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Thelaw of averagesis the commonly held belief that a particularoutcomeoreventwill, over certain periods of time, occur at afrequencythat is similar to itsprobability.[1][2]Depending on context or application it can be considered a valid common-sense observation or a misunderstanding of probability. This notion can lead... | https://en.wikipedia.org/wiki/Law_of_averages |
Inmathematics, aproof without words(orvisual proof) is an illustration of anidentityor mathematical statement which can be demonstrated asself-evidentby a diagram without any accompanying explanatory text. Such proofs can be considered moreelegantthan formal ormathematically rigorousproofs due to their self-evident nat... | https://en.wikipedia.org/wiki/Proof_without_words#Jensen.27s_inequality |
Inmathematics, more precisely in the theory of functions ofseveral complex variables, apseudoconvex setis a special type ofopen setin then-dimensional complex spaceCn. Pseudoconvex sets are important, as they allow for classification ofdomains of holomorphy.
Let
be a domain, that is, anopenconnectedsubset. One says t... | https://en.wikipedia.org/wiki/Pseudoconvexity |
AJones diagramis a type ofCartesian graphdeveloped byLoyd A. Jonesin the 1940s, where each axis represents a differentvariable. In a Jones diagram opposite directions of an axis represent different quantities, unlike in a Cartesian graph where they represent positive or negativesignsof the same quantity. The Jones diag... | https://en.wikipedia.org/wiki/Jones_diagram |
Applicative computing systems, orACSare the systems of object calculi founded oncombinatory logicandlambda calculus.[1]The only essential notion which is under consideration in these systems is the representation ofobject. Incombinatory logicthe only metaoperator isapplicationin a sense of applying one object to other.... | https://en.wikipedia.org/wiki/Applicative_computing_systems |
TheB, C, K, Wsystem is a variant ofcombinatory logicthat takes as primitive the combinatorsB, C, K, andW. This system was discovered byHaskell Curryin his doctoral thesisGrundlagen der kombinatorischen Logik, whose results are set out in Curry (1930).
It has expressive power equivalent to that ofS, K, Isystem. Both sy... | https://en.wikipedia.org/wiki/B,_C,_K,_W_system |
Thecategorical abstract machine(CAM) is amodel of computationfor programs[1]that preserves the abilities of applicative, functional, or compositional style. It is based on the techniques ofapplicative computing.
The notion of the categorical abstract machine arose in the mid-1980s. It took its place in computer scienc... | https://en.wikipedia.org/wiki/Categorical_abstract_machine |
Incomputer science,lambda calculiare said to haveexplicit substitutionsif they pay special attention to the formalization of the process ofsubstitution. This is in contrast to the standardlambda calculuswhere substitutions are performed bybeta reductionsin an implicit manner which is not expressed within the calculus; ... | https://en.wikipedia.org/wiki/Explicit_substitution |
Agraph reduction machineis a special-purposecomputerbuilt to performcombinatorcalculations bygraph reduction.
Examples include the SKIM ("S-K-I machine") computer, built at theUniversity of Cambridge Computer Laboratory,[1]the multiprocessor GRIP ("Graph Reduction In Parallel") computer, built atUniversity College Lon... | https://en.wikipedia.org/wiki/Graph_reduction_machine |
TheSKI combinator calculusis acombinatory logic systemand acomputational system. It can be thought of as a computer programming language, though it is not convenient for writing software.[citation needed]Instead, it is important in the mathematical theory ofalgorithmsbecause it is an extremely simpleTuring completelang... | https://en.wikipedia.org/wiki/SKI_combinator_calculus |
Asupercombinatoris amathematical expressionwhich isfully boundand self-contained. It may be either aconstantor acombinatorwhere all the subexpressions are supercombinators. Supercombinators are used in the implementation of functional languages.
