'''logic programming''' programming paradigm largely based formal logic program written logic programming language set sentences logical form expressing facts rules problem domain major logic programming language families include prolog answer set programming asp datalog languages rules written form ''clauses'' kbd h b sub sub b sub n sub kbd read declaratively logical implications kbd h b sub sub b sub n sub kbd kbd h kbd called ''head'' rule kbd b sub sub kbd kbd b sub n sub kbd called ''body''

Formal science
Computer science
Logic in computer science
Logic programming


facts rules body written simplified form kbd h kbd simplest case kbd h kbd kbd b sub sub kbd kbd b sub n sub kbd atomic formulae clauses called definite clauses horn clauses however many extensions simple case important one case conditions body clause also negations atomic formulas logic programming languages include extension knowledge representation capabilities non monotonic logic asp datalog logic programs declarative reading execution


performed means proof procedure model generator whose behaviour meant controlled programmer however prolog family languages logic programs also procedural interpretation goal reduction procedures solve kbd h kbd solve kbd b sub sub kbd solve kbd b sub n sub kbd consider following clause example kbd fallible x human x kbd based example used terry winograd ref name winograd ref illustrate programming language planner clause logic program used procedure test whether kbd x kbd kbd fallible kbd testing whether kbd x kbd


kbd human kbd procedure find kbd x kbd kbd fallible kbd finding kbd x kbd kbd human kbd even facts procedural interpretation example clause kbd human socrates kbd used procedure show kbd socrates kbd kbd human kbd procedure find kbd x kbd kbd human kbd assigning kbd socrates kbd kbd x kbd declarative reading logic programs used programmer verify correctness moreover logic based program transformation techniques also used transform logic programs logically equivalent programs efficient prolog


family logic programming languages programmer also use known problem solving behaviour execution mechanism improve efficiency programs history use mathematical logic represent execute computer programs also feature lambda calculus developed alonzo church however first proposal use clausal form logic representing computer programs made cordell green used axiomatization subset lisp together representation input output relation compute relation simulating execution program


lisp foster elcock 's absys hand employed combination equations lambda calculus assertional programming language places constraints order operations performed logic programming present form traced back debates late early declarative versus procedural representations knowledge artificial intelligence advocates declarative representations notably working stanford associated john mccarthy bertram raphael cordell green edinburgh john alan robinson academic visitor syracuse university pat hayes robert kowalski advocates


procedural representations mainly centered mit leadership marvin minsky seymour papert although based proof methods logic planner developed mit first language emerge within proceduralist paradigm planner featured pattern directed invocation procedural plans goals ie goal reduction backward chaining assertions ie forward chaining influential implementation planner subset planner called micro planner implemented gerry sussman eugene charniak terry winograd used implement winograd 's natural language understanding program shrdlu landmark time ref


kowalski robert kowalski http wwwdocicacuk rak papers ifip pdf predicate logic programming language memo department artificial intelligence edinburgh university also proceedings ifip congress stockholm north holland publishing co pp nbsp ref variant sl resolution showed treats implications goal reduction procedures kowalski collaborated colmerauer marseille developed ideas design implementation programming language prolog association logic programming founded promote logic programming prolog gave rise programming languages alf fril g del mercury oz ciao visual prolog xsb prolog well variety


concurrent logic programming languages constraint logic programming languages datalog concepts semantics maarten van emden robert kowalski defined three semantics horn clause logic programs model theoretic fixed point proof theoretic showed equivalent logic control logic programming viewed controlled deduction important concept logic programming separation programs logic component control component pure logic programming languages logic component alone determines solutions produced control component varied provide alternative ways executing logic program notion


captured slogan algorithm logic control logic represents logic program control represents different theorem proving strategies problem solving simplified propositional case logic program top level atomic goal contain variables backward reasoning determines tree constitutes search space solving goal top level goal root tree given node tree clause whose head matches node exists set child nodes corresponding sub goals body clause child nodes


grouped together alternative sets children corresponding alternative ways solving node grouped together search strategy used search space prolog uses sequential last first backtracking strategy one alternative one sub goal considered time search strategies parallel search intelligent backtracking best first search find optimal solution also possible general case sub goals share variables strategies used choosing subgoal highly instantiated


sufficiently instantiated one procedure applies strategies used example concurrent logic programming negation failure practical applications well applications require non monotonic reasoning artificial intelligence horn clause logic programs need extended normal logic programs negative conditions ''clause'' normal logic program form kbd h sub sub sub n sub b sub sub b sub n sub kbd read declaratively logical implication kbd h sub sub sub n sub b sub sub b sub n sub kbd kbd h kbd


kbd sub sub kbd kbd b sub sub kbd atomic formulas negation negative literals kbd b sub sub kbd commonly referred negation failure implementations negative condition kbd b sub sub kbd shown hold showing positive condition kbd b sub sub kbd fails hold example syntaxhighlight lang prolog canfly x bird x abnormal x abnormal x wounded x bird john bird mary wounded john syntaxhighlight given goal finding something fly syntaxhighlight lang prolog canfly x syntaxhighlight two candidate solutions solve first subgoal kbd bird x kbd namely kbd x john kbd kbd x mary kbd


second subgoal kbd abnormal john kbd first candidate solution fails kbd wounded john kbd succeeds therefore kbd abnormal john kbd succeeds however second subgoal kbd abnormal mary kbd second candidate solution succeeds kbd wounded mary kbd fails therefore kbd abnormal mary kbd fails therefore kbd x mary kbd solution goal micro planner construct called thnot applied expression returns value true evaluation expression fails equivalent operator typically exists modern prolog 's implementations typically written code ''goal'' code code ''goal'' code code ''goal'' code goal proposition proved


program operator differs negation first order logic negation code x code fails variable code x code bound atom code code succeeds cases including code x code unbound makes prolog 's reasoning non monotonic code x x code always fails code x x code succeed binding code x code code code depending whether code x code initially bound note standard prolog executes goals left right order logical status negation failure unresolved keith clark


showed certain natural conditions correct sometimes complete implementation classical negation respect completion program completion amounts roughly regarding set program clauses predicate left hand side say kbd h body sub sub kbd kbd kbd kbd h body sub k sub kbd definition predicate kbd h iff body sub sub body sub k sub kbd iff means writing completion also requires explicit use equality predicate inclusion set


appropriate axioms equality however implementation negation failure needs halves definitions without axioms equality example completion program kbd canfly x iff bird x abnormal x kbd kbd abnormal x iff wounded x kbd kbd bird x iff x john x mary kbd kbd x x kbd kbd john mary kbd kbd mary john kbd notion completion closely related mccarthy 's circumscription semantics default reasoning closed world assumption alternative completion semantics negation failure also interpreted epistemically


stable model semantics answer set programming interpretation b sub sub means literally b sub sub known believed epistemic interpretation advantage combined simply classical negation extended logic programming formalise phrases contrary shown contrary classical negation shown epistemic interpretation negation failure knowledge representation fact horn clauses given procedural interpretation vice versa goal reduction procedures understood horn clauses backward


reasoning means logic programs combine declarative procedural representations knowledge inclusion negation failure means logic programming kind non monotonic logic despite simplicity compared classical logic combination horn clauses negation failure proved surprisingly expressive example provides natural representation common sense laws cause effect formalised situation calculus event calculus also shown correspond quite naturally semi formal language legislation particular prakken sartor credit representation


british nationality act logic program hugely influential development computational representations legislation showing logic programming enables intuitively appealing representations directly deployed generate automatic inferences variants extensions prolog programming language prolog developed alain colmerauer emerged collaboration colmerauer marseille robert kowalski edinburgh colmerauer working natural language understanding using logic represent semantics using resolution question answering summer colmerauer kowalski discovered clausal form logic could used


represent formal grammars resolution theorem provers could used parsing observed theorem provers like hyper resolution behave bottom parsers others like sl resolution behave top parsers following summer kowalski working colmerauer developed procedural interpretation implications dual declarative procedural interpretation later became formalised prolog notation kbd h b sub sub b sub n sub kbd read used declaratively procedurally also became clear clauses could restricted definite clauses horn clauses kbd h kbd kbd b sub sub kbd


kbd b sub n sub kbd atomic predicate logic formulae sl resolution could restricted generalised lush sld resolution kowalski 's procedural interpretation lush described memo published ref name kowalski colmerauer philippe roussel used dual interpretation clauses basis prolog implemented summer autumn first prolog program also written implemented marseille french question answering system use prolog practical programming language given great momentum development compiler david warren


edinburgh experiments demonstrated edinburgh prolog could compete processing speed symbolic programming languages lisp edinburgh prolog became ' 'de facto'' standard strongly influenced definition iso standard prolog abductive logic programming abductive logic programming extension normal logic programming allows predicates declared abducible predicates open undefined clause abductive logic program form kbd h b sub sub b sub n sub sub sub sub n sub kbd kbd h kbd atomic formula abducible kbd b sub sub kbd literals whose predicates


abducible kbd sub sub kbd atomic formulas whose predicates abducible abducible predicates constrained integrity constraints form kbd false l sub sub l sub n sub kbd kbd l sub sub kbd arbitrary literals defined abducible atomic negated example syntaxhighlight lang prolog canfly x bird x normal x false normal x wounded x bird john bird mary wounded john syntaxhighlight predicate kbd normal kbd abducible problem solving achieved deriving hypotheses expressed terms abducible predicates solutions problems solved problems either observations need explained classical


