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In mathematics, thestructuretensor, also referred to as thesecond-moment matrix, is amatrixderived from thegradientof afunction. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant to the observing coordinates. The structure tensor is often used i... | https://en.wikipedia.org/wiki/Structure_tensor |
Invector calculus, thesurface gradientis avectordifferential operatorthat is similar to the conventionalgradient. The distinction is that the surface gradient takes effect along a surface.
For asurfaceS{\displaystyle S}in ascalar fieldu{\displaystyle u}, the surface gradient is defined and notated as
wheren^{\displa... | https://en.wikipedia.org/wiki/Surface_gradient |
Inmathematicsandmathematical optimization, theconvex conjugateof a function is a generalization of theLegendre transformationwhich applies to non-convex functions. It is also known asLegendre–Fenchel transformation,Fenchel transformation, orFenchel conjugate(afterAdrien-Marie LegendreandWerner Fenchel). The convex con... | https://en.wikipedia.org/wiki/Convex_duality |
Inmathematics, adualitytranslates concepts,theoremsormathematical structuresinto other concepts, theorems or structures in aone-to-onefashion, often (but not always) by means of aninvolutionoperation: if the dual ofAisB, then the dual ofBisA. In other cases the dual of the dual – the double dual or bidual – is not nece... | https://en.wikipedia.org/wiki/Duality_(mathematics) |
Inmathematical optimizationand related fields,relaxationis amodeling strategy. A relaxation is anapproximationof 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, alinear programmingrelaxation of aninteger p... | https://en.wikipedia.org/wiki/Relaxation_(approximation) |
Inoperations research, theBig M methodis a method of solvinglinear programmingproblems using thesimplex algorithm. The Big M method extends the simplex algorithm to problems that contain "greater-than" constraints. It does so by associating the constraints with large negative constants which would not be part of any o... | https://en.wikipedia.org/wiki/Big_M_method |
Inmathematical optimization,Dantzig'ssimplex algorithm(orsimplex method) is a popularalgorithmforlinear programming.[1]
The name of the algorithm is derived from the concept of asimplexand was suggested byT. S. Motzkin.[2]Simplices are not actually used in the method, but one interpretation of it is that it operates o... | https://en.wikipedia.org/wiki/Simplex_algorithm |
Interior-point methods(also referred to asbarrier methodsorIPMs) arealgorithmsfor solvinglinearandnon-linearconvex optimizationproblems. IPMs combine two advantages of previously-known algorithms:
In contrast to the simplex method which traverses theboundaryof the feasible region, and the ellipsoid method which bounds... | https://en.wikipedia.org/wiki/Interior-point_method |
In anoptimization problem, aslack variableis a variable that is added to aninequality constraintto transform it into an equality constraint. A non-negativity constraint on the slack variable is also added.[1]: 131
Slack variables are used in particular inlinear programming. As with the other variables in the augmented... | https://en.wikipedia.org/wiki/Slack_variable |
Inmathematics,Slater's condition(orSlater condition) is asufficient conditionforstrong dualityto hold for aconvex optimization problem, named after Morton L. Slater.[1]Informally, Slater's condition states that thefeasible regionmust have aninterior point(see technical details below).
Slater's condition is a specific ... | https://en.wikipedia.org/wiki/Slater%27s_condition |
Design optimizationis an engineering design methodology using a mathematical formulation of a design problem to support selection of the optimal design among many alternatives. Design optimization involves the following stages:[1][2]
The formal mathematical (standard form) statement of the design optimization problem ... | https://en.wikipedia.org/wiki/Design_Optimization |
Incomputational complexity theory, afunction problemis acomputational problemwhere a single output (of atotal function) is expected for every input, but the output is more complex than that of adecision problem. For function problems, the output is not simply 'yes' or 'no'.
