text stringlengths 11 320k | source stringlengths 26 161 |
|---|---|
Inmathematicsandclassical mechanics, thePoisson bracketis an importantbinary operationinHamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltoniandynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations,... | https://en.wikipedia.org/wiki/Poisson_bracket |
Inmathematical physics, theternary commutatoris an additionalternary operationon atriple systemdefined by
Also called theternutatororalternating ternary sum, it is a special case of then-commutatorforn= 3, whereas the 2-commutator is the ordinarycommutator.
Thisabstract algebra-related article is astub. You can help ... | https://en.wikipedia.org/wiki/Ternary_commutator |
Inmathematics, more specificallygroup theory, thethree subgroups lemmais a result concerningcommutators. It is a consequence ofPhilip HallandErnst Witt'seponymous identity.
In what follows, the following notation will be employed:
LetX,YandZbe subgroups of a groupG, and assume
Then[Z,X,Y]=1{\displaystyle [Z,X,Y]=1}.... | https://en.wikipedia.org/wiki/Three_subgroups_lemma |
Inquantum statistics,Bose–Einstein statistics(B–E statistics) describes one of two possible ways in which a collection of non-interactingidentical particlesmay occupy a set of available discreteenergy statesatthermodynamic equilibrium. The aggregation of particles in the same state, which is a characteristic of particl... | https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics |
Fermi–Dirac statisticsis a type ofquantum statisticsthat applies to thephysicsof asystemconsisting of many non-interacting,identical particlesthat obey thePauli exclusion principle. A result is the Fermi–Dirac distribution of particles overenergy states. It is named afterEnrico FermiandPaul Dirac, each of whom derived ... | https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics |
Incomputer architecture, atrace cacheorexecution trace cacheis a specializedinstruction cachewhich stores the dynamic stream ofinstructionsknown astrace. It helps in increasing the instruction fetchbandwidthand decreasing power consumption (in the case ofIntelPentium 4) by storing traces of instructions that have alrea... | https://en.wikipedia.org/wiki/Trace_cache |
Inmathematical logic,algebraic logicis the reasoning obtained by manipulating equations withfree variables.
What is now usually called classical algebraic logic focuses on the identification and algebraic description ofmodelsappropriate for the study of various logics (in the form of classes of algebras that constitut... | https://en.wikipedia.org/wiki/Algebraic_logic |
Inmathematical logic,abstract model theoryis a generalization ofmodel theorythat studies the general properties of extensions offirst-order logicand their models.[1]
Abstract model theory provides an approach that allows us to step back and study a wide range of logics and their relationships.[2]The starting point for... | https://en.wikipedia.org/wiki/Abstract_model_theory |
Inmathematics, ahierarchyis aset-theoreticalobject, consisting of apreorderdefined on aset. This is often referred to as anordered set, though that is an ambiguous term that many authors reserve forpartially ordered setsortotally ordered sets. The termpre-ordered setis unambiguous, and is always synonymous with a mathe... | https://en.wikipedia.org/wiki/Hierarchy_(mathematics) |
Inmathematical logic,model theoryis the study of the relationship betweenformal theories(a collection ofsentencesin aformal languageexpressing statements about amathematical structure), and theirmodels(thosestructuresin which the statements of the theory hold).[1]The aspects investigated include the number and size of ... | https://en.wikipedia.org/wiki/Model_theory |
Inuniversal algebra, avariety of algebrasorequational classis theclassof allalgebraic structuresof a givensignaturesatisfying a given set ofidentities. For example, thegroupsform a variety of algebras, as do theabelian groups, therings, themonoidsetc. According toBirkhoff's theorem, a class of algebraic structures of t... | https://en.wikipedia.org/wiki/Variety_(universal_algebra) |
Universal Logic is an emerging interdisciplinary field involving logic, non-classical logic, categorical logic, set theory, foundation of logic, and the philosophy and history of logic. The goal of the field is to develop an understanding of the nature of different types of logic. The expressionUniversal logicwas coine... | https://en.wikipedia.org/wiki/Universal_logic |
GF(2)(also denotedF2{\displaystyle \mathbb {F} _{2}},Z/2ZorZ/2Z{\displaystyle \mathbb {Z} /2\mathbb {Z} }) is thefinite fieldwith two elements.[1][a]
GF(2)is thefieldwith the smallest possible number of elements, and is unique if theadditive identityand themultiplicative identityare denoted respectively0and1, as usual... | https://en.wikipedia.org/wiki/GF(2) |