In mathematical terms, alambda expressionSis a supercombinator ofaritynif... | https://en.wikipedia.org/wiki/Supercombinator |
To Mock a Mockingbird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic(1985,ISBN0-19-280142-2) is a book by themathematicianandlogicianRaymond Smullyan. It contains many nontrivial recreational puzzles of the sort for which Smullyan is well known. It is also a gentle and humorous introductio... | https://en.wikipedia.org/wiki/To_Mock_a_Mockingbird |
Inmathematics, thecomposition operatorCϕ{\displaystyle C_{\phi }}with symbolϕ{\displaystyle \phi }is alinear operatordefined by the ruleCϕ(f)=f∘ϕ{\displaystyle C_{\phi }(f)=f\circ \phi }wheref∘ϕ{\displaystyle f\circ \phi }denotesfunction composition. It is also encountered in composition ofpermutationsin permutations g... | https://en.wikipedia.org/wiki/Composition_operator |
In mathematics, apolynomial decompositionexpresses apolynomialfas thefunctional compositiong∘h{\displaystyle g\circ h}of polynomialsgandh, wheregandhhavedegreegreater than 1; it is an algebraicfunctional decomposition.Algorithmsare known for decomposingunivariate polynomialsinpolynomial time.
Polynomials which are dec... | https://en.wikipedia.org/wiki/Polynomial_decomposition |
In mathematics, aCarleman matrixis a matrix used to convertfunction compositionintomatrix multiplication. It is often used in iteration theory to find the continuousiteration of functionswhich cannot be iterated bypattern recognitionalone. Other uses of Carleman matrices occur in the theory ofprobabilitygenerating fun... | https://en.wikipedia.org/wiki/Carleman_matrix |
Inmathematicsandcomputer science,curryingis the technique of translating afunctionthat takes multipleargumentsinto a sequence of families of functions, each taking a single argument.
In the prototypical example, one begins with a functionf:(X×Y)→Z{\displaystyle f:(X\times Y)\to Z}that takes two arguments, one fromX{\d... | https://en.wikipedia.org/wiki/Currying |
Inobject-oriented programming,inheritanceis the mechanism of basing anobjectorclassupon another object (prototype-based inheritance) or class (class-based inheritance), retaining similarimplementation. Also defined as deriving new classes (sub classes) from existing ones such as super class orbase classand then forming... | https://en.wikipedia.org/wiki/Implementation_inheritance |
Inobject-oriented programming,behavioral subtypingis the principle that subclasses should satisfy the expectations of clients accessing subclass objects through references of superclass type, not just as regards syntactic safety (such as the absence of "method-not-found" errors) but also as regards behavioral correctne... | https://en.wikipedia.org/wiki/Inheritance_semantics |
Infunctional programming, aniterateeis acomposableabstractionfor incrementally processing sequentially presented chunks of input data in apurely functionalfashion. With iteratees, it is possible to lazily transform how a resource will emit data, for example, by converting each chunk of the input to uppercase as they ar... | https://en.wikipedia.org/wiki/Iteratee |
InUnix-likecomputeroperating systems, apipelineis a mechanism forinter-process communicationusing message passing. A pipeline is a set ofprocesseschained together by theirstandard streams, so that the output text of each process (stdout) is passed directly as input (stdin) to the next one. The second process is started... | https://en.wikipedia.org/wiki/Pipeline_(Unix) |
Virtual inheritanceis aC++technique that ensures only one copy of abase class's member variables areinheritedby grandchild derived classes. Without virtual inheritance, if two classesBandCinherit from a classA, and a classDinherits from bothBandC, thenDwill contain two copies ofA's member variables: one viaB, and one v... | https://en.wikipedia.org/wiki/Virtual_inheritance |
ABayesian network(also known as aBayes network,Bayes net,belief network, ordecision network) is aprobabilistic graphical modelthat represents a set of variables and theirconditional dependenciesvia adirected acyclic graph(DAG).[1]While it is one of several forms ofcausal notation, causal networks are special cases of B... | https://en.wikipedia.org/wiki/Bayesian_networks |
Inmathematics, theL-functionsofnumber theoryare expected to have several characteristic properties, one of which is that they satisfy certainfunctional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural.