A functional problemP{\displaystyle P}is def... | https://en.wikipedia.org/wiki/Function_problem |
Inoperations research, theglove problem[1](also known as thecondom problem[2]) is anoptimization problemused as an example that the cheapest capital cost often leads to dramatic increase in operational time, but that the shortest operational time need not be given by the most expensive capital cost.[3]
Mdoctors are ea... | https://en.wikipedia.org/wiki/Glove_problem |
Operations research(British English:operational research) (U.S.Air Force Specialty Code: Operations Analysis), often shortened to theinitialismOR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and decision-making.[1][2]Although the termman... | https://en.wikipedia.org/wiki/Operations_research |
Satisficingis adecision-makingstrategy or cognitiveheuristicthat entails searching through the available alternatives until an acceptability threshold is met, without necessarilymaximizingany specific objective.[1]The termsatisficing, aportmanteauofsatisfyandsuffice,[2]was introduced byHerbert A. Simonin 1956,[3][4]alt... | https://en.wikipedia.org/wiki/Satisficing |
Inoptimization theory,semi-infinite programming(SIP) is anoptimization problemwith a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the constraints are typically parameterized.[1]
The problem can be stated simp... | https://en.wikipedia.org/wiki/Semi-infinite_programming |
TheFrank–Wolfe algorithmis aniterativefirst-orderoptimizationalgorithmforconstrainedconvex optimization. Also known as theconditional gradient method,[1]reduced gradient algorithmand theconvex combination algorithm, the method was originally proposed byMarguerite FrankandPhilip Wolfein 1956.[2]In each iteration, the Fr... | https://en.wikipedia.org/wiki/Frank%E2%80%93Wolfe_algorithm |
Aseparation oracle(also called acutting-plane oracle) is a concept in the mathematical theory ofconvex optimization. It is a method to describe aconvex setthat is given as an input to an optimization algorithm. Separation oracles are used as input toellipsoid methods.[1]: 87, 96, 98
LetKbe aconvexandcompactset in Rn. ... | https://en.wikipedia.org/wiki/Separation_oracle |
InBayesian statistics, acredible intervalis anintervalused to characterize aprobability distribution. It is defined such that an unobservedparametervalue has a particularprobabilityγ{\displaystyle \gamma }to fall within it. For example, in an experiment that determines the distribution of possible values of the paramet... | https://en.wikipedia.org/wiki/Credible_interval |
Inclinical trialsand other scientific studies, aninterim analysisis an analysis of data that is conducted before data collection has been completed. Clinical trials are unusual in that enrollment of subjects is a continual process staggered in time. If a treatment can be proven to be clearly beneficial or harmful com... | https://en.wikipedia.org/wiki/Interim_analysis |
Bayesian statistics(/ˈbeɪziən/BAY-zee-ənor/ˈbeɪʒən/BAY-zhən)[1]is a theory in the field ofstatisticsbased on theBayesian interpretation of probability, whereprobabilityexpresses adegree of beliefin anevent. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments... | https://en.wikipedia.org/wiki/Bayesian_statistics |
The word "probability" has been used in a variety of ways since it was first applied to the mathematical study ofgames of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answeri... | https://en.wikipedia.org/wiki/Probability_interpretations |
Mean-field particle methodsare a broad class ofinteracting typeMonte Carloalgorithms for simulating from a sequence of probability distributions satisfying a nonlinear evolution equation.[1][2][3][4]These flows of probability measures can always be interpreted as the distributions of the random states of a Markov proce... | https://en.wikipedia.org/wiki/Mean-field_particle_methods |
Metropolis light transport(MLT) is aglobal illuminationapplication of aMonte Carlo methodcalled theMetropolis–Hastings algorithmto therendering equationfor generating images from detailed physical descriptions ofthree-dimensionalscenes.[1][2]
The procedure constructs paths from the eye to a light source usingbidirecti... | https://en.wikipedia.org/wiki/Metropolis_light_transport |
Multiple-try Metropolis (MTM)is asampling methodthat is a modified form of theMetropolis–Hastingsmethod, first presented by Liu, Liang, and Wong in 2000.
It is designed to help the sampling trajectory converge faster,
by increasing both the step size and the acceptance rate.