Causal research, is the investigation of (researchinto)cause-relationships.[1][2][3]To determine causality, variation in the variable presumed to influence the difference in another variable(s) must be detected, and then the variations from the other variable(s) must be calculated (s). Otherconfounding influencesmust ... | https://en.wikipedia.org/wiki/Causal_research |
Causal inferenceis the process of determining the independent, actual effect of a particular phenomenon that is a component of a larger system. The main difference between causal inference and inference ofassociationis that causal inference analyzes the response of an effect variable when a cause of the effect variable... | https://en.wikipedia.org/wiki/Causal_inference |
Causality: Models, Reasoning, and Inference(2000;[1]updated 2009[2]) is a book byJudea Pearl.[3]It is an exposition and analysis ofcausality.[4][5]It is considered to have been instrumental in laying the foundations of the modern debate oncausal inferencein several fields includingstatistics,computer scienceandepidemio... | https://en.wikipedia.org/wiki/Causality_(book) |
Causationrefers to the existence of "cause and effect" relationships between multiple variables.[1]Causation presumes that variables, which act in a predictable manner, can produce change in related variables and that this relationship can be deduced through direct and repeated observation.[2]Theories of causation unde... | https://en.wikipedia.org/wiki/Causation_(sociology) |
In thephilosophy of religion, acosmological argumentis an argument for the existence ofGodbased uponobservationalandfactualstatements concerning theuniverse(or some general category of itsnaturalcontents) typically in the context ofcausation, change, contingency or finitude.[1][2][3]In referring toreasonand observation... | https://en.wikipedia.org/wiki/Cosmological_argument |
Adomino effectis the cumulative effect produced when one event sets off a series of similar[1]or related events, a form ofchain reaction. The term is an analogy to afalling row of dominoes. It typically refers to a linked sequence of events where the time between successive events is relatively short. The term can be u... | https://en.wikipedia.org/wiki/Domino_effect |
Timeis the continuous progression ofexistencethat occurs in an apparentlyirreversiblesuccession from thepast, through thepresent, and into thefuture.[1][2][3]It is a component quantity of various measurements used tosequenceevents, to compare the duration of events (or the intervals between them), and to quantifyrates ... | https://en.wikipedia.org/wiki/Sequence_of_events |
ThisCharles Sanders Peirce bibliographyconsolidates numerous references to the writings ofCharles Sanders Peirce, including letters, manuscripts, publications, andNachlass. For an extensive chronological list of Peirce's works (titled in English), see theChronologische Übersicht(Chronological Overview) on theSchriften(... | https://en.wikipedia.org/wiki/Charles_Sanders_Peirce_bibliography |
Statistical relational learning(SRL) is a subdiscipline ofartificial intelligenceandmachine learningthat is concerned withdomain modelsthat exhibit bothuncertainty(which can be dealt with using statistical methods) and complex,relationalstructure.[1][2]Typically, theknowledge representationformalisms developed in SRL u... | https://en.wikipedia.org/wiki/Statistical_relational_learning |
Bayesian probability(/ˈbeɪziən/BAY-zee-ənor/ˈbeɪʒən/BAY-zhən)[1]is aninterpretation of the concept of probability, in which, instead offrequencyorpropensityof some phenomenon, probability is interpreted as reasonable expectation[2]representing a state of knowledge[3]or as quantification of a personal belief.[4]
The Ba... | https://en.wikipedia.org/wiki/Bayesian_probability |
Cox's theorem, named after the physicistRichard Threlkeld Cox, is a derivation of the laws ofprobability theoryfrom a certain set ofpostulates.[1][2]This derivation justifies the so-called "logical" interpretation of probability, as the laws of probability derived by Cox's theorem are applicable to any proposition. Log... | https://en.wikipedia.org/wiki/Cox%27s_theorem |
Imprecise probabilitygeneralizesprobability theoryto allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a uniqueprobability distributionmay be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately. I... | https://en.wikipedia.org/wiki/Imprecise_probability |
Anon-monotonic logicis aformal logicwhoseentailmentrelation is notmonotonic. In other words, non-monotonic logics are devised to capture and representdefeasible inferences, i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidenc... | https://en.wikipedia.org/wiki/Non-monotonic_logic |
Possibility theoryis a mathematical theory for dealing with certain types ofuncertaintyand is an alternative toprobability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. ProfessorLotfi Zadehfirst introduced possibili... | https://en.wikipedia.org/wiki/Possibility_theory |
Most real databases contain data whose correctness is uncertain. In order to work with such data, there is a need to quantify the integrity of the data. This is achieved by using probabilistic databases.
