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In mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitted model is a mathematical model that contains more parameters than can be ju... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
For example, a model might be selected by maximizing its performance on some set of training data, and yet its suitability might be determined by its ability to perform well on unseen data; then over-fitting occurs when a model begins to "memorize" training data rather than "learning" to generalize from a trend. As an ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical modeling, resilience refers to the ability of a dynamical system to recover from perturbations and return to its original stable steady state. It is a measure of the stability and robustness of a system in the face of changes or disturbances. If a system is not resilient enough, it is more susceptible t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical morphology and digital image processing, a top-hat transform is an operation that extracts small elements and details from given images. There exist two types of top-hat transform: the white top-hat transform is defined as the difference between the input image and its opening by some structuring elemen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration (or pattern) in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or-miss transform is the set of positions where the first structuring element f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries. In the binary numeral system, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization and related fields, relaxation is a modeling strategy. A relaxation is an approximation of a difficult problem by a nearby problem that is easier to solve. A solution of the relaxed problem provides information about the original problem. For example, a linear programming relaxation of an i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
This fact is called weak duality. In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality.
In mathematical optim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, Bland's rule (also known as Bland's algorithm, Bland's anti-cycling rule or Bland's pivot rule) is an algorithmic refinement of the simplex method for linear optimization. With Bland's rule, the simplex algorithm solves feasible linear optimization problems without cycling.The original sim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
History-based rules defeat the deformed hypercube constructions because they tend to average out how many times a variable pivots. It has recently been shown by David Avis and Oliver Friedmann that there is a family of linear programs on which the simplex algorithm equipped with Cunningham's rule requires exponential t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, Himmelblau's function is a multi-modal function, used to test the performance of optimization algorithms. The function is defined by: f ( x , y ) = ( x 2 + y − 11 ) 2 + ( x + y 2 − 7 ) 2 . {\displaystyle f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}.\quad } It has one local maximum at x = − 0.27... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, Zadeh's rule (also known as the least-entered rule) is an algorithmic refinement of the simplex method for linear optimization. The rule was proposed around 1980 by Norman Zadeh (son of Lotfi A. Zadeh), and has entered the folklore of convex optimization since then.Zadeh offered a reward o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potentially including inequalities, equalities, and integer constraints. This is th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions. It has the form minimize 1 2 x T P 0 x + q 0 T x subject to 1 2 x T P i x + q i T x + r i ≤ 0 for i = 1 , … , m , A x = b , {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function, which is to b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a speci... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, neighborhood search is a technique that tries to find good or near-optimal solutions to a combinatorial optimisation problem by repeatedly transforming a current solution into a different solution in the neighborhood of the current solution. The neighborhood of a solution is a set of simil... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, the Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed by David Ackley in his 1987 PhD dissertation.On a 2-dimensional domain it is defined by: f ( x , y ) = − 20 exp − exp + e + 20 {\displaystyle {\begin{aligned}f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The generalized version was popularized by Hoffmeister & Bäck and Mühlenbein et al. Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima. On an n {\displaystyle n} -dimensional domain it is defined by: f ( x ) = A n + ∑ i = 1 n {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
