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Nanoparticle tracking analysis(NTA) is a method for visualizing and analyzing particles in liquids that relates the rate ofBrownian motionto particle size. The rate of movement is related only to theviscosityand temperature of the liquid; it is not influenced by particledensityorrefractive index. NTA allows the determi... | https://en.wikipedia.org/wiki/Nanoparticle_tracking_analysis |
Thenarrow escape problem[1][2]is a ubiquitous problem inbiology,biophysicsandcellular biology.
The mathematical formulation is the following: aBrownian particle(ion,molecule, orprotein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can es... | https://en.wikipedia.org/wiki/Narrow_escape_problem |
Osmosis(/ɒzˈmoʊsɪs/,USalso/ɒs-/)[1]is the spontaneous net movement ordiffusionofsolventmolecules through aselectively-permeable membranefrom a region of highwater potential(region of lowersoluteconcentration) to a region of low water potential (region of higher solute concentration),[2]in the direction that tends to eq... | https://en.wikipedia.org/wiki/Osmosis |
Inprobability theory, theSchramm–Loewner evolutionwith parameterκ, also known asstochastic Loewner evolution(SLEκ), is a family of random planar curves that have been proven to be thescaling limitof a variety of two-dimensional lattice models instatistical mechanics. Given a parameterκand adomainUin thecomplex plane, i... | https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution |
Single-particle trajectories(SPTs) consist of a collection of successivediscretepoints causal in time. Thesetrajectoriesare acquired from images in experimental data. In the context of cell biology, the trajectories are obtained by the transient activation by a laser of small dyes attached to a moving molecule.
Molec... | https://en.wikipedia.org/wiki/Single_particle_trajectories |
Single-particle tracking(SPT) is the observation of the motion of individual particles within a medium. The coordinates time series, which can be either in two dimensions (x,y) or in three dimensions (x,y,z), is referred to as atrajectory. The trajectory is typically analyzed using statistical methods to extract inform... | https://en.wikipedia.org/wiki/Single_particle_tracking |
Incomputational fluid dynamics, theStochastic Eulerian Lagrangian Method (SELM)[1]is an approach to capture essential features of fluid-structure interactions subject tothermal fluctuationswhile introducing approximations which facilitate analysis and the development of tractable numerical methods. SELM is a hybrid ap... | https://en.wikipedia.org/wiki/Stochastic_Eulerian_Lagrangian_method |
Stokesian dynamics[1]is a solution technique for theLangevin equation, which is the relevant form ofNewton's 2nd lawfor aBrownian particle. The method treats the suspended particles in a discrete sense while the continuum approximation remains valid for the surrounding fluid, i.e., the suspended particles are generally... | https://en.wikipedia.org/wiki/Stokesian_dynamics |
Surface diffusionis a general process involving the motion ofadatoms,molecules, and atomic clusters (adparticles) at solid materialsurfaces.[1]The process can generally be thought of in terms of particles jumping between adjacentadsorptionsites on a surface, as in figure 1. Just as in bulkdiffusion, this motion is typi... | https://en.wikipedia.org/wiki/Surface_diffusion |
Twophysical systemsare inthermal equilibriumif there is no net flow of thermal energy between them when they are connected by a path permeable toheat. Thermal equilibrium obeys thezeroth law of thermodynamics. A system is said to be in thermal equilibrium with itself if the temperature within the system is spatially un... | https://en.wikipedia.org/wiki/Thermal_equilibrium |
Thermodynamic equilibriumis a notion ofthermodynamicswithaxiomaticstatus referring to an internalstateof a singlethermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeablewalls. In thermodynamic equilibrium, there are no netmacroscopicflowsof mass nor of... | https://en.wikipedia.org/wiki/Thermodynamic_equilibrium |
TheTyndall effectislight scattering by particlesin acolloidsuch as a very finesuspension(asol). Also known asTyndall scattering, it is similar toRayleigh scattering, in that the intensity of the scattered light isinversely proportionalto the fourth power of thewavelength, soblue lightis scattered much more strongly tha... | https://en.wikipedia.org/wiki/Tyndall_effect |
Anultramicroscopeis amicroscopewith a system that lights the object in a way that allows viewing of tinyparticlesvialight scattering, and notlight reflectionorabsorption. When the diameter of a particle is below or near thewavelengthofvisible light(around 500nanometers), the particle cannot be seen in alight microscope... | https://en.wikipedia.org/wiki/Ultramicroscope |