Aprobabilistic databaseis anuncertain databasein which thepossible worldshave associatedprobabilities. Probabilisti... | https://en.wikipedia.org/wiki/Probabilistic_database |
Probabilistic Soft Logic (PSL)is astatistical relational learning (SRL)framework for modeling probabilistic and relational domains.[2]It is applicable to a variety ofmachine learningproblems, such ascollective classification,entity resolution,link prediction, andontology alignment.
PSL combines two tools:first-order lo... | https://en.wikipedia.org/wiki/Probabilistic_soft_logic |
Probabilistic causationis a concept in a group of philosophical theories that aim to characterize the relationship between cause and effect using the tools ofprobability theory. The central idea behind these theories is that causes raise the probabilities of their effects,all else being equal.
Interpretingcausationas ... | https://en.wikipedia.org/wiki/Probabilistic_causation |
Uncertain inferencewas first described byC. J. van Rijsbergen[1]as a way to formally define a query and document relationship inInformation retrieval. This formalization is alogical implicationwith an attached measure of uncertainty.
Rijsbergen proposes that the measure ofuncertaintyof a documentdto a queryqbe the pro... | https://en.wikipedia.org/wiki/Uncertain_inference |
Upper and lower probabilitiesare representations ofimprecise probability. Whereasprobability theoryuses a single number, theprobability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.
Becausefrequentist statisticsdis... | https://en.wikipedia.org/wiki/Upper_and_lower_probabilities |
Inmathematical logicandmetalogic, aformal systemis calledcompletewith respect to a particularpropertyif everyformulahaving the property can bederivedusing that system, i.e. is one of itstheorems; otherwise the system is said to beincomplete.
The term "complete" is also used without qualification, with differing meaning... | https://en.wikipedia.org/wiki/Completeness_(logic) |
ANOR gateor a NOT OR gate is a logic gate which gives a positive output only when both inputs are negative.
LikeNAND gates, NOR gates are so-called "universal gates" that can be combined to form any other kind oflogic gate. For example, the firstembedded system, theApollo Guidance Computer, was built exclusively from ... | https://en.wikipedia.org/wiki/NOR_logic |
Aone-instruction set computer(OISC), sometimes referred to as anultimatereduced instruction set computer(URISC), is anabstract machinethat uses only one instruction – obviating the need for amachine languageopcode.[1][2][3]With a judicious choice for the single instruction and given arbitrarily many resources, an OISC ... | https://en.wikipedia.org/wiki/One-instruction_set_computer |
Informal semantics,homogeneityis the phenomenon wherepluralexpressions that seem to mean "all" negate to "none" rather than "not all". For example, theEnglishsentence "Robin read the books" requires Robin to have read all of the books, while "Robin didn't read the books" requires her to have read none of them. Neither ... | https://en.wikipedia.org/wiki/Homogeneity_(semantics) |
Inlogic, thelaw of excluded middleor theprinciple of excluded middlestates that for everyproposition,eitherthis proposition or itsnegationistrue.[1][2]It is one of thethree laws of thought, along with thelaw of noncontradiction, and thelaw of identity; however, no system of logic is built on just these laws, and none o... | https://en.wikipedia.org/wiki/Law_of_excluded_middle |
Inlogic, athree-valued logic(alsotrinary logic,trivalent,ternary, ortrilean,[1]sometimes abbreviated3VL) is any of severalmany-valued logicsystems in which there are threetruth valuesindicatingtrue,false, and some third value. This is contrasted with the more commonly knownbivalentlogics (such as classical sentential o... | https://en.wikipedia.org/wiki/Trivalent_logic |
Inlogic, thecorresponding conditionalof anargument(or derivation) is amaterial conditionalwhoseantecedentis theconjunctionof the argument's (or derivation's)premisesand whoseconsequentis the argument's conclusion. An argument isvalidif and only ifits corresponding conditional is alogical truth. It follows that an argum... | https://en.wikipedia.org/wiki/Corresponding_conditional |
Counterfactual conditionals(alsocontrafactual,subjunctiveorX-marked) areconditional sentenceswhich discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted withindicatives, which are generally restricted to discus... | https://en.wikipedia.org/wiki/Counterfactual_conditional |
Dynamic semanticsis a framework inlogicandnatural language semanticsthat treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynamic semantics, knowing the meaning of a sentence means knowing "the change it ... | https://en.wikipedia.org/wiki/Dynamic_semantics |