A prototypical example, theRiemann zeta functionhas a fu... | https://en.wikipedia.org/wiki/Functional_equation_(L-function) |
Afunctional differential equationis adifferential equationwith deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values.[1]
Functional differential equations find use in mathematical models that assume a s... | https://en.wikipedia.org/wiki/Functional_differential_equation |
Algorithmic transparencyis the principle that the factors that influence the decisions made byalgorithmsshould be visible, or transparent, to the people who use, regulate, and are affected by systems that employ those algorithms. Although the phrase was coined in 2016 by Nicholas Diakopoulos and Michael Koliska about t... | https://en.wikipedia.org/wiki/Algorithmic_transparency |
In theregulationofalgorithms, particularlyartificial intelligenceand its subfield ofmachine learning, aright to explanation(orright toanexplanation) is arightto be given anexplanationfor an output of the algorithm. Such rights primarily refer toindividual rightsto be given an explanation for decisions that significantl... | https://en.wikipedia.org/wiki/Right_to_explanation |
Accumulated local effects(ALE) is amachine learninginterpretability method.[1]
ALE uses a conditional feature distribution as an input and generates augmented data, creating more realistic data than a marginal distribution.[2]
It ignores far out-of-distribution (outlier) values.[1]Unlike partial dependence plots and ... | https://en.wikipedia.org/wiki/Accumulated_local_effects |
Instatistics, amixture modelis aprobabilistic modelfor representing the presence ofsubpopulationswithin an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to themixture distributionthat re... | https://en.wikipedia.org/wiki/Mixture_model |
Inprobability theory,de Finetti's theoremstates thatexchangeableobservations areconditionally independentrelative to somelatent variable. Anepistemic probabilitydistributioncould then be assigned to this variable. It is named in honor ofBruno de Finetti, and one of its uses is in providing a pragmatic approach to de Fi... | https://en.wikipedia.org/wiki/De_Finetti_theorem |
Inmathematics, ameasurable spaceorBorel space[1]is a basic object inmeasure theory. It consists of asetand aσ-algebra, which defines thesubsetsthat will be measured.
It captures and generalises intuitive notions such as length, area, and volume with a setX{\displaystyle X}of 'points' in the space, butregionsof the spa... | https://en.wikipedia.org/wiki/Measurable_space |
Inprobability theory, aMarkov kernel(also known as astochastic kernelorprobability kernel) is a map that in the general theory ofMarkov processesplays the role that thetransition matrixdoes in the theory of Markov processes with afinitestate space.[1]
Let(X,A){\displaystyle (X,{\mathcal {A}})}and(Y,B){\displaystyle (Y... | https://en.wikipedia.org/wiki/Markov_kernel |
Incategory theory, a branch ofmathematics, amonadis a triple(T,η,μ){\displaystyle (T,\eta ,\mu )}consisting of afunctorTfrom a category to itself and twonatural transformationsη,μ{\displaystyle \eta ,\mu }that satisfy the conditions like associativity. For example, ifF,G{\displaystyle F,G}are functorsadjointto each oth... | https://en.wikipedia.org/wiki/Monad_(category_theory) |
Inmathematics, thecategory of Markov kernels, often denotedStoch, is thecategorywhoseobjectsaremeasurable spacesand whosemorphismsareMarkov kernels.[1][2][3][4]It is analogous to thecategory of sets and functions, but where thearrowscan be interpreted as beingstochastic.
Several variants of this category are used in t... | https://en.wikipedia.org/wiki/Category_of_Markov_kernels |
In mathematics, the termcategorical probabilitydenotes a collection ofcategory-theoreticapproaches toprobability theoryand related fields such asstatistics,information theoryandergodic theory.
The earliest ideas in the field were developed independently byLawvereand byChentsov, where they defined a version of what we ... | https://en.wikipedia.org/wiki/Categorical_probability |
Pruningis adata compressiontechnique inmachine learningandsearch algorithmsthat reduces the size ofdecision treesby removing sections of the tree that are non-critical and redundant to classify instances. Pruning reduces the complexity of the finalclassifier, and hence improves predictive accuracy by the reduction ofov... | https://en.wikipedia.org/wiki/Pruning_(algorithm) |
Branch and bound(BB,B&B, orBnB) is a method for solving optimization problems by breaking them down into smaller sub-problems and using a bounding function to eliminate sub-problems that cannot contain the optimal solution.
It is analgorithmdesign paradigmfordiscreteandcombinatorial optimizationproblems, as well asma... | https://en.wikipedia.org/wiki/Branch_and_bound |
Combinatorial optimizationis a subfield ofmathematical optimizationthat consists of finding an optimal object from afinite setof objects,[1]where the set offeasible solutionsisdiscreteor can be reduced to a discrete set. Typical combinatorial optimization problems are thetravelling salesman problem("TSP"), theminimum s... | https://en.wikipedia.org/wiki/Combinatorial_optimization |
Principal variation search(sometimes equated with the practically identicalNegaScout) is anegamaxalgorithm that can be faster thanalpha–beta pruning. Like alpha–beta pruning, NegaScout is a directional search algorithm for computing theminimaxvalue of a node in atree. It dominates alpha–beta pruning in the sense that i... | https://en.wikipedia.org/wiki/Principal_variation_search |
Thehistory of chessbegan nearly 1500 years ago. The introduction ofchess enginesaround 1960 and permanent improvement over time has made chess engines become an integral part of chess analysis and influenced what and how chess is played today by humans. It also lead to the problem ofcheating.