Incomputer science, theiterated logarithmofn{\displaystyle n}, writtenlog*n{\displaystyle n}(usually read "log star"), is the number of times thelogarithmfunction must beiterativelyapplied before the result is less than or equal to1{\displaystyle 1}.[1]The simplest formal definition is the result of thisrecurrence rela... | https://en.wikipedia.org/wiki/Iterated_logarithm |
Inmathematics, theWiener process(orBrownian motion, due to its historical connection withthe physical process of the same name) is a real-valuedcontinuous-timestochastic processdiscovered byNorbert Wiener.[1][2]It is one of the best knownLévy processes(càdlàgstochastic processes withstationaryindependent increments). I... | https://en.wikipedia.org/wiki/Wiener_process |
Ingraph theory,eigenvector centrality(also calledeigencentralityorprestige score[1]) is a measure of the influence of anodein a connectednetwork. Relative scores are assigned to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than... | https://en.wikipedia.org/wiki/Eigenvector_centrality |
Incondensed matter physics,Anderson localization(also known asstrong localization)[1]is the absence of diffusion of waves in adisorderedmedium. This phenomenon is named after the American physicistP. W. Anderson, who was the first to suggest that electron localization is possible in a lattice potential, provided that t... | https://en.wikipedia.org/wiki/Anderson_localization |
Inphysics,critical phenomenais the collective name associated with the
physics ofcritical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down. Critical phenomena includescalingrelations among different quantities,power-lawdivergences of some quantities (such as thema... | https://en.wikipedia.org/wiki/Critical_phenomena |
In themathematicalfield ofgraph theory, aHamiltonian path(ortraceable path) is apathin an undirected or directed graph that visits eachvertexexactly once. AHamiltonian cycle(orHamiltonian circuit) is acyclethat visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be complete... | https://en.wikipedia.org/wiki/Hamiltonian_path |
Aknight's touris a sequence of moves of aknighton achessboardsuch that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is "closed", or "re-entrant"; othe... | https://en.wikipedia.org/wiki/Knight%27s_tour |
Snakeis agenreofaction video gameswhere the player maneuvers the end of a growing line, often themed as asnake. The player must keep the snake from colliding with both other obstacles and itself, which gets harder as the snake lengthens.
The genre originated in the 1976 competitivearcade video gameBlockadefromGremlin ... | https://en.wikipedia.org/wiki/Snake_(video_game_genre) |
Instatistical mechanics,universalityis the observation that there are properties for a large class of systems that are independent of thedynamicaldetails of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together. The modern meaning of the term was introduced ... | https://en.wikipedia.org/wiki/Universality_(dynamical_systems) |
Inmathematical analysis, aspace-filling curveis acurvewhoserangereaches every point in a higher dimensional region, typically theunit square(or more generally ann-dimensional unithypercube). BecauseGiuseppe Peano(1858–1932) was the first to discover one, space-filling curves in the2-dimensional planeare sometimes calle... | https://en.wikipedia.org/wiki/Space-filling_curves |
Instatistics, theDickey–Fuller testtests thenull hypothesisthat aunit rootis present in anautoregressive(AR) time series model. Thealternative hypothesisis different depending on which version of the test is used, but is usuallystationarityortrend-stationarity. The test is named after thestatisticiansDavid DickeyandWay... | https://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test |
Instatistics, anaugmented Dickey–Fuller test(ADF) tests thenull hypothesisthat aunit rootis present in atime seriessample. Thealternative hypothesisdepends on which version of the test is used, but is usuallystationarityortrend-stationarity. It is an augmented version of theDickey–Fuller testfor a larger and more compl... | https://en.wikipedia.org/wiki/Augmented_Dickey%E2%80%93Fuller_test |
Instatisticsandeconometrics, theADF-GLS test(orDF-GLS test) is a test for aunit rootin an economictime seriessample. It was developed by Elliott, Rothenberg and Stock (ERS) in 1992 as a modification of theaugmented Dickey–Fuller test(ADF).[1]
A unit root test determines whether a time series variable is non-stationary... | https://en.wikipedia.org/wiki/ADF-GLS_test |
Instatistics, aunit root testtests whether atime seriesvariable is non-stationary and possesses aunit root. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is eitherstationarity,trend stationarityor explosive root depending on the test used.