Innatural languages, anindicative conditionalis aconditional sentencesuch as "If Leona is at home, she isn't in Paris", whose grammatical form restricts it to discussing what could be true. Indicatives are typically defined in opposition tocounterfactual conditionals, which have extra grammatical marking which allows t... | https://en.wikipedia.org/wiki/Indicative_conditional |
Logical consequence(alsoentailmentorlogical implication) is a fundamentalconceptinlogicwhich describes the relationship betweenstatementsthat hold true when one statement logicallyfollows fromone or more statements. Avalidlogicalargumentis one in which theconclusionis entailed by thepremises, because the conclusion is ... | https://en.wikipedia.org/wiki/Logical_consequence |
InBoolean algebra, thealgebraic normal form(ANF),ring sum normal form(RSNForRNF),Zhegalkin normal form, orReed–Muller expansionis a way of writingpropositional logicformulas in one of three subforms:
Formulas written in ANF are also known asZhegalkin polynomialsand Positive Polarity (or Parity)Reed–Muller expressions(... | https://en.wikipedia.org/wiki/Algebraic_normal_form |
InBoolean algebra, aformulais inconjunctive normal form(CNF) orclausal normal formif it is aconjunctionof one or moreclauses, where a clause is adisjunctionofliterals; otherwise put, it is aproduct of sumsoran AND of ORs.
In automated theorem proving, the notion "clausal normal form" is often used in a narrower sense,... | https://en.wikipedia.org/wiki/Conjunctive_normal_form |
Inboolean logic, adisjunctive normal form(DNF) is acanonical normal formof a logical formula consisting of a disjunction of conjunctions; it can also be described as anOR of ANDs, asum of products, or — inphilosophical logic— acluster concept.[1]As anormal form, it is useful inautomated theorem proving.
A logical form... | https://en.wikipedia.org/wiki/Disjunctive_normal_form |
Logic optimizationis a process of finding an equivalent representation of the specifiedlogic circuitunder one or more specified constraints. This process is a part of alogic synthesisapplied indigital electronicsandintegrated circuit design.
Generally, the circuit is constrained to a minimum chip area meeting a predef... | https://en.wikipedia.org/wiki/Logic_optimization |
Q.E.D.orQEDis aninitialismof theLatin phrasequod erat demonstrandum, meaning "that which was to be demonstrated". Literally, it states "what was to be shown".[1]Traditionally, the abbreviation is placed at the end ofmathematical proofsandphilosophicalargumentsin print publications, to indicate that the proof or the arg... | https://en.wikipedia.org/wiki/Q.E.D. |
Amathematical symbolis a figure or a combination of figures that is used to represent amathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in aformula. As formulas are entirely constituted with symbols of various types, many sy... | https://en.wikipedia.org/wiki/List_of_mathematical_symbols |
Inmathematics, aconstructive proofis a method ofproofthat demonstrates the existence of amathematical objectby creating or providing a method for creating the object. This is in contrast to anon-constructive proof(also known as anexistence prooforpure existence theorem), which proves the existence of a particular kind ... | https://en.wikipedia.org/wiki/Constructive_proof |
In thephilosophy of mathematics,constructivismasserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove theexistenceof a mathematical object without "finding" that object explicitly, ... | https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) |
Ordinary language philosophy(OLP[1]) is aphilosophical methodologythat sees traditional philosophical problems as rooted in misunderstandings philosophers develop bydistorting or forgetting how words are ordinarily usedto conveymeaninginnon-philosophical contexts. "Such 'philosophical' uses of language, on this view, c... | https://en.wikipedia.org/wiki/Ordinary_language_philosophy |
Metaphilosophy, sometimes calledthe philosophy of philosophy, is "the investigation of the nature ofphilosophy".[1]Its subject matter includes the aims of philosophy, the boundaries of philosophy, and its methods.[2][3]Thus, while philosophy characteristically inquires into the nature ofbeing, the reality of objects, t... | https://en.wikipedia.org/wiki/Metaphilosophy |
Theinternal–external distinctionis a distinction used in philosophy to divide anontologyinto two parts: an internal part concerningobservationrelated to philosophy, and an external part concerningquestionrelated to philosophy.