To converge to the global minimum, however, is difficult. The function is defined by f ( x , y ) = ( a − x ) 2 + b ( y − x 2 ) 2 {\displaystyle f(x,y)=(a-x)^{2}+b(y-x^{2})^{2}} It has a global minimum at ( x , y ) = ( a , a 2 ) {\displaystyle (x,y)=(a,a^{2})} , where f ( x , y ) = 0 {\displaystyle f(x,y)=0} . Usually, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The active set at x 0 {\displaystyle x_{0}} is made up of those constraints g i ( x 0 ) {\displaystyle g_{i}(x_{0})} that are active at the current point (Nocedal & Wright 2006, p. 308). The active set is particularly important in optimization theory, as it determines which constraints will influence the final result o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The theory of Linear Programming dictates that under mild assumptions (if the linear program has an optimal solution, and if the feasible region does not contain a line), one can always find an extreme point or a corner point that is optimal. The obtained optimum is tested for being an integer solution. If it is not, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cutting-plane methods for general convex continuous optimization and variants are known under various names: Kelley's method, Kelley–Cheney–Goldstein method, and bundle methods. They are popularly used for non-differentiable convex minimization, where a convex objective function and its subgradient can be evaluated eff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
This means that all partial derivatives should be zero, including the partial derivative with respect to λ {\displaystyle \lambda ~} . ∂ L ∂ x = 0 {\displaystyle \ {\frac {\ \partial {\mathcal {L}}\ }{\partial x}}=0\qquad } and ∂ L ∂ λ = 0 ; {\displaystyle \qquad {\frac {\ \partial {\mathcal {L}}\ }{\partial \lambda }}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, the ordered subset expectation maximization (OSEM) method is an iterative method that is used in computed tomography. In applications in medical imaging, the OSEM method is used for positron emission tomography, for single photon emission computed tomography, and for X-ray computed tomogra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, the proximal operator is an operator associated with a proper, lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle {\mathcal {X}}} to {\displaystyle } , and is defined by: prox f ( v ) = arg min x ∈ X ( f ( x ) + 1 2 ‖ x − v ‖ X 2 ) . {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The generic algorithm has a strongly polynomial O(V 2E) time complexity, which is asymptotically more efficient than the O(VE 2) Edmonds–Karp algorithm. Specific variants of the algorithms achieve even lower time complexities. The variant based on the highest label node selection rule has O(V 2√E) time complexity and i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical optimization, total dual integrality is a sufficient condition for the integrality of a polyhedron. Thus, the optimization of a linear objective over the integral points of such a polyhedron can be done using techniques from linear programming. A linear system A x ≤ b {\displaystyle Ax\leq b} , where A ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with positive-definite, as opposed to Lorentzian, metric. There are many possible generalizations of the original conception of a gravitational instanton: for example one can allow gravitational instanto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The quadratic Fourier transform extends this further to the group of all linear symplectic transformations in phase space (of which rotations are a subgroup). More specifically, for every member of the metaplectic group (which is a double cover of the symplectic group) there is a corresponding quadratic Fourier transfo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers. Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex 2 dimensional Hilbert space. I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics and probability and statistics, the Gaussian q-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel. It is a q-analog of the Gaussian or normal distribution. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A characteristic feature of the many variants of Hilbert systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if one is interested only in the derivability of tautologies, no hypothetical judgments, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, Minkowski space (or Minkowski spacetime) () combines inertial space and time manifolds (x,y) with a non-inertial reference frame of space and time (x',t') into a four-dimensional model relating a position (inertial frame of reference) to the field. A four-vector (x,y,z,t) consists of a coordina... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
It is generated by rotations, reflections and translations. When time is appended as a fourth dimension, the further transformations of translations in time and Lorentz boosts are added, and the group of all these transformations is called the Poincaré group. Minkowski's model follows special relativity where motion ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, a Gibbons–Hawking space, named after Gary Gibbons and Stephen Hawking, is essentially a hyperkähler manifold with an extra U(1) symmetry. (In general, Gibbons–Hawking metrics are a subclass of hyperkähler metrics.) Gibbons–Hawking spaces, especially ambipolar ones, find an application in the st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, a caloron is the finite temperature generalization of an instanton.