In general, the approa... | https://en.wikipedia.org/wiki/Unit_root_test |
Instatistics, thePhillips–Perron test(named afterPeter C. B. PhillipsandPierre Perron) is aunit roottest.[1]That is, it is used intime seriesanalysis to test thenull hypothesisthat a time series isintegrated of order1. It builds on theDickey–Fuller testof the null hypothesisρ=1{\displaystyle \rho =1}inΔyt=(ρ−1)yt−1+ut... | https://en.wikipedia.org/wiki/Phillips%E2%80%93Perron_test |
Ineconometrics,cointegrationis astatisticalproperty describing a long-term, stable relationship between two or moretime seriesvariables, even if those variables themselves are individuallynon-stationary(i.e., they have trends). This means that despite their individual fluctuations, the variables move together in the lo... | https://en.wikipedia.org/wiki/Cointegration |
Ineconometrics,Kwiatkowski–Phillips–Schmidt–Shin (KPSS) testsare used for testing anull hypothesisthat an observabletime seriesisstationaryaround a deterministic trend (i.e.trend-stationary) against the alternative of aunit root.[1]
Contrary to mostunit root tests, the presence of a unit root is not the null hypothesi... | https://en.wikipedia.org/wiki/KPSS_tests |
Clusteringcan refer to the following:
Incomputing:
Ineconomics:
Ingraph theory: | https://en.wikipedia.org/wiki/Clustering_(disambiguation) |
Inprobability theory, theChinese restaurant processis adiscrete-timestochastic process, analogous to seating customers at tables in a restaurant.
Imagine a restaurant with an infinite number of circular tables, each with infinite capacity. Customer 1 sits at the first table. The next customer either sits at the same ta... | https://en.wikipedia.org/wiki/Chinese_Restaurant_Process |
Instatistics,cluster analysisis the algorithmic grouping of objects into homogeneous
groups based on numerical measurements.Model-based clustering[1]based on a statistical model for the data, usually amixture model. This has several advantages, including a principledstatisticalbasis for clustering,
and ways to choose t... | https://en.wikipedia.org/wiki/Model-based_clustering |
Generative topographic map(GTM) is amachine learningmethod that is a probabilistic counterpart of theself-organizing map(SOM), is probably convergent and does not require a shrinkingneighborhoodor a decreasing step size. It is agenerative model: the data is assumed to arise by first probabilistically picking a point in... | https://en.wikipedia.org/wiki/Generative_topographic_map |
Meta-learning[1][2]is a subfield ofmachine learningwhere automatic learning algorithms are applied tometadataabout machine learning experiments. As of 2017, the term had not found a standard interpretation, however the main goal is to use such metadata to understand how automatic learning can become flexible in solving... | https://en.wikipedia.org/wiki/Meta-learning_(computer_science) |
Multivariate statisticsis a subdivision ofstatisticsencompassing the simultaneous observation and analysis of more than oneoutcome variable, i.e.,multivariate random variables.
Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and ... | https://en.wikipedia.org/wiki/Multivariate_analysis |
Weak supervision(also known assemi-supervised learning) is a paradigm inmachine learning, the relevance and notability of which increased with the advent oflarge language modelsdue to large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data(exclusi... | https://en.wikipedia.org/wiki/Weak_supervision |
Automatic taxonomy construction(ATC) is the use of software programs to generate taxonomical classifications from a body of texts called acorpus. ATC is a branch ofnatural language processing, which in turn is a branch ofartificial intelligence.
