karl banse sayed elsayed paul falkowski gotthilf hempel victor hensen uwe kils johannes krey j rgen lenz vivienne cassie cooper external links https webarchiveorg web http wwwuwmedu jrs hensen htm victor hensen biography https webarchiveorg web http arccsoduedu dp getrecord oai dc oai univoldenburgbis gotthilf hempel https webarchiveorg web http marinerutgersedu faculty pfalkowskihtml paul falkowski homepage subfields zoology


''testing wine'' english school th century '''connoisseur''' french traditional pre spelling middle french meaning 'to acquainted with' 'to know somebody something' person great deal knowledge fine arts keen appreciator cuisines fine wines gourmet products expert judge matters taste many areas term air pretension may used partly ironic sense art trade however expert connoisseurship remains crucial skill


identification attribution individual artists works style technique documentary evidence provenance lacking situation wine trade similar example assessing potential ageing young wine wine tasting connoisseurship art ability tell almost instinctively painted picture defined nbsp connoisseurship ref name arthistorynews ref connoisseurs evaluate works art basis experience style technique artists judgment informed intuition essential must grounded thorough understanding work


basis empirical evidence refinement perception technique form disciplined method analysis responsibility connoisseur attribute authorship validate authenticity appraise quality findings crucial valuation works collected organized ''catalogue raisonn '' work single artist school ''meaning visual arts'' erwin panofsky explains difference connoisseur art historian connoisseur might defined laconic art historian art historian loquacious


undervalued art history establishment ref name arthistorynews svetlana alpers confirms art historians reservations identification individual style works essentially assigned group specialists field known conoisseurs nonetheless christie 's education offers history art art market includes seminar connoisseurship covers critical skills needed look art write art research evaluate works including handling viewing art objects visiting artists' studios conservation labs museums earlier views connoisseurship th century term


often used synonym still vaguer man taste pretend critic oliver goldsmith said painting someday become sole object fashionable care title connoisseur art present safest passport every fashionable society well timed shrug admiring attitude one two exotic tones exclamation sufficient qualifications men low circumstances curry favour giovanni morelli wrote art connoisseurs say art historians write understand


many products whose consumption pleasing senses see also amateur dilettante appraiser collector enthusiast expert fan aficionado gourmet gourmand sommelier references reading beck james h ''from duccio raphael connoisseurship crisis'' fucecchio european press academic publishing friedl nder max j ''on art connoisseurship'' trans tancred borenius london b cassierer ''l'art du connaisseur art connoisseurship'' montr al mus e des beaux arts de montr al robinson terry f http wwwoxfordhandbookscom view oxfordhb oxfordhb e eighteenth century connoisseurship female body oxford handbooks online oxford university press web may scallen catherine b ''rembrandt reputation practice


connoisseurship'' amsterdam amsterdam university press trummers anna koenraad jonckheere eds ''art market connoisseurship closer look paintings rembrandt rubens contemporaries'' amsterdam amsterdam university press art history


mathematics '''projective geometry''' study geometric properties invariant respect projective transformations means compared elementary euclidean geometry projective geometry different setting projective space selective set basic geometric concepts basic intuitions projective space points euclidean space given dimension geometric transformations permitted transform extra points called points infinity euclidean points vice versa properties meaningful projective geometry respected new idea transformation radical effects

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expressed transformation matrix translations affine transformations first issue geometers kind geometry adequate novel situation possible refer angles projective geometry euclidean geometry angle example concept invariant respect projective transformations seen perspective drawing one source projective geometry indeed theory perspective another difference elementary geometry way parallel lines said meet point infinity


concept translated projective geometry 's terms notion intuitive basis railway tracks meeting horizon perspective drawing see projective plane basics projective geometry two dimensions ideas available earlier projective geometry mainly development th century included theory complex projective space coordinates used homogeneous coordinates complex numbers several major types abstract mathematics including invariant theory italian school algebraic geometry felix klein 's erlangen programme resulting study classical


groups based projective geometry also subject many practitioners sake synthetic geometry another topic developed axiomatic studies projective geometry finite geometry topic projective geometry divided many research subtopics two examples projective algebraic geometry study projective varieties projective differential geometry study differential invariants projective transformations overview fundamental theory projective geometry projective geometry elementary non metrical form geometry meaning based concept


distance two dimensions begins study configurations points lines indeed geometric interest sparse setting first established desargues others exploration principles perspective art higher dimensional spaces considered hyperplanes always meet linear subspaces exhibit principle duality simplest illustration duality projective plane statements two distinct points determine unique line ie line two distinct lines determine unique point ie point


intersection show structure propositions projective geometry also seen geometry constructions straight edge alone since projective geometry excludes compass constructions circles angles measurements parallels concept intermediacy realised theorems apply projective geometry simpler statements example different conic sections equivalent complex projective geometry theorems circles considered special cases general theorems early th century work jean victor poncelet lazare carnot


others established projective geometry independent field mathematics rigorous foundations addressed karl von staudt perfected italians giuseppe peano mario pieri alessandro padoa gino fano late th century projective geometry like affine euclidean geometry also developed erlangen program felix klein projective geometry characterized invariants transformations projective group much work large number theorems subject therefore basics projective geometry became understood incidence structure cross ratio fundamental invariants projective


transformations projective geometry modeled affine plane affine space plus line hyperplane infinity treating line hyperplane ordinary algebraic model projective geometry style analytic geometry given homogeneous coordinates hand axiomatic studies revealed existence non desarguesian planes examples show axioms incidence modelled two dimensions structures accessible reasoning homogeneous coordinate systems growth measure polar vortices based work lawrence edwards foundational sense


g rard desargues independently developed concept point infinity desargues developed alternative way constructing perspective drawings generalizing use vanishing points include case infinitely far away made euclidean geometry parallel lines truly parallel special case encompassing geometric system desargues 's study conic sections drew attention year old blaise pascal helped formulate pascal 's theorem works gaspard monge end th beginning th century important subsequent development projective geometry


permutations called bius transformations generalised circles complex plane transformations represent projectivities complex projective line study lines space julius pl cker used homogeneous coordinates description set lines viewed klein quadric one early contributions projective geometry new field called algebraic geometry offshoot analytic geometry projective ideas projective geometry instrumental validation speculations lobachevski bolyai concerning hyperbolic geometry providing models hyperbolic plane example poincar disc


italian school algebraic geometry enriques segre severi broke traditional subject matter area demanding deeper techniques later part th century detailed study projective geometry became less fashionable although literature voluminous important work done enumerative geometry particular schubert considered anticipating theory chern classes taken representing algebraic topology grassmannians paul dirac studied projective geometry used basis developing concepts quantum mechanics although published results always


algebraic form see http wwwmathcolumbiaedu woit wordpress p blog article referring article book subject also talk dirac gave general audience boston projective geometry without specifics application physics description projective geometry less restrictive either euclidean geometry affine geometry intrinsically non metrical geometry meaning facts independent metric structure projective transformations incidence structure relation projective harmonic conjugates preserved projective range one dimensional foundation projective geometry formalizes one


central principles perspective art parallel lines meet infinity therefore drawn way essence projective geometry may thought extension euclidean geometry direction line subsumed within line extra point horizon directions corresponding coplanar lines regarded line thus two parallel lines meet horizon line virtue incorporating direction idealized directions referred points infinity idealized horizons referred lines infinity


turn lines lie plane infinity however infinity metric concept purely projective geometry single points lines planes regard infinity treated like others euclidean geometry contained within projective geometry projective geometry simpler foundation general results euclidean geometry may derived transparent manner separate similar theorems euclidean geometry may handled collectively within framework projective geometry example parallel nonparallel lines need treated separate


cases rather arbitrary projective plane singled ideal plane located infinity using homogeneous coordinates additional properties fundamental importance include desargues' theorem theorem pappus projective spaces dimension greater construction allows one prove desargues' theorem dimension must separately postulated using desargues' theorem combined axioms possible define basic operations arithmetic geometrically resulting operations satisfy axioms field except commutativity multiplication requires pappus 's hexagon theorem


result points line one one correspondence given field supplemented additional element except remain undefined projective geometry also includes full theory conic sections subject also extensively developed euclidean geometry advantages able think hyperbola ellipse distinguished


thoroughly example method multi volume treatise h f baker many projective geometries may divided discrete continuous ' 'discrete'' geometry comprises set points may may ''finite'' number ''continuous'' geometry infinitely many points gaps projective geometry dimension single point projective geometry dimension consists single line containing least points geometric construction arithmetic operations cannot performed either cases dimension


rich structure virtue absence desargues' theorem fano plane projective plane fewest points lines smallest dimensional projective geometry fewest points fano plane points every line points lines following collinearities abc ade afg bdg bef cdf ceg homogeneous coordinates affine coordinates affine coordinates desarguesian plane points designated


points infinity example c e g defined several ways standard notation finite projective geometry written projective geometric dimension one less number points line called ''order'' geometry thus example points written term projective geometry used sometimes indicate generalised underlying abstract geometry sometimes indicate particular geometry wide interest metric geometry flat


space analyse use homogeneous coordinates euclidean geometry may embedded hence name extended euclidean plane fundamental property singles projective geometries ''elliptic'' incidence property two distinct lines projective plane intersect exactly one point special case analytic geometry ''parallel'' lines subsumed smoother form line ''at infinity'' lies ''line infinity'' thus line like theory way special