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons. Techniques for solving these include the inver... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, a null dust solution (sometimes called a null fluid) is a Lorentzian manifold in which the Einstein tensor is null. Such a spacetime can be interpreted as an exact solution of Einstein's field equation, in which the only mass–energy present in the spacetime is due to some kind of massless radia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Attempts to put quantum field theory on a basis of completely defined concepts have involved most branches of mathematics, including functional analysis, differential equations, probability theory, representation theory, geometry, and topology. It is known that a quantum field is inherently hard to handle using convent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The traditional basis of constructive quantum field theory is the set of Wightman axioms. Osterwalder and Schrader showed that there is an equivalent problem in mathematical probability theory. The examples with d < 4 satisfy the Wightman axioms as well as the Osterwalder–Schrader axioms. They also fall in the related ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
First proposed by Luigi Fantappiè in 1954, the theory remained obscure until it was rediscovered in 1968 by Henri Bacry and Jean-Marc Lévy-Leblond. In 1972, Freeman Dyson popularized it as a hypothetical road by which mathematicians could have guessed part of the structure of general relativity before it was discovered... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is t 2 − r 2 = T 2 {\displaystyle t^{2}-r^{2}=T^{2}} (t and r being the u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal, one-to-one transformations on coordinate space-time. They are less studied in physics because, unlike the rotations and translations of Poincaré symmetry, an object cannot be physically trans... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere (with its canonical Riemannian metric). The main application of de Sitter space is its use in general relat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Various lower limits have been claimed for the noncommutative scale, (i.e. how accurately positions can be measured) but there is currently no experimental evidence in favour of such a theory or grounds for ruling them out. One of the novel features of noncommutative field theories is the UV/IR mixing phenomenon in whi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
{\displaystyle \subset {\mathfrak {h}},\qquad \subset {\mathfrak {f}}.} (In physics, for instance, h {\displaystyle {\mathfrak {h}}} amount to vector generators and f {\displaystyle {\mathfrak {f}}} to axial ones.) There exists an open neighborhood U of the unit of G such that any element g ∈ U {\displaystyle g\in U} i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Then there is a local section s ( g σ 0 ) = exp ( F ) {\displaystyle s(g\sigma _{0})=\exp(F)} of G → G / H {\displaystyle G\to G/H} over U 0 {\displaystyle U_{0}} . With this local section, one can define the induced representation, called the nonlinear realization, of elements g ∈ U G ⊂ G {\displaystyle g\in U_{G}\s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Then a desired nonlinear realization of g {\displaystyle {\mathfrak {g}}} in f × V {\displaystyle {\mathfrak {f}}\times V} reads F α: ( σ γ F γ , v ) → ( F α ( σ γ ) F γ , F α ( v ) ) , I a: ( σ γ F γ , v ) → ( I a ( σ γ ) F γ , I a v ) , {\displaystyle F_{\alpha }:(\sigma ^{\gamma }F_{\gamma },v)\to (F_{\alpha }(\sigm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In order for F to be described in terms of a scalar potential only, any of the following equivalent statements have to be true: − ∫ a b F ⋅ d l = P ( b ) − P ( a ) , {\displaystyle -\int _{a}^{b}\mathbf {F} \cdot d\mathbf {l} =P(\mathbf {b} )-P(\mathbf {a} ),} where the integration is over a Jordan arc passing from loc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The gravity potential is the gravitational potential energy per unit mass. In electrostatics the electric potential is the scalar potential associated with the electric field, i.e., with the electrostatic force per unit charge. The electric potential is in this case the electrostatic potential energy per unit charge.
... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Examples of non-conservative forces include frictional forces, magnetic forces, and in fluid mechanics a solenoidal field velocity field. By the Helmholtz decomposition theorem however, all vector fields can be describable in terms of a scalar potential and corresponding vector potential. In electrodynamics, the electr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime. It is a vector space that a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the energy–momentum tensor that is constructed from the canonical energy–momentum tensor and the spin current so as to be symmetric yet still conserved. In a classical or quantum local field theory, the generator of Lorentz transformations ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The Belinfante–Rosenfeld tensor is a modification of the energy momentum tensor T B μ ν = T μ ν + 1 2 ∂ λ ( S μ ν λ + S ν μ λ − S λ ν μ ) {\displaystyle T_{B}^{\mu \nu }=T^{\mu \nu }+{\frac {1}{2}}\partial _{\lambda }(S^{\mu \nu \lambda }+S^{\nu \mu \lambda }-S^{\lambda \nu \mu })} that is constructed from the canonica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is not an integral in the Lebesgue sense; the word "integral" is used because ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Degasperis–Procesi equation u t − u x x t + 2 κ u x + 4 u u x = 3 u x u x x + u u x x x {\displaystyle \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}} is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs: ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Dirac algebra is the Clifford algebra Cl 1 , 3 ( C ) {\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )} . This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of the gamma matrices, which rep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
For this article we fix the signature to be mostly minus, that is, ( + , − , − , − ) {\displaystyle (+,-,-,-)} . The Dirac algebra is then the linear span of the identity, the gamma matrices γ μ {\displaystyle \gamma ^{\mu }} as well as any linearly independent products of the gamma matrices. This forms a finite-dimens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved spacetime, a general Lorentzian manifold.
