Integral Categories and Calculus Categories
|
|
- Rodger Cox
- 5 years ago
- Views:
Transcription
1 Intgral Catgoris and Calculus Catgoris Robin Cocktt 1 and Jan-Simon Lmay 2 1 Dpt. o Computr Scinc, Univrsity o Calgary, Calgary, AB, Canada robin@ucalgary.ca 2 Dpt. o Mathmatics and Statistics, Univrsity o Calgary, Calgary, AB, Canada jansimon.lmay@ucalgary.ca Abstract Dirntial catgoris ar now an stablishd abstract stting or dirntiation. Th papr prsnts th paralll dvlopmnt or intgration by axiomatizing an intgral transormation, s A : A, in a symmtric monoidal catgory with a coalgbra modality. Whn intgration is combind with dirntiation, th two undamntal thorms o calculus ar xpctd to hold (in a suitabl sns): a dirntial catgory with intgration which satisis ths two thorm is calld a calculus catgory. Modiying an approach to antidrivativs by T. Ehrhard, it is shown how xampls o calculus catgoris aris as dirntial catgoris with antidrivativs in this nw sns. Having antidrivativs amounts to dmanding that a crtain natural transormation, K :, is invrtibl. W obsrv that a dirntial catgory having antidrivativs, in this sns, is always a calculus catgory and w provid xampls o such catgoris ACM Subjct Classiication F.3.3 Studis o Program Constructs, F.4.1 Mathmatical Logic Kywords and phrass Dirntial Catgoris, Intgral Catgoris, Calculus Catgoris Digital Objct Idntiir /LIPIcs.CSL Introduction Th two undamntal thorms o calculus rlat th two most important oprations o calculus: dirntiation and intgration. Th irst thorm stats that th drivativ o th intgral o a ral unction is th original unction: d( t (x) dx) a dt (x) (x). Whil th scond stats that th intgral o th drivativ o a ral unction on a closd intrval [a, b] is qual to th dirnc o valuatd at th nd points: b d(t) a dt (x) dt (b) (a). Thy ar calld undamntal" thorms bcaus thy ar absolutly undamntal to th dvlopmnt o classical calculus. Sinc th turn o th 21 st cntury, thr has bn signiicant progrss in th abstract undrstanding o dirntiation with th study o dirntial catgoris. Th abstract ormulation o intgration, on th othr hand, has not rcivd th sam lvl o attntion. Nonthlss, on might xpct that, whn suitably adjoind to th ormulation o dirntiation, a commnsurat abstract orm or intgration should ncompass ths undamntal thorms. Th purpos o this xtndd abstract is to xplor th xtnt to which this xpctation is ralizd. This work was partially supportd by NSERC, Canada. Robin Cocktt and Jan-Simon Lmay; licnsd undr Crativ Commons Licns CC-BY 26th EACSL Annual Conrnc on Computr Scinc Logic (CSL 2017). Editors: Valntin Goranko and Mads Dam; Articl No. 20; pp. 20:1 20:17 Libniz Intrnational Procdings in Inormatics Schloss Dagstuhl Libniz-Zntrum ür Inormatik, Dagstuhl Publishing, Grmany
2 20:2 Intgral Catgoris and Calculus Catgoris In th arly 2000 s, T. Ehrhard and L. Rgnir introducd th dirntial λ-calculus [12] and dirntial proo nts [13], which ormalizd dirntiation in linar logic. A w yars latr, R. Blut, R. Cocktt and R. Sly introducd dirntial catgoris [7], which wr th appropriat catgorical structur or modlling Ehrhard and Rgnir s dirntial linar logic. Dirntial catgoris now hav a rich litratur o thir own [4, 3, 2, 6, 14, 10, 9] and thr ar many xampls which hav bn xtnsivly studid [7, 11, 5]. Howvr, as mntiond abov, littl attntion has bn givn to abstracting intgration. In 2014, T. Ehrhard obsrvd that in crtain -autonomous catgoris which had th appropriat structur to b a dirntial catgory, it was possibl with on additional assumption to produc antidrivativs [11]. Th additional assumption was that a crtain natural transormation which h calld J constructd rom th driving transormation was a natural isomorphism. With this assumption, Ehrhard constructd an intgral transormation with an invrs bhaviour to th driving transormation, in th sns that h gav ncssary and suicint conditions or a map to satisy th irst undamntal thorm o calculus by proving Poincaré s Lmma. Furthrmor, whn th driving transormation satisid an xtra non-quational condition, which h calld th Taylor Proprty", h thn showd that vry dirntiabl unction satisid th scond undamntal thorm o calculus. Whil much o th inspiration or our approach to intgration drivs rom ths obsrvations, Ehrhard mad no attmpt to axiomatiz intgration sparatly rom dirntiation. Hr w introduc (tnsor) intgral catgory as a notion which stands on its own (i.. in th absnc o dirntiation). Th inspiration or this indpndnt axiomatization o intgral catgoris coms rom th much oldr notion o a Rota-Baxtr algbra [1, 20, 15], th classical algbraic abstraction o intgration. Brily, or a commutativ ring R and λ R, a Rota-Baxtr algbra o wight λ is an R-algbra A with an R-linar morphism P : A A which satisis th Rota-Baxtr rul: P (a)p (b) P (ap (b)) + P (P (a)b) + λp (ab) or all a, b A. Th map P is calld a Rota-Baxtr oprator o wight λ. A particular xampl o a Rota-Baxtr algbra o wight zro is th R-algbra o ral