The earliest form of a – ... | https://en.wikipedia.org/wiki/History_of_chess_engines |
Draughts(British English) orcheckers(American English), also calledstraight checkersor simplydraughts,[note 1]is a form of thestrategyboard gamecheckers(or draughts). It is played on an 8×8checkerboardwith 12 pieces per side. The pieces move and capture diagonally forward, until they reach the opposite end of the board... | https://en.wikipedia.org/wiki/Computer_checkers |
Computer Gois the field ofartificial intelligence(AI) dedicated to creating acomputer programthat plays the traditionalboard gameGo. The field is sharply divided into two eras. Before 2015, the programs of the era were weak. The best efforts of the 1980s and 1990s produced only AIs that could be defeated by beginner... | https://en.wikipedia.org/wiki/Computer_Go |
Computer Othellorefers to computer architecture encompassing computer hardware and computer software capable of playing the game ofOthello. It was notably included inMicrosoft Windowsfrom1.0toXP, where it is simply known as Reversi.[citation needed]
There are many Othello programs such asNTest, Saio,Edax, Cassio, Poin... | https://en.wikipedia.org/wiki/Computer_Othello |
Computer shogiis a field ofartificial intelligenceconcerned with the creation ofcomputer programswhich can playshogi. The research and development of shogi software has been carried out mainly by freelance programmers, university research groups and private companies. By 2017, the strongest programs were outperforming ... | https://en.wikipedia.org/wiki/Computer_shogi |
Thefog of waris theuncertaintyinsituational awarenessexperienced by participants inmilitary operations.[1]The term seeks to capture the uncertainty regarding one's own capability, adversary capability, and adversaryintentduring an engagement, operation, or campaign. Military forces try to reduce the fog of war throughm... | https://en.wikipedia.org/wiki/Fog_of_war |
Anti-computer tacticsare methods used by humans to try to beat computer opponents at various games, most typicallyboard gamessuch aschessandArimaa. They are most associated with competitions against computer AIs that are playing to their utmost to win, rather than AIs merely programmed to be an interesting challenge t... | https://en.wikipedia.org/wiki/Anti-computer_tactics |
Thetrolley problemis a series ofthought experimentsinethics,psychology, andartificial intelligenceinvolving stylizedethical dilemmasof whether to sacrifice one person to save a larger number. The series usually begins with ascenarioin which arunawaytrolleyortrainis on course to collide with and kill a number of people... | https://en.wikipedia.org/wiki/Trolley_problem |
Condorcet methods
Positional voting
Cardinal voting
Quota-remainder methods
Approval-based committees
Fractional social choice
Semi-proportional representation
By ballot type
Pathological response
Strategic voting
Paradoxes ofmajority rule
Positive results
Multiwinner[1]orcommittee[2][3]votingrefers toelect... | https://en.wikipedia.org/wiki/Multiwinner_voting |
Inmechanism design, aregret-free truth-telling mechanism(RFTT, orregret-free mechanismfor short) is a mechanism in which each player who reveals his true private information does not feelregretafter seeing the mechanism outcome. A regret-free mechanism incentivizes agents who want to avoid regret to report their prefer... | https://en.wikipedia.org/wiki/Regret-free_mechanism |
Indecision theoryandmachine learning,competitive regretrefers to a performance measure that evaluates an algorithm'sregretrelative to anoracleor benchmark strategy. Unlike traditional regret, which compares against the best fixed decision in hindsight, competitive regret compares against decision-makers with different ... | https://en.wikipedia.org/wiki/Competitive_regret |
Info-gap decision theoryseeks to optimizerobustnessto failure under severeuncertainty,[1][2]in particular applyingsensitivity analysisof thestability radiustype[3]to perturbations in the value of a given estimate of the parameter of interest. It has some connections withWald's maximin model; some authors distinguish th... | https://en.wikipedia.org/wiki/Info-gap_decision_theory |
Swap regretis a concept fromgame theory. It is a generalization ofregretin a repeated,n-decision game.