In mathematical physics, the Dirac–von Neumann axioms give a mathematical formulation of quantum mechanics in terms of oper... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation within the nonlinear Schrödinger class: i ψ t + ψ x x + 2 ( | ψ | 2 ) x ψ + | ψ | 4 ψ = 0. {\displaystyle i\psi _{t}+\psi _{xx}+2\left(|\psi |^{2}\right)_{x}\,\psi +|\psi |^{4}\,\psi =0.} The equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Gordon decomposition (named after Walter Gordon) of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Peres metric is defined by the proper time d τ 2 = d t 2 − 2 f ( t + z , x , y ) ( d t + d z ) 2 − d x 2 − d y 2 − d z 2 {\displaystyle {d\tau }^{2}=dt^{2}-2f(t+z,x,y)(dt+dz)^{2}-dx^{2}-dy^{2}-dz^{2}} for any arbitrary function f. If f is a harmonic function with respect to x and y, then th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 after Haag–Ruelle scatter... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
{\displaystyle p_{w}(s)={\frac {\pi s}{2}}e^{-\pi s^{2}/4}.} Here, s = S D {\displaystyle s={\frac {S}{D}}} where S is a particular spacing and D is the mean distance between neighboring intervals.In a mixed sequence (spin and parity are different), the probability density function can be obtained by randomly superimpo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In practice, it is a good approximation for the actual distribution for real symmetric matrices of any dimension. The corresponding result for complex hermitian matrices (which is also exact in the 2 × 2 {\displaystyle 2\times 2} case and a good approximation in general) with distribution proportional to e − 1 2 T r ( ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian H = p 2 + f ( x ) {\displaystyle H=p^{2}+f(x)} with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by ( n ) = u ( n + 1 ) + u ( n − 1 ) + 2 λ cos ( 2 π ( ω + n α ) ) u ( n ) , {\displaystyle (n)=u(n+1)+u(n-1)+2\lambda \cos(2\pi (\omega +n\alpha ))u(n),\,} acting as a self-adjoint operator on the Hilber... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some vari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations. They called a generic expression of conformal symmetry a spherical wave transformation. General relativity in two spacetime dimensions also enjoys conformal symmetry.
In mathematical physics, the diagrammatic M... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In Dirac representation, the four contravariant gamma matrices are γ 0 = ( 1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1 ) , γ 1 = ( 0 0 0 1 0 0 1 0 0 − 1 0 0 − 1 0 0 0 ) , γ 2 = ( 0 0 0 − i 0 0 i 0 0 i 0 0 − i 0 0 0 ) , γ 3 = ( 0 0 1 0 0 0 0 − 1 − 1 0 0 0 0 1 0 0 ) . {\displaystyle {\begin{aligned}\gamma ^{0}&={\begin{pmatrix}1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
More compactly, γ 0 = σ 3 ⊗ I 2 , {\displaystyle \ \gamma ^{0}=\sigma ^{3}\otimes I_{2}\ ,} and γ j = i σ 2 ⊗ σ j , {\displaystyle \ \gamma ^{j}=i\sigma ^{2}\otimes \sigma ^{j}\ ,} where ⊗ {\displaystyle \ \otimes \ } denotes the Kronecker product and the σ j {\displaystyle \ \sigma ^{j}\ } (for j = 1, 2, 3) denote the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by = a b c − a c b − b a c + b c a + c a b − c b a . {\displaystyle =abc-acb-bac+bca+cab-cba.\,} Also called the ternutator or alternating ternary sum, it is a special case of the n-commutator for n = 3, where... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously defined Yang–Mills measure, meaning that the (Euclidean) path integral can be interpreted as a measure on the set of conne... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Not all concepts are interchangeable; some require other concepts as prerequisites. Conversely, competency at one skill may ease the acquisition of another through similarity. A knowledge space marks out which collections of skills are feasible: they can be learned without mastering any other skills. Under reasonable a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical representation theory, Steinberg's formula, introduced by Steinberg (1961), describes the multiplicity of an irreducible representation of a semisimple complex Lie algebra in a tensor product of two irreducible representations. It is a consequence of the Weyl character formula, and for the Lie algebra s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical representation theory, a good filtration is a filtration of a representation of a reductive algebraic group G such that the subquotients are isomorphic to the spaces of sections F(λ) of line bundles λ over G/B for a Borel subgroup B. In characteristic 0 this is automatically true as the irreducible modu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
An (n+1k) for 0 ≤ k ≤ n (exterior powers of vector representation). Quasi-minuscule: n2+2n (adjoint) Bn 1 (trivial), 2n (spin). Quasi-minuscule: 2n+1 (vector) Cn 1 (trivial), 2n (vector).