Ataxonomy(or taxonomical classification) is ascheme of classification, es... | https://en.wikipedia.org/wiki/Automatic_taxonomy_construction |
Ininformation science, anontologyencompasses a representation, formal naming, and definitions of the categories, properties, and relations between the concepts, data, or entities that pertain to one, many, or alldomains of discourse. More simply, an ontology is a way of showing the properties of a subject area and how ... | https://en.wikipedia.org/wiki/Ontology_(information_science)#Domain_ontology |
Natural language understanding(NLU) ornatural language interpretation(NLI)[1]is a subset ofnatural language processinginartificial intelligencethat deals with machinereading comprehension. NLU has been considered anAI-hardproblem.[2]
There is considerable commercial interest in the field because of its application toa... | https://en.wikipedia.org/wiki/Natural_language_understanding |
Incomputer science,canonicalization(sometimesstandardizationornormalization) is a process for convertingdatathat has more than one possible representation into a "standard", "normal", orcanonical form. This can be done to compare different representations for equivalence, to count the number of distinct data structures... | https://en.wikipedia.org/wiki/Canonicalization |
Inmathematics, acanonical basisis a basis of analgebraic structurethat is canonical in a sense that depends on the precise context:
The canonical basis for the irreducible representations of a quantized enveloping algebra of
typeADE{\displaystyle ADE}and also for the plus part of that algebra was introduced by Lusztig... | https://en.wikipedia.org/wiki/Canonical_basis |
Inmathematics, thecanonical bundleof anon-singularalgebraic varietyV{\displaystyle V}of dimensionn{\displaystyle n}over a field is theline bundleΩn=ω{\displaystyle \,\!\Omega ^{n}=\omega }, which is then{\displaystyle n}thexterior powerof thecotangent bundleΩ{\displaystyle \Omega }onV{\displaystyle V}.
Over thecomplex... | https://en.wikipedia.org/wiki/Canonical_class |
Normalizationornormalisationrefers to a process that makes something more normal or regular. | https://en.wikipedia.org/wiki/Normalization_(disambiguation) |
Standardization(American English) orstandardisation(British English) is the process of implementing and developingtechnical standardsbased on the consensus of different parties that include firms, users, interest groups, standards organizations and governments.[1]Standardization can help maximizecompatibility,interoper... | https://en.wikipedia.org/wiki/Standardization |
In mathematics, aneigenvalue perturbationproblem is that of finding theeigenvectors and eigenvaluesof a systemAx=λx{\displaystyle Ax=\lambda x}that isperturbedfrom one with known eigenvectors and eigenvaluesA0x0=λ0x0{\displaystyle A_{0}x_{0}=\lambda _{0}x_{0}}. This is useful for studying how sensitive the original sys... | https://en.wikipedia.org/wiki/Eigenvalue_perturbation |
Inmatrix theory, theFrobenius covariantsof asquare matrixAare special polynomials of it, namelyprojectionmatricesAiassociated with theeigenvalues and eigenvectorsofA.[1]: pp.403, 437–8They are named after the mathematicianFerdinand Frobenius.
Each covariant is aprojectionon theeigenspaceassociated with the eigenvalueλ... | https://en.wikipedia.org/wiki/Frobenius_covariant |
Inlinear algebra, aHouseholder transformation(also known as aHouseholder reflectionorelementary reflector) is alinear transformationthat describes areflectionabout aplaneorhyperplanecontaining the origin. The Householder transformation was used in a 1958 paper byAlston Scott Householder.[1]
TheHouseholderoperator[2]ma... | https://en.wikipedia.org/wiki/Householder_transformation |
This article lists some important classes ofmatricesused inmathematics,scienceandengineering. Amatrix(plural matrices, or less commonly matrixes) is a rectangulararrayofnumberscalledentries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is mat... | https://en.wikipedia.org/wiki/List_of_matrices |
Inmatrix theory,Sylvester's formulaorSylvester's matrix theorem(named afterJ. J. Sylvester) orLagrange−Sylvester interpolationexpresses an analyticfunctionf(A)of amatrixAas a polynomial inA, in terms of theeigenvalues and eigenvectorsofA.[1][2]It states that[3]
where theλiare the eigenvalues ofA, and the matrices
are... | https://en.wikipedia.org/wiki/Sylvester%27s_formula |
Inlinear algebra, anorthogonal matrix, ororthonormal matrix, is a realsquare matrixwhose columns and rows areorthonormalvectors.
One way to express this isQTQ=QQT=I,{\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}whereQTis thetransposeofQandIis theidentity matrix.
