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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.
In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) of a finite group is the probability that two randomly chosen elements commute. It can be used to measure how close to abelian a finite group is. It can be generalized to infinite g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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.\,}
In mathematics and physics, a scalar field is a function associating a single n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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 |
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 |
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas.
Infinite-dimensional vector spaces occur in many areas of mathematics. For example... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can also be applied to vector potentials. An equipotential of a scalar potential f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Extending this definition, an isopotential is the locus of all points that are of the same potential. Gravity is perpendicular to the equipotential surfaces of the gravity potential, and in electrostatics and steady electric currents, the electric field (and hence the current, if any) is perpendicular to the equipotent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as ∇ ⋅ ( A ( x → ϵ ) ∇ u ϵ ) = f {\displaystyle \nabla \cdot \left(A\left({\frac {\vec {x}}{\epsilon }}\right)\nabla u_{\epsilon }\right)=f} where ϵ {\displaystyle \epsilon } is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Under this assumption, materials such as fluids, solids, etc. can be treated as homogeneous materials and associated with these materials are material properties such as shear modulus, elastic moduli, etc. Frequently, inhomogeneous materials (such as composite materials) possess microstructure and therefore they are su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
{\displaystyle \nabla _{y}\cdot \left(A({\vec {y}})\nabla w_{j}\right)=-\nabla _{y}\cdot \left(A({\vec {y}}){\vec {e}}_{j}\right).} This process of replacing an equation with a highly oscillatory coefficient with one with a homogeneous (uniform) coefficient is known as homogenization.
This subject is inextricably link... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Classical results of homogenization theory were obtained for media with periodic microstructure modeled by partial differential equations with periodic coefficients. These results were later generalized to spatially homogeneous random media modeled by differential equations with random coefficients which statistical pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical enti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Einstein's general theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and physics, surface growth refers to models used in the dynamical study of the growth of a surface, usually by means of a stochastic differential equation of a field.
In mathematics and physics, the Artin billiard is a type of a dynamical billiard first studied by Emil Artin in 1924. It describes the g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The quantum mechanical version of Artin's billiard is also exactly solvable. The eigenvalue spectrum consists of a bound state and a continuous spectrum above the energy E = 1 / 4 {\displaystyle E=1/4} . The wave functions are given by Bessel functions.
In mathematics and physics, the Christoffel symbols are an array ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.
In general, there are an infinite number of metric... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Like the KdV equation, the KP equation is completely integrable. It can also be solved using the inverse scattering transform much like the nonlinear Schrödinger equation.In 2002, the regularized version of the KP equation, naturally referred to as the Benjamin–Bona–Mahony–Kadomtsev–Petviashvili equation (or simply the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state. The Poincaré recurrence time... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and physics, the diamagnetic inequality relates the Sobolev norm of the absolute value of a section of a line bundle to its covariant derivative. The diamagnetic inequality has an important physical interpretation, that a charged particle in a magnetic field has more energy in its ground state than it wo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a give... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges. Following Robert Richtmyer and John von Neumann's introduction of "artificial viscosity" methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks. Solutions of the he... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and physics, the notion of orientation entanglement is sometimes used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be simply connected.
In mathematics and physics, the plate trick, also known as Dir... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
One can see this by holding one's hands outward and together, palms up, with the thumbs out-stretched to the right and left, and the fingers making a curling motion from straight outward to pointing upward. If the curling motion of the fingers represents a movement from the first (x-axis) to the second (y-axis), then t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
For example, the vacuum expectation value of the baryon number is given by the spectral asymmetry of the Hamiltonian operator. The spectral asymmetry of the confined quark fields is an important property of the chiral bag model. For fermions, it is known as the Witten index, and can be understood as describing the Casi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and political science, the quota rule describes a desired property of a proportional apportionment or election method. It states that the number of seats that should be allocated to a given party should be between the upper or lower roundings (called upper and lower quotas) of its fractional proportional... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs. Continuu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and related subjects, understanding a mathematical expression depends on an understanding of symbols of grouping, such as parentheses (), brackets , and braces {}. These same symbols are also used in ways where they are not symbols of grouping. For example, in the expression 3(x+y) the parentheses are sy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
If two of these symbols are used, one on the left and the mirror image of it on the right, it almost always indicates a set, as in { a , b , c } {\displaystyle \{a,b,c\}} , the set containing three members, a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} . But if it is used only on the left, it grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossib... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The basic idea is that the negative frequency components of the Fourier transform (or spectrum) of a real-valued function are superfluous, due to the Hermitian symmetry of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complex-va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (the z-domain or z-plane) representation.It can be considered as a discrete-time equivalent of the Laplace transform (the s-domain or s-plane). This simi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and signal processing, the constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier transform and very closely related to the complex Morlet wavelet transform. Its design is suited for musical representation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in k-space that can be reached from the origin withou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and statistics, a piecewise linear, PL or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments.