continuous unctions Cont(R), whr th Rota-Baxtr oprator P : Cont(R) Cont(R) is dind as th intgral o th unction cntrd at zro: P ()(x) x (t) dt. Th Rota-Baxtr rul or 0 this xampl is th xprssion o th intgration by parts rul without th us o drivativs: x 0 (t) dt x 0 g(t) dt x 0 (t) ( t 0 g(u) du) dt + x 0 ( t (u) du) g(t) dt (s [15] or mor 0 dtails). This motivats th Rota-Baxtr rul as an axiom o intgration. Whn dirntiation and intgration ar combind into what w call hr a calculus catgory, w dmand that th two undamntal thorms hold. Th scond undamntal thorm is assumd to hold vrbatim. Howvr, th irst undamntal thorm, as abov, has to b intrprtd as bing on maps rathr than objcts and, undr this intrprtation, bcoms th Poincaré proprty, a conditional proprty which provids ncssary and suicint conditions or a map to b th dirntial o its intgral. Th nam o th condition coms rom th Poincaré Lmma rom cohomology [22] and dirntial topology [8], which stats an analogous rsult o giving critria or a map to b an antidrivativ. To obtain th notion o intgration as an antidrivativ, w insist that a slightly dirnt natural transormation, which w call K, should b invrtibl. W show this is quivalnt to rquiring both that Ehrhard s transormation J is invrtibl and that th Taylor Proprty which Ehrhard had suggstd was dsirabl holds. This improvmnt is asily undrstimatd: th Taylor Proprty is a conditional rquirmnt, rplacing a conditional rquirmnt by a purly quational rquirmnt is always, mathmatically, a signiicant stp. Dmanding that K is invrtibl not only producs an intgral transormation, but also scurs th irst and scond undamntal thorm o calculus. Invrting only Ehrhard s transormation, J,
3 J. R. B. Cocktt and J.-S. Lmay 20:3 dos not by itsl vn produc an intgral transormation; th Taylor Proprty is rquird, in addition to th invrtibility o J, to scur an intgral transormation. Th act that, whn K is invrtibl, J is invrtibl is usul particularly in th proo o th Poincaré s lmma. Thus, it is important to obsrv that, th antidrivativ producd by th invrs o K is prcisly th sam as th antidrivativ producd by th invrs o J whn K is alrady invrtibl. Finally, th notion o a dirntial catgory with anti-drivativs, givn by rquiring K to b invrtibl, provids a plntiul supply o calculus catgoris as w xplain. Bor bginning, w should mntion th convntions that w us in th papr. First o, w will us diagrammatic ordr or composition. Explicitly, this mans that th composit map g is th map which irst dos thn g. Scondly, to simpliy working in symmtric monoidal catgoris, w will allow ourslvs to work in strict symmtric monoidal catgoris and so will gnrally supprss th associator and unitor isomorphisms. For a symmtric monoidal catgory w will us or th tnsor product, I or th monoidal unit, and σ : A B B A or th symmtry isomorphism. Full dtaild proos o all th rsults in this xtndd abstract can b ound in th scond author s mastrs thsis [18]. 2 Coalgbra Modalitis Tnsor intgral and dirntial catgoris ar structurs ovr additiv symmtric monoidal catgoris with a coalgbra modality. W bgin by rcalling th componnts o this structur starting with th notion o an additiv catgory. Hr w man additiv in th sns o bing commutativ monoid nrichd. Thus, w do not assum ngativs nor do w assum biproducts (this dirs rom th usag in [19] or xampl). This allows many important xampls such as th catgory o sts and rlation or th catgory o moduls o a commutativ rig 1. Dinition 1. An additiv catgory is a commutativ monoid nrichd catgory, that is a catgory in which ach hom-st is a commutativ monoid with addition opration + and zro 0 and in which composition prsrvs addition that is: [Add.1] k( +g) k +kg and 0 0; [Add.2] ( +g)hh+gh and 0 0. An additiv symmtric monoidal catgory is an additiv catgory with a tnsor product which is compatibl with th additiv structur in th sns that: [Add.1] ( +g) h h+g h and 0 h0; [Add.2] k ( +g)k +k g and h 00. In any additiv catgory thir is a notion o scalar multiplication o maps by th natural numbrs N. Th scalar multiplication o a map : A B by n N, is th map n : A B dind by summing n copis o togthr. I n 0, thn 0 is simply th zro map rom A to B. Furthrmor, or additiv symmtric monoidal catgoris, on thn has that (n ) g n ( g) (n g). Dinition 2. A coalgbra modality [7, 3] on a symmtric monoidal catgory is a quintupl (!, δ,,, ) consisting o a comonad (!, δ, ), a natural transormation with 1 Rigs ar also known as a smirings: thy ar rings without ngativs. C S L