InMarkov chain Monte Carlo, theMetropolis–H... | https://en.wikipedia.org/wiki/Multiple-try_Metropolis |
Parallel tempering, inphysicsandstatistics, is a computer simulation method typically used to find the lowest energy state of a system of many interacting particles. It addresses the problem that at high temperatures, one may have a stable state different from low temperature, whereas simulations at low temperatures ma... | https://en.wikipedia.org/wiki/Parallel_tempering |
Inmathematics,Weyl's lemma, named afterHermann Weyl, states that everyweak solutionofLaplace's equationis asmoothsolution. This contrasts with thewave equation, for example, which has weak solutions that are not smooth solutions. Weyl's lemma is a special case ofellipticorhypoelliptic regularity.
LetΩ{\displaystyle \O... | https://en.wikipedia.org/wiki/Weyl%27s_lemma_(Laplace_equation) |
Autocorrelation, sometimes known asserial correlationin thediscrete timecase, measures thecorrelationof asignalwith a delayed copy of itself. Essentially, it quantifies the similarity between observations of arandom variableat different points in time. The analysis of autocorrelation is a mathematical tool for identify... | https://en.wikipedia.org/wiki/Autocorrelation_function |
Acorrelation functionis afunctionthat gives the statisticalcorrelationbetweenrandom variables, contingent on the spatial or temporal distance between those variables.[1]If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often ... | https://en.wikipedia.org/wiki/Correlation_function |
Inprobability theoryandstatistics, acovariance matrix(also known asauto-covariance matrix,dispersion matrix,variance matrix, orvariance–covariance matrix) is a squarematrixgiving thecovariancebetween each pair of elements of a givenrandom vector.
Intuitively, the covariance matrix generalizes the notion of variance to... | https://en.wikipedia.org/wiki/Covariance_matrix |
Inprobability theory, for aprobability measurePon aHilbert spaceHwithinner product⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle }, thecovarianceofPis thebilinear formCov:H×H→Rgiven by
for allxandyinH. Thecovariance operatorCis then defined by
(from theRiesz representation theorem, such operator exists if Cov isbou... | https://en.wikipedia.org/wiki/Covariance_operator |
Inoperator theory, a branch of mathematics, apositive-definite kernelis a generalization of apositive-definite functionor apositive-definite matrix. It was first introduced byJames Mercerin the early 20th century, in the context of solvingintegral operator equations. Since then, positive-definite functions and their va... | https://en.wikipedia.org/wiki/Positive-definite_kernel |
Inphysicsandmathematics, arandom fieldis a random function over an arbitrary domain (usually a multi-dimensional space such asRn{\displaystyle \mathbb {R} ^{n}}). That is, it is a functionf(x){\displaystyle f(x)}that takes on a random value at each pointx∈Rn{\displaystyle x\in \mathbb {R} ^{n}}(or some other domain). I... | https://en.wikipedia.org/wiki/Random_field |
Inprobability theoryand related fields, astochastic(/stəˈkæstɪk/) orrandom processis amathematical objectusually defined as afamilyofrandom variablesin aprobability space, where theindexof the family often has the interpretation oftime.Stochasticprocesses are widely used asmathematical modelsof systems and phenomena th... | https://en.wikipedia.org/wiki/Stochastic_process |
Inspatial statisticsthe theoreticalvariogram, denoted2γ(s1,s2){\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})}, is a function describing the degree ofspatial dependenceof a spatialrandom fieldorstochastic processZ(s){\displaystyle Z(\mathbf {s} )}. Thesemivariogramγ(s1,s2){\displaystyle \gamma (\mathbf {s} _... | https://en.wikipedia.org/wiki/Variogram |
Inmathematics, adifferential equationis anequationthat relates one or more unknownfunctionsand theirderivatives.[1]In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations... | https://en.wikipedia.org/wiki/Differential_equation |
Inmathematics, anintegro-differential equationis anequationthat involves bothintegralsandderivativesof afunction.
The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form
As is typical withdifferential equations, obtaining a closed-form solution... | https://en.wikipedia.org/wiki/Integro-differential_equation |
Inactuarial scienceandapplied probability,ruin theory(sometimesrisk theory[1]orcollective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit ... | https://en.wikipedia.org/wiki/Ruin_theory |
Inmathematics, theVolterra integral equationsare a special type ofintegral equations.[1]They are divided into two groups referred to as the first and the second kind.
