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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. | Wikipedia - Lagrange duality - Summary |
In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968. | Wikipedia - Linear complementarity problem - Summary |
In mathematical optimization theory, the mixed linear complementarity problem, often abbreviated as MLCP or LMCP, is a generalization of the linear complementarity problem to include free variables. | Wikipedia - Mixed linear complementarity problem - Summary |
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... | Wikipedia - Bland's rule - Summary |
It gets stuck at a basic feasible solution (a corner of the feasible polytope) and changes bases in a cyclic way without decreasing the minimization target. Such cycles are avoided by Bland's rule for choosing a column to enter and a column to leave the basis. Bland's rule was developed by Robert G. Bland, now an Emeri... | Wikipedia - Bland's rule - Summary |
In mathematical optimization, Cunningham's rule (also known as least recently considered rule or round-robin rule) is an algorithmic refinement of the simplex method for linear optimization. The rule was proposed 1979 by W. H. Cunningham to defeat the deformed hypercube constructions by Klee and Minty et al. (see, e.g.... | Wikipedia - Cunningham's rule - Summary |
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... | Wikipedia - Cunningham's rule - Summary |
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operat... | Wikipedia - Simplex algorithm - Summary |
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... | Wikipedia - Himmelblau's function - Summary |
In mathematical optimization, Wolfe duality, named after Philip Wolfe, is type of dual problem in which the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle. | Wikipedia - Wolfe duality - Summary |
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... | Wikipedia - Zadeh's rule - Summary |
In mathematical optimization, Zermelo's navigation problem, proposed in 1931 by Ernst Zermelo, is a classic optimal control problem that deals with a boat navigating on a body of water, originating from a point A {\displaystyle A} to a destination point B {\displaystyle B} . The boat is capable of a certain maximum spe... | Wikipedia - Zermelo's navigation problem - Summary |
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... | Wikipedia - Solution space - Summary |
The feasible set of the problem is separate from the objective function, which states the criterion to be optimized and which in the above example is x 2 + y 4 . {\displaystyle x^{2}+y^{4}.} In many problems, the feasible set reflects a constraint that one or more variables must be non-negative. | Wikipedia - Solution space - Summary |
In pure integer programming problems, the feasible set is the set of integers (or some subset thereof). In linear programming problems, the feasible set is a convex polytope: a region in multidimensional space whose boundaries are formed by hyperplanes and whose corners are vertices. Constraint satisfaction is the proc... | Wikipedia - Solution space - Summary |
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... | Wikipedia - Quadratically constrained quadratic program - Summary |
In mathematical optimization, affine scaling is an algorithm for solving linear programming problems. Specifically, it is an interior point method, discovered by Soviet mathematician I. I. Dikin in 1967 and reinvented in the U.S. in the mid-1980s. | Wikipedia - Affine scaling - Summary |
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... | Wikipedia - Constrained minimisation - Summary |
In mathematical optimization, cost functions and non-linear components within can be linearized in order to apply a linear solving method such as the Simplex algorithm. The optimized result is reached much more efficiently and is deterministic as a global optimum. | Wikipedia - Linearization - Optimization |
In mathematical optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio of two functions that are in general nonlinear. The ratio to be optimized often describes some kind of efficiency of a system. | Wikipedia - Fractional programming - Summary |
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... | Wikipedia - Linear-fractional programming - Summary |
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... | Wikipedia - Very large-scale neighborhood search - Summary |
If neighborhood searched is limited to just one or a very small number of changes from the current solution, then it can be difficult to escape from local minima, even with additional meta-heuristic techniques such as Simulated Annealing or Tabu search. In large neighborhood search techniques, the possible changes from... | Wikipedia - Very large-scale neighborhood search - Summary |
In mathematical optimization, oracle complexity is a standard theoretical framework to study the computational requirements for solving classes of optimization problems. It is suitable for analyzing iterative algorithms which proceed by computing local information about the objective function at various points (such as... | Wikipedia - Oracle complexity (optimization) - Summary |
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... | Wikipedia - Ackley function - Summary |
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequali... | Wikipedia - KKT conditions - Summary |
In mathematical optimization, the Klee–Minty cube is an example that shows the worst-case computational complexity of many algorithms of linear optimization. It is a deformed cube with exactly 2D corners in dimension D. Klee and Minty showed that Dantzig's simplex algorithm visits all corners of a (perturbed) cube in d... | Wikipedia - Klee–Minty cube - Worst case |
In mathematical optimization, the Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function. It was first proposed in 1974 by Rastrigin as a 2-dimensional function and has been generalized by Rudolph. | Wikipedia - Rastrigin function - Summary |
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... | Wikipedia - Rastrigin function - Summary |
In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Rosenbrock's banana function. The global minimum is inside a long, narrow, parab... | Wikipedia - Rosenbrock function - Summary |
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, ... | Wikipedia - Rosenbrock function - Summary |
In mathematical optimization, the active-set method is an algorithm used to identify the active constraints in a set of inequality constraints. The active constraints are then expressed as equality constraints, thereby transforming an inequality-constrained problem into a simpler equality-constrained subproblem. An opt... | Wikipedia - Active set - Summary |