distinguished later spirit erlangen programme one could point way group transformations move line ''line infinity'' parallel properties elliptic euclidean hyperbolic geometries contrast follows given line point line ''elliptic'' exists line meet ''euclidean'' exists exactly one line meet ''hyperbolic'' exists one line


meet parallel property elliptic geometry key idea leads principle projective duality possibly important property projective geometries common duality joseph gergonne noted principle duality characterizing projective plane geometry given theorem definition geometry substituting ''point'' ''line'' ''lie on'' ''pass through'' ''collinear'' ''concurrent'' ''intersection'' ''join'' vice versa results another theorem valid definition dual first similarly dimensions duality relation holds points planes allowing theorem


transformed swapping ''point'' ''plane'' ''is contained by'' ''contains'' generally projective spaces dimension n duality subspaces dimension r dimension n r n specializes commonly known form duality points lines duality principle also discovered independently jean victor poncelet establish duality requires establishing theorems dual versions axioms dimension question thus dimensional spaces one needs show every point lies distinct planes


every two planes intersect unique line dual version effect intersection plane p q coplanar intersection plane r respective intersections planes p r q assuming planes p distinct q r practice principle duality allows us set ' 'dual correspondence'' two geometric constructions famous polarity reciprocity two figures conic curve dimensions


quadric surface dimensions commonplace example found reciprocation symmetrical polyhedron concentric sphere obtain dual polyhedron another example brianchon 's theorem dual already mentioned pascal 's theorem one whose proofs simply consists applying principle duality pascal 's comparative statements two theorems cases within framework projective plane '''pascal ''' six vertices hexagon lie conic intersections opposite sides '' regarded full lines since projective plane


thing line segment '' three collinear points line joining called '''pascal line''' hexagon '''brianchon ''' six sides hexagon tangent conic diagonals ie lines joining opposite vertices three concurrent lines point intersection called '''brianchon point''' hexagon conic degenerates two straight lines pascal 's becomes pappus 's theorem interesting dual since brianchon point trivially becomes two lines' intersection point axioms projective geometry given


one incidence relation points lines three axioms g every line contains least points g every two distinct points b lie unique line ab g lines ab cd intersect lines ac bd assumed distinct b c reason line assumed contain least points eliminate degenerate cases spaces satisfying three axioms either one line projective spaces dimension


division ring non desarguesian planes additional axioms one add axioms restricting dimension coordinate ring example coxeter 's ''projective geometry'' references veblen three axioms together axioms make dimension coordinate ring commutative field characteristic nbsp axioms using ternary relation one pursue axiomatization postulating ternary relation abc denote three points necessarily distinct collinear axiomatization may written terms relation well c aba c b


two points abc abd bdc c b two points third point c abc c c two points b also bce ade abe point f acf bdf two different points b line ab defined consisting points c abc axioms c c provide formalization g c g c g concept line


generalizes planes higher dimensional subspaces subspace abxy may thus recursively defined terms subspace abx containing points lines yz z ranges abx collinearity generalizes relation independence set b z points independent abz b z minimal generating subset subspace abz projective axioms may supplemented axioms postulating limits dimension space minimum dimension determined existence independent set required


size lowest dimensions relevant conditions may stated equivalent form follows projective space l least dimension least point l least dimension least distinct points therefore line l least dimension least non collinear points two lines line point line l least dimension least non coplanar points maximum dimension may also determined similar fashion


lowest dimensions take following forms projective space dimension point dimension line dimension plane general theorem consequence axiom coplanar lines intersect principle projective geometry originally intended embody therefore property may equivalently stated lines intersect one another generally assumed projective


spaces least dimension cases focus projective planes variant may postulated axioms eves instance include l axiom becomes vacuously true therefore needed context axioms projective planes incidence geometry authors give treatment embraces fano plane pg nbsp smallest finite projective plane axiom system achieves follows p two distinct points lie unique line p two


distinct lines meet unique point p exist least four points three collinear coxeter 's ''introduction geometry'' gives list five axioms restrictive concept projective plane attributed bachmann adding pappus 's theorem list axioms eliminates non desarguesian planes excluding projective planes fields characteristic n't satisfy fano 's axiom restricted planes given manner closely resemble real projective plane perspectivity projectivity given three non collinear points three lines connecting four


one plane yields configuration another applies configuration complete quadrangle thus harmonic quadruples preserved perspectivity one perspectivity follows another configurations follow along composition two perspectivities longer perspectivity '''projectivity''' corresponding points perspectivity converge point convergence ''not'' true projectivity ''not'' perspectivity projective geometry intersection lines formed corresponding points projectivity plane particular interest set


line ring joseph wedderburn grassmann cayley algebra notes references cite id refhughes cite santal luis ''geometr proyectiva'' editorial universitaria de buenos aires external links http citeseerxistpsuedu viewdoc summarydoi projective geometry machine vision tutorial joe mundy andrew zisserman http xahleeorg projective geometry projective geometryhtml notes based coxeter 's ''the real projective plane'' http learinrialpesfr people triggs pubs isprs isprs html projective geometry image analysis free tutorial roger mohr bill triggs http wwwgeometerorg mathcircles projectivepdf projective geometry free tutorial tom davis http neo classical physicsinfo uploads burali forti grassman proj geompdf grassmann method projective geometry compilation three notes cesare burali forti application exterior algebra projective geometry http neo classical physicsinfo uploads burali forti diff geom following grassmannpdf c burali forti introduction


differential geometry following method h grassmann english translation book http neo classical physicsinfo uploads kummer rectilinear ray systemspdf e kummer general theory rectilinear ray systems english translation http neo classical physicsinfo uploads pasch focal singularity surfacespdf pasch focal surfaces ray systems singularity surfaces complexes english translation


point p l exactly one line parallel l passes p fundamental affine geometry comparisons figures affine geometry made affine transformations mappings preserve alignment points parallelism lines affine geometry developed two ways essentially equivalent synthetic geometry affine space set ''points'' associated set lines satisfy axioms playfair 's axiom affine geometry also developed basis linear algebra context

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Affine geometry


vector give another point operations required satisfy number axioms notably two successive translations effect translation sum vector choosing point origin points one one correspondence vectors preferred choice origin thus affine space may viewed obtained associated vector space forgetting origin zero vector idea forgetting metric applied theory manifolds developed article affine connection history


leonhard euler introduced term ''affine'' latin ''affinis'' related book ''introductio analysin infinitorum'' volume chapter xviii august bius wrote affine geometry ''der barycentrische calcul'' chapter felix klein 's erlangen program affine geometry recognized generalization euclidean geometry hermann weyl referred affine geometry text ''space time matter'' used affine geometry introduce vector addition subtraction earliest stages development mathematical physics later e whittaker wrote weyl 's geometry interesting historically


first affine geometries worked detail based special type parallel transport using worldlines light signals four dimensional space time short element one world lines may called ''null vector'' parallel transport question carries null vector one point position null vector neighboring point systems axioms several axiomatic approaches affine geometry put forward pappus' law pappus 's law red lines parallel blue lines parallel dotted black


lines must parallel affine geometry deals parallel lines one properties parallels noted pappus alexandria taken premise suppose math b c math one line math a' b' c' math another lines math ab' math math a'b math parallel lines math bc' math math b'c math parallel lines math ca' math math c'a math parallel full axiom system proposed ''point'' ''line'' ''line containing point'' primitive notions two points contained one line line ''l'' point ''p''


built axioms ordered geometry addition two additional axioms affine axiom parallelism given point line one line meet desargues given seven distinct points math a' b b' c c' math math aa' math math bb' math math cc' math distinct lines math math math ab math parallel math a'b' math math bc math parallel math b'c' math math ac math parallel math a'c' math affine concept parallelism forms equivalence relation lines since axioms


ternary ring plane said minor affine desargues property two triangles parallel perspective two parallel sides must also third sides parallel property holds affine plane defined ternary ring equivalence relation vectors defined pairs points plane furthermore vectors form abelian group addition ternary ring linear satisfies right distributivity ''a'' ''b'' ''c'' ''ac'' ''bc'' affine transformations geometrically affine transformations affinities preserve collinearity transform parallel lines


parallel lines preserve ratios distances along parallel lines identify ''affine theorems'' geometric result invariant affine group felix klein 's erlangen programme underlying group symmetry transformations affine geometry consider vector space ''v'' general linear group gl ''v'' whole ''affine group'' must allow also translations vectors ''v'' ''v'' translation maps ''w'' ''v'' ''w v'' affine group generated general linear group translations


fact semidirect product math v rtimes mathrm gl v math think ''v'' group operation addition use defining representation gl ''v'' ''v'' define semidirect product example theorem plane geometry triangles concurrence lines joining vertex midpoint opposite side ''centroid'' ''barycenter'' depends notions ''mid point'' ''centroid'' affine invariants examples include theorems ceva menelaus affine invariants also assist calculations example lines divide


area triangle two equal halves form envelope inside triangle ratio area envelope area triangle affine invariant needs calculated simple case unit isosceles right angled triangle give math tfrac log e tfrac math ie less triangles familiar formulas half base times height area triangle third base times height volume pyramid likewise affine invariants


latter less obvious former general case easily seen one sixth unit cube formed face area midpoint cube height hence holds pyramids even slanting ones whose apex directly center base base parallelogram instead square formula generalizes pyramids whose base dissected parallelograms including cones allowing infinitely many parallelograms due attention convergence approach