A linear Volterra equation of the first kind is
wherefis a given function andxis an unknown function to be solved for. A linear Volterra equation of t... | https://en.wikipedia.org/wiki/Volterra_integral_equation |
Inmathematics, specificallyfunctional analysis,Mercer's theoremis a representation of a symmetricpositive-definitefunction on a square as a sum of a convergent sequence of product functions. This theorem, presented in (Mercer 1909), is one of the most notable results of the work ofJames Mercer(1883–1932). It is an imp... | https://en.wikipedia.org/wiki/Mercer%27s_theorem |
The termkernelis used instatistical analysisto refer to awindow function. The term "kernel" has several distinct meanings in different branches of statistics.
In statistics, especially inBayesian statistics, the kernel of aprobability density function(pdf) orprobability mass function(pmf) is the form of the pdf or pmf... | https://en.wikipedia.org/wiki/Kernel_(statistics) |
Akernel smootheris astatisticaltechnique to estimate a real valuedfunctionf:Rp→R{\displaystyle f:\mathbb {R} ^{p}\to \mathbb {R} }as theweighted averageof neighboring observed data. The weight is defined by thekernel, such that closer points are given higher weights. The estimated function is smooth, and the level of s... | https://en.wikipedia.org/wiki/Kernel_smoothing |
Instatistics,adaptiveor"variable-bandwidth" kernel density estimationis a form ofkernel density estimationin which the size of the kernels used in the estimate are varied
depending upon either the location of the samples or the location of the test point.
It is a particularly effective technique when the sample space i... | https://en.wikipedia.org/wiki/Variable_kernel_density_estimation |
Head/tail breaksis aclustering algorithmfor data with aheavy-tailed distributionsuch aspower lawsandlognormal distributions. The heavy-tailed distribution can be simply referred to the scaling pattern of far more small things than large ones, or alternatively numerous smallest, a very few largest, and some in between t... | https://en.wikipedia.org/wiki/Head/tail_Breaks |
Inmachine learning,kernel machinesare a class of algorithms forpattern analysis, whose best known member is thesupport-vector machine(SVM). These methods involve using linear classifiers to solve nonlinear problems.[1]The general task ofpattern analysisis to find and study general types of relations (for examplecluster... | https://en.wikipedia.org/wiki/Kernel_methods |
Learning to rank[1]ormachine-learned ranking(MLR) is the application ofmachine learning, typicallysupervised,semi-supervisedorreinforcement learning, in the construction ofranking modelsforinformation retrievalsystems.[2]Training datamay, for example, consist of lists of items with somepartial orderspecified between it... | https://en.wikipedia.org/wiki/Learning_to_rank |
Ingeometry, anintersectionis a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case inEuclidean geometryis theline–line intersectionbetween two distinctlines, which either is onepoint(sometimes called avertex) or does not exist (if the lines areparallel). ... | https://en.wikipedia.org/wiki/Line_segment_intersection |
Inmathematics, aprojective planeis a geometric structure that extends the concept of aplane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equip... | https://en.wikipedia.org/wiki/Projective_plane#Lines_joining_points_and_intersection_of_lines_.28using_duality.29 |
Thedistancebetween twoparallellinesin theplaneis the minimum distance between any two points.
Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the distance. Given the equations of two non-vertical parallel lines
the distance ... | https://en.wikipedia.org/wiki/Distance_between_two_parallel_lines |
Thedistance(orperpendicular distance)from a point to a lineis the shortestdistancefrom a fixedpointto any point on a fixed infinitelineinEuclidean geometry. It is the length of theline segmentwhich joins the point to the line and isperpendicularto the line. The formula for calculating it can be derived and expressed in... | https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line |
In analyticgeometry, theintersection of alineand aplaneinthree-dimensional spacecan be theempty set, apoint, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.