{\displaystyle g_{i}(x_{0})>0.} Equality constraints are always active. | Wikipedia - Active set - Summary |
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... | Wikipedia - Active set - Summary |
In mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and nonlinear objective functions; there are criss-cross algorithms for linear-fractional programming... | Wikipedia - Criss-cross algorithm - Summary |
In mathematical optimization, the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, a... | Wikipedia - Cutting plane - Summary |
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... | Wikipedia - Cutting plane - Summary |
Finding such an inequality is the separation problem, and such an inequality is a cut. A cut can be added to the relaxed linear program. | Wikipedia - Cutting plane - Summary |
Then, the current non-integer solution is no longer feasible to the relaxation. This process is repeated until an optimal integer solution is found. | Wikipedia - Cutting plane - Summary |
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... | Wikipedia - Cutting plane - Summary |
In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a number of steps that is polynomial in the input size.... | Wikipedia - Ellipsoidal algorithm - Summary |
In mathematical optimization, the firefly algorithm is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behavior of fireflies. | Wikipedia - Firefly algorithm - Summary |
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... | Wikipedia - Lagrange multiplier - Summary |
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 }}... | Wikipedia - Lagrange multiplier - Summary |
The solution corresponding to the original constrained optimization is always a saddle point of the Lagrangian function, which can be identified among the stationary points from the definiteness of the bordered Hessian matrix.The great advantage of this method is that it allows the optimization to be solved without exp... | Wikipedia - Lagrange multiplier - Summary |
In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. The network simplex method works very well in practice, typically 200 to 300 times faster than the simplex method applied... | Wikipedia - Network simplex algorithm - Summary |
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... | Wikipedia - Ordered subset expectation maximization - Summary |
In mathematical optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact that any such function defines a perturbation of the initial problem. In many cases this takes the form of shifting the constraints.In some texts the value function is called t... | Wikipedia - Perturbation function - Summary |
In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are not allowed to become negative. That is, given a matrix A and a (column) vector of response variables y, the goal is to find a r g m i n x ‖ A x − y ‖ 2 2 {\displays... | Wikipedia - Non-negative least squares - Summary |
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... | Wikipedia - Proximal operator - Summary |
In mathematical optimization, the push–relabel algorithm (alternatively, preflow–push algorithm) is an algorithm for computing maximum flows in a flow network. The name "push–relabel" comes from the two basic operations used in the algorithm. Throughout its execution, the algorithm maintains a "preflow" and gradually c... | Wikipedia - Push–relabel maximum flow algorithm - Summary |
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... | Wikipedia - Push–relabel maximum flow algorithm - Summary |
In mathematical optimization, the revised simplex method is a variant of George Dantzig's simplex method for linear programming. The revised simplex method is mathematically equivalent to the standard simplex method but differs in implementation. Instead of maintaining a tableau which explicitly represents the constrai... | Wikipedia - Revised simplex algorithm - Summary |
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 ... | Wikipedia - Total dual integrality - Summary |
Further, Giles and Pulleyblank showed that if P {\displaystyle P} is a polytope whose vertices are all integer valued, then P {\displaystyle P} is the solution set of some TDI system A x ≤ b {\displaystyle Ax\leq b} , where b {\displaystyle b} is integer valued. Note that TDI is a weaker sufficient condition for integr... | Wikipedia - Total dual integrality - Summary |
In mathematical or statistical modeling a threshold model is any model where a threshold value, or set of threshold values, is used to distinguish ranges of values where the behaviour predicted by the model varies in some important way. A particularly important instance arises in toxicology, where the model for the eff... | Wikipedia - Liability threshold model - Summary |
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... | Wikipedia - Prime filter - Summary |
In mathematical physics Linear transport theory is the study of equations describing the migration of particles or energy within a host medium when such migration involves random absorption, emission and scattering events. Subject to certain simplifying assumptions, this is a common and useful framework for describing ... | Wikipedia - Linear transport theory - Summary |
In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory. In accordance with this analogy with s... | Wikipedia - Gravitational instantons - Summary |
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... | Wikipedia - Gravitational instantons - Summary |
In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construction of Instantons." | Wikipedia - ADHM construction - Summary |
In mathematical physics and harmonic analysis, the quadratic Fourier transform is an integral transform that generalizes the fractional Fourier transform, which in turn generalizes the Fourier transform.Roughly speaking, the Fourier transform corresponds to a change of variables from time to frequency (in the context o... | Wikipedia - Quadratic Fourier transform - Summary |
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... | Wikipedia - Quadratic Fourier transform - Summary |
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. These matrices are named after the... | Wikipedia - Pauli matrices - Summary |
In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal / vertical polarization, 45 degree polarization (right/left), and circul... | Wikipedia - Pauli matrices - Summary |
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... | Wikipedia - Pauli matrices - Summary |
{\displaystyle \ \mathbb {R} ^{3}~.} The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 form a basis for the real Lie algebra s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} , which exponentiates to the special ... | Wikipedia - Pauli matrices - Summary |