4 20:4 Intgral Catgoris and Calculus Catgoris componnts A :, and a natural transormation with componnts A : I such that or ach objct A: (i) (, A, A ) is a cocommutativ comonoid, that is, th ollowing diagrams commut: σ (ii) δ A prsrvs th comultiplication, that is, th ollowing diagram commuts: δ! δ δ!! Whn combind with th additiv structur, this nsurs that is a coalgbra in th classical algbraic sns. Furthrmor, on can prov that δ so δ is actually a comonoid homomorphism. Th coklisli maps or th comonad ar important: ths maps ar o th orm : B: amongst ths ar th linar maps g : B whr g : A B. Not that w do not assum that th coalgbra modality,!, is a monoidal unctor: to do so would put us in th ralm o Sly catgoris [3, 14] which is mor than w rquir or this basic thory. 3 Intgral Catgoris Intgral catgoris ar th intgral analogu o dirntial catgoris, thus, th main ingrdint o an intgral catgory is an intgral transormation, s A : A, a natural transormation opposit in orintation to a driving transormation which must satisy just thr quations: Dinition 3. An additiv symmtric monoidal catgory with a coalgbra modality is an intgral catgory i thr is a natural transormation s A : A, calld th intgral transormation, satisying th ollowing quations: [s.1] Constants Rul: s( 1) [s.2] Rota-Baxtr Rul: (s s) s( 1)(s 1 1) + s( 1)(1 σ)(1 1 s) [s.3] Intrchang Rul: s(s 1) s(s 1)(1 σ) Th intgral o a map : A B is dind as th composition S[] : s A : B. This should b thought o as th classical intgral o valuatd rom 0 to x as a unction o x: S[](x) : x (t) dt. To intrprt this as S[] on must rgard as bing a unction o 0 two variabls t and dt, which is linar in dt. Classically, is rgardd as a unction o on (on dimnsional) variabl, t, and to obtain th intrprtation as a unction o two argumnts on simply multiplis by th variabl dt. This allows a simpl intrprtation o th intgral notation or on dimnsional unctions: it lavs opn th intrprtation or multidimnsional unctions an issu to which w shall rturn. Th additiv structur o th catgory nsurs th intgral o a sum o maps is qual to th sum o th intgral o ach map, that is, S[ + g] S[] + S[g] and S[0] 0. Th irst axiom [s.1] stats that th intgral o a constant map is a linar map (in th sns discussd abov). Th scond axiom [s.2] is th Rota-Baxtr rul [15], which is an xprssion
5 J. R. B. Cocktt and J.-S. Lmay 20:5 o intgration by parts using only intgrals. In classical calculus notation, th Rota-Baxtr rul is xprssd as: x 0 (t) dt x 0 g(t) dt x 0 (t) ( t 0 g(u) du) dt + x 0 ( t (u) du) g(t) dt. 0 Th third axiom [s.3] nsurs th indpndnc o th ordr o intgration th intrchang law that is intgrating with rspct to u thn t is th sam as intgrating with rspct to t thn u. It may b tmpting to think this is rlatd to Fubini s thorm. In act, it is not closly rlatd at all: w discuss this at th nd o this sction. [s.3] can b xprssd in classical notation as: x 0 ( t 0 (u) du dt) x 0 ( u (t) dt du). 0 Just lik dirntial catgoris, intgral catgoris hav a graphical calculus (s [21] or an introduction to th graphical calculus in monoidal catgoris and its variations). W rprsnt th intgral transormation in string diagrams as ollows (which should b rad rom top to bottom): s : A Th intgral axioms [s.1], [s.2] and [s.3] ar thn rprsntd in th graphical calculus as ollows (w omit writing th objcts at th nd o th wirs). [s.1] Constants Rul: [s.2] Rota-Baxtr Rul: + [s.3] Intrchang Rul: With th graphical calculus, w ar now in a position to xplor polynomial intgration. Prhaps th irst ormula larnt in irst yar calculus is x 0 xn dx 1 n+1 xn+1. Howvr this ormula cannot b xprssd in a gnral additiv catgory simply bcaus thr may not b ractions. That said, w will soon s that in vry intgral catgory thr is a notion o scalar multiplication by positiv rationals, that is, crtain hom-sts ar Q 0 -moduls, whr Q 0 is th rig o non-ngativ rationals. Th intgral o monomials idntity can b r-xprss as th rquirmnt that (n + 1) x 0 xn dx x n+1 and this idntity dos hold in any intgral catgory! To xprss this idntity in an intgral catgory, w will nd th n-old comultiplication n : n which is dind as n ( 1)( 1 1)...( 1 n 2 ). By convntion w st 0, 1 1 and 2. C S L
6 20:6 Intgral Catgoris and Calculus Catgoris Thorm 4. For vry n N, th intgral transormation satisis th polynomial idntity: [s. Poly] (n + 1) s( n 1)( n 1) n+1 ( n+1 ) n+1 (n + 1) n Proo. Th bauty o this proo is that it uss vry intgral transormation axiom. Th proo is much smoothr using th graphical calculus, which is quivalnt to proos don algbraically as shown in [16]. W will prov th quality or th intgral transormation by induction on n. For th bas cas o n 0, this quality holds dirctly by th constant rul [s.1]. Assum th induction hypothsis [s. Poly] holds or n, w now show it or n + 1: n+2 n+1 (n + 1) n ( ) ( ) (n + 1) n + n (n + 1) n + n n+1 + (n + 1) n+1 n+1 + (n + 1) 2 n n+1 + (n + 1) 2 n+1 + (n + 1) n+1 n An important consqunc o polynomial intgration is that crtain hom-sts ar Q 0 - moduls. In an additiv catgory, or vry objct A and or vry natural numbr n N, din th map n A : A A by summing n copis o th idntitis: n A n 1 A. W will now prov that in any intgral catgory that or vry objct A and n 2, th map n is invrtibl. Thorm 5. In an intgral catgory, or vry natural numbr n N, n 2, and vry objct A X, th map n : is an isomorphism. Rmark. Notic that th cas n 1 is also tru sinc th idntity map is an isomorphism.