Di... | https://en.wikipedia.org/wiki/Line%E2%80%93plane_intersection |
Ingeometry, theparallel postulateis the fifth postulate inEuclid'sElementsand a distinctiveaxiominEuclidean geometry. It states that, in two-dimensional geometry:
If aline segmentintersects two straightlinesforming two interior angles on the same side that are less than tworight angles, then the two lines, if extended... | https://en.wikipedia.org/wiki/Parallel_postulate |
Incomputer vision,triangulationrefers to the process of determining a point in 3D space given its projections onto two, or more, images. In order to solve this problem it is necessary to know the parameters of the camera projection function from 3D to 2D for the cameras involved, in the simplest case represented by th... | https://en.wikipedia.org/wiki/Triangulation_(computer_vision) |
Ingeometry, anintersectionis a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case inEuclidean geometryis theline–line intersectionbetween two distinctlines, which either is onepoint(sometimes called avertex) or does not exist (if the lines areparallel). ... | https://en.wikipedia.org/wiki/Intersection_(Euclidean_geometry)#Two_line_segments |
Linear least squares(LLS) is theleast squares approximationoflinear functionsto data.
It is a set of formulations for solving statistical problems involved inlinear regression, including variants forordinary(unweighted),weighted, andgeneralized(correlated)residuals.Numerical methods for linear least squaresinclude inve... | https://en.wikipedia.org/wiki/Linear_least_squares |
Segmented regression, also known aspiecewise regressionorbroken-stick regression, is a method inregression analysisin which theindependent variableis partitioned into intervals and a separate line segment is fit to each interval. Segmented regression analysis can also be performed on multivariate data by partitioning t... | https://en.wikipedia.org/wiki/Linear_segmented_regression |
Least-squares support-vector machines (LS-SVM)forstatisticsand instatistical modeling, areleast-squaresversions ofsupport-vector machines(SVM), which are a set of relatedsupervised learningmethods that analyze data and recognize patterns, and which are used forclassificationandregression analysis. In this version one ... | https://en.wikipedia.org/wiki/Least_squares_support_vector_machine |
Inmathematics,nonlinear programming(NLP) is the process of solving anoptimization problemwhere some of the constraints are notlinear equalitiesor the objective function is not alinear function. Anoptimization problemis one of calculation of the extrema (maxima, minima or stationary points) of anobjective functionover a... | https://en.wikipedia.org/wiki/Nonlinear_programming |
Mathematical optimization(alternatively spelledoptimisation) ormathematical programmingis the selection of a best element, with regard to some criteria, from some set of available alternatives.[1][2]It is generally divided into two subfields:discrete optimizationandcontinuous optimization. Optimization problems arise i... | https://en.wikipedia.org/wiki/Optimization_(mathematics) |
Thepartial least squares path modelingorpartial least squares structural equation modeling(PLS-PM,PLS-SEM)[1][2][3]is a method forstructural equation modelingthat allows estimation of complex cause-effect relationships in path models withlatent variables.
PLS-PM[4][5]is a component-based estimation approach that diffe... | https://en.wikipedia.org/wiki/Partial_least_squares_path_modeling |
Instatistics,projection pursuit regression (PPR)is astatistical modeldeveloped byJerome H. FriedmanandWerner Stuetzlethat extendsadditive models. This model adapts the additive models in that it first projects thedata matrixofexplanatory variablesin the optimal direction before applying smoothing functions to these exp... | https://en.wikipedia.org/wiki/Projection_pursuit_regression |
Inmathematics, ahomogeneous functionis afunction of several variablessuch that the following holds: If each of the function's arguments is multiplied by the samescalar, then the function's value is multiplied by some power of this scalar; the power is called thedegree of homogeneity, or simply thedegree. That is, ifkis... | https://en.wikipedia.org/wiki/Homogeneous_function |
Collective intelligenceCollective actionSelf-organized criticalityHerd mentalityPhase transitionAgent-based modellingSynchronizationAnt colony optimizationParticle swarm optimizationSwarm behaviour
Social network analysisSmall-world networksCentralityMotifsGraph theoryScalingRobustnessSystems biologyDynamic networks
... | https://en.wikipedia.org/wiki/Nonlinear_system |
Inmathematics, apiecewise linearorsegmented functionis areal-valued functionof a real variable, whosegraphis composed of straight-line segments.[1]
A piecewise linear function is a function defined on a (possibly unbounded)intervalofreal numbers, such that there is a collection of intervals on each of which the functi... | https://en.wikipedia.org/wiki/Piecewise_linear_function |
Inmathematics,linear mapsform an important class of "simple"functionswhich preserve the algebraic structure oflinear spacesand are often used as approximations to more general functions (seelinear approximation). If the spaces involved are alsotopological spaces(that is,topological vector spaces), then it makes sense ... | https://en.wikipedia.org/wiki/Discontinuous_linear_map |
Open innovationis a term used to promote anInformation Agemindset toward innovation that runs counter to thesecrecyandsilo mentalityof traditional corporate research labs. The benefits and driving forces behind increased openness have been noted and discussed as far back as the 1960s, especially as it pertains to inter... | https://en.wikipedia.org/wiki/Open_innovation |
Aninnovation competitionis a method or process of theindustrial process,productorbusiness development. It is a form ofsocial engineering, which focuses to the creation and elaboration of the best and sustainable ideas, coming from the bestinnovators.