This leads to the equivalent characterization: ... | https://en.wikipedia.org/wiki/Orthogonal_matrix |
Source separation,blind signal separation(BSS) orblind source separation, is the separation of a set of sourcesignalsfrom a set of mixed signals, without the aid of information (or with very little information) about the source signals or the mixing process. It is most commonly applied indigital signal processingand in... | https://en.wikipedia.org/wiki/Signal_separation |
Instatistics, avarimax rotationis used to simplify the expression of a particular sub-space in terms of just a few major items each. The actual coordinate system is unchanged, it is theorthogonalbasis that is being rotated to align with those coordinates. The sub-space found withprincipal component analysisorfactor a... | https://en.wikipedia.org/wiki/Varimax_rotation |
In themathematicalfield ofFourier analysis, theconjugate Fourier seriesarises by realizing the Fourier series formally as the boundary values of thereal partof aholomorphic functionon theunit disc. Theimaginary partof that function then defines the conjugate series.Zygmund (1968)studied the delicate questions of conve... | https://en.wikipedia.org/wiki/Conjugate_Fourier_series |
Ageneralized Fourier seriesis the expansion of asquare integrablefunction into a sum of square integrableorthogonal basis functions. The standardFourier seriesuses anorthonormal basisoftrigonometric functions, and the series expansion is applied to periodic functions. In contrast, a generalized Fourier series uses any ... | https://en.wikipedia.org/wiki/Generalized_Fourier_series |
Inmathematics,Fourier–Bessel seriesis a particular kind ofgeneralized Fourier series(aninfinite seriesexpansion on a finite interval) based onBessel functions.
Fourier–Bessel series are used in the solution topartial differential equations, particularly incylindrical coordinatesystems.
The Fourier–Bessel series of a ... | https://en.wikipedia.org/wiki/Fourier%E2%80%93Bessel_series |
Inmathematics, theLaplace transform, named afterPierre-Simon Laplace(/ləˈplɑːs/), is anintegral transformthat converts afunctionof arealvariable(usuallyt{\displaystyle t}, in thetime domain) to a function of acomplexvariables{\displaystyle s}(in the complex-valuedfrequency domain, also known ass-domain, ors-plane).
Th... | https://en.wikipedia.org/wiki/Laplace_transform |
Inmathematics, thetwo-sided Laplace transformorbilateral Laplace transformis anintegral transformequivalent toprobability'smoment-generating function. Two-sided Laplace transforms are closely related to theFourier transform, theMellin transform, theZ-transformand the ordinary or one-sidedLaplace transform. Iff(t) is a... | https://en.wikipedia.org/wiki/Two-sided_Laplace_transform |
Inmathematics, theMellin transformis anintegral transformthat may be regarded as themultiplicativeversion of thetwo-sided Laplace transform. This integral transform is closely connected to the theory ofDirichlet series, and is
often used innumber theory,mathematical statistics, and the theory ofasymptotic expansions; i... | https://en.wikipedia.org/wiki/Mellin_transform |
In applied mathematics, thenon-uniform discrete Fourier transform(NUDFTorNDFT) of a signal is a type ofFourier transform, related to adiscrete Fourier transformordiscrete-time Fourier transform, but in which the input signal is not sampled at equally spaced points or frequencies (or both). It is a generalization of the... | https://en.wikipedia.org/wiki/Non-uniform_discrete_Fourier_transform |
Inquantum computing, thequantum Fourier transform (QFT)is alinear transformationonquantum bits, and is the quantum analogue of thediscrete Fourier transform. The quantum Fourier transform is a part of manyquantum algorithms, notablyShor's algorithmfor factoring and computing thediscrete logarithm, thequantum phase esti... | https://en.wikipedia.org/wiki/Quantum_Fourier_transform |
Inmathematics, asetBof elements of avector spaceVis called abasis(pl.:bases) if every element ofVcan be written in a unique way as a finitelinear combinationof elements ofB. The coefficients of this linear combination are referred to ascomponentsorcoordinatesof the vector with respect toB. The elements of a basis are ... | https://en.wikipedia.org/wiki/Basis_vector |
Inprobability theoryandstatistics, thecharacteristic functionof anyreal-valuedrandom variablecompletely defines itsprobability distribution. If a random variable admits aprobability density function, then the characteristic function is theFourier transform(with sign reversal) of the probability density function. Thus i... | https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory) |
Inmathematics,orthogonal functionsbelong to afunction spacethat is avector spaceequipped with abilinear form. When the function space has anintervalas thedomain, the bilinear form may be theintegralof the product of functions over the interval:
The functionsf{\displaystyle f}andg{\displaystyle g}areorthogonalwhen this... | https://en.wikipedia.org/wiki/Orthogonal_functions |