7 J. R. B. Cocktt and J.-S. Lmay 20:7 Proo. W will simply din th invrs o n. For ach objct A and n 2, din n 1 :, as: n 1 δ As ( n 1 1)( n 1 1)(1 n 1 ), writtn in th graphical calculus as: δ n 1 n 1 This implis, in an intgral catgory, hom-sts with domain ar Q 0 -moduls. Th scalar multiplication o a map : B with a non-ngativ rational p q Q 0 is th map p q : B dind as p q q 1 (p ). Finally, w discuss th intrprtation o Fubini s thorm. Th thorm rquirs that th coalgbra modality is monoidal and, thus, that thr is a Sly isomorphism [2, 14]: χ :!B!(A B). Fubini s thorm concrns th doubl intgration o a unction o th orm :!(A B) A B C whos typ nsurs it is bilinar in th scond two occurrncs o A and B. Functions o this orm can b intgratd with rspct to ithr A or B, or both A and B: th lattr, th doubl intgral o, is obtaind as ollows:!(a B) χ 1!B s s A!B B 1 σ 1!B A B χ 1 1!(A B) A B C Fubini s thorm assrts that th ordr o intgration in this doubl intgral dos not mattr. At this lvl o gnrality this ordr indpndnc is an immdiat consqunc o th biunctoriality o _ _. 4 Calculus Catgoris In this sction w wish to put intgration togthr with dirntiation and to discuss how thy should intract. W start by brily rcalling th dinition o a dirntial catgory [7] bor introducing calculus catgoris whos structur is inducd by th undamntal thorms o calculus. Dinition 6. An additiv symmtric monoidal catgory with a coalgbra modality is a dirntial catgory i th coalgbra modality coms quippd with a driving transormation [7], that is, a natural transormation d with componnts d A : A, satisying th ollowing quations: [d.1] Constant Rul: d 0 [d.2] Libniz Rul: d ( 1)(1 σ)(d 1) + ( 1)(1 d) [d.3] Linar Rul: d ( 1)λ [d.4] Chain Rul: dδ ( 1)(δ 1 1)(1 d)d [d.5] Intrchang Rul: (d 1)d (1 σ)(d 1)d Th drivativ o a map : B is th composition D[] : d A : A B. Th irst axiom, [d.1], stats that th drivativ o a constant map is zro. Th scond axiom [d.2] is th Libniz rul or dirntiation also calld th product rul. Th third axiom [d.3] says that th drivativ o a linar map is a constant. Th ourth axiom [d.4] is th chain rul. Th last axiom [d.5] is th indpndnc o dirntiation or th intrchang law, which naivly stats that dirntiating with rspct to x thn y is th sam as dirntiation with rspct to y thn x. It should b notd that [d.5] was not a rquirmnt in [7] but C S L
8 20:8 Intgral Catgoris and Calculus Catgoris was latr addd to th dinition [3, 3] to nsur that th coklisli catgory o a dirntial catgory was a Cartsian dirntial catgory. As prviously statd, dirntial catgoris hav a graphical calculus. Th driving transormation is rprsntd as ollows: A d Th string diagram rprsntations o [d.1] to [d.5] ar as ollows: [d.1] Constant Rul: 0 [d.2] Libniz Rul: + [d.3] Linar Rul: [d.4] Chain Rul: δ δ [d.5] Intrchang Rul: W ar now rady to tackl th intraction btwn intgration and dirntiation. W start with th scond undamntal thorm o calculus and rturn to discuss th irst undamntal thorm o calculus: Dinition 7. Lt X b a dirntial catgory and an intgral catgory with driving transormation d and intgrating transormation s on th sam coalgbra modality (!, δ,,, ).