abductive reasoning goals solved normal logic programming example hypothesis kbd normal mary kbd explains observation kbd canfly mary kbd moreover hypothesis entails solution kbd x mary kbd goal finding something fly syntaxhighlight lang prolog canfly x syntaxhighlight abductive logic programming used fault diagnosis planning natural language processing machine learning also used interpret negation failure form abductive reasoning metalogic programming mathematical logic long tradition distinguishing object language metalanguage logic programming also allows metalevel programming


simplest metalogic program called vanilla meta interpreter syntaxhighlight lang prolog solve true solve ab solve solve b solve clause ab solve b syntaxhighlight true represents empty conjunction clause ab means object level clause form b metalogic programming allows object level metalevel representations combined natural language also used implement logic specified inference rules metalogic used logic programming implement metaprograms manipulate programs databases knowledge bases axiomatic theories data constraint logic programming constraint logic programming combines horn clause logic programming constraint


predefined domain specific model theoretic structure theory procedurally subgoals whose predicates defined program solved goal reduction ordinary logic programming constraints checked satisfiability domain specific constraint solver implements semantics constraint predicates initial problem solved reducing satisfiable conjunction constraints following constraint logic program represents toy temporal database kbd john 's kbd history teacher syntaxhighlight lang prolog teaches john hardware teaches john software teaches john logic


rank john instructor rank john professor syntaxhighlight kbd kbd kbd kbd constraint predicates usual intended semantics following goal clause queries database find kbd john kbd taught kbd logic kbd kbd professor kbd kbd teaches john logic rank john professor kbd solution kbd kbd constraint logic programming used solve problems fields civil engineering mechanical engineering digital circuit verification automated timetabling air traffic control


finance closely related abductive logic programming concurrent logic programming concurrent logic programming integrates concepts logic programming concurrent programming development given big impetus choice systems programming language japanese fifth generation project fgcs concurrent logic program set guarded horn clauses form kbd h g sub sub g sub n sub b sub sub b sub n sub kbd conjunction kbd g sub sub g sub n sub kbd called guard clause kbd kbd commitment operator declaratively guarded horn clauses read


ordinary logical implications kbd h g sub sub g sub n sub b sub sub b sub n sub kbd however procedurally several clauses whose heads kbd h kbd match given goal clauses executed parallel checking whether guards kbd g sub sub g sub n sub kbd hold guards one clause hold committed choice made one clauses execution proceeds subgoals kbd b sub sub b sub n sub kbd chosen clause subgoals also executed parallel thus concurrent logic programming implements form


n't care nondeterminism rather n't know nondeterminism example following concurrent logic program defines predicate kbd shuffle left right merge kbd used shuffle two lists kbd left kbd kbd right kbd combining single list kbd merge kbd preserves ordering two lists kbd left kbd kbd right kbd syntaxhighlight lang prolog shuffle shuffle left right merge left first rest merge first shortmerge shuffle rest right shortmerge shuffle left right merge right first rest merge first shortmerge shuffle left rest shortmerge syntaxhighlight kbd kbd represents empty list


kbd head tail kbd represents list first element kbd head kbd followed list kbd tail kbd prolog notice first occurrence kbd kbd second third clauses list constructor whereas second occurrence kbd kbd commitment operator program used example shuffle lists kbd ace queen king kbd kbd kbd invoking goal clause syntaxhighlight lang prolog shuffle ace queen king merge syntaxhighlight program non deterministically generate single solution example kbd merge ace queen king


kbd arguably concurrent logic programming based message passing subject indeterminacy concurrent message passing systems actors see indeterminacy concurrent computation carl hewitt argued concurrent logic programming based logic sense computational steps cannot logically deduced ref name hewitt ref however concurrent logic programming result terminating computation logical consequence program partial result partial computation logical consequence program residual goal process network thus indeterminacy computations


linear logic programming one use ambient linear logic support state change early designs logic programming languages based linear logic include lo andreoli pareschi lolli acl forum miller forum provides goal directed interpretation linear logic object oriented logic programming f logic extends logic programming objects frame syntax logtalk extends prolog programming language support objects protocols oop concepts supports standard compliant prolog systems backend compilers transaction logic programming transaction logic extension logic programming logical theory


ehud shapiro editor ''concurrent prolog'' mit press james slagle https dlacmorg doi abs experiments deductive question answering program cacm december gabbay dov hogger christopher john robinson ja eds https wwwworldcatorg title handbook logic artificial intelligence logic programming oclc handbook logic artificial intelligence logic programming vols oxford university press reading carl hewitt https wwwijcaiorg proceedings papers apdf procedural embedding knowledge planner ijcai carl hewitt https webarchiveorg web https pdfssemanticscholarorg ec faaab da b e ad eb bpdf repeated demise logic programming reincarnated ''aaai spring symposium went wrong lessons ai research applications'' evgeny dantsin thomas georg gottlob andrei voronkov http cmpeemuedutr bayram courses forpresentation p dantsinpdf complexity expressive


power logic programming acm comput surv ulf nilsson jan maluszynski http wwwidaliuse ulfni lpp logic programming prolog external links https webarchiveorg web http vlfmnetinfo logic prog ''logic programming'' virtual library entry http liinwwwiraukade bibliography logicprogramming bibliographies logic programming http wwwlogicprogrammingorg association logic programming alp '' http wwwcskuleuvenbe dtai projects alp tplp theory practice logic programming '' journal https webarchiveorg web http wwwmpprogrammingcom cpp logic programming c castor http wwwmozart ozorg documentation tutorial node html logic programming oz http wwwpdcdk prolog development center http docsracket langorg racklog racklog logic programming racket computer related introductions programming paradigms logic


mathematics '''lie algebra''' pronounced lee vector space math mathfrak g math together operation called '''lie bracket''' alternating bilinear map math mathfrak g times mathfrak g rightarrow mathfrak g x mapsto x math satisfies jacobi identity represent bilinear operation often commutator ''x'' ''y'' ''y'' ''x'' associative product vector space necessarily vector space math mathfrak g math together operation non associative algebra meaning lie bracket necessarily associative lie algebras closely

Formal science
Mathematics
Pure mathematics
Algebra
Lie algebra


related lie groups groups also smooth manifolds lie group gives rise lie algebra tangent space identity conversely finite dimensional lie algebra real complex numbers corresponding connected lie group unique finite coverings lie 's third theorem correspondence allows one study structure classification lie groups terms lie algebras physics lie groups appear symmetry groups physical systems lie algebras tangent vectors near identity may thought infinitesimal symmetry


math v math velocity equal magnitude math v math lie bracket measure non commutativity two rotations since rotation commutes alternating property math xx x times x math history lie algebras introduced study concept infinitesimal transformations marius sophus lie independently discovered wilhelm killing name ''lie algebra'' given hermann weyl older texts term ''infinitesimal group'' used definitions definition lie algebra lie algebra


vector space math mathfrak g math field math f math together binary operation math cdot cdot mathfrak g times mathfrak g mathfrak g math called lie bracket satisfying following axioms allows generally module commutative ring article called lie ring bilinearity math x b z x z b z math math z x b z x b z math scalars math math math b math math f math elements math x math '' math math '' '' math z math '' math mathfrak g math alternativity math xx math math x math math mathfrak g math jacobi identity


h b n math lie algebra associated lie group algebra denoted fraktur version group example lie algebra su ''n'' math mathfrak su n math generators dimension elements lie algebra math mathfrak g math said generate smallest subalgebra containing elements math mathfrak g math ' 'dimension'' lie algebra dimension vector space '' math f math '' cardinality minimal generating set lie algebra always less equal dimension see classification low dimensional real lie


algebras small examples subalgebras ideals homomorphisms lie bracket required associative meaning math xy z math need equal math x yz math however flexible nonetheless much terminology associative rings algebras commonly applied lie algebras ''lie subalgebra'' subspace math mathfrak h subseteq mathfrak g math closed lie bracket ''ideal'' math mathfrak subseteq mathfrak g math subalgebra satisfying stronger condition math mathfrak g mathfrak subseteq mathfrak math lie algebra ''homomorphism'' linear map compatible respective lie brackets math phi mathfrak g mathfrak g' quad phi xy phi x phi


mid x text math centralizer math mathfrak g math ''center'' math mathfrak z mathfrak g math similarly subspace ''s'' normalizer subalgebra '' math math '' math mathfrak n mathfrak g x mathfrak g mid xs text math equivalently math math lie subalgebra math mathfrak n mathfrak g math largest subalgebra math math ideal math mathfrak n mathfrak g math examples math mathfrak subset mathfrak gl math commutator two elements blockquote math begin align left begin bmatrix b c end bmatrix begin bmatrix x end bmatrix


right begin bmatrix ax cx dy end bmatrix begin bmatrix ax bx cy dy end bmatrix begin bmatrix b x c x end bmatrix end align math blockquote shows math mathfrak math subalgebra ideal fact every one dimensional linear subspace lie algebra induced abelian lie algebra structure generally ideal simple lie algebra abelian lie algebras never ideals direct sum semidirect product two lie algebras math mathfrak g math math mathfrak g' math direct sum lie algebra vector space