shows four dimensional pyramid hypervolume one quarter volume parallelepiped base times height higher dimensions kinematics two types affine transformation used kinematics classical modern velocity v described using length direction length presumed unbounded variety kinematics styled galilean newtonian uses coordinates absolute space time shear mapping plane axis represents coordinate change observer moving velocity v resting frame reference finite light speed


first noted delay appearance moons jupiter requires modern kinematics method involves rapidity instead velocity substitutes squeeze mapping shear mapping used earlier affine geometry developed synthetically express special theory relativity affine plane associated lorentzian vector space ''l'' sup sup described graciela birman katsumi nomizu article entitled trigonometry lorentzian geometry affine space affine geometry viewed geometry affine space given dimension ''n'' coordinatized field ''k''


also two dimensions combinatorial generalization coordinatized affine space developed synthetic finite geometry projective geometry ''affine space'' means complement hyperplane infinity projective space ''affine space'' also viewed vector space whose operations limited linear combinations whose coefficients sum one example ''x'' nbsp nbsp ''y'' ''x'' nbsp nbsp ''y'' nbsp nbsp ''z'' ''x'' nbsp nbsp ''y'' nbsp nbsp ''z'' '''i'''''x'' nbsp nbsp nbsp nbsp '''i''' ''y'' etc synthetically affine planes dimensional affine geometries defined terms relations points lines sometimes higher dimensions hyperplanes defining affine projective geometries configurations


points lines hyperplanes instead using coordinates one gets examples coordinate fields major property examples dimension finite examples dimension finite affine planes valuable study configurations infinite affine spaces group theory combinatorics despite less general configurational approach approaches discussed successful illuminating parts geometry related symmetry projective view traditional geometry affine geometry considered study euclidean geometry


projective geometry one hand affine geometry euclidean geometry congruence left hand affine geometry may obtained projective geometry designation particular line plane represent points infinity affine geometry metric structure parallel postulate hold affine geometry provides basis euclidean structure perpendicular lines defined basis minkowski geometry notion hyperbolic orthogonality viewpoint affine transformation projective transformation permute finite points points


infinity affine transformation geometry study geometrical properties action group affine transformations see also non euclidean geometry references reading emil artin https archiveorg details geometricalgebra mbp page n mode upview theater ''geometric algebra'' chapter affine projective geometry via internet archive vg ashkinuse isaak yaglom ''ideas methods affine projective geometry'' russian ministry education moscow k bennett ''affine projective geometry'' john wiley sons h coxeter affine plane scripta mathematica lecture delivered forum society friends ''scripta


mathematica'' monday april felix klein ''elementary mathematics advanced standpoint geometry'' translated e r hedrick c noble pp macmillan company bruce e meserve ''fundamental concepts geometry'' chapter affine geometry pp addison wesley peter scherk rolf lingenberg ''rudiments plane affine geometry'' mathematical expositions university toronto press wanda szmielew ''from affine euclidean geometry axiomatic approach'' reidel oswald veblen ''projective geometry'' volume chapter affine group plane pp ginn company external links


peter cameron 's http wwwmathsqmulacuk pjc pps pps pdf projective affine geometries university london jean h gallier ''geometric methods applications computer science engineering'' chapter http wwwcisupennedu cis geombchap pdf basics affine geometry pdf springer texts applied mathematics chapter online university pennsylvania


mathematics '''multilinear algebra''' extends methods linear algebra linear algebra built concept vector develops theory vector spaces multilinear algebra builds concepts ''p'' vectors multivectors grassmann algebras origin vector space dimension ''n'' one usually considers vectors according hermann grassmann others presumption misses complexity considering structures pairs triples general multivectors since several combinatorial possibilities space multivectors turns sup ''n'' sup dimensions abstract formulation determinant

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algebra applied studies multivariate calculus manifolds jacobian matrix comes play infinitesimal differentials single variable calculus become differential forms multivariate calculus manipulation done exterior algebra grassmann developments multilinear algebra made victor schlegel published first part ''system der raumlehre'' elwin bruno christoffel major advance multilinear algebra came work gregorio ricci curbastro tullio levi civita see references ''absolute differential calculus'' form multilinear algebra marcel grossmann


michele besso introduced albert einstein publication einstein general relativity explanation precession perihelion mercury established multilinear algebra tensors physically important mathematics use algebraic topology around middle th century study tensors reformulated abstractly bourbaki group 's treatise ''multilinear algebra'' especially influential fact term ''multilinear algebra'' may originated one reason time new area application homological algebra development algebraic topology gave additional incentive development


purely algebraic treatment tensor product computation homology groups product two topological spaces involves tensor product simplest cases torus directly calculated fashion see k nneth theorem topological phenomena subtle enough need better foundational concepts technically speaking tor functors defined material organise quite extensive including also ideas going back hermann grassmann ideas theory differential forms led de rham cohomology well elementary ideas


wedge product generalises cross product resulting rather severe write topic bourbaki entirely rejected one approach vector calculus quaternion route general case relation lie groups instead applied novel approach using category theory lie group approach viewed separate matter since leads much cleaner treatment probably going back purely mathematical terms strictly universal property approach invoked somewhat general category theory relationship two


approach principle abstract approach recover everything done via traditional approach practice may seem simple hand notion ''naturality'' consistent ''general covariance'' principle general relativity latter deals tensor fields tensors varying point point manifold covariance asserts language tensors essential proper formulation general relativity decades later rather abstract view coming category theory tied approach developed hermann weyl


working general relativity via abstract tensor analysis additionally book ''the classical groups'' way took theory full circle connecting content old new viewpoints topics multilinear algebra subject matter multilinear algebra evolved less presentation years pages centrally relevant bilinear operator component free treatment tensors cramer 's rule dual space einstein notation exterior algebra exterior derivative inner product kronecker delta levi civita symbol metric tensor mixed tensor multilinear map multilinear form symmetric algebra symmetric power symmetric


tensor tensor tensor algebra free algebra tensor contraction also glossary tensor theory applications ways multilinear algebra concepts applied classical treatment tensors dyadic tensor bra ket notation geometric algebra clifford algebra pseudoscalar pseudovector spinor outer product hypercomplex number multilinear subspace learning references hermann grassmann ''extension theory'' american mathematical society translation lloyd kannenberg ''ausdehnungslehre'' ronald shaw multilinear algebra group representations volume ''linear algebra group representations'' academic press multilinear algebra


includes ring theory representation theory theory banach algebras overview commutative algebra essentially study rings occurring algebraic number theory algebraic geometry algebraic number theory rings algebraic integers dedekind rings constitute therefore important class commutative rings considerations related modular arithmetic led notion valuation ring restriction algebraic field extensions subrings led notions integral extensions integrally closed domains well notion ramification extension valuation rings notion

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localization ring particular localization respect prime ideal localization consisting inverting single element total quotient ring one main differences commutative algebra theory non commutative rings leads important class commutative rings local rings one maximal ideal set prime ideals commutative ring naturally equipped topology zariski topology notions widely used algebraic geometry basic technical tools definition scheme theory


grounded things complex analysis classical invariant theory turn hilbert strongly influenced emmy noether recast many earlier results terms ascending chain condition known noetherian condition another important milestone work hilbert 's student emanuel lasker introduced primary ideals proved first version lasker noether theorem main figure responsible birth commutative algebra mature subject wolfgang krull introduced fundamental notions localization completion ring well regular local rings established


concept krull dimension ring first noetherian rings moving expand theory cover general valuation rings krull rings day krull 's principal ideal theorem widely considered single important foundational theorem commutative algebra results paved way introduction commutative algebra algebraic geometry idea would revolutionize latter subject much modern development commutative algebra emphasizes modules ideals ring ''r'' ''r'' algebras special cases ''r'' modules module theory encompasses ideal


theory theory ring extensions though already incipient kronecker 's work modern approach commutative algebra using module theory usually credited krull noether main tools results noetherian rings mathematics specifically area modern algebra known ring theory '''noetherian ring''' named emmy noether ring every non empty set ideals maximal element equivalently ring noetherian satisfies ascending chain condition ideals given chain math subseteq cdots k subseteq k subseteq k subseteq cdots math exists ''n''


ring noetherian satisfies descending chain condition ''prime ideals'' property suggests deep theory dimension noetherian rings beginning notion krull dimension hilbert 's basis theorem hilbert 's basis theorem immediate corollaries induction see math r x dotsc x n math also noetherian since affine variety math r n math ie locus set collection polynomials may written locus ideal math mathfrak subset r x dotsc x n math locus generators follows every affine variety locus


finitely many polynomials ie intersection finitely many hypersurfaces math math finitely generated math r math algebra know math simeq r x dotsc x n mathfrak math math mathfrak math ideal basis theorem implies math mathfrak math must finitely generated say math mathfrak p dotsc p n math ie math math finitely presented primary decomposition ideal ''q'' ring said ''primary'' ''q'' proper whenever ''xy'' ''q'' either ''x'' ''q'' ''y sup n sup '' ''q'' positive integer ''n'' '''z''' primary ideals


rad ''q sub j sub '' ''i'' ''j'' furthermore math bigcap k p math decomposition ''i'' rad ''p sub sub '' rad ''p sub j sub '' ''i'' ''j'' decompositions ''i'' ''irredundant'' meaning proper subset either ''q'' sub sub ''q sub sub '' ''p'' sub sub ''p sub k sub '' yields intersection equal ''i'' ''t'' ''k'' possibly renumbering ''q sub sub '' rad ''q sub sub '' rad ''p sub sub '' ''i'' primary decomposition ''i'' set radicals set rad ''q'' sub sub rad ''q sub sub '' remains lasker noether theorem fact turns noetherian ring