9 J. R. B. Cocktt and J.-S. Lmay 20:9 (i) d and s ar said to satisy th Scond Fundamntal Thorm o Calculus i: sd +!0 1, writtn in th graphical calculus as: +!0 (ii) d and s ar said to b compatibl i: dsd d, writtn in th graphical calculus as: (iii) d is said to b Taylor i or vry pair o maps, g : C B, such that (1 d) (1 d)g thn + (1!(0))g g + (1!(0)). Th irst part o th dinition xprsss th scond undamntal thorm o calculus. Compatibility is a wakr vrsion o th scond undamntal thorm. Th Taylor proprty (s [11]) is th proprty that i two maps hav th sam drivativ thn thy dir by constants. Thorm 8. For a driving transormation d and an intgral transormation s on th sam coalgbra modality, th ollowing ar quivalnt: (i) d and s satisy th Scond Fundamntal Thorm o Calculus; (ii) d and s ar compatibl and d is Taylor. Rmark. This is an xtnsion o Proposition 14 o [11] which provd (ii) (i) or Ehrhard s original intgral using J 1, howvr th notion o compatibility was not idntiid although it was usd in th proo. Proo. (i) (ii): Suppos d and s satisy th Scond Fundamntal Thorm o Calculus. For Taylor, suppos that (1 d) (1 d)g. Thn w hav th ollowing quality: +(1!0)g (1 s)(1 d) +(1!0) +(1!0)g (1 s)(1 d)g+(1!0) +(1!0)g g+(1!0) For compatibility, by naturality, w hav th ollowing quality: d dsd + d!(0) dsd + (!(0) 0)d dsd + 0 dsd. (i) (ii): Suppos d and s ar Compatibl and d is Taylor. Notic by Compatibility w hav: dsd d, and thn by Taylor (whr sd and g 1) w hav th ollowing quality: sd +!(0) 1 +!(0)sd. Howvr, using naturality, w hav: sd +!(0) 1 +!(0)sd 1 + s(!(0) 0)d Th intrprtation o th irst undamntal thorm o calculus, unlik th scond undamntal thorm, is as a proprty o a map: C S L
10 20:10 Intgral Catgoris and Calculus Catgoris Dinition 9. A map : C A B satisis th First Fundamntal Thorm (in th last two argumnts) i (1 (d A s A )), writtn in th graphical calculus as: Thus, i satisis th First Fundamntal thorm, it may b viwd as th dirntial o a map namly th dirntial o its intgral. Clarly not all maps will satisy th First Fundamntal thorm calculus, a ncssary condition is: Lmma 10. I a map, : C A B, satisis th First Fundamntal Thorm, thn: (1 1 σ)(1 d 1) (1 d 1). Proo. As (1 ds), th intrchang rul or th driving transormation [d.5] givs: (1 1 σ)(1 d 1) (1 1 σ)(1 d 1)(1 (ds)) (1 d 1)(1 (ds)) (1 d 1) W shall us th convrs o this lmma as an axiom and call it th Poincaré condition: Dinition 11. A dirntial catgory with an intgral transormation satisis th Poincaré condition i any map : C A B or which: (1 1 σ)(1 d 1) (1 d 1), satisis th First Fundamntal Thorm that is: (1 ds). Th Poincaré condition and Lmma 10 imply th ollowing quivalnc: Th Poincaré condition also implis compatibility o th driving transormation and intgrating transormation. Thorm 12. Th intgral and driving transormation ar compatibl in any dirntial catgory with an intgral transormation which satisis th Poincaré condition. Proo. By [d.5], th driving transormation d satisis th Poincaré pr-condition that (1 σ)(d 1)d (d 1)d. Thror, d satisis th First undamntal thorm o Calculus, which is simply th statmnt o compatibility: dsd d. Corollary 13. A driving and intgral transormation which satisy th Poincaré condition such that d is Taylor, satisis th Scond Fundamntal Thorm o Calculus. This suggsts th ollowing basic dinition: Dinition 14. A calculus catgory is a dirntial catgory and an intgral catgory on th sam coalgbra modality such that th driving transormation and th intgral transormation satisy th Scond Fundamntal Thorm o Calculus and th Poincaré condition.