Rudolf Carnapintroduced the idea of a 'linguistic framework' or a 'form of language' that us... | https://en.wikipedia.org/wiki/Internal%E2%80%93external_distinction |
Inphilosophical logic,metaphysics, and thephilosophy of language, theproblem of absolute generalityis the problem of referring to absolutely everything.[1]Historically, philosophers have assumed that some of their statements are absolutely general, referring to truly everything.[1]In recent years, logicians working in ... | https://en.wikipedia.org/wiki/Absolute_generality |
Inmathematics, aclassification theoremanswers theclassificationproblem: "What are the objects of a given type, up to someequivalence?". It gives a non-redundantenumeration: each object is equivalent to exactly one class.
A few issues related to classification are the following.
There exist manyclassification theorems... | https://en.wikipedia.org/wiki/Classification_theorem |
In mathematics, the termmodulo("with respect to a modulus of", theLatinablativeofmoduluswhich itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced intomathema... | https://en.wikipedia.org/wiki/Modulo_(jargon) |
Inmathematics, more specifically incategory theory, auniversal propertyis a property that characterizesup toanisomorphismthe result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of theinte... | https://en.wikipedia.org/wiki/Universal_property |
Inset theory, a branch ofmathematics, anurelementorur-element(from theGermanprefixur-, 'primordial') is an object that is not aset(has no elements), but that may be anelementof a set. It is also referred to as anatomorindividual. Ur-elements are also not identical with the empty set.
There are several different but es... | https://en.wikipedia.org/wiki/Urelement |
Inmathematics, thehorizontal line testis a test used to determine whether afunctionisinjective(i.e., one-to-one).[1]
Ahorizontal lineis a straight, flat line that goes from left to right. Given a functionf:R→R{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }(i.e. from thereal numbersto the real numbers), we can dec... | https://en.wikipedia.org/wiki/Horizontal_line_test |
Inmathematics, especially in the area ofabstract algebraknown asmodule theory, aninjective moduleis amoduleQthat shares certain desirable properties with theZ-moduleQof allrational numbers. Specifically, ifQis asubmoduleof some other module, then it is already adirect summandof that module; also, given a submodule of a... | https://en.wikipedia.org/wiki/Injective_module |
In themathematicalfield ofnumerical analysis,monotone cubic interpolationis a variant ofcubic interpolationthat preservesmonotonicityof thedata setbeing interpolated.
Monotonicity is preserved bylinear interpolationbut not guaranteed bycubic interpolation.
Monotone interpolation can be accomplished usingcubic Hermite... | https://en.wikipedia.org/wiki/Monotone_cubic_interpolation |
Inmathematics, apseudo-monotone operatorfrom areflexiveBanach spaceinto itscontinuous dual spaceis one that is, in some sense, almost aswell-behavedas amonotone operator. Many problems in thecalculus of variationscan be expressed using operators that are pseudo-monotone, and pseudo-monotonicity in turn implies the exis... | https://en.wikipedia.org/wiki/Pseudo-monotone_operator |
Instatistics,Spearman's rank correlation coefficientorSpearman'sρ, named afterCharles Spearman[1]and often denoted by the Greek letterρ{\displaystyle \rho }(rho) or asrs{\displaystyle r_{s}}, is anonparametricmeasure ofrank correlation(statistical dependencebetween therankingsof twovariables). It assesses how well the ... | https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient |
Inreal analysis, a branch ofmathematics,Bernstein's theoremstates that everyreal-valuedfunctionon the half-line[0, ∞)that istotally monotoneis a mixture ofexponential functions. In one important special case the mixture is aweighted average, orexpected value.
