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wiki_1100_chunk_0 | Method of complements | In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (or mechanism) for addition throughout the whole range. For a given number of places half of the possible representations of numbers encode t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1101_chunk_0 | Method of complements | Changing the sign of any number is encoded by generating its complement, which can be done by a very simple and efficient algorithm. This method was commonly used in mechanical calculators and is still used in modern computers. The generalized concept of the radix complement (as described below) is also valuable in num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1102_chunk_0 | Method of complements | The leftmost digit '1' of the result is then discarded. Discarding the leftmost '1' is especially convenient on calculators or computers that use a fixed number of digits: there is nowhere for it to go so it is simply lost during the calculation. The nines' complement plus one is known as the ten's complement. The meth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1103_chunk_0 | Universal hashing | In mathematics and computing, universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below). This guarantees a low number of collisions in expectation, even if the data is chosen b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1104_chunk_0 | Corner solution | In mathematics and economics, a corner solution is a special solution to an agent's maximization problem in which the quantity of one of the arguments in the maximized function is zero. In non-technical terms, a corner solution is when the chooser is either unwilling or unable to make a trade-off between goods. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1105_chunk_0 | Arc elasticity | In mathematics and economics, the arc elasticity is the elasticity of one variable with respect to another between two given points. It is the ratio of the percentage change of one of the variables between the two points to the percentage change of the other variable. It contrasts with the point elasticity, which is th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1106_chunk_0 | Envelope Theorem | In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1107_chunk_0 | Cavity Perturbation Theory | In mathematics and electronics, Cavity perturbation theory describes methods for derivation of perturbation formulae for performance changes of a cavity resonator. These performance changes are assumed to be caused by either introduction of a small foreign object into the cavity, or a small deformation of its boundary.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1108_chunk_0 | Quantification (science) | In mathematics and empirical science, quantification (or quantitation) is the act of counting and measuring that maps human sense observations and experiences into quantities. Quantification in this sense is fundamental to the scientific method. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1109_chunk_0 | Kobayashi metric | In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the property that the Kobayashi pseudometric i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1110_chunk_0 | Kähler manifold | In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1111_chunk_0 | Kähler manifold | Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics. Ev... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1112_chunk_0 | Stanley symmetric function | In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by Richard Stanley (1984) in his study of the symmetric group of permutations. Formally, the Stanley symmetric function Fw(x1, x2, ...) indexed by a permutation w is defined as a sum ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1113_chunk_0 | Stanley symmetric function | They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation w0 = n(n − 1)...21 (written here in one-line notation) has exactly ( n 2 ) ! 1 n − 1 ⋅ 3 n − 2 ⋅ 5 n − 3 ⋯ ( 2 n − 3 ) 1 {\displaystyle {\frac {{\binom {n}{2}}! }{1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1114_chunk_0 | Plane partition | In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers π i , j {\displaystyle \pi _{i,j}} (with positive integer indices i and j) that is nonincreasing in both indices. This means that π i , j ≥ π i , j + 1 {\displaystyle \pi _{i,j}\geq \pi _{i,j+1}} and π i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1115_chunk_0 | Plane partition | The image has matrix form 4 4 3 2 1 4 3 1 1 3 2 1 1 {\displaystyle {\begin{matrix}4&4&3&2&1\\4&3&1&1\\3&2&1\\1\end{matrix}}} Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset P {\displaystyle {\mathcal {P}}} of posi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1116_chunk_0 | Plane partition | The number of plane partitions with sum n is denoted by PL(n). For example, there are six plane partitions with sum 3 3 2 1 1 1 1 2 1 1 1 1 1 1 1 {\displaystyle {\begin{matrix}3\end{matrix}}\qquad {\begin{matrix}2&1\end{matrix}}\qquad {\begin{matrix}1&1&1\end{matrix}}\qquad {\begin{matrix}2\\1\end{matrix}}\qquad {\begi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1117_chunk_0 | Sum of reciprocals | In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1118_chunk_0 | Decomposable operator | In mathematics and functional analysis a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1119_chunk_0 | Decomposable operator | Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings. Results