set precisely assassinator module ''r'' ''i'' set annihilators ''r'' ''i'' viewed module ''r'' prime localization localization formal way introduce denominators given ring module introduces new ring module existing one consists fractions math frac math denominators ' 's'' range given subset ''s'' ''r'' archetypal example construction ring '''q''' rational numbers ring '''z''' integers completion completion


several related functors rings modules result complete topological rings modules completion similar localization together among basic tools analysing commutative rings complete commutative rings simpler structure general ones hensel 's lemma applies zariski topology prime ideals zariski topology defines topology spectrum ring set prime ideals formulation zariski closed sets taken sets math v p operatorname spec mid subseteq p math ''a''


coordinate ring maximal form thus ''v'' ''s'' maximal ideals containing ''s'' grothendieck 's innovation defining spec replace maximal ideals prime ideals formulation natural simply generalize observation definition closed set spectrum ring examples fundamental example commutative algebra ring integers math mathbb z math existence primes unique factorization theorem laid foundations concepts noetherian rings primary decomposition


important examples polynomial rings math r x x n math p adic integers rings algebraic integers connections algebraic geometry commutative algebra form polynomial rings quotients used definition algebraic varieties always part algebraic geometry however late algebraic varieties subsumed alexander grothendieck 's concept scheme local objects affine schemes prime spectra locally ringed spaces form category antiequivalent dual category commutative unital rings extending duality category affine algebraic varieties


field ''k'' category finitely generated reduced ''k'' algebras gluing along zariski topology one glue within category locally ringed spaces also using yoneda embedding within abstract category presheaves sets category affine schemes zariski topology set theoretic sense replaced zariski topology sense grothendieck topology grothendieck introduced grothendieck topologies mind exotic geometrically finer sensitive examples crude zariski topology namely tale topology two flat grothendieck topologies fppf


fpqc nowadays examples become prominent including nisnevich topology sheaves furthermore generalized stacks sense grothendieck usually additional representability conditions leading artin stacks even finer deligne mumford stacks often called algebraic stacks see also list commutative algebra topics glossary commutative algebra combinatorial commutative algebra gr bner basis homological algebra notes references michael atiyah ian g macdonald ''introduction commutative algebra'' massachusetts addison wesley publishing bourbaki nicolas ''commutative algebra chapters '' translated french reprint


english translation elements mathematics berlin springer verlag berlin xxiv pp bourbaki nicolas '' l ments de math matique alg bre commutative chapitres et '' elements mathematics commutative algebra chapters reprint original springer berlin ii pp r mi goblot alg bre commutative cours et exercices corrig e dition dunod ernst kunz introduction commutative algebra algebraic geometry birkhauser matsumura hideyuki ''commutative algebra'' second edition mathematics lecture note series benjamin cummings publishing co inc reading mass xv pp matsumura hideyuki ''commutative ring theory'' second edition translated japanese cambridge studies advanced mathematics


cambridge uk cambridge university press nagata masayoshi ''local rings'' interscience tracts pure applied mathematics interscience publishers division john wiley sons new york london xiii pp miles reid ''undergraduate commutative algebra london mathematical society student texts '' cambridge uk cambridge university press jean pierre serre ''local algebra'' translated french cheewhye chin revised author original title ''alg bre locale multiplicit s'' springer monographs mathematics springer verlag berlin xiv pp sharp r ''steps commutative algebra'' second edition london mathematical society student texts cambridge university press cambridge


xii pp zariski oscar samuel pierre ''commutative algebra'' vol cooperation cohen corrected reprinting edition graduate texts mathematics springer verlag new york heidelberg berlin


mathematics '''k theory''' roughly speaking study ring generated vector bundles topological space scheme algebraic topology cohomology theory known topological k theory algebra algebraic geometry referred algebraic k theory also fundamental tool field operator algebras seen study certain kinds invariants large matrices k theory involves construction families ''k'' functors map topological spaces schemes associated rings rings reflect aspects structure

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original spaces schemes functors groups algebraic topology reason functorial mapping easier compute topological properties mapped rings original spaces schemes examples results gleaned k theory approach include grothendieck riemann roch theorem bott periodicity atiyah singer index theorem adams operations high energy physics k theory particular twisted k theory appeared type ii string theory conjectured classify branes ramond ramond field strengths also certain spinors generalized complex manifolds


condensed matter physics k theory used classify topological insulators superconductors stable fermi surfaces details see k theory physics grothendieck completion grothendieck completion abelian monoid abelian group necessary ingredient defining k theory since definitions start constructing abelian monoid suitable category turning abelian group universal construction given abelian monoid math ' math let math sim math relation math times math defined math sim b b math exists math c math math ' b ' c


math math math math identity element math g math first math sim nn math math n math since set math c math apply equation equivalence relation get math n n math implies math ab ba ba b math hence additive inverse element math g math give us hint thinking equivalence classes math ab math formal differences math b math another useful observation invariance equivalence classes scaling math ab sim kb k math math k math grothendieck


example see scaling invariance math sim sim sim sim math general math k min ab math math ab sim kb k math form math c math math math shows think math math positive integers math b math negative integers definitions number basic definitions k theory two coming topology two algebraic geometry grothendieck group compact hausdorff spaces given compact hausdorff space math x math consider set isomorphism classes finite dimensional vector bundles math x math


denoted math text vect x math let isomorphism class vector bundle math pi e x math denoted math e math since isomorphism classes vector bundles behave well respect direct sums write operations isomorphism classes math e oplus e' e oplus e' math clear math text vect x oplus math abelian monoid unit given trivial vector bundle math r times x x math apply grothendieck completion get abelian group abelian monoid called k theory math x math denoted math k x math use serre swan theorem


algebra get alternative description vector bundles ring continuous complex valued functions math c x complex math projective modules identified idempotent matrices ring matrices math n times n c x complex math define equivalence classes idempotent matrices form abelian monoid math textbf idem x math grothendieck completion also called math k x math one main techniques computing grothendieck group topological spaces comes atiyah hirzebruch spectral sequence makes accessible required computations understanding spectral sequences computing group math k math spheres math n math sup pg


sup grothendieck group vector bundles algebraic geometry analogous construction considering vector bundles algebraic geometry noetherian scheme math x math set math text vect x math isomorphism classes algebraic vector bundles math x math direct sum math oplus math isomorphisms classes vector bundles well defined giving abelian monoid math text vect x oplus math grothendieck group math k x math defined application grothendieck construction abelian monoid grothendieck group coherent sheaves algebraic geometry algebraic geometry construction applied


algebraic vector bundles smooth scheme alternative construction noetherian scheme math x math look isomorphism classes coherent sheaves math operatorname coh x math mod relation math mathcal e mathcal e ' mathcal e '' math short exact sequence math mathcal e ' mathcal e mathcal e '' math gives grothendieck group math k x math isomorphic math k x math math x math smooth group math k x math special also ring structure define math mathcal e cdot mathcal e ' sum k left operatorname tor k mathcal x mathcal e mathcal e ' right math using grothendieck riemann roch theorem


math operatorname ch k x otimes q x otimes q math isomorphism rings hence use math k x math intersection theory early history subject said begin alexander grothendieck used formulate grothendieck riemann roch theorem takes name german ''klasse'' meaning class grothendieck needed work coherent sheaves algebraic variety ''x'' rather working directly sheaves defined group using isomorphism classes sheaves generators group subject relation identifies extension two sheaves


sum resulting group called ''k'' ''x'' locally free sheaves used ''g'' ''x'' coherent sheaves either two constructions referred grothendieck group ''k'' ''x'' cohomological behavior ''g'' ''x'' homological behavior ''x'' smooth variety two groups smooth affine variety extensions locally free sheaves split group alternative definition topology applying construction vector bundles michael atiyah friedrich hirzebruch defined ''k'' ''x'' topological space ''x''


corresponding constructions involving auxiliary quadratic form received general name l theory major tool surgery theory string theory k theory classification ramond ramond field strengths charges stable branes first proposed examples properties k sub sub field easiest example grothendieck group grothendieck group point math text spec mathbb f math field math mathbb f math since vector bundle space finite dimensional vector space free object category coherent sheaves hence projective monoid


isomorphism classes math n math corresponding dimension vector space easy exercise show grothendieck group math z math k sub sub artinian algebra field one important property grothendieck group noetherian scheme math x math invariant reduction hence math k x k x text red math hence grothendieck group artinian math mathbb f math algebra direct sum copies math z math one connected component spectrum example math display block k left text spec left frac mathbb f x x times mathbb f right right mathbb z oplus mathbb z math k sub sub projective space one


commonly used computations grothendieck group computation math k mathbb p n math projective space field intersection numbers projective math x math computed embedding math x hookrightarrow mathbb p n math using push pull formula math mathcal e cdot mathcal f math makes possible concrete calculations elements math k x math without explicitly know structure since math display block k mathbb p n frac mathbb z n math one technique determining grothendieck group math mathbb p n math comes stratification math display block mathbb p n mathbb n coprod mathbb n coprod cdots coprod mathbb math since grothendieck group