Total monotonicity (sometimes alsocomplete monotonicity) o... | https://en.wikipedia.org/wiki/Total_monotonicity |
Inmathematics,cyclical monotonicityis a generalization of the notion ofmonotonicityto the case ofvector-valued function.[1][2]
Let⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle }denote the inner product on aninner product spaceX{\displaystyle X}and letU{\displaystyle U}be anonemptysubset ofX{\displaystyle X}. Acorre... | https://en.wikipedia.org/wiki/Cyclical_monotonicity |
In linear algebra, theoperator monotone functionis an important type ofreal-valued function, fully classified byCharles Löwnerin 1934.[1]It is closely allied to the operator concave and operator concave functions, and is encountered inoperator theoryand inmatrix theory, and led to theLöwner–Heinz inequality.[2][3]
A f... | https://en.wikipedia.org/wiki/Operator_monotone_function |
In mathematics, asubmodular set function(also known as asubmodular function) is aset functionthat, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit (diminishing returns). The naturaldiminishing returnsproperty which makes th... | https://en.wikipedia.org/wiki/Monotone_set_function |
In mathematics, the notions of anabsolutely monotonic functionand acompletely monotonic functionare two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its de... | https://en.wikipedia.org/wiki/Absolutely_and_completely_monotonic_functions_and_sequences |
In themathematical theoryof functions ofoneormore complex variables, and also incomplex algebraic geometry, abiholomorphismorbiholomorphic functionis abijectiveholomorphic functionwhoseinverseis alsoholomorphic.
Formally, abiholomorphic functionis a functionϕ{\displaystyle \phi }defined on anopen subsetUof then{\displ... | https://en.wikipedia.org/wiki/Biholomorphic_mapping |
Incomplex analysis,de Branges's theorem, or theBieberbach conjecture, is a theorem that gives anecessary conditionon aholomorphic functionin order for it to map theopen unit diskof thecomplex planeinjectivelyto the complex plane. It was posed byLudwig Bieberbach(1916) and finally proven byLouis de Branges(1985).
The s... | https://en.wikipedia.org/wiki/De_Branges%27s_theorem |
Incomplex analysis, a branch ofmathematics, theKoebe 1/4 theoremstates the following:
Koebe Quarter Theorem.The image of an injective analytic functionf:D→C{\displaystyle f:\mathbf {D} \to \mathbb {C} }from theunit diskD{\displaystyle \mathbf {D} }onto asubsetof thecomplex planecontains the disk whose center isf(0){\d... | https://en.wikipedia.org/wiki/Koebe_quarter_theorem |
Incomplex analysis, theRiemann mapping theoremstates that ifU{\displaystyle U}is anon-emptysimply connectedopen subsetof thecomplex number planeC{\displaystyle \mathbb {C} }which is not all ofC{\displaystyle \mathbb {C} }, then there exists abiholomorphicmappingf{\displaystyle f}(i.e. abijectiveholomorphicmapping whose... | https://en.wikipedia.org/wiki/Riemann_mapping_theorem |
Inmathematics,Nevanlinna's criterionincomplex analysis, proved in 1920 by the Finnish mathematicianRolf Nevanlinna, characterizesholomorphicunivalent functionson theunit diskwhich arestarlike. Nevanlinna used this criterion to prove theBieberbach conjecturefor starlike univalent functions.
A univalent functionhon the ... | https://en.wikipedia.org/wiki/Nevanlinna%27s_criterion |
Inmathematics, anembedding(orimbedding[1]) is one instance of somemathematical structurecontained within another instance, such as agroupthat is asubgroup.
When some objectX{\displaystyle X}is said to be embedded in another objectY{\displaystyle Y}, the embedding is given by someinjectiveand structure-preserving mapf:... | https://en.wikipedia.org/wiki/Embedding |
Instatistical mechanicsandmathematics, theBethe lattice(also called aregular tree) is an infinitesymmetricregulartreewhere all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature byHans Bethein 1935. In such a graph, each node is connected tozneighbors; the numberzis... | https://en.wikipedia.org/wiki/Bethe_lattice |
In themathematicaldiscipline ofgraph theory, agraphCis acovering graphof another graphGif there is acovering mapfrom thevertexset ofCto the vertex set ofG. A covering mapfis asurjectionand a localisomorphism: theneighbourhoodof a vertexvinCis mappedbijectivelyonto the neighbourhood off(v){\displaystyle f(v)}inG.