In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one. The positions (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness.
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose uncond... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A trend stationary process is not strictly stationary, but can easily be transformed into a stationary process by removing the underlying trend, which is solely a function of time. Similarly, processes with one or more unit roots can be made stationary through differencing. An important type of non-stationary process t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and statistics, an error term is an additive type of error. Common examples include: errors and residuals in statistics, e.g. in linear regression the error term in numerical integration
In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. Random projection methods are known for their power, simplicity, and low error rates when compared to other methods. According to experimental results, random projection pre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Euler's sum of powers conjecture (disproved) concerns situations in which the sum of n integers, each a kth power of an integer, equals another kth power. The Fermat-Catalan conjecture asks whether there are an infinitude of examples in which the sum of two coprime integers, each a power of an integer, with the powers ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Waring's problem asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers. The successive powers of the golden ratio φ obey the Fibonacci recurrence: φ n + 1 = φ n + φ n − 1 . {\displaystyle \varphi ^{n+1}=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The Erdős–Moser equation, 1 k + 2 k + ⋯ + m k = ( m + 1 ) k {\displaystyle 1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k}} where m {\displaystyle m} and k {\displaystyle k} are positive integers, is conjectured to have no solutions other than 11 + 21 = 31. The sums of three cubes cannot equal 4 or 5 modulo 9, but it is unknown wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and statistics, the arithmetic mean ( arr-ith-MET-ik), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux compactifications of string theory, the base is usually a five-dimensiona... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
This model is considered to be pioneering and the origin of continuum percolation. Network models based on geometric probability were later proposed and used in the late 1970s and continued throughout the 1980s for examining packet radio networks. Later their use increased significantly for studying a number of wireles... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and the field of transportation theory, the transport functions J(n,x) are defined by J ( n , x ) = ∫ 0 x t n e t ( e t − 1 ) 2 d t . {\displaystyle J(n,x)=\int _{0}^{x}t^{n}{\frac {e^{t}}{(e^{t}-1)^{2}}}\,dt.} Note that e t ( e t − 1 ) 2 = ∑ k = 0 ∞ k e k t . {\displaystyle {\frac {e^{t}}{(e^{t}-1)^{2}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or more of its immediate predecessors. A constant-recursive sequence is also known as a linear recurrence sequence, linear-rec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Formally, a sequence of numbers s 0 , s 1 , s 2 , s 3 , … {\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots } is constant-recursive if it satisfies a recurrence relation where c i {\displaystyle c_{i}} are constants. For example, the Fibonacci sequence satisfies the recurrence relation F n = F n − 1 + F n − 2 , {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical computer science, a k-synchronized sequence is an infinite sequence of terms s(n) characterized by a finite automaton taking as input two strings m and n, each expressed in some fixed base k, and accepting if m = s(n). The class of k-synchronized sequences lies between the classes of k-au... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical computer science, a set constraint is an equation or an inequation between sets of terms. Similar to systems of (in)equations between numbers, methods are studied for solving systems of set constraints. Different approaches admit different operators (like "∪", "∩", "\", and function appli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical computer science, entropy compression is an information theoretic method for proving that a random process terminates, originally used by Robin Moser to prove an algorithmic version of the Lovász local lemma.
In mathematics and theoretical physics (especially twistor string theory), an a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation ∇ A ′ ( A Ω B ) = 0 {\displaystyle \nabla _{A'}^{(A}\Omega _{^{}}^{B)}=0} . It was described in the 1960s by Roger Penrose and Malcolm MacCallum. According to Andrew Hodges,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ( E ≥ 0 ) {\displaystyle ~(~E\geq 0~)~} energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since this group is noncompact, these unitary representation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
For the first case Note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with P = ( m , 0 , 0 , 0 ) {\displaystyle ~P=(m,0,0,0)~} is a representation of SO(3).In the ray interpretation, one can go over to Spin(3) instead. So, massive states are classified by an irreducible Spin(3) uni... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics, a bifundamental representation is a representation obtained as a tensor product of two fundamental or antifundamental representations. For example, the MN-dimensional representation (M,N) of the group S U ( M ) × S U ( N ) {\displaystyle SU(M)\times SU(N)} is a bifundamental repr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, …, en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on R n {\displaystyle \mathbb {R} ^{n}} , functions on a manifold, vector valued functions, vector fields, or, more generally, se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics, braid statistics is a generalization of the spin statistics of bosons and fermions based on the concept of braid group. While for fermions (Bosons) the corresponding statistics is associated to a phase gain of π {\displaystyle \pi } ( 2 π {\displaystyle 2\pi } ) under the exchang... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies.That is, if we imagine that the phase space is modelled by a torus T (that is, the variables are periodic like angles),... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold.