' 'derivation'' lie algebra math mathfrak g math non associative algebra linear map math delta colon mathfrak g rightarrow mathfrak g math obeys leibniz law math delta xy delta x x delta math math xy mathfrak g math ''inner derivation'' associated math x mathfrak g math adjoint mapping math mathrm ad x math defined math mathrm ad x xy math derivation consequence jacobi identity '''outer derivations''' derivations come adjoint representation lie algebra math mathfrak g math semisimple every derivation inner derivations form vector space math mathrm der mathfrak g math lie subalgebra


math mathfrak gl mathfrak g math bracket commutator inner derivations form lie subalgebra math mathrm der mathfrak g math examples example given lie algebra ideal math mathfrak subset mathfrak g math adjoint representation math mathfrak ad mathfrak g math math mathfrak g math acts outer derivations math mathfrak math since math xi subset mathfrak math math x mathfrak g math math mathfrak math lie algebra math mathfrak b n math upper triangular matrices math mathfrak gl n math ideal math mathfrak n n math strictly upper triangular matrices non zero elements diagonal matrix instance commutator elements math mathfrak b math math mathfrak n math gives blockquote math begin align left


begin bmatrix b c e f end bmatrix begin bmatrix x z end bmatrix right begin bmatrix ax ay bz dz end bmatrix begin bmatrix dx ex yf fz end bmatrix begin bmatrix x f ex bz f z


end bmatrix end align math blockquote shows exist outer derivations math mathfrak b math math text der mathfrak n math split lie algebra let ''v'' finite dimensional vector space field ''f'' math mathfrak gl v math lie algebra linear transformations math mathfrak g subseteq mathfrak gl v math lie subalgebra math mathfrak g math said '' 'split''' roots characteristic polynomials linear transformations math mathfrak g math base field ''f'' generally finite dimensional lie algebra math mathfrak g math said split cartan subalgebra whose image adjoint representation math operatorname ad mathfrak g mathfrak gl mathfrak g math split lie algebra


theory defining lie algebra terms linear maps morphisms category vector spaces without considering individual elements section field algebra defined supposed characteristic different two category theoretic definition lie algebras two braiding isomorphisms needed vector space ''interchange isomorphism'' math tau otimes otimes math defined math tau x otimes otimes x math ''cyclic permutation braiding'' math sigma otimes otimes otimes otimes math defined math sigma mathrm id otimes tau circ tau otimes mathrm id math math mathrm id math identity


morphism equivalently math sigma math defined math sigma x otimes otimes z otimes z otimes x math notation lie algebra defined object math math category vector spaces together morphism math cdot cdot otimes rightarrow math satisfies two morphism equalities math cdot cdot circ mathrm id tau math math cdot cdot circ cdot cdot otimes mathrm id circ mathrm id sigma sigma math examples vector spaces vector space math v math endowed identically zero lie bracket becomes lie algebra lie algebras called abelian cf one dimensional lie algebra field abelian alternating property lie bracket


associative algebra commutator bracket associative algebra math math field math f math multiplication math x mapsto xy math lie bracket may defined commutator math xy xy yx math bracket math math lie algebra associative algebra ''a'' called ''enveloping algebra'' lie algebra math cdot cdot math every lie algebra embedded one arises associative algebra fashion see universal enveloping algebra associative algebra endomorphisms ''f'' vector space math v math lie bracket


denoted math mathfrak gl v math finite dimensional vector space math v f n math previous example exactly lie algebra ''n'' ''n'' matrices denoted math mathfrak gl n f math math mathfrak gl n f math bracket math xy xy yx math adjacency indicates matrix multiplication lie algebra general linear group consisting invertible matrices special matrices two important subalgebras math mathfrak gl n f math matrices trace zero form special linear lie algebra math mathfrak sl n f math lie algebra special linear group math mathrm sl n f math skew hermitian matrices form unitary lie algebra math mathfrak u n math lie algebra unitary group nbsp ''u'' ''n''


matrix lie algebras complex matrix group lie group consisting matrices math g subset n mathbb c math multiplication ''g'' matrix multiplication corresponding lie algebra math mathfrak g math space matrices tangent vectors ''g'' inside linear space math n mathbb c math consists derivatives smooth curves ''g'' identity blockquote math mathfrak g x c' n mathbb c mid text smooth c mathbb r g c math blockquote lie bracket math mathfrak g math given commutator matrices math xy xy yx math given lie algebra one


recover lie group image matrix exponential mapping math exp n mathbb c n mathbb c math defined math exp x x tfrac x cdots math converges every matrix math x math math g exp mathfrak g math following examples lie algebras matrix lie groups special linear group math rm sl n mathbb c math consisting matrices determinant lie algebra math mathfrak sl n mathbb c math consists matrices complex entries trace similarly one define corresponding real lie group math rm sl n mathbb r math lie algebra math mathfrak sl n mathbb r math unitary group math u n math consists ''n'' nbsp nbsp ''n'' unitary matrices satisfying math u u math


lie algebra math mathfrak u n math consists skew self adjoint matrices math x x math special orthogonal group math mathrm n math consisting real determinant one orthogonal matrices math mathrm math lie algebra math mathfrak n math consists real skew symmetric matrices math x rm x math full orthogonal group math mathrm n math without determinant one condition consists math mathrm n math separate connected component ' 'same'' lie algebra math mathrm n math similarly one define complex version group algebra simply allowing complex matrix entries two dimensions field math f math isomorphism single two dimensional nonabelian lie algebra generators ''x y''


bracket defined math left x right math generates affine group one dimension realized matrices math x left begin array cc end array right qquad left begin array cc end array right math since math left begin array cc c end array right n left begin array cc c end array right math natural number math n math math c math one sees resulting lie group elements upper triangular matrices unit lower diagonal


math exp cdot x b cdot left begin array cc e tfrac b e end array right tfrac e left cdot x b cdot right math three dimensions heisenberg algebra math rm h mathbb r math three dimensional lie algebra generated elements lie brackets math xy z quad xz quad yz math usually realized space times strictly upper triangular matrices commutator lie bracket basis math x left begin array ccc end array right quad left begin array ccc end array right quad z left begin array ccc end array right quad math element


heisenberg group representation product group generators ie matrix exponentials lie algebra generators math left begin array ccc c b end array right e e cz e ax math lie algebra math mathfrak math group spanned three matrices math f left begin array ccc end array right quad f left begin array ccc end array right quad f left begin array ccc end array right quad math commutation relations among generators math f f f math math f f f math math f f f math three dimensional euclidean space math mathbb r math lie bracket given


cross product vectors commutation relations thus isomorphic math mathfrak math lie algebra unitarily equivalent usual spin physics angular momentum component operators spin particles quantum mechanics infinite dimensions important class infinite dimensional real lie algebras arises differential topology space smooth vector fields differentiable manifold ''m'' forms lie algebra lie bracket defined commutator vector fields one way expressing lie bracket formalism lie derivatives identifies


representation vector space math mathfrak g math called adjoint representation goals representation theory one important aspect study lie algebras especially semisimple lie algebras study representations indeed books listed references section devote substantial fraction pages representation theory although ado 's theorem important result primary goal representation theory find faithful representation given lie algebra math mathfrak g math indeed semisimple case adjoint representation already faithful rather goal understand ''all'' possible


representation math mathfrak g math natural notion equivalence semisimple case field characteristic zero weyl 's theorem says every finite dimensional representation direct sum irreducible representations nontrivial invariant subspaces irreducible representations turn classified theorem highest weight representation theory physics representation theory lie algebras plays important role various parts theoretical physics one considers operators space states satisfy certain natural commutation relations commutation relations typically come symmetry problem specifically


relations lie algebra relevant symmetry group example would angular momentum operators whose commutation relations lie algebra math mathfrak math rotation group typically space states far irreducible pertinent operators one attempt decompose irreducible pieces one needs know irreducible representations given lie algebra study quantum hydrogen atom example quantum mechanics textbooks give without calling classification irreducible representations


lie algebra math mathfrak math structure theory classification lie algebras classified extent particular application classification lie groups abelian nilpotent solvable analogously abelian nilpotent solvable groups defined terms derived subgroups one define abelian nilpotent solvable lie algebras lie algebra math mathfrak g math ''abelian'' lie bracket vanishes ie ''x''''y'' ''x'' ''y'' math mathfrak g math abelian lie algebras correspond commutative abelian connected lie groups vector spaces math mathbb k n math tori


math mathbb n math form math mathfrak k n math meaning ''n'' dimensional vector space trivial lie bracket general class lie algebras defined vanishing commutators given length lie algebra math mathfrak g math ''nilpotent'' lower central series math mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g cdots math becomes zero eventually engel 's theorem lie algebra nilpotent every ''u'' math mathfrak g math adjoint endomorphism math operatorname ad u mathfrak g mathfrak g quad operatorname ad u v uv math nilpotent generally still lie algebra math mathfrak g math said ' 'solvable''


derived series math mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g mathfrak g cdots math becomes zero eventually every finite dimensional lie algebra unique maximal solvable ideal called radical lie correspondence nilpotent respectively solvable connected lie groups correspond nilpotent respectively solvable lie algebras simple semisimple lie algebra simple non trivial ideals abelian implies one dimensional necessarily abelian lie algebra definition simple even though nontrivial ideals lie algebra math mathfrak g math called ' 'semisimple'' isomorphic direct