coherent sheaves affine spaces isomorphic math mathbb z math intersection math mathbb n k mathbb n k math generically math display block mathbb n k cap mathbb n k mathbb n k k math math k k leq n math k sub sub projective bundle another important formula grothendieck group projective bundle formula given rank r vector bundle math mathcal e math noetherian scheme math x math grothendieck group projective bundle math mathbb p mathcal e operatorname proj operatorname sym bullet mathcal e vee math free math k x math module rank ''r'' basis math xi dots xi n math formula allows one compute grothendieck group math mathbb p n mathbb f math make possible compute math k math hirzebruch surfaces


addition used compute grothendieck group math k mathbb p n math observing projective bundle field math mathbb f math k sub sub singular spaces spaces isolated quotient singularities one recent technique computing grothendieck group spaces minor singularities comes evaluating difference math k x math math k x math comes fact every vector bundle equivalently described coherent sheaf done using grothendieck group singularity category math sg x math ref name ref derived noncommutative algebraic geometry gives long exact sequence starting math display block cdots


k x k x k sg x math higher terms come higher k theory note vector bundles singular math x math given vector bundles math e x sm math smooth locus math x sm hookrightarrow x math makes possible compute grothendieck group weighted projective spaces since typically isolated quotient singularities particular singularities isotropy groups math g math map math display block k x k x math injective cokernel annihilated math text lcm g ldots g k n math math n dim x math ref name sup pg sup k sub sub smooth projective curve


chow ring math x math essentially giving computation math k c math note math c math codimension math math points nontrivial parts spectral sequence math e q e q math hence math display block begin align e infty cong e cong text ch c e infty cong e cong text ch c end align math coniveau filtration used determine math k c math desired explicit direct sum since gives exact sequence math display block f k x k x k x f k x math left hand term isomorphic math text ch c cong text pic c math right hand term isomorphic math ch c cong mathbb z math


since math text ext text ab mathbb z g math sequence abelian groups splits giving isomorphism note math c math smooth projective curve genus math g math math mathbb c math math display block k c cong mathbb z oplus mathbb c g mathbb z g math moreover techniques using derived category singularities isolated singularities extended isolated cohen macaulay singularities giving techniques computing grothendieck group singular algebraic curve reduction gives generically smooth curve singularities cohen macaulay applications virtual bundles one useful application grothendieck group define virtual vector bundles example


embedding smooth spaces math hookrightarrow x math short exact sequence math omega omega x c x math math c x math conormal bundle math math math x math singular space math math embedded smooth space math x math define virtual conormal bundle math omega x omega math another useful application virtual bundles definition virtual tangent bundle intersection spaces let math subset x math projective subvarieties smooth projective variety define virtual tangent bundle


e x dots e x n sum infty frac x dots x n math chern character useful part facilitates computation chern class tensor product chern character used hirzebruch riemann roch theorem equivariant k theory equivariant algebraic k theory algebraic k theory associated category math operatorname coh g x math equivariant coherent sheaves algebraic scheme math x math action linear algebraic group math g math via quillen 's q construction thus definition math k g x pi b operatorname coh g x math particular math k g c math grothendieck group math operatorname coh g x math theory developed


r w thomason specifically proved equivariant analogs fundamental theorems localization theorem see also bott periodicity kk theory kr theory list cohomology theories algebraic k theory topological k theory operator k theory grothendieck riemann roch theorem notes references external links http abelharvardedu theses senior patrick patrickpdf grothendieck riemann roch https webusersimj prgfr maxkaroubi max karoubi 's page http wwwmathuiucedu k theory k theory preprint archive


''trichonephila clavipes'' '''arachnology''' scientific study arachnids comprise spiders related animals scorpions pseudoscorpions harvestmen study spiders arachnids '''arachnologists''' narrowly study spiders alone order araneae known araneology word arachnology derives greek ''arachn '' spider '' logia'' study particular subject arachnology science arachnologists primarily responsible classifying arachnids studying aspects biology popular imagination sometimes referred spider experts disciplines within arachnology include naming species determining

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Biology
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Arachnology


evolutionary relationships one another taxonomy systematics studying interact members species environment behavioural ecology distributed different regions habitats faunistics arachnologists perform research anatomy physiology arachnids including venom spiders scorpions others study impact spiders agricultural ecosystems whether used biological control agents subdisciplines arachnology broken several specialties topics include '''acarology''' study ticks mites '''araneology''' study spiders '' 'scorpiology'''


study scorpions arachnological societies arachnologists served number scientific societies national international scope main roles encourage exchange ideas researchers organise meetings congresses number cases publish academic journals also involved science outreach programs ''european spider year'' raise awareness animals among general public '''international''' international society arachnology isa http arachnologyorg website '''africa''' african arachnological society afras http afrasufsacza website '''asia''' arachnological society japan asj http wwwarachnologyjp index ephp website


asian society arachnology asa http asianarachnologycom website indian society arachnology http indianarachnologycom website iranian arachnological society ias https webarchiveorg web http arachnologyir website '''australasia''' australasian arachnological society http wwwaustralasian arachnologyorg website '''europe''' aracnofilia associazione italiana di aracnologia http wwwaracnofiliaorg website arachnologia belgica belgian arachnological society arabel http wwwarabelugentbe website arachnologische gesellschaft arages http aragesde website association francaise d'arachnologie asfra http wwwasfrafr site main publichtml website british arachnological society bas https britishspidersorguk website czech arachnological society http arachnologycz jazyk en website european society arachnology esa http wwweuropean arachnologyorg website grupo ib rico de aracnologia iberian peninsula http wwwsea entomologiaorg gia website magyar arachnolgia hungarian arachnology '''north america''' american arachnological society aas http wwwamericanarachnologyorg website arachnological journals scientific journals devoted


study arachnids include ''acarologia'' ''acta arachnologica'' published arachnological society japan ''arachnida rivista aracnologica italiana'' ''arachnology'' published british arachnological society ''arachnology letters'' published arachnologische gesellschaft ''international journal acarology'' ''journal arachnology'' published american arachnological society ''revista ib rica de aracnolog a'' published grupo ib rico de aracnolog ''revue arachnologique'' ''serket'' popular arachnology arachnids particularly tarantulas started become popular exotic pets many tarantulas thus become widely known common names mexican redknee tarantula ''brachypelma smithi''


various societies focus husbandry care study captive breeding tarantulas arachnids also typically produce journals newsletters articles advice subjects british tarantula society bts https wwwthebtscouk website deutsche arachnologische gesellschaft dearge http wwwdeargede website american tarantula society ats http atshqorg website see also cultural depictions spiders entomology arachnologists references external information links http wwwarachnologyorg international society arachnology https webarchiveorg web http wwwwashingtonedu burkemuseum spidermyth myths easyhtml spider myths spiders easy identify spiders humans subfields zoology subfields arthropodology


'''philosophy psychology''' refers many issues lie theoretical foundations modern psychology overview issues studied philosophy psychology epistemological concerns methodology psychological investigation example appropriate methodology psychology mentalism behaviorism compromise self reports reliable data gathering method conclusions drawn null hypothesis tests first person experiences emotions desires beliefs etc measured objectively scientific decision making practices laboratory routines used psychology cognitive neuroscience issues philosophy psychology

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philosophical questions nature mind brain cognition perhaps commonly thought part cognitive science philosophy mind cognitive module humans rational creatures psychological phenomena come standard required calling knowledge innateness philosophy psychology also closely monitors contemporary work conducted cognitive neuroscience evolutionary psychology artificial intelligence example questioning whether psychological phenomena explained using methods neuroscience evolutionary theory computational modeling respectively although closely related fields concerns still


arise appropriateness importing methods psychology concerns whether psychology study individuals information processing systems see donald broadbent autonomous happens brain even psychologists largely agree brain sense causes behavior see supervenience whether mind hard wired enough evolutionary investigations fruitful whether computational models anything offer possible implementations cognitive theories tell us nothing mind fodor pylyshyn philosophy psychology relatively young field scientific psychology


debate dualism monism discussed philosophy many centuries related philosophy psychology philosophical epistemological inquiries clinical psychiatry psychopathology philosophy psychiatry mainly concerned role values psychiatry derived philosophical value theory phenomenology values based practice aimed improving humanizing clinical decision making highly complex environment mental health care philosophy psychopathology mainly involved epistemological reflection implicit philosophical foundations psychiatric classification evidence based psychiatry aims unveil constructive activity underlying


description mental phenomena see also philosophy psychiatry philosophy social science moral psychology references reading http wwwuclacuk philosophy lpsg london philosophy study guide offers many suggestions read depending student 's familiarity subject http wwwuclacuk philosophy lpsg psychologyhtm philosophy psychology j stacy adams ''advances experimental social psychology'' academic press leonard berkowitz ''social psychology'' scott foresman co ned block ''readings philosophy psychology'' volume harvard university press stuart c brown royal institute philosophy macmillan original university michigan joseph margolis


algebra '''ring theory''' study rings algebraic structures addition multiplication defined similar properties operations defined integers ring theory studies structure rings representations different language modules special classes rings group rings division rings universal enveloping algebras well array properties proved interest within theory applications homological properties polynomial identities commutative rings much better understood noncommutative ones algebraic geometry algebraic number theory provide