Inmathematics,Schwartz spaceS{\displaystyle {\mathcal {S}}}is thefunction spaceof allfunctionswhosederivativesarerapidly decreasing. This space has the important property that theFourier transformis anautomorphismon this space. This property enables one, by duality, to define the Fourier transform for elements in the ... | https://en.wikipedia.org/wiki/Schwartz_space |
Insignal processing, the power spectrumSxx(f){\displaystyle S_{xx}(f)}of acontinuous timesignalx(t){\displaystyle x(t)}describes the distribution ofpowerinto frequency componentsf{\displaystyle f}composing that signal.[1]According toFourier analysis, any physical signal can be decomposed into a number of discrete frequ... | https://en.wikipedia.org/wiki/Spectral_density |
Spectral musicuses theacousticproperties of sound – orsound spectra– as a basis forcomposition.[1]
Defined in technical language, spectral music is an acoustic musical practice wherecompositionaldecisions are often informed bysonographicrepresentations andmathematicalanalysis of sound spectra, or by mathematically gen... | https://en.wikipedia.org/wiki/Spectral_music |
Inmathematics, more specifically inharmonic analysis,Walsh functionsform acomplete orthogonal setoffunctionsthat can be used to represent any discrete function—just liketrigonometric functionscan be used to represent anycontinuous functioninFourier analysis.[1]They can thus be viewed as a discrete, digital counterpart ... | https://en.wikipedia.org/wiki/Walsh_function |
Innumerical analysis, thecondition numberof afunctionmeasures how much the output value of the function can change for a small change in the input argument. This is used to measure howsensitivea function is to changes or errors in the input, and how much error in the output results from an error in the input. Very freq... | https://en.wikipedia.org/wiki/Condition_number |
Inlinear algebraandfunctional analysis, themin-max theorem, orvariational theorem, orCourant–Fischer–Weyl min-max principle, is a result that gives a variational characterization ofeigenvaluesofcompactHermitian operators onHilbert spaces. It can be viewed as the starting point of many results of similar nature.
This a... | https://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_interlacing_theorem |
Inmathematics, thePoincaré separation theorem, also known as theCauchy interlacing theorem, gives some upper and lower bounds ofeigenvaluesof a realsymmetric matrixBTABthat can be considered as theorthogonal projectionof a larger real symmetric matrixAonto a linear subspace spanned by the columns ofB. The theorem is n... | https://en.wikipedia.org/wiki/Poincar%C3%A9_separation_theorem |
Inmathematics, particularlylinear algebra, theSchur–Horn theorem, named afterIssai SchurandAlfred Horn, characterizes the diagonal of aHermitian matrixwith giveneigenvalues. It has inspired investigations and substantial generalizations in the setting ofsymplectic geometry. A few important generalizations areKostant's ... | https://en.wikipedia.org/wiki/Schur%E2%80%93Horn_theorem |
Inmachine learning,feature hashing, also known as thehashing trick(by analogy to thekernel trick), is a fast and space-efficient way of vectorizingfeatures, i.e. turning arbitrary features into indices in a vector or matrix.[1][2]It works by applying ahash functionto the features and using their hash values as indices ... | https://en.wikipedia.org/wiki/Feature_hashing |
Geohashis apublic domaingeocode systeminvented in 2008 by Gustavo Niemeyer[2]which encodes a geographic location into a short string of letters and digits. Similar ideas were introduced by G.M. Morton in 1966.[3]It is a hierarchical spatial data structure which subdivides space into buckets ofgridshape, which is one of... | https://en.wikipedia.org/wiki/Geohash |
Incomputer science,locality of reference, also known as theprinciple of locality,[1]is the tendency of a processor to access the same set of memory locations repetitively over a short period of time.[2]There are two basic types of reference locality – temporal and spatial locality. Temporal locality refers to the reus... | https://en.wikipedia.org/wiki/Locality_of_reference |
In mathematics, theCartan decompositionis a decomposition of asemisimpleLie grouporLie algebra, which plays an important role in their structure theory andrepresentation theory. It generalizes thepolar decompositionorsingular value decompositionof matrices. Its history can be traced to the 1880s work ofÉlie Cartanand... | https://en.wikipedia.org/wiki/Cartan_decomposition |
In themathematicaldiscipline oflinear algebra, amatrix decompositionormatrix factorizationis afactorizationof amatrixinto a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
Innumerical analysis, different decompositions are used to implement effi... | https://en.wikipedia.org/wiki/Matrix_decomposition#Algebraic_polar_decomposition |
Inmathematics, specificallymeasure theory, acomplex measuregeneralizes the concept ofmeasureby letting it havecomplexvalues.[1]In other words, one allows forsetswhose size (length, area, volume) is acomplex number.