There are few major works, like Terwiesch and Ulrich,[1][2]who exclu... | https://en.wikipedia.org/wiki/Innovation_competition |
Aninducement prize contest(IPC) is a competition that awards a cash prize for the accomplishment of a feat, usually of engineering. IPCs are typically designed to extend the limits of human ability. Some of the most famous IPCs include theLongitude prize(1714–1765), theOrteig Prize(1919–1927) and prizes from enterprise... | https://en.wikipedia.org/wiki/Inducement_prize_contest |
Kaggleis adata science competition platformand online community fordata scientistsandmachine learningpractitioners underGoogle LLC. Kaggle enables users to find and publish datasets, explore and build models in a web-based data science environment, work with other data scientists and machine learning engineers, and ent... | https://en.wikipedia.org/wiki/Kaggle |
Thislist of computer science awardsis an index to articles on notable awards related tocomputer science. It includes lists of awards by theAssociation for Computing Machinery, theInstitute of Electrical and Electronics Engineers, other computer science andinformation scienceawards, and a list of computer science compet... | https://en.wikipedia.org/wiki/List_of_computer_science_awards |
Theblack boxmodel of power converteralso called behavior model, is a method ofsystem identificationto represent the characteristics ofpower converter, that is regarded as a black box. There are two types of black box model of power converter - when the model includes the load, it is called terminated model, otherwise u... | https://en.wikipedia.org/wiki/Black_box_model_of_power_converter |
Data-driven control systemsare a broad family ofcontrol systems, in which theidentificationof the process model and/or the design of the controller are based entirely onexperimental datacollected from the plant.[1]
In many control applications, trying to write a mathematical model of the plant is considered a hard tas... | https://en.wikipedia.org/wiki/Data-driven_control_system |
Inmathematics,statistics, andcomputational modelling, agrey box model[1][2][3][4]combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of results, to models that need only parameter values from data or existing literature.[5]Thus,... | https://en.wikipedia.org/wiki/Grey_box_completion_and_validation |
Hysteresisis the dependence of the state of a system on its history. For example, amagnetmay have more than one possiblemagnetic momentin a givenmagnetic field, depending on how the field changed in the past. Plots of a single component of the moment often form a loop or hysteresis curve, where there are different valu... | https://en.wikipedia.org/wiki/Hysteresis |
Insystem analysis, among other fields of study, alinear time-invariant(LTI)systemis asystemthat produces an output signal from any input signal subject to the constraints oflinearityandtime-invariance; these terms are briefly defined in the overview below. These properties apply (exactly or approximately) to many impor... | https://en.wikipedia.org/wiki/LTI_system_theory |
Intime seriesmodeling, anonlinear autoregressive exogenous model(NARX) is anonlinearautoregressive modelwhich hasexogenousinputs. This means that the model relates the current value of a time series to both:
In addition, the model contains an error term which relates to the fact that knowledge of other terms will not ... | https://en.wikipedia.org/wiki/Nonlinear_autoregressive_exogenous_model |
Anopen systemis a system that has external interactions. Such interactions can take the form of information, energy, or material transfers into or out of the system boundary, depending on the discipline which defines the concept. An open system is contrasted with the concept of anisolated systemwhich exchanges neither ... | https://en.wikipedia.org/wiki/Open_system_(systems_theory) |
Estimation theoryis a branch ofstatisticsthat deals with estimating the values ofparametersbased on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. Anestimatorattempts to approximat... | https://en.wikipedia.org/wiki/Parameter_estimation |