sum simple algebras several equivalent characterizations semisimple algebras nonzero solvable ideals concept semisimplicity lie algebras closely related complete reducibility semisimplicity representations ground field ''f'' characteristic zero finite dimensional representation semisimple lie algebra semisimple ie direct sum irreducible representations general lie algebra called reductive adjoint representation semisimple thus semisimple lie algebra reductive cartan 's criterion cartan 's criterion gives conditions lie algebra nilpotent solvable semisimple


zero furthermore semisimple lie algebras algebraically closed field completely classified root systems relation lie groups tangent space sphere point math x math math x math identity element tangent space also lie algebra although lie algebras often studied right historically arose means study lie groups briefly outline relationship lie groups lie algebras lie group gives rise canonically determined lie algebra concretely ''the tangent space identity'' conversely


finite dimensional lie algebra math mathfrak g math exists corresponding connected lie group math g math lie algebra math mathfrak g math lie 's third theorem see baker campbell hausdorff formula lie group determined uniquely however two lie groups lie algebra ''locally isomorphic'' particular universal cover instance special orthogonal group special unitary group su give rise lie algebra isomorphic math mathbb r math cross product su simply connected twofold cover consider ' 'simply connected'' lie


groups however one one correspondence finite dimensional real lie algebra math mathfrak g math unique simply connected lie group math g math lie algebra math mathfrak g math correspondence lie algebras lie groups used several ways including classification lie groups related matter representation theory lie groups every representation lie algebra lifts uniquely representation corresponding connected simply connected lie group conversely every representation lie group induces representation group 's lie algebra representations


one one correspondence therefore knowing representations lie algebra settles question representations group classification shown connected lie group given lie algebra isomorphic universal cover mod discrete central subgroup classifying lie groups becomes simply matter counting discrete subgroups center classification lie algebras known solved cartan et al semisimple case lie algebra infinite dimensional issue subtle many instances exponential map


math mathfrak su math ref name fulton given semisimple finite dimensional complex lie algebra math mathfrak g math ' 'split form'' real form splits ie cartan subalgebra acts via adjoint representation real eigenvalues split form exists unique isomorphisms ref name fulton ''compact form'' real form lie algebra compact lie group compact form exists also unique ref name fulton lie algebra additional structures lie algebra equipped additional structures


assumed compatible bracket example graded lie algebra lie algebra graded vector space structure also comes differential underlying graded vector space chain complex called differential graded lie algebra simplicial lie algebra simplicial object category lie algebras words obtained replacing underlying set simplicial set might better thought family lie algebras lie ring ''lie ring'' arises


generalisation lie algebras study lower central series groups lie ring defined nonassociative ring multiplication anticommutative satisfies jacobi identity specifically define lie ring math l math abelian group operation math cdot cdot math following properties bilinearity math x z x z z quad z x z x z math ''x'' ''y'' ''z'' isin ''l'' ''jacobi identity'' math x yz zx z xy


quad math ''x'' ''y'' ''z'' ''l'' ''x'' ''l'' math xx quad math lie rings need lie groups addition lie algebra example lie ring associative ring made lie ring defining bracket operator math xy xy yx math conversely lie algebra corresponding ring called universal enveloping algebra lie rings used study finite p groups ''lazard correspondence'' lower central factors ''p'' group


finite abelian ''p'' groups modules '''z''' ''p'''''z''' direct sum lower central factors given structure lie ring defining bracket commutator two coset representatives lie ring structure enriched another module homomorphism ''p''th power map making associated lie ring called restricted lie ring lie rings also useful definition p adic analytic groups endomorphisms studying lie algebras rings integers p adic integers definition finite groups lie type due


chevalley involves restricting lie algebra complex numbers lie algebra integers reducing modulo ''p'' get lie algebra finite field examples lie algebra general ring instead field example lie ring lie rings ''not'' lie groups addition despite name associative ring made lie ring defining bracket operator math xy xy yx math example lie ring arising study groups


appropriate lie theoretic properties see also adjoint representation lie algebra anyonic lie algebra chiral lie algebra free lie algebra index lie algebra lie algebra cohomology lie algebra extension lie algebra representation lie bialgebra lie coalgebra lie operad particle physics representation theory lie superalgebra poisson algebra pre lie algebra quantum groups moyal algebra quasi frobenius lie algebra quasi lie algebra restricted lie algebra serre relations symmetric lie algebra gelfand fuks cohomology remarks references sources erdmann karin wildon mark ''introduction lie algebras'' st edition springer external links lie groups


library book shelf hong kong arranged using dewey classification '''library classification''' system knowledge organization library resources arranged ordered systematically library classifications notational system represents order topics classification allows items stored order library classification systems group related materials together typically arranged hierarchical tree structure different kind classification system called faceted classification system also widely used allows assignment multiple classifications object enabling classifications ordered

Applied science
Library and museum studies
Classification
Library classification


many ways description library classification aspect library information science distinct scientific classification goal provide useful ordering documents rather theoretical organization knowledge although practical purpose creating physical ordering documents generally attempt adhere accepted scientific knowledge library classification helps accommodate newly published literature already created order arrangement filiatory sequence library classification


defined arrangement books shelves description manner useful read ultimate aim grouping similar things together library classification meant achieve four purposes like ordering fields knowledge systematic way bring related items together helpful sequence provide orderly access shelve provide location item shelf library classification distinct application subject headings classification organizes knowledge systematic order subject headings provide


access intellectual materials vocabulary terms may may organized knowledge system ref name chan ref characteristics bibliographic classification demands sake reaching purposes useful sequence subjects levels concise memorable notation host techniques devices number synthesis history library classifications preceded classifications used bibliographers conrad gessner earliest library classification schemes organized books broad subject categories earliest known library classification scheme pinakes callimachus scholar library alexandria


third century bc renaissance reformation era libraries organized according whims knowledge individuals charge changed format various materials classified collections classified language others printed printing revolution sixteenth century increase available printed materials made broad classification unworkable granular classifications library materials developed nineteenth century ref name shera ref gabriel naud published book called ''advice establishing library''


time working private library president henri de mesmes ii mesmes around printed books many greek latin french written manuscripts although private library scholars references could access purpose ''advice establishing library'' identify rules private book collectors organize collections orderly way increase collection 's usefulness beauty naud developed classification system based seven different classes theology medicine jurisprudence history philosophy mathematics humanities seven classes would later increased


covers subjects eg dewey decimal classification ddc universal decimal classification udc library congress classification lcc colon classification cc specific classification schemes covers particular subjects types materials eg iconclass art british catalogue music classification dickinson classification music nlm classification medicine national schemes specially created certain countries eg swedish library classification system sab sveriges allm nna biblioteksf rening terms functionality classification systems often described enumerative subject headings listed alphabetically numbers assigned heading alphabetical order hierarchical subjects divided hierarchically


cataloging classifying library materials called ''cataloger'' ''catalog librarian'' library classification systems one two tools used facilitate subject access consists alphabetical indexing languages thesauri subject headings systems library classification piece work consists two steps firstly subject topic material ascertained next call number essentially book 's address based classification system use particular library assigned work using notation system important note


unlike subject heading thesauri multiple terms assigned work library classification systems work placed one class due shelving purposes book one physical place however classified catalogs one may main entries well added entries classification systems like dewey decimal classification ddc library congress classification also add cutter number work adds code main entry primary access point work eg author classification systems libraries generally play


two roles firstly facilitate subject access allowing user find works documents library certain subject secondly provide known location information source located eg shelved th century libraries closed stacks library classification served organize subject catalog th century libraries opened stacks public started shelve library material according library classification simplify subject browsing classification systems suitable aiding subject access


rather shelf location example universal decimal classification uses complicated notation pluses colons difficult use purpose shelf arrangement expressive compared ddc terms showing relationships subjects similarly faceted classification schemes difficult use shelf arrangement unless user knowledge citation order depending size library collection libraries might use classification systems solely one purpose extreme cases public library small collection might use


classification system location resources might use complicated subject classification system instead resources might put couple wide classes travel crime magazines etc known mark park classification method formally called reader interest classification comparing library classification systems result differences notation history use enumeration hierarchy facets classification systems differ following ways type notation notation pure consisting numerals example mixed consisting letters numerals letters numerals


symbols expressiveness degree notation express relationship concepts structure whether support mnemonics example number ddc notation often means concerns aspect france example dewey classification concerns birds france signifies geographic division represents france hospitality degree system able accommodate new subjects brevity length notation express concept speed updates degree support better classification systems frequently reviewed consistency simplicity usability


lines planes rotations also functional analysis branch mathematical analysis may viewed application linear algebra spaces functions linear algebra also used sciences fields engineering allows modeling many natural phenomena computing efficiently models nonlinear systems cannot modeled linear algebra often used dealing first order approximations using fact differential multivariate function point linear map best approximates function near point history procedure solving