on direct integrals can be viewed as generalizations of results about finite-dimensional C*-algebras of matrices; in this case the resu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1120_chunk_0 | Bulgarian solitaire | In mathematics and game theory, Bulgarian solitaire is a card game that was introduced by Martin Gardner. In the game, a pack of N {\displaystyle N} cards is divided into several piles. Then for each pile, remove one card; collect the removed cards together to form a new pile (piles of zero size are ignored). If N {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1121_chunk_0 | Discrete symmetry | In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1122_chunk_0 | Variadic functions | In mathematics and in computer programming, a variadic function is a function of indefinite arity, i.e., one which accepts a variable number of arguments. Support for variadic functions differs widely among programming languages. The term variadic is a neologism, dating back to 1936–1937. The term was not widely used u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1123_chunk_0 | Lehmer code | In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of n numbers. It is an instance of a scheme for numbering permutations and is an example of an inversion table. The Lehmer code is named in reference to Derrick Henry Lehmer, but the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1124_chunk_0 | Measurable function | In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topologic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1125_chunk_0 | Hurwitz's theorem (complex analysis) | In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit. The theorem is named after Adolf Hurwitz. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1126_chunk_0 | Stone–von Neumann theorem | In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1127_chunk_0 | Sigma-martingale | In mathematics and information theory of probability, a sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. In financial mathematics, sigma-martingales appear in the fundamental theorem of asset pricing as an equivalent cond... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1128_chunk_0 | Sanov's theorem | In mathematics and information theory, Sanov's theorem gives a bound on the probability of observing an atypical sequence of samples from a given probability distribution. In the language of large deviations theory, Sanov's theorem identifies the rate function for large deviations of the empirical measure of a sequence... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1129_chunk_0 | Sanov's theorem | Let A be a set of probability distributions over an alphabet X, and let q be an arbitrary distribution over X (where q may or may not be in A). Suppose we draw n i.i.d. samples from q, represented by the vector x n = x 1 , x 2 , … , x n {\displaystyle x^{n}=x_{1},x_{2},\ldots ,x_{n}} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1130_chunk_0 | Sanov's theorem | Then, we have the following bound on the probability that the empirical measure p ^ x n {\displaystyle {\hat {p}}_{x^{n}}} of the samples falls within the set A: q n ( p ^ x n ∈ A ) ≤ ( n + 1 ) | X | 2 − n D K L ( p ∗ | | q ) {\displaystyle q^{n}({\hat {p}}_{x^{n}}\in A)\leq (n+1)^{|X|}2^{-nD_{\mathrm {KL} }(p^{*}||q)}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1131_chunk_0 | Sturm–Liouville eigenproblem | In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form: for given functions p ( x ) {\displaystyle p(x)} , q ( x ) {\displaystyle q(x)} and w ( x ) {\displaystyle w(x)} , together with some boundary conditions at extreme values of x {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1132_chunk_0 | Sturm–Liouville eigenproblem | This theory is important in applied mathematics, where Sturm–Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations. For example, in quantum mechanics, the one-dimensional time-independent Schrödinger equation is a Sturm–Liouville problem. Sturm–Liouvill... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1133_chunk_0 | Stefan problem | In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1134_chunk_0 | Stefan problem | To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogous problems occur, f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1135_chunk_0 | Mean square | In mathematics and its applications, the mean square is normally defined as the arithmetic mean of the squares of a set of numbers or of a random variable.It may also be defined as the arithmetic mean of the squares of the deviations between a set of numbers and a reference value (e.g., may be a mean or an assumed mean... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1136_chunk_0 | Root Mean Square | In mathematics and its applications, the root mean square of a set of numbers x i {\displaystyle x_{i}} (abbreviated as RMS, RMS or rms and denoted in formulas as either x R M S {\displaystyle x_{\mathrm {RMS} }} or R M S x {\displaystyle \mathrm {RMS} _{x}} ) is defined as the square root of the mean square (the arith... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1137_chunk_0 | Signed distance function | In mathematics and its applications, the signed distance function (or oriented distance function) is the orthogonal distance of a given point x to the boundary of a set Ω in a metric space, with the sign determined by whether or not x is in the interior of Ω. The function has positive values at points x inside Ω, it de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1138_chunk_0 | Axiom of heredity | In mathematics and logic, Ackermann set theory (AST) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.AST differs from Zermelo–Fraenkel set theory (ZF) in that it allows proper classes, that is, objects that are not sets, including a class of all sets. It replaces several of the standard ZF axioms for c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1139_chunk_0 | Axiom of heredity | In its use of classes, AST differs from other alternative set theories such as Morse–Kelley set theory and Von Neumann–Bernays–Gödel set theory in that a class may be an element of another class. William N. Reinhardt established in 1970 that AST is effectively equivalent in strength to ZF, putting it on equal foundatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1140_chunk_0 | Corollary | In mathematics and logic, a corollary ( KORR-ə-lerr-ee, UK: korr-OL-ər-ee) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1141_chunk_0 | Direct proof | In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1142_chunk_0 | Direct proof | Common proof rules used are modus ponens and universal instantiation.In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example, instead of showing directly p ⇒ q, one prov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1143_chunk_0 | Ordered logic | In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less we... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1144_chunk_0 | Vacuous truth | In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. It is sometimes said that a statement is vacuously true because it does not really say anything. For example... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1145_chunk_0 | Vacuous truth | In essence, a conditional statement, that is based on the material conditional, is true when the antecedent ("Tokyo is in France" in the example) is false regardless of whether the conclusion or consequent ("the Eiffel Tower is in Bolivia" in the example) is true or false because the material conditional is defined in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1146_chunk_0 | Axiomatic proof | In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1147_chunk_0 | Effectiveness | In heat transfer, effectiveness is a measure of the performance of a heat exchanger when using the NTU method. In medicine, effectiveness relates to how well a treatment works in practice, especially as shown in pragmatic clinical trials, as opposed to efficacy, which measures how well it works in explanatory clinical ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1148_chunk_0 | Plural quantification | In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc. for x, we may substitute both Alice and Bob, or all the numbers between... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1149_chunk_0 | Mackey–Glass equations | In mathematics and mathematical biology, the Mackey–Glass equations, named after Michael Mackey and Leon Glass, refer to a family of delay differential equations whose behaviour manages to mimic both healthy and pathological behaviour in certain biological contexts, controlled by the equation's parameters. Originally, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1150_chunk_0 | Boolean value | In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1151_chunk_0 | Boolean value | Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1152_chunk_0 | Boolean value | Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854). According to Huntington, the term "Boolean algebra" was first suggested by Henry M. Sheffer in 1913, although Charles Sanders Peirce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1153_chunk_0 | Infimal convolution | In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1154_chunk_0 | Holonomic basis | In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold M is a set of basis vector fields {e1, ..., en} defined at every point P of a region of the manifold as e α = lim δ x α → 0 δ s δ x α , {\displaystyle \mathbf {e} _{\alpha }=\lim _{\delta x^{\alpha }\to 0}{\frac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1155_chunk_0 | Holonomic basis | Since we have that u = uαeα, the identification is often made between a coordinate basis vector eα and the partial derivative operator ∂/∂xα, under the interpretation of vectors as operators acting on functions.A local condition for a basis {e1, ..., en} to be holonomic is that all mutual Lie derivatives vanish: = L e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1156_chunk_0 | Factorization algebra | In mathematics and mathematical physics, a factorization algebra is an algebraic structure first introduced by Beilinson and Drinfel'd in an algebro-geometric setting as a reformulation of chiral algebras, and also studied in a more general setting by Costello to study quantum field theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1157_chunk_0 | Probabilistic potential theory | In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1158_chunk_0 | Probabilistic potential theory | For example, a result about the singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of the Laplace equation. This is not a hard and fast distinction, and in practice there is considerable o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1159_chunk_0 | Probabilistic potential theory | In the continuous case, this is closely related to analytic theory. In the finite state space case, this connection can be introduced by introducing an electrical network on the state space, with resistance between points inversely proportional to transition probabilities and densities proportional to potentials. Even ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1160_chunk_0 | Raising and lowering indices | In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1161_chunk_0 | Euler–Rodrigues formula | In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization. The rotation is described by four Euler parameters due to Leonhard Euler. The Rodrigues formula (named after Olinde Rodrigue... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1162_chunk_0 | Centering matrix | In mathematics and multivariate statistics, the centering matrix is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components of the vector from every component of that vector. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1163_chunk_0 | Adaptive stepsize | In mathematics and numerical analysis, an adaptive step size is used in some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration) in order to control the errors of the method and to ensure stability properties such as A-stability. Using an adaptive ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1164_chunk_0 | Mexican hat wavelet | In mathematics and numerical analysis, the Ricker wavelet ψ ( t ) = 2 3 σ π 1 / 4 ( 1 − ( t σ ) 2 ) e − t 2 2 σ 2 {\displaystyle \psi (t)={\frac {2}{{\sqrt {3\sigma }}\pi ^{1/4}}}\left(1-\left({\frac {t}{\sigma }}\right)^{2}\right)e^{-{\frac {t^{2}}{2\sigma ^{2}}}}} is the negative normalized second derivative of a Gau... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1165_chunk_0 | Mexican hat wavelet | It is also known as the Marr wavelet for David Marr. ψ ( x , y ) = 1 π σ 4 ( 1 − 1 2 ( x 2 + y 2 σ 2 ) ) e − x 2 + y 2 2 σ 2 {\displaystyle \psi (x,y)={\frac {1}{\pi \sigma ^{4}}}\left(1-{\frac {1}{2}}\left({\frac {x^{2}+y^{2}}{\sigma ^{2}}}\right)\right)e^{-{\frac {x^{2}+y^{2}}{2\sigma ^{2}}}}} The multidimensional ge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1166_chunk_0 | Van Wijngaarden transformation | In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series. One algorithm to compute Euler's transform runs as follows: Compute a row of partial sums and form rows of averages between neighbors The first colum... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1167_chunk_0 | Van Wijngaarden transformation | If a 0 , a 1 , … , a 12 {\displaystyle a_{0},a_{1},\ldots ,a_{12}} are available, then s 8 , 4 {\displaystyle s_{8,4}} is almost always a better approximation to the sum than s 12 , 0 {\displaystyle s_{12,0}} . In many cases the diagonal terms do not converge in one cycle so process of averaging is to be repeated with ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1168_chunk_0 | Pseudo-Boolean function | In mathematics and optimization, a pseudo-Boolean function is a function of the form f: B n → R , {\displaystyle f:\mathbf {B} ^{n}\to \mathbb {R} ,} where B = {0, 1} is a Boolean domain and n is a nonnegative integer called the arity of the function. A Boolean function is then a special case, where the values are also... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1169_chunk_0 | Consistent and inconsistent equations | In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1170_chunk_0 | Initial condition | In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value,: pp. 160 is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1171_chunk_0 | Initial condition | In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1172_chunk_0 | Circumgon | In mathematics and particularly in elementary geometry, a circumgon is a geometric figure which circumscribes some circle, in the sense that it is the union of the outer edges of non-overlapping triangles each of which has a vertex at the center of the circle and opposite side on a line that is tangent to the circle. :... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1173_chunk_0 | Circumgon | All circumgons have common properties regarding area–perimeter ratios and centroids. It is these properties that make circumgons interesting objects of study in elementary geometry. The concept and the terminology of a circumgon were introduced and their properties investigated first by Tom M. Apostol and Mamikon A. Mn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1174_chunk_0 | Pairwise Stone space | In mathematics and particularly in topology, pairwise Stone space is a bitopological space ( X , τ 1 , τ 2 ) {\displaystyle \scriptstyle (X,\tau _{1},\tau _{2})} which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional. Pairwise Stone spaces are a bitopological version of the Stone spaces. Pairwise ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1175_chunk_0 | Spherical harmonic function | In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1176_chunk_0 | Spherical harmonic function | Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1177_chunk_0 | Spherical harmonic function | Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree ℓ {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1178_chunk_0 | Spherical harmonic function | The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence r ℓ {\displaystyle r^{\ell }} from the above-mentioned polynomial of degree ℓ {\displaystyle \ell } ; the remaining factor can be regarded as a function of the spherical angular coordinates ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1179_chunk_0 | Spherical harmonic function | Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). A specific set of spherical harmonics, denoted Y ℓ m ( θ , φ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} or Y ℓ m ( r ) {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} , ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1180_chunk_0 | Spherical harmonic function | These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configura... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1181_chunk_0 | Canonical commutation relation algebra | In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively. They play a prominent role in quantum statistical mechanics and quantum field theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1182_chunk_0 | Laplace's equation | In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as or where Δ = ∇ ⋅ ∇ = ∇ 2 {\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}} is the Laplace operator, ∇ ⋅ {\displaystyle \nabla... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1183_chunk_0 | Laplace's equation | Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation. The general theory of solutions to Laplace's equation is known as potential theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1184_chunk_0 | Laplace's equation | The twice continuously differentiable solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1185_chunk_0 | Lieb–Thirring conjecture | In mathematics and physics, Lieb–Thirring inequalities provide an upper bound on the sums of powers of the negative eigenvalues of a Schrödinger operator in terms of integrals of the potential. They are named after E. H. Lieb and W. E. Thirring. The inequalities are useful in studies of quantum mechanics and differenti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1186_chunk_0 | Tensor diagram notation | In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several shapes linked together by lines. The notation widely appears in modern quantu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1187_chunk_0 | Hamiltonian flow | In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1188_chunk_0 | Global mode | In mathematics and physics, a global mode of a system is one in which the system executes coherent oscillations in time. Suppose a quantity y ( x , t ) {\displaystyle y(x,t)} which depends on space x {\displaystyle x} and time t {\displaystyle t} is governed by some partial differential equation which does not have an ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1189_chunk_0 | Global mode | Philip Drazin introduced the concept of a global mode in his 1974 paper, and gave a technique for finding the normal modes of a linear PDE problem in which the coefficients or geometry vary slowly in x {\displaystyle x} . This technique is based on the WKBJ approximation, which is a special case of multiple-scale analy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1190_chunk_0 | Non-perturbative | In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function f ( x ) = e − 1 / x 2 , {\displaystyle f(x)=e^{-1/x^{2}},} which does not have a Taylor series at x = 0. Every coefficient of the Taylor expansion around x = 0 is exactly... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1191_chunk_0 | Non-perturbative | For not too strong fields, the rate per unit volume of this process is given by, Γ = ( e E ) 2 4 π 3 e − π m 2 e E {\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}}}\mathrm {e} ^{-{\frac {\pi m^{2}}{eE}}}} which cannot be expanded in a Taylor series in the electric charge e {\displaystyle e} , or the electric field s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1192_chunk_0 | Exact solutions of nonlinear partial differential equations | In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1193_chunk_0 | Quantum graph | In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge. An example would be a power network consisting of power lines (edges) co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1194_chunk_0 | Recurrent tensor | In mathematics and physics, a recurrent tensor, with respect to a connection ∇ {\displaystyle \nabla } on a manifold M, is a tensor T for which there is a one-form ω on M such that ∇ T = ω ⊗ T . {\displaystyle \nabla T=\omega \otimes T.\,} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1195_chunk_0 | Scalar fields | In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (with units). In a physical context, scalar fields are required to be independent of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1196_chunk_0 | Soliton wave | In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is strongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1197_chunk_0 | Soliton wave | The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation". The term soliton was coined by Zabusky and Kruskal to describe localized, strongly stable ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1198_chunk_0 | Tensor analysis | In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the phy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1199_chunk_0 | Traveling plane wave | In mathematics and physics, a traveling plane wave is a special case of plane wave, namely a field whose evolution in time can be described as simple translation of its values at a constant wave speed c {\displaystyle c} , along a fixed direction of propagation n → {\displaystyle {\vec {n}}} . Such a field can be writt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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