A player'sswap-regretis defined to be the following:
Intuitively, it is how much a player could improve by switching each occurrence of decisionito the best decisionjpossible inhindsight. The swap regret is always ... | https://en.wikipedia.org/wiki/Swap_regret |
Leela Chess Zero(abbreviated asLCZero,lc0) is afree, open-sourcechess engineandvolunteer computingproject based onGoogle'sAlphaZeroengine. It was spearheaded byGary Linscott, a developer for theStockfish chess engine, and adapted from theLeela ZeroGoengine.[1]
Like Leela Zero and AlphaGo Zero, early iterations of Leel... | https://en.wikipedia.org/wiki/Leela_Chess_Zero |
In competitive two-player games, thekiller heuristicis a move-ordering method based on the observation that a strong move or small set of such moves in a particular position may be equally strong in similar positions at the same move (ply) in the game tree.
Retaining such moves obviates the effort of rediscovering them... | https://en.wikipedia.org/wiki/Killer_heuristic |
Inmathematics– and in particular the study of games on the unit square –Parthasarathy's theoremis a generalization ofVon Neumann's minimax theorem. It states that a particular class of games has a mixed value, provided that at least one of the players has astrategythat is restricted to absolutely continuous distributi... | https://en.wikipedia.org/wiki/Parthasarathy%27s_theorem |
Thedualof a givenlinear program(LP) is another LP that is derived from the original (theprimal) LP in the following schematic way:
Theweak duality theoremstates that the objective value of the dual LP at any feasible solution is always a bound on the objective of the primal LP at any feasible solution (upper or lower ... | https://en.wikipedia.org/wiki/Dual_linear_program |
Incomputational complexity theory,Yao's principle(also calledYao's minimax principleorYao's lemma) relates the performance ofrandomized algorithmsto deterministic (non-random) algorithms. It states that, for certain classes of algorithms, and certain measures of the performance of the algorithms, the following two quan... | https://en.wikipedia.org/wiki/Yao%27s_principle |
Insocial psychology,group polarizationrefers to the tendency for a group to make decisions that are more extreme than the initial inclination of its members. These more extreme decisions are towards greater risk if individuals' initial tendencies are to be risky and towards greater caution if individuals' initial tende... | https://en.wikipedia.org/wiki/Attitude_polarization |
Thegame of chicken, also known as thehawk-dove gameorsnowdrift game,[1]is a model ofconflictfor two players ingame theory. The principle of the game is that while the ideal outcome is for one player to yield (to avoid the worst outcome if neither yields), individuals try to avoid it out of pride, not wanting to look li... | https://en.wikipedia.org/wiki/Chicken_(game) |
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TheChristmas truce(German:Weihnachtsfrieden; French:Trêve de Noël;Dutch:Kerstbestand) was a series of widespread unofficialceasefiresalong theWestern Frontof theFirst World Wararound Christmas 1914.
Thetruceoccurred five months after hostilities had begun. Lulls occurred in... | https://en.wikipedia.org/wiki/Christmas_truce |
Deterrence theoryrefers to the scholarship and practice of how threats of using force by one party can convince another party to refrain from initiating some other course of action.[1]The topic gained increased prominence as a military strategy during theCold Warwith regard to the use ofnuclear weaponsand is related to... | https://en.wikipedia.org/wiki/Deterrence_theory |
"An eye for an eye" (Biblical Hebrew:עַיִן תַּחַת עַיִן,ʿayīn taḥaṯ ʿayīn)[a]is a commandment found in theBook of Exodus21:23–27 expressing the principle of reciprocal justice measure for measure. The earliest known use of the principle appears in theCode of Hammurabi, which predates the writing of the Hebrew Bible but... | https://en.wikipedia.org/wiki/Eye_for_an_eye |
TheGolden Ruleis theprincipleof treating others as one would want to be treated by them. It is sometimes called an ethics of reciprocity, meaning that one should reciprocate to others how one would like them to treat the person (not necessarily how they actually treat them). Various expressions of this rule can be foun... | https://en.wikipedia.org/wiki/Golden_Rule |
Mutual assured destruction(MAD) is adoctrineofmilitary strategyandnational security policywhich posits that a full-scale use ofnuclear weaponsby an attacker on a nuclear-armed defender withsecond-strike capabilitieswould result in thecomplete annihilationof both the attacker and the defender.[1]It is based on the theor... | https://en.wikipedia.org/wiki/Mutual_assured_destruction |