11 J. R. B. Cocktt and J.-S. Lmay 20:11 5 Antidrivativs In this last sction, w xplor how on obtains a calculus catgory rom a dirntial catgory with antidrivativs. A dirntial catgory has antidrivativs whn a crtain natural transormation, K which is prsnt in all dirntial catgoris is a natural isomorphism. This is a strngthning o Ehrhard s original ida in [11], which rquird a dirnt natural transormation, J, to b a natural isomorphism. Invrting J by itsl dos not appar to giv vn an intgral catgory: to obtain an intgral catgory and th scond undamntal thorm o calculus Ehrhard also dmandd th Taylor proprty. Invrting K, as w shall s, gts all ths proprtis and, thus, a calculus catgory in on stp. In an additiv symmtric monoidal catgory with a coalgbra modality, th codriving transormation is th natural transormation d A : A(1 A ) : A (this calld th annihilation oprator in [14]). W rprsnt th codriving transormation as an upsid down driving transormation in th graphical calculus: d : Th codriving transormation would probably b on s irst attmpt at constructing an intgrating transormation in dirntial catgory. Th ollowing thorm indicats how clos th codriving transormation is to bing an intgrating transormation: Thorm 15. Th codriving transormation d satisis th ollowing proprtis: [cd.1] d ( 1) [cd.2] d ( 1) ( ) [cd.3] d ( 1) (1 d ) [cd.4] d ( 1)(1 σ) (d 1) [cd.5] 2 (d d ) d ( 1)(d 1 1) + d ( 1)(1 σ)(1 1 d ) [cd.6] d (δ 1) δd (1 ) [cd.7] d (d 1) d (d 1)(1 σ) Notic in particular [cd.1], [cd.5] and [cd.7]. I w lt s d, thn [cd.1] and [cd.7] ar prcisly th sam as [s.1] and [s.3]. Howvr, th codriving transormation ails to satisy [s.2], th Rota-Baxtr rul, sinc [cd.5] has an xtra actor o 2. O cours, i th dirntial catgory is in act nrichd ovr idmpotnt commutativ monoids, so that , th codriving transormation would b an intgral transormation: this happns, or xampl, in th catgory o sts and rlations. Anothr important proprty th codriving transormation satisis is its rlation with th driving transormation. Thorm 16. Th driving and codriving transormations satisy th ollowing quality: d A d A W A + 1 A whr W is th natural transormation with componnts W A (d A 1 A)(1 σ)(d A 1 A ). C S L
12 20:12 Intgral Catgoris and Calculus Catgoris Th notation W was introducd by Ehrhard s in [11] whr a proo can b ound. In th graphical calculus, th abov idntity is xprssd as ollows: + In a dirntial catgory thr ar two important natural transormations K and J dind by K A : d A d A +!0 : and J A : d A d A + 1 :, writtn in th graphical calculus as: K +!0 J + K and J satisy a long list o vry similar proprtis which dscrib thir intraction with th dirntial structur. W giv som o th mor important ons in th ollowing thorm: Thorm 17. K and J satisy th ollowing proprtis: [K.1] K!(0)!(0)!(0)K; [K.2] K ; [K.3] K ; [K.4] K ((dd ) 1) + (1 (dd )) + (!(0)!(0)); [K.5] Kδ δd (1 (dd ))d + δ!(!(0)); [K.6] (K 1)W W(K 1); [K.7] (K 1)dd dd (K 1). [J.1] J!(0)!(0)!(0)J; [J.2] J ; [J.3] J 2 ; [J.4] J (J 1) + (1 (dd )) ((dd ) 1) + (1 J) ; [J.5] Jδ δd (1 (dd ))d + δ; [J.6] (J 1)d dk; [J.7] d (J 1) Kd ; [J.8] (J 1)W W(J 1); [J.9] (J 1)dd dd (J 1). Rcall that Ehrhard s original ida was to obtain intgration by rquiring that J b a natural isomorphism. Howvr, Ehrhard s intgral transormation, using only that J is invrtibl, appars to ail th Rota-Baxtr rul [s.2]. This is why w hav strngthnd Ehrhard s approach by rquiring instad that K b a natural isomorphism. W obsrv: Thorm 18. For a dirntial catgory, K is a natural isomorphism i and only i J is a natural isomorphism and th driving transormation is Taylor.
13 J. R. B. Cocktt and J.-S. Lmay 20:13 Proo. W giv th dinitions o and J 1 : (i) (ii) I K is a natural isomorphism, thn: J 1 A th graphical calculus as: : δ A d (!( A) A )ρ, writtn in δ J 1 (ii) (i) I J is a natural isomorphism and th driving transormation is Taylor, thn A : d A (J 1 A 1 A)(J 1 A 1 A)d A +!0, whr 0 : A A, and writtn in th graphical calculus as: J 1 J 1 +!0 Dinition 19. A dirntial catgory has antidrivativs i K is a natural isomorphism. Equivalntly, o cours, a dirntial catgory has antidrivativs i J is a natural isomorphism and th driving transormation is Taylor. Whil our dinition o antidrivativs dirs only slightly rom Ehrhard s, [11], our dinition dos imply Ehrhard s dinition and, at th sam tim, scurs th proprty o bing an intgral catgory or which, as ar as w can s, invrting J is insuicint. Thorm 20. In a dirntial catgory with antidrivativs, and J 1 satisy th ollowing proprtis: [.1]!(0)!(0)!(0) ; [.2] ; [.3] ; [.4] ( ) + (!(0)) + (!(0) ) ( 1) + (1 ) + (!(0)!(0)); [.5] ( 1)W W( 1); [.6] ( 1)dd dd ( 1); [J 1.1] J 1!(0)!(0)!(0)J 1 ; [J 1.2] J 1 ; [J 1.3] 2 J 1 ; [J 1.4] (J 1 1)d d ; [J 1.5] d (J 1 1) d ; [J 1.6] (J 1 1)W W(J 1 1); [J 1.7] (J 1 1)dd dd (J 1 1); In particular, [J 1.5] will imply that th intgral transormation constructd using ithr or J 1 ar qual to on anothr. Finally, with ths proprtis o and J 1, w obtain th main rsult o this sction, namly that a dirntial catgory with antidrivativs is a calculus catgory: C S L