In the area ofsystem identification, adynamical systemisstructurally identifiableif it is possible to infer its unknown parameters by measuring its output over time. This problem arises in many branch of applied mathematics, sincedynamical systems(such as the ones described byordinary differential equations) are common... | https://en.wikipedia.org/wiki/Structural_identifiability |
System dynamics(SD) is an approach to understanding thenonlinearbehaviour ofcomplex systemsover time usingstocks, flows, internalfeedback loops, table functions and time delays.[1]
System dynamics is amethodologyandmathematical modelingtechnique to frame, understand, and discuss complex issues and problems. Originally... | https://en.wikipedia.org/wiki/System_dynamics |
Insystems theory, arealizationof astate spacemodel is an implementation of a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of (time-varying)matrices[A(t),B(t),C(t),D(t)]{\displaystyle [A(t),B(t),C(t),D(t)]}such that
with(u(t),y(t)){\displaystyle (u(t),y(t))}de... | https://en.wikipedia.org/wiki/System_realization |
Inmathematics, particularlydifferential geometry, aFinsler manifoldis adifferentiable manifoldMwhere a (possiblyasymmetric)Minkowski normF(x, −)is provided on each tangent spaceTxM, that enables one to define the length of anysmooth curveγ: [a,b] →Mas
Finsler manifolds are more general thanRiemannian manifoldssince th... | https://en.wikipedia.org/wiki/Finsler_manifold |
Inmathematics, twometricson the same underlyingsetare said to beequivalentif the resulting metric spaces share certain properties. Equivalence is a weaker notion thanisometry; equivalent metrics do not have to be literally the same. Instead, it is one of several ways of generalizingequivalence of normsto general metr... | https://en.wikipedia.org/wiki/Equivalence_of_metrics |
Infunctional analysis, anF-spaceis avector spaceX{\displaystyle X}over therealorcomplexnumbers together with ametricd:X×X→R{\displaystyle d:X\times X\to \mathbb {R} }such that
The operationx↦‖x‖:=d(0,x){\displaystyle x\mapsto \|x\|:=d(0,x)}is called anF-norm, although in general an F-norm is not required to be homogen... | https://en.wikipedia.org/wiki/F-space |
Infunctional analysisand related areas ofmathematics,Fréchet spaces, named afterMaurice Fréchet, are specialtopological vector spaces.
They are generalizations ofBanach spaces(normed vector spacesthat arecompletewith respect to themetricinduced by thenorm).
AllBanachandHilbert spacesare Fréchet spaces.
Spaces ofinfi... | https://en.wikipedia.org/wiki/Fr%C3%A9chet_space |
Inmathematics, the concept of ageneralised metricis a generalisation of that of ametric, in which the distance is not areal numberbut taken from an arbitraryordered field.
In general, when we definemetric spacethe distance function is taken to be a real-valuedfunction. The real numbers form an ordered field which isAr... | https://en.wikipedia.org/wiki/Generalised_metric |
Inmathematics, more specifically infunctional analysis, aK-spaceis anF-spaceV{\displaystyle V}such that every extension of F-spaces (or twisted sum) of the form0→R→X→V→0.{\displaystyle 0\rightarrow \mathbb {R} \rightarrow X\rightarrow V\rightarrow 0.\,\!}is equivalent to the trivial one[1]0→R→R×V→V→0.{\displaystyle 0\r... | https://en.wikipedia.org/wiki/K-space_(functional_analysis) |
Infunctional analysisand related areas ofmathematics,locally convex topological vector spaces(LCTVS) orlocally convex spacesare examples oftopological vector spaces(TVS) that generalizenormed spaces. They can be defined astopologicalvector spaces whose topology isgeneratedby translations ofbalanced,absorbent,convex set... | https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space |
Inmathematics, ametric spaceis asettogether with a notion ofdistancebetween itselements, usually calledpoints. The distance is measured by afunctioncalled ametricordistance function.[1]Metric spaces are a general setting for studying many of the concepts ofmathematical analysisandgeometry.