Formal science
Mathematics
Pure mathematics
Algebra
Linear algebra


simultaneous linear equations called gaussian elimination appears ancient chinese mathematical text chapter eight ''rectangular arrays'' ''the nine chapters mathematical art'' use illustrated eighteen problems two five equations systems linear equations arose europe introduction ren descartes coordinates geometry fact new geometry called cartesian geometry lines planes represented linear equations computing intersections amounts solving systems linear equations first systematic methods solving linear systems used determinants first considered leibniz


line segments math overline w z math math overline w z math length direction segments equipollent four dimensional system math mathbb h math quaternions started term ' 'vector'' introduced ''v'' ''x'' ''y'' j ''z'' k representing point space quaternion difference ''p'' ''q'' also produces segment equipollent math overline p q math hypercomplex number systems also used idea linear space basis arthur cayley introduced matrix multiplication inverse matrix making possible general linear group mechanism group


representation became available describing complex hypercomplex numbers crucially cayley used single letter denote matrix thus treating matrix aggregate object also realized connection matrices determinants wrote would many things say theory matrices seems precede theory determinants ref name vitulli marie benjamin peirce published ''linear associative algebra'' son charles sanders peirce extended work later telegraph required explanatory system publication treatise electricity magnetism instituted


field theory forces required differential geometry expression linear algebra flat differential geometry serves tangent spaces manifolds electromagnetic symmetries spacetime expressed lorentz transformations much history linear algebra history lorentz transformations first modern precise definition vector space introduced peano ref name vitulli marie theory linear transformations finite dimensional vector spaces emerged linear algebra took modern form first half twentieth century many ideas methods previous


equipped two binary operations satisfying following axioms elements called ' 'vectors'' elements ''f'' called ' 'scalars'' first operation ' 'vector addition'' takes two vectors outputs third vector second operation ' 'scalar multiplication'' takes scalar vector outputs new axioms addition scalar multiplication must satisfy following list arbitrary elements arbitrary scalars field border style width '''axiom'''


'''signification''' associativity addition style background f f ff commutativity addition identity element addition exists element called ''zero vector'' simply ''zero'' style background f f ff inverse elements addition every exists element called ''additive inverse'' distributivity scalar multiplication respect vector addition style background f f ff distributivity scalar multiplication respect field addition compatibility scalar


multiplication field multiplication style background f f ff identity element scalar multiplication denotes multiplicative identity first four axioms mean abelian group addition element specific vector space may various nature example could sequence function polynomial matrix linear algebra concerned properties objects common vector spaces linear maps '''linear maps''' mappings vector spaces preserve vector space structure given two vector spaces


field linear map also called contexts linear transformation linear mapping map math v w math compatible addition scalar multiplication math mathbf u mathbf v mathbf u mathbf v quad mathbf v mathbf v math vectors scalar implies vectors scalars one math mathbf u b mathbf v mathbf u b mathbf v mathbf u bt mathbf v math


vector space linear map math v v math also known ''linear operator'' bijective linear map two vector spaces every vector second space associated exactly one first isomorphism isomorphism preserves linear structure two isomorphic vector spaces essentially linear algebra point view sense cannot distinguished using vector space properties essential question linear algebra testing whether linear map isomorphism


isomorphism finding range image set elements mapped zero vector called kernel map questions solved using gaussian elimination variant algorithm subspaces span basis study subsets vector spaces vector spaces induced operations fundamental similarly many mathematical structures subsets called linear subspaces precisely linear subspace vector space field subset


every coefficient math math set vectors spans vector space called spanning set generating set spanning set ''linearly dependent'' linearly independent element span elements span would remain one remove one may continue remove elements getting ''linearly independent spanning set'' linearly independent set spans vector space called basis importance bases lies


fact simultaneously minimal generating sets maximal independent sets precisely linearly independent set spanning set math subseteq math basis math subseteq b subseteq math two bases vector space cardinality called dimension dimension theorem vector spaces moreover two vector spaces field isomorphic dimension basis therefore every basis


finite number elements ''finite dimensional vector space'' subspace case finite dimensional equality dimensions implies ''u'' sub sub ''u'' sub sub subspaces ''v'' math dim u u dim u dim u dim u cap u math math u u math denotes span math u cup u math matrices matrices allow explicit manipulation finite dimensional vector spaces linear maps theory thus essential part linear algebra let finite dimensional vector space field


basis thus dimension definition basis map math begin align ldots mapsto mathbf v cdots mathbf v f v end align math bijection math f math set sequences elements onto isomorphism vector spaces math f math equipped standard structure vector space vector addition scalar multiplication done component component isomorphism allows representing vector inverse image isomorphism coordinates vector math ldots math


column matrix math begin bmatrix vdots end bmatrix math another finite dimensional vector space possibly basis math mathbf w ldots mathbf w n math linear map well defined values basis elements math f mathbf w ldots f mathbf w n math thus well represented list corresponding column matrices math f w j j v cdots mj v math represented matrix math begin bmatrix cdots n vdots ddots vdots cdots mn end bmatrix math rows columns matrix multiplication defined way product


two matrices matrix composition corresponding linear maps product matrix column matrix column matrix representing result applying represented linear map represented vector follows theory finite dimensional vector spaces theory matrices two different languages expressing exactly concepts two matrices encode linear transformation different bases called similar proved two matrices similar one transform one


elementary row column operations matrix representing linear map row operations correspond change bases column operations correspond change bases every matrix similar identity matrix possibly bordered zero rows zero columns terms vector spaces means linear map bases part basis mapped bijectively part basis remaining basis elements


linear systems example let numblk math begin alignat x z x z x z end alignat math linear system system one may associate matrix math left begin array rrr


end array right math right member vector math mathbf v begin bmatrix end bmatrix math let linear transformation associated matrix solution system vector math mathbf x begin bmatrix x z end bmatrix math math mathbf x mathbf v math element preimage let associated homogeneous system right hand sides equations put zero numblk math begin alignat x z


x z x z end alignat math solutions exactly elements kernel equivalently gaussian elimination consists performing elementary row operations augmented matrix math left begin array c c mathbf v end array right left begin array rrr r end array right math putting


reduced row echelon form row operations change set solutions system equations example reduced echelon form math left begin array c c mathbf v end array right left begin array rrr r end array right math showing system unique solution math begin align x z end align math follows matrix interpretation linear systems methods applied solving linear systems many operations matrices linear transformations include computation ranks kernels matrix inverses endomorphisms


square matrices linear endomorphism linear map maps vector space basis elements endomorphism represented square matrix size respect general linear maps linear endomorphisms square matrices specific properties make study important part linear algebra used many parts mathematics including geometric transformations coordinate changes quadratic forms many part mathematics determinant ' 'determinant'' square matrix defined math sum sigma n sigma


sigma cdots n sigma n math math n math group permutations elements math sigma math permutation math sigma math parity permutation matrix invertible determinant invertible ie nonzero scalars belong field cramer 's rule closed form expression terms determinants solution n unknowns cramer 's rule useful reasoning solution except rarely used computing solution since gaussian elimination faster algorithm ' 'determinant endomorphism''


determinant matrix representing endomorphism terms ordered basis definition makes sense since determinant independent choice basis eigenvalues eigenvectors linear endomorphism vector space field ''eigenvector'' nonzero vector scalar scalar ''eigenvalue'' dimension finite basis chosen may represented respectively square matrix


column matrix equation defining eigenvectors eigenvalues becomes math mz az math using identity matrix whose entries zero except main diagonal equal one may rewritten math ai z math supposed nonzero means singular matrix thus determinant math det ai math equals zero eigenvalues thus roots polynomial math det xi math dimension monic polynomial degree called characteristic polynomial matrix endomorphism


diagonalizable non diagonalizable matrices simplest math begin bmatrix end bmatrix math cannot diagonalizable since square zero matrix square nonzero diagonal matrix never zero endomorphism diagonalizable bases simple form although simple diagonal form frobenius normal form need extending field scalars makes characteristic polynomial immediately readable matrix jordan normal form requires extend field scalar containing eigenvalues differs diagonal form


entries main diagonal equal duality linear form linear map vector space math v math field math f math field scalars math f math viewed vector space equipped pointwise addition multiplication scalar linear forms form vector space called '' 'dual space''' math v math usually denoted math v math math v' math math mathbf v ldots mathbf v n math basis math v math implies finite dimensional one define linear map math v math


math v mathbf e math math v mathbf e j math linear maps form basis math v math called dual basis math mathbf v ldots mathbf v n math finite dimensional math v math may defined similarly linearly independent form basis math mathbf v math math v math map math f f mathbf v math linear form math v math defines canonical linear map math v math math v math dual math v math called '''bidual''' math v math canonical map isomorphism math v math finite dimensional allows identifying math v math


finite dimensional matrix terms ordered bases matrix math f math dual bases transpose math mathsf math obtained exchanging rows columns elements vector spaces duals represented column vectors duality may expressed bra ket notation math langle h mathsf mathbf v rangle langle h mathsf mathbf v rangle math highlighting symmetry two members equality sometimes written math langle h mathsf mid mid mathbf v rangle math inner product spaces besides basic concepts


leq math call quantity cosine angle two vectors two vectors orthogonal math langle mathbf u mathbf v rangle math orthonormal basis basis basis vectors length orthogonal given finite dimensional vector space orthonormal basis could found gram schmidt procedure orthonormal bases particularly easy deal since math langle mathbf v mathbf v rangle math inner product facilitates construction many useful concepts instance given transform ''t''