14 20:14 Intgral Catgoris and Calculus Catgoris Thorm 21. A dirntial catgory with antidrivativs is a calculus catgory with th intgral transormation dind by s A : A d A d A (J 1 A 1 A), xprssd in th graphical calculus as: s J 1 Proo. To prov th intgral transormation axioms and th scond undamntal thorm w us th orm o th intgrating transormation. Whil to prov th Poincaré condition w us Ehrhard s J 1. W will us th graphical calculus to hlp us. W irst show that our intgral transormation satisis [s.1] to [s.3]. [s.1]: Hr w us [cd.1] and [.2]: [s.2]: Hr w us [.4], [cd.3], [cd.4] and naturality o th codriving transormation: +!0 +!0 } {{ } 0 } {{ } 0 + +!0!0 + } {{ } 0 [s.3]: Hr w us [J 1.5] and [cd.7]: J 1 J 1 J 1 J 1 Nxt w show th scond undamntal thorm o calculus. Hr w us [.1]: +!0 +!0 K
15 J. R. B. Cocktt and J.-S. Lmay 20:15 Finally w prov th Poincaré Condition. Lt : C A B satisy th Poincaré pr-condition, that is, (1 1 σ)(1 d 1) (1 d 1). First notic that by Thorm 16 and th Poincaré pr-condition, satisis th ollowing idntity: + + J Thn using [J 1.4] and th abov idntity w gt th ollowing quality: J 1 J 1 J Which complts th proo that antidrivativs giv a calculus catgory. Th convrs o Thorm 21 is tru i th coalgbra is monoidal (in th sns xplaind at th nd o Sction 3 whn discussing Fubini s thorm) and th intgral transormation is compatibl with monoidal strngth, that is, a calculus catgory with a monoidal coalgbra modality and a monoidal intgral transormation is a dirntial catgory with antidrivativs. Mor dtails and a proo o this will b givn in a subsqunt papr. W ar now ar in a position to giv two xampls o dirntial catgoris which hav antidrivativs, and thror, two xampls o calculus catgoris: Exampl 22. Th catgory o sts and rlations, REL, is a dirntial catgory [7] with antidrivativs. Th symmtric monoidal strucur is givn by th Cartsian product o sts whil th additiv structur is givn by th union o sts. Th coalgbra modality is givn by th init bag/multist comonad (s [7] or mor dtails), whr or a st X,!X is th st o bags/multists o X. Th driving transormation d X :!X X!X is th rlation which adds an xtra lmnt to th bag: d X {((B, x), B x) x X, B!X} Th additiv idmpotncy o REL maks both K and J th idntity and thus trivially isomorphisms. Thror, th intgral transormation is th codriving transormation :!X!X X, which is th rlation which rmovs an lmnts rom th bag: d X d X {(B, (B {x}, x)) x X, B!X} Exampl 23. Th catgory o vctor spacs ovr a ild K o charactristic o 0, VEC K, is a co-dirntial catgory [7] with antidrivativs, so that, VEC op K is a calculus catgory. Whil having a ild o charactristic zro is not rquird to obtain dirntial structur, it is rquird or antidrivativs. Th additiv symmtric monoidal structur is givn by th standard tnsor product and additiv nrichmnt o vctor spacs. Th algbra modality is givn by th r symmtric algbra monad whr or a vctor spac V,!V is th r commutativ algbra ovr V (s [17] or mor dtails). Equivalntly, i X {x 1, x 2,...} is C S L
16 20:16 Intgral Catgoris and Calculus Catgoris a basis o V, thn!v is isomorphic to th polynomial ring ovr X:!V K[X] [17]. Thn th driving transormation d V :!V!V V (rcall VEC op K is th calculus catgory) on monomials is givn by th sum o partial drivativs o th monomial: d V (x r1 1...xrn n ) n i1 (x r1 1...xri 1...x rn ) x i i On monomials, K multiplis th non-constant monomials by thir dgr and multiplis th constants by on, whil J multiplis monomials by thir dgr plus on. As th rationals ar mbddd in our ild, both ar isomorphisms, and th rsulting intgral transormation s V :!V V!V is dind on monomials by: 1 s V ((x r1 1...xrn n ) x i ) 1 + n j1 r x r1 1...xri+1 i...x rn j At irst glanc this may sm bizarr. On might xpct th intgrating transormation 1 to intgrat a monomial with rspct to th variabl x i and thus only multiply by 1+r i. Howvr, this classical ida o intgration ails th Rota-Baxtr rul [s.2] or any vctor spac o dimnsion gratr than on. 6 Conclusion and Futur Work Th thory o dirntial catgoris was dvlopd in stags: (tnsor) dirntial catgoris [7], cartsian