The most familiar example o... | https://en.wikipedia.org/wiki/Metric_space |
Inmathematics, apseudometric spaceis ageneralizationof ametric spacein which the distance between two distinct points can be zero. Pseudometric spaces were introduced byĐuro Kurepa[1][2]in 1934. In the same way as everynormed spaceis ametric space, everyseminormed spaceis a pseudometric space. Because of this analogy, ... | https://en.wikipedia.org/wiki/Pseudometric_space |
In mathematics, particularly infunctional analysisandconvex analysis, theUrsescu theoremis a theorem that generalizes theclosed graph theorem, theopen mapping theorem, and theuniform boundedness principle.
The following notation and notions are used, whereR:X⇉Y{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}is aset... | https://en.wikipedia.org/wiki/Ursescu_theorem |
Infunctional analysisand related areas ofmathematics, ametrizable(resp.pseudometrizable)topological vector space(TVS) is a TVS whose topology is induced by a metric (resp.pseudometric). AnLM-spaceis aninductive limitof a sequence oflocally convexmetrizable TVS.
Apseudometricon a setX{\displaystyle X}is a mapd:X×X→R{\d... | https://en.wikipedia.org/wiki/Metrizable_topological_vector_space |
Inpsychology,number senseis the term used for the hypothesis that some animals, particularly humans, have a biologically determined ability that allows them to represent and manipulate large numerical quantities. The term was popularized byStanislas Dehaenein his 1997 book "The Number Sense," but originally named by t... | https://en.wikipedia.org/wiki/Number_sense |
Inmathematics, thecardinalityof asetis the number of its elements. The cardinality of a set may also be called itssize, when no confusion with other notions of size is possible.[a]Beginning in the late 19th century, this concept of size was generalized toinfinite sets, allowing one to distinguish between different type... | https://en.wikipedia.org/wiki/Set_size |
Infunctional analysis, thedual normis a measure of size for acontinuouslinear functiondefined on anormed vector space.
LetX{\displaystyle X}be anormed vector spacewith norm‖⋅‖{\displaystyle \|\cdot \|}and letX∗{\displaystyle X^{*}}denote itscontinuous dual space. Thedual normof a continuouslinear functionalf{\displays... | https://en.wikipedia.org/wiki/Dual_norm |
In mathematics, thelogarithmic normis a real-valuedfunctionalonoperators, and is derived from either aninner product, a vector norm, or its inducedoperator norm. The logarithmic norm was independently introduced byGermund Dahlquist[1]and Sergei Lozinskiĭ in 1958, for squarematrices. It has since been extended to nonlin... | https://en.wikipedia.org/wiki/Logarithmic_norm |
Infunctional analysis, a branch of mathematics, two methods of constructingnormed spacesfromdiskswere systematically employed byAlexander Grothendieckto definenuclear operatorsandnuclear spaces.[1]One method is used if the diskD{\displaystyle D}is bounded: in this case, theauxiliary normed spaceisspanD{\displaystyle \... | https://en.wikipedia.org/wiki/Auxiliary_normed_space |
Cauchy's functional equationis thefunctional equation:f(x+y)=f(x)+f(y).{\displaystyle f(x+y)=f(x)+f(y).\ }
A functionf{\displaystyle f}that solves this equation is called anadditive function. Over therational numbers, it can be shown usingelementary algebrathat there is a single family of solutions, namelyf:x↦cx{\dis... | https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation |
Infunctional analysisand related areas ofmathematics,locally convex topological vector spaces(LCTVS) orlocally convex spacesare examples oftopological vector spaces(TVS) that generalizenormed spaces. They can be defined astopologicalvector spaces whose topology isgeneratedby translations ofbalanced,absorbent,convex set... | https://en.wikipedia.org/wiki/Finest_locally_convex_topology |
Inintegral geometry(otherwise called geometric probability theory),Hadwiger's theoremcharacterises thevaluationsonconvex bodiesinRn.{\displaystyle \mathbb {R} ^{n}.}It was proved byHugo Hadwiger.
LetKn{\displaystyle \mathbb {K} ^{n}}be the collection of all compact convex sets inRn.{\displaystyle \mathbb {R} ^{n}.}Ava... | https://en.wikipedia.org/wiki/Hadwiger%27s_theorem |
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