The... | https://en.wikipedia.org/wiki/Covering_graph |
Indiscrete mathematics, particularly ingraph theory, agraphis a structure consisting of asetof objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions calledvertices(also callednodesorpoints) and each of the related pairs of vertices is called anedge(also calledl... | https://en.wikipedia.org/wiki/Undirected_graph |
Ingraph theory, thebipartite double coverof anundirected graphGis abipartite,covering graphofG, with twice as manyverticesasG. It can be constructed as thetensor product of graphs,G×K2. It is also called theKronecker double cover,canonical double coveror simply thebipartite doubleofG.
It should not be confused with ac... | https://en.wikipedia.org/wiki/Bipartite_double_cover |
Inmathematics, acovering groupof atopological groupHis acovering spaceGofHsuch thatGis a topological group and the covering mapp:G→His acontinuousgroup homomorphism. The mappis called thecovering homomorphism. A frequently occurring case is adouble covering group, atopological double coverin whichHhasindex2 inG; exampl... | https://en.wikipedia.org/wiki/Covering_group |
Inmathematics, especially inorder theory, aGalois connectionis a particular correspondence (typically) between twopartially ordered sets(posets). Galois connections find applications in various mathematical theories. They generalize thefundamental theorem of Galois theoryabout the correspondence betweensubgroupsandsub... | https://en.wikipedia.org/wiki/Galois_connection |
Intopologyand related areas ofmathematics, thequotient spaceof atopological spaceunder a givenequivalence relationis a new topological space constructed by endowing thequotient setof the original topological space with thequotient topology, that is, with thefinest topologythat makescontinuousthecanonical projection map... | https://en.wikipedia.org/wiki/Quotient_space_(topology) |
Inset theory, anordinal number, orordinal, is a generalization ofordinal numerals(first, second,nth, etc.) aimed to extendenumerationtoinfinite sets.[1]
A finite set can be enumerated by successively labeling each element with the leastnatural numberthat has not been previously used. To extend this process to variousi... | https://en.wikipedia.org/wiki/Ordinal_number |
Anenumerative definitionof a concept or term is a special type ofextensional definitionthat gives an explicit and exhaustive listing of all theobjectsthat fall under the concept or term in question. Enumerative definitions are only possible for finite sets and only practical for relatively small sets.
An example of a... | https://en.wikipedia.org/wiki/Enumerative_definition |
Inmathematics, asequenceis an enumerated collection ofobjectsin which repetitions are allowed andordermatters. Like aset, it containsmembers(also calledelements, orterms). The number of elements (possiblyinfinite) is called thelengthof the sequence. Unlike a set, the same elements can appear multiple times at different... | https://en.wikipedia.org/wiki/Sequence |
In mathematics, anaffine bundleis afiber bundlewhose typical fiber, fibers, trivialization morphisms and transition functions are affine.[1]
Letπ¯:Y¯→X{\displaystyle {\overline {\pi }}:{\overline {Y}}\to X}be avector bundlewith a typical fiber avector spaceF¯{\displaystyle {\overline {F}}}. Anaffine bundlemodelled on ... | https://en.wikipedia.org/wiki/Affine_bundle |
Inmathematics, analgebra bundleis afiber bundlewhosefibersarealgebrasandlocal trivializationsrespect the algebra structure. It follows that thetransition functionsarealgebra isomorphisms. Since algebras are alsovector spaces, every algebra bundle is avector bundle.
Examples include thetensor-algebra bundle,exterior bu... | https://en.wikipedia.org/wiki/Algebra_bundle |
Inmathematics, acharacteristic classis a way of associating to eachprincipal bundleofXacohomologyclass ofX. The cohomology class measures the extent to which the bundle is "twisted" and whether it possessessections. Characteristic classes are globalinvariantsthat measure the deviation of alocalproduct structure from a ... | https://en.wikipedia.org/wiki/Characteristic_class |
Ingeometryandtopology, given agroupG(which may be a topological or Lie group), anequivariant bundleis afiber bundleπ:E→B{\displaystyle \pi \colon E\to B}such that the total spaceE{\displaystyle E}and the base spaceB{\displaystyle B}are bothG-spaces(continuous or smooth, depending on the setting) and the projection mapπ... | https://en.wikipedia.org/wiki/Equivariant_bundle |
Indifferential geometry, in the category ofdifferentiable manifolds, afibered manifoldis asurjectivesubmersionπ:E→B{\displaystyle \pi :E\to B\,}that is, a surjective differentiable mapping such that at each pointy∈E{\displaystyle y\in E}the tangent mappingTyπ:TyE→Tπ(y)B{\displaystyle T_{y}\pi :T_{y}E\to T_{\pi (y)}B}is... | https://en.wikipedia.org/wiki/Fibered_manifold |
The notion of afibrationgeneralizes the notion of afiber bundleand plays an important role inalgebraic topology, a branch of mathematics.