Formal science
Mathematics
Pure mathematics
Algebra
Ring theory


many natural examples commutative rings driven much development commutative ring theory name ''commutative algebra'' major area modern mathematics three fields algebraic geometry algebraic number theory commutative algebra intimately connected usually difficult meaningless decide field particular result belongs example hilbert 's nullstellensatz theorem fundamental algebraic geometry stated proved terms commutative algebra similarly fermat 's last theorem stated terms elementary arithmetic part


understanding noncommutative rings especially noncommutative noetherian rings definitions ring basic concepts properties see ''ring mathematics '' definitions terms used throughout ring theory may found ''glossary ring theory'' commutative rings ring called ''commutative'' multiplication commutative commutative rings resemble familiar number systems various definitions commutative rings designed formalize properties integers commutative rings also important algebraic geometry commutative ring theory numbers often replaced ideals definition prime ideal tries capture


essence prime numbers integral domains non trivial commutative rings two non zero elements multiply give zero generalize another property integers serve proper realm study divisibility principal ideal domains integral domains every ideal generated single element another property shared integers euclidean domains integral domains euclidean algorithm carried important examples commutative rings constructed rings polynomials factor rings summary euclidean domain principal ideal domain unique factorization domain integral


domain commutative ring algebraic geometry algebraic geometry many ways mirror image commutative algebra correspondence started hilbert 's nullstellensatz establishes one one correspondence points algebraic variety maximal ideals coordinate ring correspondence enlarged systematized translating proving geometrical properties algebraic varieties algebraic properties associated commutative rings alexander grothendieck completed introducing schemes generalization algebraic varieties may built commutative ring precisely spectrum commutative ring space


prime ideals equipped zariski topology augmented sheaf rings objects affine schemes generalization affine varieties general scheme obtained gluing together purely algebraic methods several affine schemes analogy way constructing manifold gluing together charts atlas noncommutative rings noncommutative rings resemble rings matrices many respects following model algebraic geometry attempts made recently defining noncommutative geometry based noncommutative rings noncommutative rings associative algebras rings also vector spaces


transformations vector spaces studies modules abstract algebraic structures essence representation makes abstract algebraic object concrete describing elements matrices algebraic operations terms matrix addition matrix multiplication non commutative algebraic objects amenable description include groups associative algebras lie algebras prominent historically first representation theory groups elements group represented invertible matrices way group operation matrix multiplication relevant theorems general


isomorphism theorems rings nakayama 's lemma structure theorems artin wedderburn theorem determines structure semisimple rings jacobson density theorem determines structure primitive rings goldie 's theorem determines structure semiprime goldie rings zariski samuel theorem determines structure commutative principal ideal ring hopkins levitzki theorem gives necessary sufficient conditions noetherian ring artinian ring morita theory consists theorems determining two rings equivalent module categories cartan brauer hua theorem gives insight structure division rings wedderburn 's little theorem states finite domains fields skolem noether


theorem characterizes automorphisms simple rings structures invariants rings dimension commutative ring section ''r'' denotes commutative ring krull dimension ''r'' supremum lengths ''n'' chains prime ideals math mathfrak p subsetneq mathfrak p subsetneq cdots subsetneq mathfrak p n math turns polynomial ring math k cdots n math field ''k'' dimension ''n'' fundamental theorem dimension theory states following numbers coincide noetherian local ring math r mathfrak math krull dimension ''r'' minimum number generators


noetherian rings appear applications catenary ratliff proved noetherian local integral domain ''r'' catenary every prime ideal math mathfrak p math math operatorname dim r operatorname ht mathfrak p operatorname dim r mathfrak p math math operatorname ht mathfrak p math height math mathfrak p math ''r'' integral domain finitely generated ''k'' algebra dimension transcendence degree field fractions ''k'' ''s'' integral extension commutative ring ''r'' ''s'' ''r'' dimension closely related concepts depth global dimension general ''r''


equivalent category left modules ''r'' equivalent category left modules ''s'' fact two commutative rings morita equivalent must isomorphic notion add anything new category commutative rings however commutative rings morita equivalent noncommutative rings morita equivalence coarser isomorphism morita equivalence especially important algebraic topology functional analysis finitely generated projective module ring picard group let ''r'' commutative ring math mathbf p r math set isomorphism classes finitely generated projective


regular domain ie regular prime ideal pic r precisely divisor class group ''r'' example ''r'' principal ideal domain pic ''r'' vanishes algebraic number theory ''r'' taken ring integers dedekind thus regular follows pic ''r'' finite group finiteness class number measures deviation ring integers pid one also consider group completion math mathbf p r math results commutative ring k sub sub r note k sub sub r k sub sub two commutative rings


''r'' ''s'' morita equivalent structure noncommutative rings structure noncommutative ring complicated commutative ring example exist simple rings containing non trivial proper two sided ideals contain non trivial proper left right ideals various invariants exist commutative rings whereas invariants noncommutative rings difficult find example nilradical ring set nilpotent elements need ideal unless ring commutative specifically set nilpotent elements ring ''n'' x ''n'' matrices


division ring never forms ideal irrespective division ring chosen however analogues nilradical defined noncommutative rings coincide nilradical commutativity assumed concept jacobson radical ring intersection right left annihilators simple right left modules ring one example fact jacobson radical viewed intersection maximal right left ideals ring shows internal structure ring reflected modules also fact


intersection maximal right ideals ring intersection maximal left ideals ring context rings whether commutative noncommutative noncommutative rings serve active area research due ubiquity mathematics instance ring ''n'' ''n'' matrices field noncommutative despite natural occurrence geometry physics many parts mathematics generally endomorphism rings abelian groups rarely commutative simplest example endomorphism ring klein four group one best known noncommutative rings


division ring quaternions applications ring integers number field coordinate ring algebraic variety ''x'' affine algebraic variety set regular functions ''x'' forms ring called coordinate ring ''x'' projective variety analogous ring called homogeneous coordinate ring rings essentially things varieties correspond essentially unique way may seen via either hilbert 's nullstellensatz scheme theoretic constructions ie spec proj ring invariants basic perhaps


integers algebraic number fields algebraic function fields rings polynomials two variables noncommutative ring theory began attempts extend complex numbers various hypercomplex number systems genesis theories commutative noncommutative rings dates back early th century maturity achieved third decade th century precisely william rowan hamilton put forth quaternions biquaternions james cockle presented tessarines coquaternions william kingdon clifford enthusiast split biquaternions called ''algebraic motors'' noncommutative


algebras non associative lie algebras studied within universal algebra subject divided particular mathematical structure types one sign organization use direct sums describe algebraic structure various hypercomplex numbers identified matrix rings joseph wedderburn emil artin wedderburn 's structure theorems formulated finite dimensional algebras field artin generalized artinian rings emmy noether collaboration w schmeidler published paper theory ideals defined left right ideals ring following


year published landmark paper called ''idealtheorie ringbereichen'' analyzing ascending chain conditions regard mathematical ideals noted algebraist irving kaplansky called work revolutionary publication gave rise term noetherian ring several mathematical objects called ''noetherian'' notes references vol ii pure applied mathematics de ringtheorie ka ro inel algebr


riemann zeta function '' '' ' 's'' complex plane color point ' 's'' encodes value '' '' ' 's'' colors close black denote values close zero hue encodes value 's argument mathematics '''analytic number theory''' branch number theory uses methods mathematical analysis solve problems integers often said begun peter gustav lejeune dirichlet 's introduction dirichlet ''l'' functions give first proof dirichlet 's theorem arithmetic progressions well known results prime numbers involving prime number theorem

Formal science
Mathematics
Pure mathematics
Number theory
Analytic number theory


riemann zeta function additive number theory goldbach conjecture waring 's problem branches analytic number theory analytic number theory split two major parts divided type problems attempt solve fundamental differences technique multiplicative number theory deals distribution prime numbers estimating number primes interval includes prime number theorem dirichlet 's theorem primes arithmetic progressions additive number theory concerned additive structure integers goldbach 's conjecture


every even number greater sum two primes one main results additive number theory solution waring 's problem history precursors much analytic number theory inspired prime number theorem let ''x'' prime counting function gives number primes less equal ''x'' real number nbsp ''x'' example nbsp nbsp four prime numbers less equal prime number theorem states ''x'' ln ''x'' good approximation ''x''


sense limit ''quotient'' two functions ''x'' ''x'' ln ''x'' ''x'' approaches infinity math lim x infty frac pi x x ln x math known asymptotic law distribution prime numbers adrien marie legendre conjectured ''a'' approximated function ''a'' ''a'' ln ''a'' nbsp nbsp ''b'' ''a'' ''b'' unspecified constants second edition book number theory made precise conjecture ''a'' nbsp nbsp ''b'' nbsp nbsp minus carl friedrich gauss considered question im jahr oder according recollection nearly sixty years


later letter encke wrote logarithm table short note primzahlen unter math infty frac ln math gauss never published conjecture peter gustav lejeune dirichlet came approximating function logarithmic integral li ''x'' slightly different form series communicated gauss legendre 's dirichlet 's formulas imply conjectured asymptotic equivalence ''x'' ''x'' nbsp nbsp ln ''x'' stated although turned dirichlet 's approximation considerably better one considers differences instead


quotients dirichlet johann peter gustav lejeune dirichlet credited creation analytic number theory ref name princeton ref field found several deep results proving introduced fundamental tools many later named published dirichlet 's theorem arithmetic progressions using mathematical analysis concepts tackle algebraic problem thus creating branch analytic number theory proving theorem introduced dirichlet characters l functions ref name princeton ref name kanemitsu ref generalized arithmetic progressions theorem integers ring gaussian integers math mathbb z math ref name elstrodt ref


