals are transformed by filters, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output in a given time t cannot depend on an output which is a higher value of t. In population ecology, the argument t often represents spatial displaceme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7436_chunk_0 | Uncertainty exponent | In mathematics, the uncertainty exponent is a method of measuring the fractal dimension of a basin boundary. In a chaotic scattering system, the invariant set of the system is usually not directly accessible because it is non-attracting and typically of measure zero. Therefore, the only way to infer the presence of mem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7437_chunk_0 | Banach–Steinhaus theorem | In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7438_chunk_0 | Uniform limit theorem | In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7439_chunk_0 | Low dimensional topology | In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature. This classifies Riemannian surfaces as ellip... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7440_chunk_0 | Uniformization theorem | In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7441_chunk_0 | Uniformization theorem | It further follows that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. The uniformization theorem also yields a similar classification of closed orientable Riemannian 2-manifolds into ell... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7442_chunk_0 | Unit doublet | In mathematics, the unit doublet is the derivative of the Dirac delta function. It can be used to differentiate signals in electrical engineering: If u1 is the unit doublet, then ( x ∗ u 1 ) ( t ) = d x ( t ) d t {\displaystyle (x*u_{1})(t)={\frac {dx(t)}{dt}}} where ∗ {\displaystyle *} is the convolution operator.The ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7443_chunk_0 | Unit doublet | The function can be thought of as the limiting case of two rectangles, one in the second quadrant, and the other in the fourth. The length of each rectangle is k, whereas their breadth is 1/k2, where k tends to zero. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7444_chunk_0 | Unitary symmetry | In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7445_chunk_0 | Unitary symmetry | In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication. All the unitary groups contain copies of this group. The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermiti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7446_chunk_0 | Universal enveloping algebra | In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7447_chunk_0 | Universal enveloping algebra | Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7448_chunk_0 | Universal enveloping algebra | In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka–Krein duality between co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7449_chunk_0 | Zeta function universality | In mathematics, the universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well. The universality of the Riemann zeta function was first proven by Serge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7450_chunk_0 | Unknotting problem | In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to determine if the problem admits a polynomial time algorithm; that is, whether t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7451_chunk_0 | Upper incomplete gamma function | In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incompl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7452_chunk_0 | Upper bound theorem | In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics. Originally known as the upper bound conjecture, this statement was formul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7453_chunk_0 | Lorentz signature | In mathematics, the usual convention for any Riemannian manifold is to use a positive-definite metric tensor (meaning that after diagonalization, elements on the diagonal are all positive). In theoretical physics, spacetime is modeled by a pseudo-Riemannian manifold. The signature counts how many time-like or space-lik... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7454_chunk_0 | Value distribution theory | In mathematics, the value distribution theory of holomorphic functions is a division of mathematical analysis. It tries to get quantitative measures of the number of times a function f(z) assumes a value a, as z grows in size, refining the Picard theorem on behaviour close to an essential singularity. The theory exists... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7455_chunk_0 | Value distribution theory | In the case of one variable the term Nevanlinna theory, after Rolf Nevanlinna, is also common. The now-classical theory received renewed interest, when Paul Vojta suggested some analogies with the problem of integral solutions to Diophantine equations. These turned out to involve some close parallels, and to lead to fr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7456_chunk_0 | Van der Corput inequality | In mathematics, the van der Corput inequality is a corollary of the Cauchy–Schwarz inequality that is useful in the study of correlations among vectors, and hence random variables. It is also useful in the study of equidistributed sequences, for example in the Weyl equidistribution estimate. Loosely stated, the van der... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7457_chunk_0 | Vector flow | In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory. These related concepts are explored in a spectrum of articles: exponential map ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7458_chunk_0 | Vertex enumeration problem | In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7459_chunk_0 | Vertical and horizontal bundles | In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π: E → B {\displaystyle \pi \colon E\to B} , the vertical bundle V E {\displaystyle VE} and horizontal bundle H E {\displaystyle HE} are subbundles of the tan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7460_chunk_0 | Vertical and horizontal bundles | Also note that arbitrary choices of horizontal space at each point will not, in general, form a smooth vector bundle; they must also vary in an appropriately smooth way. The horizontal bundle is one way to formulate the notion of an Ehresmann connection on a fiber bundle. Thus, for example, if E is a principal G-bundle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7461_chunk_0 | Vertical line test | In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x. If a vertical line intersects a curve on an xy-plane more than once then for one value of x the curve has more than one value of y, and so, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7462_chunk_0 | Viscosity solution | In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7463_chunk_0 | Viscosity solution | Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either D u {\displaystyle Du} or D 2 u {\displaystyle D^{2}u} does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7464_chunk_0 | Von Neumann conjecture | Although von Neumann's name is popularly attached to the conjecture, its first written appearance seems to be due to Mahlon Marsh Day in 1957. The Tits alternative is a fundamental theorem which, in particular, establishes the conjecture within the class of linear groups. The historically first potential counterexample... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7465_chunk_0 | Walk-on-spheres method | In mathematics, the walk-on-spheres method (WoS) is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value problem for partial differential equations (PDEs). The WoS method was first introduced by Mervin E. Muller in 1956 to solve La... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7466_chunk_0 | Well-ordering principle | {\displaystyle 1,3,5,...} ). The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem". On other occasions it is understood to be the proposition that the set of integers { … , − 2 , − 1 , 0 , 1 , 2 , 3 , … } {\displaystyle \{\ldots ,-2,-1,0,1,2,3,\ldots \}} contains a we... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7467_chunk_0 | Zermelo's well-ordering theorem | In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7468_chunk_0 | Zermelo's well-ordering theorem | Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. One famous consequence of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7469_chunk_0 | Wholeness axiom | In mathematics, the wholeness axiom is a strong axiom of set theory introduced by Paul Corazza in 2000. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7470_chunk_0 | Winding number | In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be non-integer. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7471_chunk_0 | Witch of Agnesi | The graph of the derivative of the arctangent function forms an example of the witch of Agnesi. As the probability density function of the Cauchy distribution, the witch of Agnesi has applications in probability theory. It also gives rise to Runge's phenomenon in the approximation of functions by polynomials, has been ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7472_chunk_0 | Constant (mathematics) | In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings: A fixed and well-defined number or other non-changing mathematical object. The terms mathematical constant or physical con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7473_chunk_0 | Constant (mathematics) | Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question.For example, a general quadratic function is commonly written as: a x 2 + b x + c , {\displaystyle ax^{2}+bx+c\,,} where a, b and c are constants (coefficients or parameters), and x a variable—a placeholder f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7474_chunk_0 | Constant (mathematics) | Since c occurs in a term that does not involve x, it is called the constant term of the polynomial and can be thought of as the coefficient of x0. More generally, any polynomial term or expression of degree zero (no variable) is a constant. : 18 | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7475_chunk_0 | Null (mathematics) | In mathematics, the word null (from German: null meaning "zero", which is from Latin: nullus meaning "none") is often associated with the concept of zero or the concept of nothing. It is used in varying context from "having zero members in a set" (e.g., null set) to "having a value of zero" (e.g., null vector).In a vec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7476_chunk_0 | Null (mathematics) | A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element). For example, in linear algebra, the null space of a linear mapping, also known as kernel, is the set of vectors which map to the null vector under that mapping. In statistics, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7477_chunk_0 | Zero vector | In mathematics, the zero tensor is a tensor, of any order, all of whose components are zero. The zero tensor of order 1 is sometimes known as the zero vector. Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7478_chunk_0 | ℓ-adic cohomology | In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, whi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7479_chunk_0 | Śleszyński–Pringsheim theorem | In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century.It states that if a n {\displaystyle a_{n}} , b n {\displaystyle b_{n}} , for n = 1 , 2 , 3 , … {\displaystyle n=1,2,3,\l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7480_chunk_0 | Ξ function | In mathematics, the Ξ function (named for the Greek letter Ξ or Xi) may refer to: Riemann Xi function, a variant of the Riemann zeta function with a simpler functional equation Harish-Chandra's Ξ function, a special spherical function on a semisimple Lie group | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7481_chunk_0 | Ky Fan inequality (game theory) | In mathematics, there are different results that share the common name of the Ky Fan inequality. The Ky Fan inequality presented here is used in game theory to investigate the existence of an equilibrium. Another Ky Fan inequality is an inequality involving the geometric mean and arithmetic mean of two sets of real num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7482_chunk_0 | Modes of convergence | In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, see Modes of convergence (annotated index) Note that each of the following o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7483_chunk_0 | Outline of algebraic structures | In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7484_chunk_0 | Outline of algebraic structures | From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition. Concrete examples of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7485_chunk_0 | Construction of the real numbers | In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7486_chunk_0 | Kummer's function | In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7487_chunk_0 | Kummer's function | Kummer's function is defined by Λ n ( z ) = ∫ 0 z log n − 1 | t | 1 + t d t . {\displaystyle \Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.} The duplication formula is Λ n ( z ) + Λ n ( − z ) = 2 1 − n Λ n ( − z 2 ) {\displaystyle \Lambda _{n}(z)+\Lambda _{n}(-z)=2^{1-n}\Lambda _{n}(-z^{2})} .Compare... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7488_chunk_0 | Kummer's function | {\displaystyle \operatorname {Li} _{n}(z)+\operatorname {Li} _{n}(-z)=2^{1-n}\operatorname {Li} _{n}(z^{2}).} An explicit link to the polylogarithm is given by Li n ( z ) = Li n ( 1 ) + ∑ k = 1 n − 1 ( − ) k − 1 log k | z | k ! Li n − k ( z ) + ( − ) n − 1 ( n − 1 ) ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7489_chunk_0 | Kummer's function | . {\displaystyle \operatorname {Li} _{n}(z)=\operatorname {Li} _{n}(1)\;\;+\;\;\sum _{k=1}^{n-1}(-)^{k-1}\;{\frac {\log ^{k}|z|}{k! }}\;\operatorname {Li} _{n-k}(z)\;\;+\;\;{\frac {(-)^{n-1}}{(n-1)!}}\;\left.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7490_chunk_0 | Dirichlet integral | In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line: This integral is not absolutely convergent, meaning | sin x x | {\displaystyle \left|{\fra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7491_chunk_0 | Dirichlet integral | It is a good illustration of special techniques for evaluating definite integrals. The sine integral, an antiderivative of the sinc function, is not an elementary function. However the improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the inte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7492_chunk_0 | Basic theorems in algebraic K-theory | In mathematics, there are several theorems basic to algebraic K-theory. Throughout, for simplicity, we assume when an exact category is a subcategory of another exact category, we mean it is strictly full subcategory (i.e., isomorphism-closed.) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7493_chunk_0 | Ky Fan inequality | In mathematics, there are two different results that share the common name of the Ky Fan inequality. One is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval. The result was published on page 5 of the book Inequalities by Edwin F. Beckenbach and Richard E. B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7494_chunk_0 | Affine Grassmannian (manifold) | In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all k-dimensional affine subspaces of Rn (described on this page), while in the other the affine Grassmannian is a quotient of a group-ring based on formal Laurent series. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7495_chunk_0 | Place-permutation action | In mathematics, there are two natural interpretations of the place-permutation action of symmetric groups, in which the group elements act on positions or places. Each may be regarded as either a left or a right action, depending on the order in which one chooses to compose permutations. There are just two interpretati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7496_chunk_0 | Euler integral | In mathematics, there are two types of Euler integral: The Euler integral of the first kind is the beta function The Euler integral of the second kind is the gamma function For positive integers m and n, the two integrals can be expressed in terms of factorials and binomial coefficients: | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7497_chunk_0 | Hyperfinite type II-1 factor | In mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1 factor. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7498_chunk_0 | Tensor norm | In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces), but for general Banach spaces or locally convex topologi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7499_chunk_0 | Srinivasa Ramanujan | In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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