Nice Guys Finish First(BBCHorizontelevision series) is a 1986 documentary byRichard Dawkinswhich discusses selfishness and cooperation, arguing that evolution often favors co-operative behaviour, and focusing especially on thetit for tatstrategy of theprisoner's dilemmagame. The film is approximately 50 minutes long an... | https://en.wikipedia.org/wiki/Nice_Guys_Finish_First |
Richard Dawkins(born 26 March 1941)[3]is a Britishevolutionary biologist,zoologist,science communicatorand author.[4]He is anemeritus fellowofNew College, Oxford, and wasProfessor for Public Understanding of Scienceat theUniversity of Oxfordfrom 1995 to 2008, and is on the advisory board of theUniversity of Austin.[5][... | https://en.wikipedia.org/wiki/Richard_Dawkins |
Peace war gameis aniterated gameoriginally played in academic groups and bycomputer simulationfor years to study possible strategies ofcooperationandaggression.[1]As peace makers became richer over time it became clear that making war had greater costs than initially anticipated. The onlystrategythat acquired wealth mo... | https://en.wikipedia.org/wiki/Peace_war_game |
Quid pro quo(Latin: "something for something"[2]) is aLatin phraseused inEnglishto mean an exchange of goods or services, in which one transfer is contingent upon the other; "a favor for a favor". Phrases with similar meanings include: "give and take", "tit for tat", "you scratch my back, and I'll scratch yours", "this... | https://en.wikipedia.org/wiki/Quid_pro_quo |
Ingame theory, atrigger strategyis any of a class of strategies employed in a repeated non-cooperative game. A player using a trigger strategy initially cooperates but punishes the opponent if a certain level of defection (i.e., the trigger) is observed.
The level ofpunishmentand the sensitivity of the trigger vary w... | https://en.wikipedia.org/wiki/Trigger_strategy |
Zero-sum gameis amathematical representationingame theoryandeconomic theoryof a situation that involves two competing entities, where the result is an advantage for one side and an equivalent loss for the other.[1]In other words, player one's gain is equivalent to player two's loss, with the result that the net improve... | https://en.wikipedia.org/wiki/Zero-sum_game |
Instatistics,kernel Fisher discriminant analysis (KFD),[1]also known asgeneralized discriminant analysis[2]andkernel discriminant analysis,[3]is a kernelized version oflinear discriminant analysis(LDA). It is named afterRonald Fisher.
Intuitively, the idea of LDA is to find a projection where class separation is maxi... | https://en.wikipedia.org/wiki/Kernel_Fisher_discriminant_analysis |
Multiple Discriminant Analysis(MDA) is a multivariatedimensionality reductiontechnique. It has been used to predict signals as diverse asneuralmemory traces and corporate failure.[1]
MDA is not directly used to perform classification. It merely supportsclassificationby yielding acompressed signalamenable to classifica... | https://en.wikipedia.org/wiki/Multiple_discriminant_analysis |
Multidimensional scaling(MDS) is a means of visualizing the level ofsimilarityof individual cases of a data set. MDS is used to translate distances between each pair ofn{\textstyle n}objects in a set into a configuration ofn{\textstyle n}points mapped into an abstractCartesian space.[1]
More technically, MDS refers t... | https://en.wikipedia.org/wiki/Multidimensional_scaling |
Preference regressionis a statistical technique used by marketers to determine consumers’preferredcore benefits. It usually supplementsproduct positioningtechniques likemulti dimensional scalingorfactor analysisand is used to create ideal vectors onperceptual maps.
Starting with raw data from surveys, researchers appl... | https://en.wikipedia.org/wiki/Preference_regression |
Inphysicsandmathematics, theHelmholtz decomposition theoremor thefundamental theorem of vector calculus[1][2][3][4][5][6][7]states that certain differentiablevector fieldscan be resolved into the sum of anirrotational(curl-free) vector field and asolenoidal(divergence-free) vector field. Inphysics, often only the decom... | https://en.wikipedia.org/wiki/Helmholtz_decomposition |
In mathematics,Hiptmair–Xu (HX) preconditioners[1]are preconditioners for solvingH(curl){\displaystyle H(\operatorname {curl} )}andH(div){\displaystyle H(\operatorname {div} )}problems based on the auxiliary space preconditioning framework.[2]An important ingredient in the derivation of HX preconditioners in two and th... | https://en.wikipedia.org/wiki/Hiptmair%E2%80%93Xu_preconditioner |
This is a list of somevector calculusformulae for working with commoncurvilinearcoordinate systems.