Formally, acomplex measureμ{\displaystyle \mu }on ameasurable space(X,Σ){\displaystyle (X,\Sigma )}is a ... | https://en.wikipedia.org/wiki/Complex_measure#Variation_of_a_complex_measure_and_polar_decomposition |
Inmathematics,Lie group decompositionsare used to analyse the structure ofLie groupsand associated objects, by showing how they are built up out ofsubgroups. They are essential technical tools in therepresentation theoryof Lie groups andLie algebras; they can also be used to study thealgebraic topologyof such groups an... | https://en.wikipedia.org/wiki/Lie_group_decomposition |
Inquantum information theory,quantum state purificationrefers to the process of representing amixed stateas apure quantum stateof higher-dimensionalHilbert space. The purification allows the original mixed state to be recovered by taking thepartial traceover the additional degrees of freedom. The purification is not un... | https://en.wikipedia.org/wiki/Purification_of_quantum_state |
Inalgebra, theelementary divisorsof amoduleover aprincipal ideal domain(PID) occur in one form of thestructure theorem for finitely generated modules over a principal ideal domain.
IfR{\displaystyle R}is a PID andM{\displaystyle M}afinitely generatedR{\displaystyle R}-module, thenMisisomorphicto a finitedirect sumof t... | https://en.wikipedia.org/wiki/Elementary_divisors |
Theinvariant factorsof amoduleover aprincipal ideal domain(PID) occur in one form of thestructure theorem for finitely generated modules over a principal ideal domain.
IfR{\displaystyle R}is aPIDandM{\displaystyle M}afinitely generatedR{\displaystyle R}-module, then
for some integerr≥0{\displaystyle r\geq 0}and a (po... | https://en.wikipedia.org/wiki/Invariant_factors |
Inmathematics, in the field ofabstract algebra, thestructure theorem for finitely generated modules over a principal ideal domainis a generalization of thefundamental theorem of finitely generated abelian groupsand roughly states thatfinitely generatedmodulesover aprincipal ideal domain(PID) can be uniquely decomposed ... | https://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain |
Inlinear algebra, theFrobenius normal formorrational canonical formof asquarematrixAwith entries in afieldFis acanonical formfor matrices obtained by conjugation byinvertible matricesoverF. The form reflects a minimal decomposition of thevector spaceintosubspacesthat are cyclic forA(i.e.,spannedby some vector and its r... | https://en.wikipedia.org/wiki/Frobenius_normal_form |
Inlinear algebra, theHermite normal formis an analogue ofreduced echelon formformatricesover theintegersZ{\displaystyle \mathbb {Z} }. Just asreduced echelon formcan be used to solve problems about the solution to the linear systemAx=b{\displaystyle Ax=b}wherex∈Rn{\displaystyle x\in \mathbb {R} ^{n}}, the Hermite norma... | https://en.wikipedia.org/wiki/Hermite_normal_form |
Inmathematicsandphysics,Lieb–Thirring inequalitiesprovide an upper bound on the sums of powers of the negativeeigenvaluesof aSchrödinger operatorin terms of integrals of the potential. They are named afterE. H. LiebandW. E. Thirring.
The inequalities are useful in studies ofquantum mechanicsanddifferential equationsan... | https://en.wikipedia.org/wiki/Lieb%E2%80%93Thirring_inequality |
Inmathematics, atrace identityis anyequationinvolving thetraceof amatrix.
Trace identities are invariant under simultaneousconjugation.
They are frequently used in theinvariant theoryofn×n{\displaystyle n\times n}matrices to find thegeneratorsandrelationsof thering of invariants, and therefore are useful in answering... | https://en.wikipedia.org/wiki/Trace_identity |
Inphysics, thevon Neumann entropy, named afterJohn von Neumann, is a measure of the statistical uncertainty within a description of a quantum system. It extends the concept ofGibbs entropyfrom classicalstatistical mechanicstoquantum statistical mechanics, and it is the quantum counterpart of theShannon entropyfrom clas... | https://en.wikipedia.org/wiki/Von_Neumann_entropy |
Abinomial QMF– properly anorthonormal binomial quadrature mirror filter– is anorthogonal waveletdeveloped in 1990.