chebyshev two papers russian mathematician pafnuty l'vovich chebyshev attempted prove asymptotic law distribution prime numbers work notable use zeta function ' 's'' real values argument works leonhard euler early predating riemann 's celebrated memoir succeeded proving slightly weaker form asymptotic law namely limit ''x'' ''x'' ln ''x'' ''x'' goes infinity exists necessarily equal one able prove


unconditionally ratio bounded two explicitly given constants near ''x'' although chebyshev 's paper prove prime number theorem estimates ''x'' strong enough prove bertrand 's postulate exists prime number ''n'' ''n'' integer ''n'' nbsp nbsp riemann br br probable roots real course one would wish rigorous proof time fleeting vain attempts provisionally put aside search appears


subject number theory investigated riemann zeta function established importance understanding distribution prime numbers made series conjectures properties zeta function one well known riemann hypothesis hadamard de la vall e poussin extending ideas riemann two proofs prime number theorem obtained independently jacques hadamard charles jean de la vall e poussin appeared year proofs used methods complex analysis establishing main step proof riemann zeta function ' 's''


approximation transcendence theory expanded point techniques applied mordell conjecture problems results theorems results within analytic number theory tend exact structural results integers algebraic geometrical tools appropriate instead give approximate bounds estimates various number theoretical functions following examples illustrate multiplicative number theory euclid showed infinitely many prime numbers important question determine asymptotic distribution prime numbers rough description


many primes smaller given number gauss amongst others computing large list primes conjectured number primes less equal large number ''n'' close value integral math int n frac log dt math bernhard riemann used complex analysis special meromorphic function known riemann zeta function derive analytic expression number primes less equal real number nbsp ''x'' remarkably main term riemann 's formula exactly integral lending substantial


weight gauss 's conjecture riemann found error terms expression hence manner primes distributed closely related complex zeros zeta function using riemann 's ideas getting information zeros zeta function jacques hadamard charles jean de la vall e poussin managed complete proof gauss 's conjecture particular proved math pi x text number primes leq x math math lim x infty frac pi x x log x math remarkable result known ''prime number theorem''


central result analytic number theory loosely speaking states given large number ''n'' number primes less equal ''n'' ''n'' log ''n'' generally question asked number primes arithmetic progression ''a nq'' integer ''n'' one first applications analytic techniques number theory dirichlet proved arithmetic progression ''a'' ''q'' coprime contains infinitely many primes prime number theorem generalised problem letting math pi x q text number primes


leq x text p text arithmetic progression nq n mathbf z math ''a'' ''q'' coprime math lim x infty frac pi xaq phi q x log x math also many deep wide ranging conjectures number theory whose proofs seem difficult current techniques twin prime conjecture asks whether infinitely many primes ''p'' ''p'' nbsp nbsp prime assumption elliott halberstam conjecture proven recently infinitely many primes ''p'' ''p'' nbsp nbsp ''k'' prime


positive even ''k'' nbsp also proven unconditionally ie depending unproven conjectures infinitely many primes ''p'' ''p'' nbsp nbsp ''k'' prime positive even ''k'' nbsp additive number theory one important problems additive number theory waring 's problem asks whether possible ''k'' nbsp nbsp write positive integer sum bounded number ''k''th powers math n x k cdots x ell k math case squares ''k'' nbsp nbsp answered lagrange proved every positive integer


sum four squares general case proved hilbert using algebraic techniques gave explicit bounds important breakthrough application analytic tools problem hardy littlewood techniques known circle method give explicit upper bounds function ''g'' ''k'' smallest number ''k''th powers needed vinogradov 's bound math g k leq k log k math diophantine problems diophantine problems concerned integer solutions polynomial equations one may study distribution solutions counting solutions according measure


math e r r math general ''o'' ''r'' error term would possible unit circle properly closed unit disk replaced dilates bounded planar region piecewise smooth boundary furthermore replacing unit circle unit square error term general problem large linear function nbsp ''r'' therefore getting error bound form math r delta math math delta math case circle significant improvement first attain sierpi ski


showed math e r r math hardy landau showed one ''not'' math e r r math since goal show fixed math epsilon math exists real number math c epsilon math math e r leq c epsilon r epsilon math huxley showed math e r r math best published result methods analytic number theory dirichlet series one useful tools multiplicative number theory dirichlet series functions complex variable defined infinite


series form math f sum n infty nn math depending choice coefficients math n math series may converge everywhere nowhere half plane many cases even series converge everywhere holomorphic function defines may analytically continued meromorphic function entire complex plane utility functions like multiplicative problems seen formal identity math left sum n infty nn right left sum n infty b nn right sum n infty left sum k ell n kb ell right n math hence coefficients product two dirichlet series multiplicative convolutions original coefficients furthermore techniques partial summation tauberian theorems


used get information coefficients analytic information dirichlet series thus common method estimating multiplicative function express dirichlet series product simpler dirichlet series using convolution identities examine series complex function convert analytic information back information original function riemann zeta function euler showed fundamental theorem arithmetic implies least formally ''euler product'' math sum n infty frac n prod p infty frac p text math product taken


prime numbers ''p'' euler 's proof infinity prime numbers makes use divergence term left hand side ' 's'' called harmonic series purely analytic result euler also first use analytical arguments purpose studying properties integers specifically constructing generating power series beginning analytic number theory later riemann considered function complex values ' 's'' showed function extended meromorphic function entire plane simple pole


non trivial zeros lie line math math never provided proof statement famous long standing conjecture known ''riemann hypothesis'' many deep implications number theory fact many important theorems proved assumption hypothesis true example assumption riemann hypothesis error term prime number theorem nowrap math x varepsilon math early th century g h hardy littlewood proved many results zeta function attempt


prove riemann hypothesis fact hardy proved infinitely many zeros zeta function critical line math z math led several theorems describing density zeros critical line see also maier 's matrix method automorphic l function automorphic form notes references references reading ayoub ''introduction analytic theory numbers'' h l montgomery r c vaughan ''multiplicative number theory i'' ''classical theory'' h iwaniec e kowalski ''analytic number theory'' j newman ''analytic number theory'' springer specialized aspects


following books become especially well known h halberstam h e richert ''sieve methods'' r c vaughan ''the hardy littlewood method'' nd edn certain topics yet reached book form depth examples montgomery 's pair correlation conjecture work initiated ii new results goldston pintz yilidrim small gaps primes iii green tao theorem showing arbitrarily long arithmetic progressions primes exist


'''food science''' basic science applied science food scope starts overlap agricultural science nutritional science leads scientific aspects food safety food processing informing development food technology food science brings together multiple scientific disciplines incorporates concepts fields chemistry physics physiology microbiology biochemistry food technology incorporates concepts chemical engineering example activities food scientists include development new food products design processes produce foods choice packaging materials shelf life studies sensory evaluation products using

Applied science
Agriculture
Food science


survey panels potential consumers well microbiological chemical testing ref name iftorg food science activity guides ref food scientists may study fundamental phenomena directly linked production food products properties definition institute food technologists defines food science discipline engineering biological physical sciences used study nature foods causes deterioration principles underlying food processing improvement foods consuming public textbook ''food science'' defines food science simpler terms application basic


sciences engineering study physical chemical biochemical nature foods principles food processing disciplines subdisciplines food science described food chemistry food chemistry study chemical processes interactions biological non biological components foods biological substances include items meat poultry lettuce beer milk similar biochemistry main components carbohydrates lipids protein also includes areas water vitamins minerals enzymes food additives flavors colors discipline also encompasses products


change certain food processing techniques ways either enhance prevent happening food physical chemistry food physical chemistry study physical chemical interactions foods terms physical chemical principles applied food systems well application physicochemical techniques instrumentation study analysis foods food engineering pizza factory germany example food engineering food engineering industrial processes used manufacture food food microbiology food microbiology laboratory faculty food technology latvia university


life sciences technologies food microbiology study microorganisms inhabit create contaminate food including study microorganisms causing food spoilage ref name fratamicobayles ref good bacteria however probiotics becoming increasingly important food science ref name tannockpro ref ref name ljunghwadstrom ref ref name mayob ref addition microorganisms essential production foods cheese yogurt bread beer wine fermented foods food technology food technology technological aspects early scientific research food technology concentrated food preservation nicolas appert 's development canning process decisive event


process n't called canning appert really know principle process worked canning major impact food preservation techniques foodomics foodomics defined discipline studies food nutrition domains application integration advanced omics technologies improve consumer 's well health knowledge foodomics requires combination food chemistry biological sciences data analysis foodomics greatly helps scientists area food science nutrition gain better access data used analyze effects


food human health etc believed another step towards better understanding development application technology food moreover study foodomics leads omics sub disciplines including nutrigenomics integration study nutrition gene omics molecular gastronomy molecular gastronomy subdiscipline food science seeks investigate physical chemical transformations ingredients occur cooking program includes three axes cooking recognized three components social artistic technical quality control technician performing quality control quality


control involves causes prevention communication dealing food borne illness quality control also ensures product meets specs ensure customer receives expect packaging physical properties product sensory analysis sensory analysis study consumers' senses perceive food careers food science five common college degrees leading career food science food science technology biological sciences business marketing nutrition chemistry ref name ref careers available food scientists include food technologist research development r quality control flavor chemistry


science united states united states food science typically studied land grant universities country 's pioneering food scientists women attended chemistry programs land grant universities state run largely state mandates allow sex blind admission graduated difficulty finding jobs due widespread sexism chemistry industry late th early th centuries finding conventional career paths blocked found alternative employment instructors home economics departments used base launch foundation many modern food science programs


main us organization regarding food science food technology institute food technologists ift headquartered chicago illinois us member organisation international union food science technology iufost see also publications books food science academic topic food science books textbooks journals notes references reading wanucha genevieve february https wwwtechnologyreviewcom two happy clams two happy clams friendship forged food science ''mit technology review'' external links learn http wwwiftorg knowledge center learn food scienceaspx food science applied sciences


biological machines '''biological engineering''' '''bioengineering''' application principles biology tools engineering create usable tangible economically viable products biological engineering employs knowledge expertise number pure applied sciences mass heat transfer kinetics biocatalysts biomechanics bioinformatics separation purification processes bioreactor design surface science fluid mechanics thermodynamics polymer science used design medical devices diagnostic equipment biocompatible materials renewable energy ecological engineering agricultural engineering process engineering catalysis areas improve living standards societies examples