Note that the operationarctan(AB){\displaystyle \arctan \left({\frac {A}{B}}\right)}must be interpreted as the two-argument inverse tangent,atan2.
(∇2Aρ−Aρρ2−2ρ2∂Aφ∂φ)ρ^+(∇2Aφ−Aφρ2+2ρ2∂Aρ∂φ)φ^+∇2Azz^{\displaystyle {\b... | https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates |
Incontinuum mechanics,vorticityis apseudovector(or axial vector)fieldthat describes the localspinningmotion of a continuum near some point (the tendency of something to rotate[1]), as would be seen by an observer located at that point and traveling along with theflow. It is an important quantity inthe dynamical theoryo... | https://en.wikipedia.org/wiki/Vorticity |
Invector calculus, thedivergence theorem, also known asGauss's theoremorOstrogradsky's theorem,[1]is atheoremrelating thefluxof avector fieldthrough a closedsurfaceto thedivergenceof the field in the volume enclosed.
More precisely, the divergence theorem states that thesurface integralof a vector field over a closed ... | https://en.wikipedia.org/wiki/Divergence_theorem |
Inspecial relativity, afour-vector(or4-vector, sometimesLorentz vector)[1]is an object with four components, which transform in a specific way underLorentz transformations. Specifically, a four-vector is an element of a four-dimensionalvector spaceconsidered as arepresentation spaceof thestandard representationof theLo... | https://en.wikipedia.org/wiki/Four-vector |
Inspecial relativity, afour-vector(or4-vector, sometimesLorentz vector)[1]is an object with four components, which transform in a specific way underLorentz transformations. Specifically, a four-vector is an element of a four-dimensionalvector spaceconsidered as arepresentation spaceof thestandard representationof theLo... | https://en.wikipedia.org/wiki/Four-position |
Inphysics, in particular inspecial relativityandgeneral relativity, afour-velocityis afour-vectorin four-dimensionalspacetime[nb 1]that represents the relativistic counterpart ofvelocity, which is athree-dimensionalvectorin space.
Physicaleventscorrespond to mathematical points in time and space, the set of all of the... | https://en.wikipedia.org/wiki/Four-velocity |
In thetheory of relativity,four-accelerationis afour-vector(vector in four-dimensionalspacetime) that is analogous to classicalacceleration(a three-dimensional vector, seethree-acceleration in special relativity). Four-acceleration has applications in areas such as the annihilation ofantiprotons, resonance ofstrange p... | https://en.wikipedia.org/wiki/Four-acceleration |
Inspecial relativity,four-momentum(also calledmomentum–energyormomenergy[1]) is the generalization of theclassical three-dimensional momentumtofour-dimensional spacetime. Momentum is a vector inthree dimensions; similarly four-momentum is afour-vectorinspacetime. Thecontravariantfour-momentum of a particle with relati... | https://en.wikipedia.org/wiki/Four-momentum |
In thespecial theory of relativity,four-forceis afour-vectorthat replaces the classicalforce.
The four-force is defined as the rate of change in thefour-momentumof a particle with respect to the particle'sproper time. Hence,:
F=dPdτ.{\displaystyle \mathbf {F} ={\mathrm {d} \mathbf {P} \over \mathrm {d} \tau }.}
For... | https://en.wikipedia.org/wiki/Four-force |
Inspecialandgeneral relativity, thefour-current(technically thefour-current density)[1]is thefour-dimensionalanalogue of thecurrent density, with units of charge per unit time per unit area. Also known asvector current, it is used in the geometric context offour-dimensionalspacetime, rather than separating time fromthr... | https://en.wikipedia.org/wiki/Four-current |
Anelectromagnetic four-potentialis arelativisticvector functionfrom which theelectromagnetic fieldcan be derived. It combines both anelectric scalar potentialand amagnetic vector potentialinto a singlefour-vector.[1]
As measured in a givenframe of reference, and for a givengauge, the first component of the electromagn... | https://en.wikipedia.org/wiki/Four-potential |
Thefour-frequencyof amassless particle, such as aphoton, is afour-vectordefined by
whereν{\displaystyle \nu }is the photon'sfrequencyandn^{\displaystyle {\hat {\mathbf {n} }}}is aunit vectorin the direction of the photon's motion. The four-frequency of a photon is always a future-pointing andnull vector. An observer m... | https://en.wikipedia.org/wiki/Four-frequency |
Inspecial relativity, afour-vector(or4-vector, sometimesLorentz vector)[1]is an object with four components, which transform in a specific way underLorentz transformations. Specifically, a four-vector is an element of a four-dimensionalvector spaceconsidered as arepresentation spaceof thestandard representationof theLo... | https://en.wikipedia.org/wiki/Four-wavevector |
Inphysics,relativistic angular momentumrefers to the mathematical formalisms and physical concepts that defineangular momentuminspecial relativity(SR) andgeneral relativity(GR). The relativistic quantity is subtly different from thethree-dimensionalquantity inclassical mechanics.