The binomial QMF bank with perfect reconstruction(PR)was designed byAli Akansu, and published in 1990, using the family of binomial polynomials for subband decomposition of discrete-time signals.[1][2][3]... | https://en.wikipedia.org/wiki/Binomial_QMF |
TheDaubechies wavelets, based on the work ofIngrid Daubechies, are a family oforthogonal waveletsdefining adiscrete wavelet transformand characterized by a maximal number of vanishingmomentsfor some givensupport. With each wavelet type of this class, there is a scaling function (called thefather wavelet) which generate... | https://en.wikipedia.org/wiki/Daubechies_wavelet |
Inapplied mathematics,biorthogonal nearly coiflet basesarewaveletbases proposed by Lowell L. Winger. The wavelet is based onbiorthogonalcoifletwavelet bases, but sacrifices its regularity to increase the filter'sbandwidth, which might lead to betterimage compressionperformance.
Nowadays, a large amount of information ... | https://en.wikipedia.org/wiki/Biorthogonal_nearly_coiflet_basis |
Insignal processing, thechirplet transformis aninner productof an input signal with a family of analysis primitives calledchirplets.[2][3]
Similar to thewavelet transform, chirplets are usually generated from (or can be expressed as being from) a singlemother chirplet(analogous to the so-calledmother waveletof wavelet... | https://en.wikipedia.org/wiki/Chirplet_transform |
Thecomplex wavelet transform(CWT) is acomplex-valuedextension to the standarddiscrete wavelet transform(DWT). It is a two-dimensionalwavelettransform which providesmultiresolution, sparse representation, and useful characterization of the structure of an image. Further, it purveys a high degree of shift-invariance in i... | https://en.wikipedia.org/wiki/Complex_wavelet_transform |
In mathematics andsignal processing, theconstant-Q transformandvariable-Q transform, simply known asCQTandVQT, transforms a data series to thefrequency domain. It is related to theFourier transform[1]and very closely related to the complexMorlet wavelettransform.[2]Its design is suited for musical representation.
The ... | https://en.wikipedia.org/wiki/Constant-Q_transform |
Inmathematics, thecontinuous wavelet transform(CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of thewaveletsvary continuously.
The continuous wavelet transform of a functionx(t){\displaystyle x(t)}at a scalea∈R+∗{\disp... | https://en.wikipedia.org/wiki/Continuous_wavelet_transform |
Innumerical analysisandfunctional analysis, adiscrete wavelet transform(DWT) is anywavelet transformfor which thewaveletsare discretely sampled. As with other wavelet transforms, a key advantage it has overFourier transformsis temporal resolution: it captures both frequencyandlocation information (location in time).
T... | https://en.wikipedia.org/wiki/Discrete_wavelet_transform |
DjVu[a]is acomputerfile formatdesigned primarily to storescanned documents, especially those containing a combination of text, line drawings,indexed color images, and photographs. It uses technologies such as image layer separation of text and background/images,progressive loading,arithmetic coding, andlossy compressio... | https://en.wikipedia.org/wiki/DjVu |
Inmathematics, adual waveletis thedualto awavelet. In general, thewavelet seriesgenerated by asquare-integrablefunctionwill have a dual series, in the sense of theRiesz representation theorem. However, the dual series is not itself in general representable by a square-integrable function.
Given a square-integrable fun... | https://en.wikipedia.org/wiki/Dual_wavelet |
ECW(Enhanced Compression Wavelet) is aproprietarywavelet compressionimage formatused foraerial photographyandsatellite imagery. It was developed by Earth Resource Mapping, which is now owned byIntergraph, part ofHexagon AB.[1]It is alossy compressionformat for images.
In 1998 Earth Resource Mapping Ltd inPerth, Wester... | https://en.wikipedia.org/wiki/ECW_(file_format) |
Geographic data and informationis defined in theISO/TC 211series of standards as data and information having an implicit or explicit association with a location relative toEarth(ageographic locationorgeographic position).[1][2]It is also calledgeospatial data and information,[citation needed]georeferenced data and info... | https://en.wikipedia.org/wiki/Geospatial |
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