Natural science
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bioengineering research include bacteria engineered produce chemicals new medical imaging technology portable rapid disease diagnostic devices prosthetics biopharmaceuticals tissue engineered organs bioengineering overlaps substantially biotechnology biomedical sciences way analogous various forms engineering technology relate various sciences aerospace engineering space technology kinetics astrophysics general biological engineers attempt either mimic biological systems create products modify control biological systems working doctors clinicians researchers bioengineers use traditional engineering principles techniques address biological


engineering begun recognized branch engineering new concept people post wwii grew rapidly term bioengineering coined british scientist broadcaster heinz wolff national institute medical research wolff graduated year became director division biological engineering university first time bioengineering recognized branch university electrical engineering early focus discipline due work medical devices machinery time ref name ref engineers


protein chemistry cytology immunology neurobiology neuroscience cellular tissue based systems including devices sensors whole macroscopic organisms plants animals even biomes ecosystems education average length study three five years completed degree signified bachelor engineering bs engineering fundamental courses include thermodynamics biomechanics biology genetic engineering fluid mechanical dynamics chemical enzyme kinetics electronics materials properties ref name linsenmeier linsenmeier ra http wwwvanthorg curriculum def bme currpdf defining undergraduate biomedical engineering curriculum ref ref name johnson ref sub disciplines cellular automata nearest neighbor interactions depending institution particular definitional boundaries


employed major branches bioengineering may categorized note may overlap biomedical engineering application engineering principles design concepts medicine biology healthcare purposes ref name ref tissue engineering genetic engineering neural engineering pharmaceutical engineering clinical engineering bioinformatics biomechanics biochemical engineering fermentation engineering application engineering principles microscopic biological systems used create new products synthesis including production protein suitable raw materials ref name biological systems engineering application engineering principles design concepts agriculture food sciences ecosystems ref name bioprocess engineering develops technology monitor


conditions process making pharmaceuticals takes place ref name ex bioprocess design biocatalysis bioseparation bioinformatics bioenergy environmental health engineering application engineering principles control environment health comfort safety human beings includes field life support systems exploration outer space ocean ref name human factors engineering application engineering physiology psychology optimization human machine relationship ref name biotechnology use living systems organisms develop make products ex pharmaceuticals ref name biomimetics imitation models systems


elements nature purpose solving complex human problems ex velcro designed george de mestral noticed easily burs stuck dog 's hair bioelectrical engineering biomechanical engineering bionics integration biomedical focused robotics assisted technologies ex prosthetics ref name bioprinting utilizing biomaterials print organs new tissues biorobotics ex prosthetics systems biology study biological systems organizations accreditation board engineering technology abet ref name abet http wwwabetorg accreditation abet accreditation accessed ref us based accreditation board engineering bs programs makes distinction biomedical engineering biological engineering though


much overlap see american institute medical biological engineering aimbe made members main goal educate public value biological engineering world well invest research programs advance field give awards dedicated innovation field awards achievement field direct contribution biological engineering recognize encourage public continue forward movement institute biological engineering ibe


non profit organization run donations alone aim encourage public learn continue advancements biological engineering like aimbe perform research directly however offer scholarships students show promise field society biological engineering sbe technological community associated american institute chemical engineers aiche sbe hosts international conferences global organization leading engineers scientists dedicated advancing integration biology engineering references external links http wwwbioengineerorg bioengineering society http bmesorg biomedical engineering society http wwwibeorg institute biological engineering https webarchiveorg web http wwwsysbioengcom stes sysbiology html


benjoe institute systems biological engineering http wwwaimbeorg american institute medical biological engineering http wwwasabeorg american society agricultural biological engineers http wwwaicheorg sbe indexaspx society biological engineering part aiche http wwwjbiolengorg journal biological engineering jbe https webarchiveorg web http wwwasabeorg publications publications periodicalsaspx biological engineering transactions https wwwbioengineeringtumde en munich school bioengineering


land cover surrounding madison wi fields colored yellow brown water colored blue urban surfaces colored red impervious surfaces surrounding madison wi canopy cover surrounding madison wi '''landscape ecology''' science studying improving relationships ecological processes environment particular ecosystems done within variety landscape scales development spatial patterns organizational levels research policy ref name wu ref concisely landscape ecology described science landscape diversity synergetic result biodiversity geodiversity highly interdisciplinary field systems science landscape

Social science
Geography
Physical geography
Landscape ecology


ecology integrates biophysical analytical approaches humanistic holistic perspectives across natural sciences social sciences landscapes spatially heterogeneous geographic areas characterized diverse interacting patches ecosystems ranging relatively natural terrestrial aquatic systems forests grasslands lakes human dominated environments including agricultural urban settings ref name wu ref name turner ref ref name forman ref salient characteristics landscape ecology emphasis relationship among pattern process scale focus broad scale ecological environmental issues necessitate coupling biophysical socioeconomic sciences key research topics landscape ecology


include ecological flows landscape mosaics land use land cover change scaling relating landscape pattern analysis ecological processes landscape conservation sustainability landscape ecology also studies role human impacts landscape diversity development spreading new human pathogens could trigger epidemics terminology german term ''landschafts kologie'' thus ''landscape ecology'' coined german geographer carl troll developed terminology many early concepts landscape ecology part early work consisted applying aerial photograph interpretation studies interactions environment vegetation


explanation heterogeneity measure parts landscape differ one another landscape ecology looks spatial structure affects organism abundance landscape level well behavior functioning landscape whole includes studying influence pattern internal order landscape process continuous operation functions organisms landscape ecology also includes geomorphology applied design architecture landscapes ref name allaby ref geomorphology study geological formations responsible structure


included human caused landscape changes theory application concepts ref name sanderson ref landscape ecology discrete established discipline marked organization http wwwlandscape ecologyorg international association landscape ecology iale landmark book publications defined scope goals discipline including naveh lieberman ref name naveh ref forman godron ref name forman ref ref name ryszkowski ref forman ref name forman wrote although study ecology spatial configuration human scale barely decade old strong potential theory development application conceptual framework today theory application


landscape ecology continues develop need innovative applications changing landscape environment landscape ecology relies advanced technologies remote sensing gis models associated development powerful quantitative methods examine interactions patterns processes ref name turner example would determining amount carbon present soil based landform landscape derived gis maps vegetation types rainfall data region remote sensing work used extend landscape ecology field predictive vegetation mapping


instance janet franklin definitions conceptions landscape ecology nowadays least six different conceptions landscape ecology identified one group tending toward disciplinary concept ecology subdiscipline biology conceptions another group characterized interdisciplinary study relations human societies environment inclined toward integrated view geography conceptions ref name kirchhoff following quotations descriptions come source ref ''interdisciplinary analysis subjectively defined landscape units'' eg neef school ref name neef ref ref name haase ref


landscapes' characteristic patterns ecosystems investigates flux energy mineral nutrients species among component ecosystems providing important knowledge addressing land use issues ''organism centered multi scale topological ecology'' eg john wiens ref name wiens ref ref name wiens ref explicitly rejecting views expounded troll zonneveld naveh forman godron etc landscape landscape ecology defined independently human perceptions interests modifications nature 'landscape' defined regardless scale 'template' spatial patterns influence ecological processes humans rather respective species studied


societies' cultural social economic dimensions regarded integral part global ecological hierarchy landscapes claimed manifest systems 'total human ecosystem' naveh encompasses physical 'geospheric' mental 'noospheric' spheres ''ecology guided cultural meanings lifeworldly landscapes'' frequently pursued practice defined see eg hard trepl ref name kirchhoff landscape ecology defined ecology guided external aim namely maintain develop lifeworldly landscapes provides ecological knowledge necessary achieve goals investigates


sustain develop populations ecosystems material 'vehicles' lifeworldly aesthetic symbolic landscapes time ii meet societies' functional requirements including provisioning regulating supporting ecosystem services thus landscape ecology concerned mainly populations ecosystems resulted traditional regionally specific forms land use relationship ecological theory research programmes landscape ecology theory namely standing european tradition may slightly outside classical preferred domain scientific disciplines large heterogeneous areas


thus defined anthropocentric relativistic way according richard forman michel godron ref name forman landscape heterogeneous land area composed cluster interacting ecosystems repeated similar form throughout whereby list woods meadows marshes villages examples landscape 's ecosystems state landscape area least kilometres wide john wiens ref name wiens ref name wiens opposes traditional view expounded carl troll isaak zonneveld zev naveh richard forman michel godron others