define hermitian conjugate ''t'' linear transform satisfying math langle mathbf u mathbf v rangle langle mathbf u mathbf v rangle math ''t'' satisfies ''tt'' ''tt'' call ''t'' normal turns normal matrices precisely matrices orthonormal system eigenvectors span ''v'' relationship geometry strong relationship linear algebra geometry started introduction ren descartes cartesian coordinates new time geometry called cartesian geometry points represented cartesian coordinates


sequences three real numbers case usual three dimensional space basic objects geometry lines planes represented linear equations thus computing intersections lines planes amounts solving systems linear equations one main motivations developing linear algebra geometric transformation translations rotations reflections rigid motions isometries projections transform lines lines follows defined specified studied terms linear maps also case homographies bius transformations considered transformations


projective space end th century geometric spaces defined axioms relating points lines planes synthetic geometry around date appeared one may also define geometric spaces constructions involving vector spaces see example projective space affine space shown two approaches essentially equivalent classical geometry involved vector spaces vector spaces reals constructions may extended vector spaces field allowing considering geometry arbitrary fields including finite fields presently textbooks introduce geometric spaces


linear algebra geometry often presented elementary level subfield linear algebra usage applications linear algebra used almost areas mathematics thus making relevant almost scientific domains use mathematics applications may divided several wide categories geometry ambient space modeling ambient space based geometry sciences concerned space use geometry widely case mechanics robotics describing rigid body dynamics geodesy describing earth shape perspectivity computer vision computer graphics


describing relationship scene plane representation many scientific domains applications synthetic geometry often used general descriptions qualitative approach study explicit situations one must compute coordinates requires heavy use linear algebra functional analysis functional analysis studies function spaces vector spaces additional structure hilbert spaces linear algebra thus fundamental part functional analysis applications include particular quantum mechanics wave functions study complex systems physical


phenomena modeled partial differential equations solve one usually decomposes space solutions searched small mutually interacting cells linear systems interaction involves linear functions nonlinear systems interaction often approximated linear functions cases large matrices generally involved weather forecasting typical example whole earth atmosphere divided cells say nbsp km width height scientific computation nearly scientific computations involve linear algebra consequently linear algebra algorithms highly optimized blas


axioms defining vector space one may thus replace field scalars ring gives structure called '''module''' module concepts linear independence span basis linear maps also called module homomorphisms defined modules exactly vector spaces essential difference field modules basis modules basis free modules spanned finite set finitely generated modules module homomorphisms finitely generated


free modules may represented matrices theory matrices ring similar matrices field except determinants exist ring commutative square matrix commutative ring invertible determinant multiplicative inverse ring vector spaces completely characterized dimension isomorphism general complete classification modules even one restricts oneself finitely generated modules however every module cokernel homomorphism free modules modules


integers identified abelian groups since multiplication integer may identified repeated addition theory abelian groups may extended modules principal ideal domain particular principal ideal domain every submodule free module free fundamental theorem finitely generated abelian groups may extended straightforwardly finitely generated modules principal ring many rings algorithms solving linear equations systems linear equations however algorithms generally computational complexity


much higher similar algorithms field details see linear equation ring multilinear algebra tensors multilinear algebra one considers multivariable linear transformations mappings linear number different variables line inquiry naturally leads idea dual space vector space ''v'' sup sup consisting linear maps ''f'' field scalars multilinear maps described via tensor products elements ''v'' sup sup addition vector addition scalar multiplication


bilinear vector product vector space called algebra instance associative algebras algebras associate vector product like algebra square matrices algebra polynomials topological vector spaces vector spaces finite dimensional often require additional structure tractable normed vector space vector space along function called norm measures size elements norm induces metric measures distance elements induces topology allows definition continuous maps metric also allows


definition limits completeness metric space complete known banach space complete metric space along additional structure inner product conjugate symmetric sesquilinear form known hilbert space sense particularly well behaved banach space functional analysis applies methods linear algebra alongside mathematical analysis study various function spaces central objects study functional analysis l sup ''p'' sup spaces banach spaces especially ''l'' sup sup space square integrable functions hilbert space


among functional analysis particular importance quantum mechanics theory partial differential equations digital signal processing electrical engineering also provides foundation theoretical framework underlies fourier transform related methods homological algebra see also fundamental matrix computer vision geometric algebra linear programming linear regression statistical estimation method list linear algebra topics numerical linear algebra transformation matrix notes references sources reading history fearnley sander desmond https wwwjstororg stable pdf pdfcasa token oivqgvkhcreaaaaa dkktimlbzdkc f tb gjnx rbbc ocra jm ksw js bs qhvccbjzbg kd zq gr mle w hqiqiblgv hbtkyij hnle hyoh fujb nsy hermann grassmann creation linear algebra american mathematical monthly ''' ''' pp nbsp introductory textbooks murty katta g '' http wwwworldscientificcom worldscibooks computational algorithmic


linear algebra n dimensional geometry '' world scientific publishing '' http wwwworldscientificcom doi suppl suppl file chap pdf chapter systems simultaneous linear equations '' manga guide linear algebra shin takahashi iroha inoue trend pro co ltd advanced textbooks study guides outlines external links online resources https ocwmitedu courses mathematics linear algebra spring mit linear algebra video lectures series recorded lectures professor gilbert strang spring https wwwmathtechnionacil iic international linear algebra society https mathworldwolframcom topics linearalgebrahtml linear algebra mathworld http wwweconomicssotonacuk staff aldrich matriceshtm matrix linear algebra terms http jeff tripodcom mathwordhtml earliest known uses words mathematics http jeff tripodcom matriceshtml earliest uses symbols matrices vectors http jeff tripodcom mathsymhtml


earliest uses various mathematical symbols https wwwyoutubecom playlistlist plzhqobowtqdpd mizzm xvfitgf ab essence linear algebra video presentation blue brown basics linear algebra emphasis relationship geometric matrix abstract points view online books sharipov ruslan '' https arxivorg abs mathho course linear algebra multidimensional geometry '' treil sergei '' https wwwmathbrownedu treil papers ladw ladwhtml linear algebra done wrong '' numerical analysis


lieutenant debriefing new york city police department nypd officers times square nypd largest police force united states uniformed force almost officers us customs border protection officers boarding ship indonesian national police officers foot patrol ''' https theplatecarriernet best body armor law enforcement law enforcement ''' activity members government act organized manner enforce law discovering deterring rehabilitating punishing people violate rules norms governing society term encompasses police courts corrections modern state legal codes use


enforcement agency typical agency fulfilling role police social investment enforcement organizations massive terms resources invested activity number people professionally engaged perform functions ref name ile k ren hess christine hess orthmann ''introduction law enforcement criminal justice'' p ref law enforcement agencies tend limited operating within specified jurisdiction cases jurisdiction may overlap organizations example united states state statewide law enforcement arms


federal bureau investigation able act certain types crimes occurring state various specialized segments society may internal law enforcement arrangements example military organizations may military police today law enforcement department professions dominantly served caucasian males america even growing number organizations emphasizing recruitment females minorities see also outline law enforcement structured list topics related law enforcement organized subject area law enforcement country vigilantism criminal law police abolition movement references external links https theplatecarriernet best body armor law enforcement best plate


'''lexicology''' branch linguistics analyzes lexicon specific language word smallest meaningful unit language stand made small components called morphemes even smaller elements known phonemes distinguishing sounds lexicology examines every feature word including formation spelling origin usage definition ref name ref lexicology also considers relationships exist words linguistics lexicon language composed lexemes abstract units meaning correspond set related forms

Humanities
Languages and literature
Linguistics
Lexicology


explores history development word etymologists analyze related languages using comparative method set techniques allow linguists recover ancestral phonological morphological syntactic etc components modern languages comparing cognate material ref name ref means many word roots different branches indo european language family traced back single words proto indo european language english language instance contains borrowed words loan words vocabulary native words examples include ''parkour'' french ''karaoke'' japanese ''coconut'' portuguese ''mango'' hindi etc


lot music terminology like ''piano'' ' 'solo'' ''opera'' borrowed italian words classified according linguistic element borrowed phonemes morphemes semantics ref name approach '''general lexicology''' broad study words regardless language 's specific properties concerned linguistic features common among languages phonemes morphemes '''special lexicology''' hand looks particular language contributes vocabulary grammars ref name altogether lexicological studies approached two ways '''diachronic


historical''' lexicology devoted evolution words word formation time investigates origins word ways structure meaning usage since changed ref name ref '''synchronic descriptive''' lexicology examines words language within certain time frame could period language 's early stages development current state given interval ref name ref complementary perspectives proposed swiss linguist ferdinand de saussure ref name lexicology comparative contrastive methodologies '''comparative lexicology''' searches similar features


shared among two languages '''contrastive lexicology''' identifies linguistic characteristics distinguish related unrelated languages ref name semantics subfield semantics pertains especially lexicological work called lexical semantics brief lexical semantics contemplates significance words meanings several lenses including synonymy antonymy hyponymy polysemy among others semantic analysis lexical material may involve contextualization word syntactic ambiguity semasiology onomasiology relevant linguistic disciplines associated lexical semantics ref name word two kinds meaning grammatical


lexical '''grammatical meaning''' refers word 's function language tense plurality deduced affixes '''lexical meaning''' limited single form word rather word denotes base word example verb ''to walk'' become ''walks'' ''walked'' ''walking '' word different grammatical meaning lexical meaning move one 's feet regular pace phraseology another focus lexicology phraseology studies multi word expressions idioms like 'raining cats dogs' meaning


phrase whole different meaning word often unpredictable considering components individually phraseology examines meanings exist analyzes laws govern word combinations idioms phraseological units classified according content meaning difficult translate word word one language another lexicography lexicologists maso alonso october january spanish poet literary critic philologist roland barthes november march french writer


linguistic terms teaching material http www ku eichstaettde slf engluvglsw ononmon pdf ''english general historical lexicology'' joachim grzega marion sch ner journals http lexisuniv lyon fr ''lexis e journal english lexicology'' denis jamet