Angular momentum is an important dynam... | https://en.wikipedia.org/wiki/Four-spin |
Inmathematics,Ricci calculusconstitutes the rules of index notation and manipulation fortensorsandtensor fieldson adifferentiable manifold, with or without ametric tensororconnection.[a][1][2][3]It is also the modern name for what used to be called theabsolute differential calculus(the foundation of tensor calculus),te... | https://en.wikipedia.org/wiki/Ricci_calculus |
Inmathematicsandcomputer programming,index notationis used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the elements of a list, avector, or amatrix, depending on whether one is writing ... | https://en.wikipedia.org/wiki/Index_notation |
Inmathematicsandtheoretical physics, atensorisantisymmetricoralternating on(orwith respect to)an index subsetif it alternatessign(+/−) when any two indices of the subset are interchanged.[1][2]The index subset must generally either be allcovariantor allcontravariant.
For example,Tijk…=−Tjik…=Tjki…=−Tkji…=Tkij…=−Tikj…{... | https://en.wikipedia.org/wiki/Antisymmetric_tensor |
Inmathematics, especially the usage oflinear algebrainmathematical physicsanddifferential geometry,Einstein notation(also known as theEinstein summation conventionorEinstein summation notation) is a notational convention that impliessummationover a set of indexed terms in a formula, thus achieving brevity. As part of m... | https://en.wikipedia.org/wiki/Einstein_notation |
Inmathematics—more specifically, indifferential geometry—themusical isomorphism(orcanonical isomorphism) is anisomorphismbetween thetangent bundleTM{\displaystyle \mathrm {T} M}and thecotangent bundleT∗M{\displaystyle \mathrm {T} ^{*}M}of aRiemannianorpseudo-Riemannian manifoldinduced by itsmetric tensor. There are sim... | https://en.wikipedia.org/wiki/Raising_and_lowering_indices |
Abstract index notation(also referred to as slot-naming index notation)[1]is a mathematical notation fortensorsandspinorsthat uses indices to indicate their types, rather than their components in a particular basis.[2]The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thu... | https://en.wikipedia.org/wiki/Abstract_index_notation |
Inphysics, especially inmultilinear algebraandtensor analysis,covarianceandcontravariancedescribe how the quantitative description of certain geometric or physical entities changes with achange of basis.[2]Briefly, a contravariant vector is a list of numbers that transforms oppositely to a change of basis, and a covari... | https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors |
Thegrade(US) orgradient(UK) (also calledstepth,slope,incline,mainfall,pitchorrise) of a physical feature, landform or constructed line is either theelevation angleof that surface to thehorizontalor its tangent. It is a special case of theslope, where zero indicateshorizontality. A larger number indicates higher or stee... | https://en.wikipedia.org/wiki/Grade_(slope) |
Animage gradientis a directional change in the intensity or color in an image. The gradient of the image is one of the fundamental building blocks inimage processing. For example, theCanny edge detectoruses image gradient foredge detection. Ingraphics softwarefordigital image editing, the term gradient orcolor gradient... | https://en.wikipedia.org/wiki/Image_gradient |
Atime derivativeis aderivativeof a function with respect totime, usually interpreted as the rate of change of the value of the function.[1]The variable denoting time is usually written ast{\displaystyle t}.
A variety of notations are used to denote the time derivative. In addition to the normal (Leibniz's) notation,
... | https://en.wikipedia.org/wiki/Time_derivative |
Incontinuum mechanics, thematerial derivative[1][2]describes the timerate of changeof some physical quantity (likeheatormomentum) of amaterial elementthat is subjected to a space-and-time-dependentmacroscopic velocity field. The material derivative can serve as a link betweenEulerianandLagrangiandescriptions of continu... | https://en.wikipedia.org/wiki/Material_derivative |
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