Quasi-minuscule: 2n2–n–1 if n>1 Dn 1 (trivial), 2n (vector), 2n−1 (half spin), 2n−1 (half spin). Quasi-minuscule: 2n2–n (adjoint) E... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical representation theory, coherence is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. The general notion of coherence was developed by Feit (1960, 1962), as a generalization of the proof by Frobenius of the e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical representation theory, two representations of a group on topological vector spaces are called Naimark equivalent (named after Mark Naimark) if there is a closed bijective linear map between dense subspaces preserving the group action.
In mathematical set theory, Baumgartner's axiom (BA) can be one of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Baumgartner's axiom is a consequence of the proper forcing axiom. It is consistent with a combination of ZFC, Martin's axiom, and the negation of the continuum hypothesis, but not implied by those hypotheses.Another axiom introduced by Baumgartner (1975) states that Martin's axiom for partially ordered sets MAP(κ) is t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for German mathematician Georg Cantor, who first stated and proved it at the end ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical set theory, a Cohen algebra, named after Paul Cohen, is a type of Boolean algebra used in the theory of forcing. A Cohen algebra is a Boolean algebra whose completion is isomorphic to the completion of a free Boolean algebra (Koppelberg 1993).
In mathematical set theory, a permutation model is a model ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first-order formula. Ordinal definable sets were introduced by Gödel (1965). A drawback to this informal definition is that it requires quantification over all first-order f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The class of hereditarily ordinal definable sets is denoted by HOD, and is a transitive model of ZFC, with a definable well ordering. It is consistent with the axioms of set theory that all sets are ordinal definable, and so hereditarily ordinal definable. The assertion that this situation holds is referred to as V = O... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical set theory, a transitive model is a model of set theory that is standard and transitive. Standard means that the membership relation is the usual one, and transitive means that the model is a transitive set or class.
In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical set theory, an Ulam matrix is an array of subsets of a cardinal number with certain properties. Ulam matrices were introduced by Stanislaw Ulam in his 1930 work on measurable cardinals: they may be used, for example, to show that a real-valued measurable cardinal is weakly inaccessible.
In mathematical... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical set theory, the Cantor tree is either the full binary tree of height ω + 1, or a topological space related to this by joining its points with intervals, that was introduced by Robert Lee Moore in the late 1920s as an example of a non-metrizable Moore space (Jones 1966).
In mathematical set theory, the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical set theory, the Mostowski model is a model of set theory with atoms where the full axiom of choice fails, but every set can be linearly ordered. It was introduced by Mostowski (1939). The Mostowski model can be constructed as the permutation model corresponding to the group of all automorphisms of the o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bernays (1937, page 68, axiom II (2)) introduced the axiom of adjunction as one of the axioms for a system of set theory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as general set theory or finitary set theory. The adjunction operation is also used as one of the op... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical statistics, the Darmois–Skitovich theorem characterizes the normal distribution (the Gaussian distribution) by the independence of two linear forms from independent random variables. This theorem was proved independently by G. Darmois and V. P. Skitovich in 1953.
In mathematical statistics, the Fisher ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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