''allegory music'' fran ois boucher '''music''' art arranging sounds time elements melody harmony rhythm timbre one universal cultural aspects human societies general definitions music include common elements pitch governs melody harmony rhythm associated concepts tempo meter articulation dynamics loudness softness sonic qualities timbre texture sometimes termed color musical sound different styles types music may emphasize de emphasize omit elements music


performed vast range instruments vocal techniques ranging singing rapping solely instrumental pieces solely vocal pieces songs without instrumental accompaniment pieces combine singing instruments word derives greek ''mousik '' art muses ref name perseustuftsedu ref general form activities describing music art form cultural activity include creation works music songs tunes symphonies criticism music study history music aesthetic examination music ancient greek indian philosophers


defined music two parts melodies tones ordered horizontally harmonies tones ordered vertically common sayings harmony spheres music ears point notion music often ordered pleasant listen however th century composer john cage thought sound music saying example noise sound creation performance significance even definition music vary according culture social context indeed throughout history new forms styles music criticized


music including beethoven 's ''grosse fuge'' string quartet early jazz beginning hardcore punk many types music including popular music traditional music art music music written religious ceremonies work songs chanteys music ranges strictly organized compositions classical music symphonies spontaneously played improvisational music jazz avant garde styles chance based contemporary music th st centuries music divided genres eg country music genres divided subgenres


opera may recorded listened radio mp player cd player smartphone film score tv show many cultures music important part people 's way life plays key role religious rituals rite passage ceremonies eg graduation marriage social activities eg dancing cultural activities ranging amateur karaoke singing playing amateur funk band singing community choir people may make music hobby like teen playing cello youth orchestra work


professional musician singer music industry includes individuals create new songs musical pieces songwriters composers individuals perform music include orchestra jazz band rock band musicians singers conductors individuals record music music producers sound engineers individuals organize concert tours individuals sell recordings sheet music scores customers even song piece performed music critics music journalists music scholars may assess evaluate piece performance etymology greek mythology nine muses inspiration many


creative endeavors including arts word 'music' derived ancient greek ' art muses' ref name perseustuftsedu greek mythology nine muses goddesses inspired literature science arts source knowledge embodied poetry song lyrics myths greek culture according ''online etymology dictionary'' term ''music'' derived mid c old french c directly latin 'the art music' also including poetry also source spanish italian old high german german


dutch danish derived greek ''techne'' ' art muses' fem 'pertaining muses' 'muse' see muse n modern spelling dates classical greece term 'music' refers art muses presided especially music lyric poetry ref name etymonline ref art entertainment khatia buniatishvili playing grand piano music composed performed many purposes ranging aesthetic pleasure religious ceremonial purposes entertainment product marketplace music available sheet music scores


classical romantic eras music lovers would buy sheet music favourite pieces songs could perform home piano advent phonograph records popular songs rather sheet music became dominant way music lovers would enjoy favourite songs advent home tape recorders digital music music lovers could make tapes playlists favourite songs take portable cassette player mp player music


variety settings often many links amateur professional musicians beginning amateur musicians take lessons professional musicians community settings advanced amateur musicians perform professional musicians variety ensembles community concert bands community orchestras distinction often made music performed live audience music performed studio recorded distributed music retail system broadcasting system however also many cases live performance front audience also recorded


distributed live concert recordings popular classical music popular music forms rock illegally taped live concerts prized music lovers jam band scene live improvised jam sessions preferred studio recordings composition people composing music using electronic keyboards computers french baroque music composer michel richard delalande pen hand composition act practice creating song instrumental music piece work singing instruments another type music many cultures including western classical music act


may use notation instead compose song mind play record memory jazz popular music notable recordings influential performers given weight written scores play classical music even music notated relatively precisely classical music many decisions performer make notation specify elements music precisely process deciding perform music previously composed notated termed interpretation different performers' interpretations


work music vary widely terms tempos chosen playing singing style phrasing melodies composers songwriters present music interpreting songs much perform music others standard body choices techniques present given time given place referred performance practice whereas interpretation generally used mean individual choices performer although musical composition often uses musical notation single author always


case work music multiple composers often occurs popular music band collaborates write song musical theatre one person writes melodies second person writes lyrics third person orchestrates songs styles music blues composer songwriter may create perform record new songs pieces without ever writing music notation piece music also composed words images computer programs explain notate singer musician create


musical sounds examples range avant garde music uses graphic notation text compositions ''aus den sieben tagen'' computer programs select sounds musical pieces music makes heavy use randomness chance called aleatoric music associated contemporary composers active th century john cage morton feldman witold lutos awski commonly known example chance based music sound wind chimes jingling breeze study composition traditionally dominated examination methods practice western classical music


definition composition broad enough include creation popular music traditional music songs instrumental pieces well spontaneously improvised works like free jazz performers african percussionists ewe drummers notation sheet music written representation music homorhythmic ie hymn style arrangement traditional adeste fideles standard two staff format mixed voicesleft music notation typically means written expression music notes rhythms paper using symbols music written pitches rhythm music notes


melody notated music notation also often provides instructions perform music example sheet music song may state song slow blues fast swing indicates tempo genre read music notation person must understanding music theory harmony performance practice associated particular song piece 's genre written notation varies style period music notated music produced sheet music individuals computer scorewriter programs


image computer screen ancient times music notation put onto stone clay tablets perform music notation singer instrumentalist requires understanding rhythmic pitch elements embodied symbols performance practice associated piece music genre genres requiring musical improvisation performer often plays music chord changes form song written requiring performer great understanding music 's structure harmony styles particular genre eg jazz


country music western art music common types written notation scores include music parts ensemble piece parts music notation individual performers singers popular music jazz blues standard musical notation lead sheet notates melody chords lyrics vocal piece structure music fake books also used jazz may consist lead sheets simply chord charts permit rhythm section members improvise accompaniment part jazz songs scores


parts also used popular music jazz particularly large ensembles jazz big bands popular music guitarists electric bass players often read music notated tablature often abbreviated tab indicates location notes played instrument using diagram guitar bass fingerboard tablature also used baroque era notate music lute stringed fretted instrument improvisation improvisation plays central role jazz musicians learn progressions using scale chord tones pictured johnny hodges musical improvisation


creation spontaneous music often within based pre existing harmonic framework chord progression improvisers use notes chord various scales associated chord chromatic ornaments passing tones may neither chord tones typical scales associated chord musical improvisation done without preparation improvisation major part types music blues jazz jazz fusion instrumental performers improvise solos melody lines accompaniment parts western art music tradition improvisation important


skill baroque era classical era baroque era performers improvised ornaments basso continuo keyboard players improvised chord voicings based figured bass notation well top soloists expected able improvise pieces preludes classical era solo performers singers improvised virtuoso cadenzas concerts however th early st century common practice western art music performance became institutionalized symphony orchestras opera houses ballets improvisation played smaller role music notated scores parts


musicians play time th st century art music composers increasingly included improvisation creative work indian classical music improvisation core component essential criterion performances theory music theory encompasses nature mechanics music often involves identifying patterns govern composers' techniques examining language notation music grand sense music theory distills analyzes parameters elements music rhythm harmony harmonic function melody structure form texture broadly music theory may include statement belief


conception music people study properties known music theorists typically work professors colleges universities music conservatories applied acoustics human physiology psychology explanation music perceived music theorists publish research music theory journals university press books elements music many different fundamentals elements depending definition element used include pitch beat pulse tempo rhythm melody harmony texture style allocation voices timbre color dynamics expression articulation form structure


elements music feature prominently music curriculums australia uk us three curriculums identify pitch dynamics timbre texture elements identified elements music far universally agreed upon list three official versions elements music australia pitch timbre texture dynamics expression rhythm form structure uk pitch timbre texture dynamics duration tempo structure usa pitch timbre texture dynamics rhythm form harmony style articulation relation uk curriculum term appropriate musical notations added list


elements title list changed elements music inter related dimensions music inter related dimensions music listed pitch duration dynamics tempo timbre texture structure appropriate musical notations phrase elements music used number different contexts two common contexts differentiated describing rudimentary elements music perceptual elements music rudimentary phrases elements music rudiments music used interchangeably elements described


documents refer aspects music needed order become musician recent writers espie estrella seem using phrase elements music similar manner definition accurately reflects usage rudimentary principles art science etc elements grammar uk 's curriculum switch inter related dimensions music seems move back using rudimentary elements music perceptual since emergence study psychoacoustics lists elements music related