In mathematics ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The even dimensional space of dimension d {\displaystyle d} can be described using complex coordinates w i ∈ C d / 2 {\displaystyle w_{i}\in \mathbb {C} ^{d/2}} with a metric g i j ¯ = ( 1 + ρ d r d ) 2 / d , {\displaystyle g_{i{\bar {j}}}={\bigg (}1+{\frac {\rho ^{d}}{r^{d}}}{\bigg )}^{2/d}{\bigg },} where ρ {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback. It may be determined using the following formula (using the Einstein summation convention), which is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and, in particular, differential geometry a circular surface is the image of a map ƒ: I × S1 → R3, where I ⊂ R is an open interval and S1 is the unit circle, defined by f ( t , θ ) := γ ( t ) + r ( t ) u ( t ) cos θ + r ( t ) v ( t ) sin θ , {\displaystyle f(t,\theta ):=\gamma (t)+r(t){\mathbf {u} }(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions. The upper Dini derivative, which is also called an upper right-hand derivative, of a co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics applied to analysis of social structures, homogeneity blockmodeling is an approach in blockmodeling, which is best suited for a preliminary or main approach to valued networks, when a prior knowledge about these networks is not available. This is due to the fact, that homogeneity blockmodeling emphasizes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.
In mathematics complex analysis, the Sarason interpolation theorem, introduced by Sarason (1967), is a generalization o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics education at primary school level, a number bond (sometimes alternatively called an addition fact) is a simple addition sum which has become so familiar that a child can recognise it and complete it almost instantly, with recall as automatic as that of an entry from a multiplication table in multiplicati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
As a result, it is often considered to be a more intuitive, but a less systematic approach to divisions – where the efficiency is highly dependent upon one's numeracy skills. To calculate the whole number quotient of dividing a large number by a small number, the student repeatedly takes away "chunks" of the large numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
However, it is argued that chunking, rather than moving straight to short division, gives a better introduction to division, in part because the focus is always holistic, focusing throughout on the whole calculation and its meaning, rather than just rules for generating successive digits. The more freeform nature of ch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts through developmentally appropriate hands-on experience. The use of manipulatives in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Notable collections of virtual manipulatives include The National Library of Virtual Manipulatives and the Ubersketch. Multiple experiences with manipulatives provide children with the conceptual foundation to understand mathematics at a conceptual level and are recommended by the NCTM.Some of the manipulatives are now... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Examples of such notations are: 3 + 4 {\displaystyle 3+4}: refers to the process of adding as well as the outcome of the process. ∑ n = 0 ∞ ( a n ) {\displaystyle \sum _{n=0}^{\infty }(a_{n})}: refers to the process of summing an infinite sequence, and to the outcome of the process. f ( x ) = 3 x + 2 {\displaystyle f(x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics education, ethnomathematics is the study of the relationship between mathematics and culture. Often associated with "cultures without written expression", it may also be defined as "the mathematics which is practised among identifiable cultural groups". It refers to a broad cluster of ideas ranging from ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pierre van Hiele published Structure and Insight in 1986, further describing his theory. The model has greatly influenced geometry curricula throughout the world through emphasis on analyzing properties and classification of shapes at early grade levels. In the United States, the theory has influenced the geometry stra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics graph theory, a single-entry single-exit (SESE) region in a given graph is an ordered edge pair. For example, with the ordered edge pair, (a, b) of distinct control-flow edges a and b where: a dominates b b postdominates a Every cycle containing a also contains b and vice versa.where a node x is said to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
On the probability space we define the space X = { X } {\displaystyle {\mathcal {X}}=\{X\}} of random variables with values in measurable metric space ( U , d u ) {\displaystyle (U,d_{u})} and the space Y = { Y } {\displaystyle {\mathcal {Y}}=\{Y\}} of random variables with values in measurable metric space ( V , d v )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
C = F − 1 B , {\displaystyle X\in {\mathcal {A}},\mathbf {F} X\in {\mathcal {B}}\Leftrightarrow X\in {\mathcal {C}},i.e.{\mathcal {C}}=\mathbf {F} ^{-1}{\mathcal {B}},} where F − 1 B {\displaystyle \mathbf {F} ^{-1}{\mathcal {B}}} denotes the complete inverse image of B {\displaystyle {\mathcal {B}}} in A {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics in the branch of differential geometry, the cocurvature of a connection on a manifold is the obstruction to the integrability of the vertical bundle.
In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k generated by the d-by-d minors of a generic d-by-n matrix (x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics numerical analysis, the Nyström method or quadrature method seeks the numerical solution of an integral equation by replacing the integral with a representative weighted sum. The continuous problem is broken into n {\displaystyle n} discrete intervals; quadrature or numerical integration determines the w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematics of special functions, the Neuman–Sándor mean M, of two positive and unequal numbers a and b, is defined as: M ( a , b ) = a − b 2 arsinh ( a − b a + b ) {\displaystyle M(a,b)={\frac {a-b}{2\operatorname {arsinh} \left({\frac {a-b}{a+b}}\right)}}} This mean interpolates the inequality of the unweighted ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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