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processes, such as Brownian motion. Indeed, according to Donsker's theorem, the di... | Wikipedia - Continuum limit - Summary |
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... | Wikipedia - Gaussian q-distribution - Summary |
In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form u ( x , t ) = e − i ω t ϕ ( x ) {\displaystyle u(x,t)=e^{-i\omega t}\phi (x)} is said to be orbitally stable if any solution with the initial data sufficiently close to ϕ ( x ) {\displaystyle \phi (x)} forev... | Wikipedia - Orbital stability - Summary |
In mathematical physics higher gauge theory is the general study of counterparts of gauge theory that involve higher-degree differential forms instead of the traditional connection forms of gauge theories. | Wikipedia - Higher gauge theory - Summary |
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... | Wikipedia - KZ equation - Summary |
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where ... | Wikipedia - Clebsch–Gordan coefficients for SU(3) - Summary |
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... | Wikipedia - Gleason's theorem - Summary |
In mathematical physics, Hilbert system is an infrequently used term for a physical system described by a C*-algebra.In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attribute... | Wikipedia - Hilbert-Ackermann system - Summary |
Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemes. The most commonly studied Hilbert systems have either just one rule of inference – modus ponens, for propositional logics – or two – with generalisation, to handle predicate logics, as well – and severa... | Wikipedia - Hilbert-Ackermann system - Summary |
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... | Wikipedia - Hilbert-Ackermann system - Summary |
In mathematical physics, Kundt spacetimes are Lorentzian manifolds admitting a geodesic null congruence with vanishing optical scalars (expansion, twist and shear). A well known member of Kundt class is pp-wave. Ricci-flat Kundt spacetimes in arbitrary dimension are algebraically special. | Wikipedia - Kundt spacetime - Summary |
In four dimensions Ricci-flat Kundt metrics of Petrov type III and N are completely known. All VSI spacetimes belong to a subset of the Kundt spacetimes. == References == | Wikipedia - Kundt spacetime - Summary |
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... | Wikipedia - Geometry of special relativity - Summary |
The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz, Henri Poincaré, and others, and said it "was grown on experimental physical grounds." Minkowsk... | Wikipedia - Geometry of special relativity - Summary |
While the individual components in Euclidean space and time might differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events. Minkowski space differs from four-dimensional Euclidean space insofar as it treats time diff... | Wikipedia - Geometry of special relativity - Summary |
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... | Wikipedia - Geometry of special relativity - Summary |
Spacetime is equipped with an indefinite non-degenerate bilinear form, variously called the Minkowski metric, the Minkowski norm squared or Minkowski inner product depending on the context. The Minkowski inner product is defined so as to yield the spacetime interval between two events when given their coordinate differ... | Wikipedia - Geometry of special relativity - Summary |
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(n), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagne... | Wikipedia - Yang–Mills action - Summary |
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... | Wikipedia - Gibbons–Hawking space - Summary |
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express... | Wikipedia - Grassman variable - Summary |
In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions. | Wikipedia - Pöschl–Teller potential - Summary |
In mathematical physics, a caloron is the finite temperature generalization of an instanton. | Wikipedia - Caloron - Summary |
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 Willem Jacob van Stockum in 1937 and later confirmed by Kurt Gödel in 1949, who discovered a s... | Wikipedia - Closed timelike loop - Summary |
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lat... | Wikipedia - Lattice model (physics) - Summary |
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... | Wikipedia - Lattice model (physics) - Summary |
The solution of these models has given insights into the nature of phase transitions, magnetization and scaling behaviour, as well as insights into the nature of quantum field theory. Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory t... | Wikipedia - Lattice model (physics) - Summary |
However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information, aka Holographic principle. More generally, lattice gauge theory and lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics ... | Wikipedia - Lattice model (physics) - Summary |
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... | Wikipedia - Null dust solution - Summary |
In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance in superstring theory: the Ramond algebra (named after Pierre Ramond) and the Neveu–Schwarz algebra (named after André ... | Wikipedia - Neveu–Schwarz boundary conditions - Summary |
In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise mathematical structures. This demonstration requires new mathematics, in a sense analogous to classical real analysis, putting calculus on a mathematically rigorous fou... | Wikipedia - Constructive quantum field theory - Summary |
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... | Wikipedia - Constructive quantum field theory - Summary |
The existence theorems for quantum fields can be expected to be very difficult to find, if indeed they are possible at all. One discovery of the theory that can be related in non-technical terms, is that the dimension d of the spacetime involved is crucial. Notable work in the field by James Glimm and Arthur Jaffe show... | Wikipedia - Constructive quantum field theory - Summary |
Along with work of their students, coworkers, and others, constructive field theory resulted in a mathematical foundation and exact interpretation to what previously was only a set of recipes, also in the case d < 4. Theoretical physicists had given these rules the name "renormalization," but most physicists had been s... | Wikipedia - Constructive quantum field theory - Summary |
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