''hear'' music learn play study ce seashore book ''psychology music'' identified four psychological attributes sound pitch loudness time timbre p nbsp call elements music referred elemental components p nbsp nonetheless elemental components link precisely four common musical elements pitch timbre match exactly loudness links dynamics time links time based elements rhythm duration tempo usage phrase elements music links


closely ''webster 's new th century dictionary'' definition element substance cannot divided simpler form known methods educational institutions' lists elements generally align definition well although writers lists rudimentary elements music vary lists depending personal institutional priorities perceptual elements music consist established proven list discrete elements independently manipulated achieve intended musical effect seems stage still research done


area slightly different way approaching identification elements music identify elements sound pitch duration loudness timbre sonic texture spatial location ref name searchinformitcomau burton rl https searchinformitcomau documentsummary dn res ielhss elements music cares j rosevear harding eds asme xxth national conference proceedings paper presented music educating life asme xxth national conference pp parkville victoria australian society music education inc ref define elements music sound structure artistic intent ref name searchinformitcomau descriptions pitch melody


often contain notes chords used song melodies simple folk songs traditional songs may use notes single scale scale associated tonic note key given song example folk song key c also referred c major may melody uses notes c major scale individual notes c e f g b c white notes piano keyboard hand bebop era jazz


contemporary music th st centuries may use melodies many chromatic notes ie notes addition notes major scale piano chromatic scale would include notes keyboard including white notes black notes unusual scales whole tone scale whole tone scale key c would contain notes c e f g low deep musical line played bass instruments double bass electric bass tuba called bassline harmony


chords notes time creates chord western music including classical music pop music rock music many related styles common chords triads three notes usually played time commonly used chords major chord minor chord example major chord three pitches c e g example minor chord three pitches c e pictured guitar player performing chord guitar harmony refers vertical sounds pitches


around home note tonic key simple classical pieces many pop traditional music songs written music single key complex classical pop traditional music songs pieces may two keys cases three keys classical music romantic era written often contains multiple keys jazz especially bebop jazz key home note song may change every four bars even every two bars rhythm rhythm


arrangement sounds silences time meter animates time regular pulse groupings called measures bars western classical popular traditional music often group notes sets two eg time three eg time also known waltz time time four eg time meters made easier hear songs pieces often always place emphasis first beat grouping notable exceptions exist backbeat used much western pop rock song uses measure


song texture piece song determined melodic rhythmic harmonic materials combined composition thus determining overall nature sound piece texture often described regard density thickness range width lowest highest pitches relative terms well specifically distinguished according number voices parts relationship voices see common types example thick texture contains many 'layers' instruments one layers could string


section another brass thickness also affected amount richness instruments texture commonly described according number relationship parts lines music monophony single melody tune neither instrumental accompaniment harmony part mother singing lullaby baby would example heterophony two instruments singers playing singing melody performer slightly varying rhythm speed melody adding different ornaments melody two bluegrass fiddlers playing traditional fiddle


tune together typically vary melody degree add different ornaments polyphony multiple independent melody lines interweave together sung played time choral music written renaissance music era typically written style round song row row row boat different groups singers start sing different time example polyphony homophony clear melody supported chordal accompaniment western popular music songs th century onward written


texture music contains large number independent parts eg double concerto accompanied orchestral instruments many interweaving melodic lines generally said thicker denser texture work parts eg solo flute melody accompanied single cello timbre tone color timbre sometimes called color tone color quality sound voice instrument timbre makes particular musical sound different another even pitch loudness example nbsp hz note sounds different


played oboe piano violin electric guitar even different players instrument play note notes might sound different due differences instrumental technique eg different embouchures different types accessories eg mouthpieces brass players reeds oboe bassoon players strings made different materials string players eg gut strings versus steel strings even two instrumentalists playing note instrument one may sound different due different ways playing instrument eg two string players might hold


bow differently physical characteristics sound determine perception timbre include spectrum envelope overtones note musical sound electric instruments developed th century electric guitar electric bass electric piano performer also change tone adjusting equalizer controls tone controls instrument using electronic effects units distortion pedals tone electric hammond organ controlled adjusting drawbars expression singers add expression melodies sing using many methods including changing tone singing adding


vibrato certain notes emphasizing important words lyrics expressive qualities elements music create change music without changing main pitches substantially changing rhythms melody accompaniment performers including singers instrumentalists add musical expression song piece adding phrasing adding effects vibrato voice instruments guitar violin brass instruments woodwinds dynamics loudness softness piece section tempo fluctuations eg ritardando accelerando respectively slowing speeding


tempo adding pauses fermatas cadence changing articulation notes eg making notes pronounced accented making notes legato means smoothly connected making notes shorter expression achieved manipulation pitch inflection vibrato slides etc volume dynamics accent tremolo etc duration tempo fluctuations rhythmic changes changing note duration legato staccato etc timbre eg changing vocal timbre light resonant voice sometimes even texture eg doubling bass note richer effect


piano piece expression therefore seen manipulation elements order convey indication mood spirit character etc cannot included unique perceptual element music although considered important rudimentary element music form sheet music notation chorus refrain christmas song jingle bells left music form describes overall structure plan song piece music describes layout composition divided sections early th century tin pan


alley songs broadway musical songs often aaba bar form sections repeated eight bar melody variation b section provided contrasting melody harmony eight bars onward western pop rock songs often verse chorus form based around sequence verse chorus refrain sections new lyrics verses repeating lyrics choruses popular music often makes use strophic form sometimes conjunction twelve bar blues tenth edition ''the oxford


companion music'' percy scholes defines musical form series strategies designed find successful mean opposite extremes unrelieved repetition unrelieved alteration examples common forms western music include fugue invention sonata allegro canon strophic theme variations rondo scholes states european classical music six stand alone forms simple binary simple ternary compound binary rondo air variations fugue although musicologist alfred mann emphasized fugue primarily method composition sometimes taken certain structural conventions piece cannot readily


broken sectional units though might borrow form poem story programme said composed often case fantasia prelude rhapsody etude study symphonic poem bagatelle impromptu etc professor charles keil classified forms formal detail sectional developmental variational analysis styles funk places emphasis rhythm groove entire songs based around vamp single chord pictured influential funk musicians george clinton parliament funkadelic styles music place emphasis certain


fundamentals others place less emphasis certain elements give one example bebop era jazz makes use complex chords including altered dominants challenging chord progressions chords changing two times per bar keys changing several times tune funk places emphasis rhythm groove entire songs based around vamp single chord romantic era classical music mid late makes great use dramatic changes dynamics whispering pianissimo sections thunderous fortissimo sections entire baroque dance suites


contemporary music genre ancient egyptian music preserved many features rhythms instruments asian cultures indian women dressed regional attire playing variety musical instruments popular different parts india class wikitable style text align right float left margin right px align center gangubai hangal br small durga small style text align left background silver px asian music covers vast swath music cultures surveyed articles arabia central asia east asia south asia southeast asia several traditions reaching antiquity indian classical music one oldest musical traditions world indus valley civilization


nbsp nbsp nbsp nbsp european influenced music ancient greece music important part social cultural life ancient greece fact one main subjects taught children musical education considered important development individual 's soul musicians singers played prominent role greek theater ref name grove savage roger http wwwoxfordmusiconlinecom subscriber article grove music incidental music grove music online oxford music online accessed august ref ones received musical education seen nobles perfect harmony read republic plato mixed gender choruses performed


entertainment celebration spiritual ceremonies ref name west ref holy ancient greek music considered example perfection purity instruments included double reed ''aulos'' plucked string instrument ''lyre'' principally special kind called ''kithara'' music important part education boys taught music starting age six greek musical literacy created flowering music development greek music theory included greek musical modes eventually became basis western religious classical music later influences roman empire eastern europe byzantine empire changed greek music seikilos epitaph


oldest surviving example complete musical composition including musical notation anywhere world oldest surviving work written subject music theory ''harmonika stoicheia'' aristoxenus ref name aristoxenus henry stewart macran ref western classical middle ages class wikitable style text align right float left margin right px align center l onin p rotin br small breves dies hominis small style text align left background dcdcdc px musical notation catholic missal c medieval era took place middle ages started introduction monophonic single melodic line chanting roman catholic church services musical notation


used since ancient times greek culture middle ages notation first introduced catholic church chant melodies could written facilitate use melodies religious music across entire catholic empire european medieval repertory found written form monophonic liturgical plainsong chant roman catholic church central tradition called gregorian chant alongside traditions sacred church music existed vibrant tradition secular song non religious songs examples


musical styles quickly across larger area musicians singers often worked church courts towns church choirs grew size church remained important patron music middle th century composers wrote richly polyphonic sacred music different melody lines interwoven simultaneously prominent composers era include guillaume dufay giovanni pierluigi da palestrina thomas morley orlande de lassus musical activity shifted church aristocratic courts kings queens princes competed finest composers many leading important composers came


era include johann sebastian bach ''cello suites'' george frideric handel ''messiah'' georg philipp telemann antonio lucio vivaldi ''the four seasons'' classicism class wikitable style text align right float left margin right px align center wa mozart br small symphony nbsp g minor small style text align left background px wolfgang amadeus mozart seated keyboard child prodigy virtuoso performer piano violin even became celebrated composer widely known gifted performer improviser music classical period aimed imitate seen key elements art philosophy


ancient greece rome ideals balance proportion disciplined expression note music classical period confused classical music general term refers western art music th century includes classical period one number periods music classical period lighter clearer considerably simpler texture baroque music preceded main style homophony prominent melody subordinate chordal accompaniment part clearly distinct classical instrumental melodies tended