Fibrations are used, for example, inPostnikov systemsorobstruction theory.
In this article, all mappings arecontinuousmappings betweentopological spaces.
A mappingp:E→B{\displayst... | https://en.wikipedia.org/wiki/Fibration |
Inphysics, agauge theoryis a type offield theoryin which theLagrangian, and hence the dynamics of the system itself, does not change underlocal transformationsaccording to certain smooth families of operations (Lie groups). Formally, the Lagrangian isinvariantunder these transformations.
The term "gauge" refers to any... | https://en.wikipedia.org/wiki/Gauge_theory |
Indifferential topology, theHopf fibration(also known as theHopf bundleorHopf map) describes a3-sphere(ahypersphereinfour-dimensional space) in terms ofcirclesand an ordinarysphere. Discovered byHeinz Hopfin 1931, it is an influential early example of afiber bundle. Technically, Hopf found a many-to-onecontinuous funct... | https://en.wikipedia.org/wiki/Hopf_bundle |
In mathematics, anI-bundleis afiber bundlewhose fiber is anintervaland whose base is amanifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, evenrays, can be the fiber. An I-bundle is said to be twisted if it is not trivial.
Two simple examples of I-bundles are theannulusand theM... | https://en.wikipedia.org/wiki/I-bundle |
Indifferential geometry, a field inmathematics, anatural bundleis anyfiber bundleassociated to thes-frame bundleFs(M){\displaystyle F^{s}(M)}for somes≥1{\displaystyle s\geq 1}. It turns out that its transition functions depend functionally on local changes of coordinates in the basemanifoldM{\displaystyle M}together wi... | https://en.wikipedia.org/wiki/Natural_bundle |
Inmathematics, aprincipal bundle[1][2][3][4]is a mathematical object that formalizes some of the essential features of theCartesian productX×G{\displaystyle X\times G}of a spaceX{\displaystyle X}with agroupG{\displaystyle G}. In the same way as with the Cartesian product, a principal bundleP{\displaystyle P}is equipped... | https://en.wikipedia.org/wiki/Principal_bundle |
Inmathematics, aprojective bundleis afiber bundlewhose fibers areprojective spaces.
By definition, aschemeXover aNoetherian schemeSis aPn-bundle if it is locally a projectiven-space; i.e.,X×SU≃PUn{\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}}and transition automorphisms are linear. Over a regular schemeSsuch... | https://en.wikipedia.org/wiki/Projective_bundle |
Inmathematics, apullback bundleorinduced bundle[1][2][3]is thefiber bundlethat is induced by a map of its base-space. Given a fiber bundleπ:E→Band acontinuous mapf:B′ →Bone can define a "pullback" ofEbyfas a bundlef*EoverB′. The fiber off*Eover a pointb′inB′is just the fiber ofEoverf(b′). Thusf*Eis thedisjoint unionof ... | https://en.wikipedia.org/wiki/Pullback_bundle |
Inalgebraic topology, aquasifibrationis a generalisation offibre bundlesandfibrationsintroduced byAlbrecht DoldandRené Thom. Roughly speaking, it is a continuous mapp:E→Bhaving the same behaviour as a fibration regarding the (relative)homotopy groupsofE,Bandp−1(x). Equivalently, one can define a quasifibration to be a ... | https://en.wikipedia.org/wiki/Quasifibration |
Inmathematics, theuniversal bundlein the theory offiber bundleswith structure group a giventopological groupG, is a specific bundle over aclassifying spaceBG, such that every bundle with the givenstructure groupGoverMis apullbackby means of acontinuous mapM→BG.
When the definition of the classifying space takes place ... | https://en.wikipedia.org/wiki/Universal_bundle |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.