dirntial catgoris [3], dirntial rstriction catgoris [10], and tangnt catgoris [9]. Th dvlopmnt o intgral catgoris, bing vry closly rlatd, has paralll stags. Hr w hav brily introducd th irst stag o this dvlopmnt: tnsor intgral catgoris. Th nxt stag, Cartsian intgral catgoris, is actually wll in hand. Th coklisli catgory o an intgral catgory is a Cartsian intgral catgory. Furthrmor, Cartsian intgral catgoris hav a trm logic which has a mor classic l: w borrowd parts o this trm logic to hlp motivat this papr. Th study o intgration in rstriction catgoris and tangnt catgoris is, by comparison, in its arlist stags. Acknowldgmnts. Th authors would lik to thank Rick Blut or drawing both authors attntion to Rota-Baxtr algbras. Intgral catgoris simply would not hav dvlopd so rapidly without this basic inspiration. Jonathan Gallaghr rmindd us o Ehrhard s work at xactly th right momnt, whil Kristin Baur providd continual constructiv criticism during th volution o our thoughts. Rrncs 1 Gln Baxtr t al. An analytic problm whos solution ollows rom a simpl algbraic idntity. Paciic J. Math, 10(3): , R. Blut, J. R. B. Cocktt, and R. A. G. Sly. Cartsian dirntial storag catgoris. Thory and Applications o Catgoris, 30(18): , R. F. Blut, J. Robin B. Cocktt, and R. A. G. Sly. Cartsian dirntial catgoris. Thory and Applications o Catgoris, 22(23): , Richard Blut, J. R. B. Cocktt, Timothy Portr, and R. A. G. Sly. Kählr catgoris. Cahirs d Topologi t Géométri Diérntill Catégoriqus, 52(4): , Richard Blut, Thomas Ehrhard, and Christin Tasson. A convnint dirntial catgory. arxiv prprint arxiv: , 2010.
17 J. R. B. Cocktt and J.-S. Lmay 20:17 6 Richard Blut, Rory B. B. Lucyshyn-Wright, and Kith O Nill. Drivations in codirntial catgoris. arxiv prprint arxiv: , Richard F. Blut, J. Robin B. Cocktt, and Robrt A. G. Sly. Dirntial catgoris. Mathmatical structurs in computr scinc, 16(06): , Raoul Bott and Loring W. Tu. Dirntial orms in algbraic topology, volum 82. Springr Scinc & Businss Mdia, J. Robin B. Cocktt and Go S. H. Cruttwll. Dirntial Structur, Tangnt Structur, and SDG. Applid Catgorical Structurs, 22(2): , J. R. B. Cocktt, G. S. H. Cruttwll, and J. D. Gallaghr. Dirntial rstriction catgoris. Thory and Applications o Catgoris, 25(21): , Thomas Ehrhard. An introduction to dirntial linar logic: proo-nts, modls and antidrivativs. Mathmatical Structurs in Computr Scinc, pags 1 66, Thomas Ehrhard and Laurnt Rgnir. Th dirntial lambda-calculus. Thortical Computr Scinc, 309(1):1 41, Thomas Ehrhard and Laurnt Rgnir. Dirntial intraction nts. Thortical Computr Scinc, 364(2): , Marclo P. Fior. Dirntial structur in modls o multiplicativ biadditiv intuitionistic linar logic. In Intrnational Conrnc on Typd Lambda Calculi and Applications, pags Springr, Li Guo. An introduction to Rota-Baxtr algbra, volum 2. Intrnational Prss Somrvill, André Joyal and Ross Strt. Th gomtry o tnsor calculus, I. Advancs in Mathmatics, 88(1):55 112, Srg Lang. Algbra rvisd third dition. Graduat Txts in Mathmatics, 1(211):ALL ALL, J.-S. P. Lmay. Intgral Catgoris and Calculus Catgoris. Univrsity o Calgary, Saundrs Mac Lan. Catgoris or th working mathmatician, volum 5. Springr Scinc & Businss Mdia, Gian-Carlo Rota. Baxtr algbras and combinatorial idntitis. I. Bulltin o th Amrican Mathmatical Socity, 75(2): , Ptr Slingr. A survy o graphical languags or monoidal catgoris. In Nw structurs or physics, pags Springr, Charls A. Wibl. An introduction to homological algbra. Cambridg univrsity prss, C S L
NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationThere Is Only One Notion of Differentiation
Thr Is Only On Notion of Diffrntiation J. Robin B. Cocktt 1 and Jan-Simon Lmay 2 1 Dpartmnt of Comptr Scinc, Univrsity of Calgary, Calgary, AB, Canada robin@calgary.ca 2 Dpartmnt of Mathmatics and Statistics,
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationMATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
More informationPROBLEM SET Problem 1.
PROLEM SET 1 PROFESSOR PETER JOHNSTONE 1. Problm 1. 1.1. Th catgory Mat L. OK, I m not amiliar with th trminology o partially orr sts, so lt s go ovr that irst. Dinition 1.1. partial orr is a binary rlation
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationThere is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.
Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual
More informationCHAPTER 1. Introductory Concepts Elements of Vector Analysis Newton s Laws Units The basis of Newtonian Mechanics D Alembert s Principle
CHPTER 1 Introductory Concpts Elmnts of Vctor nalysis Nwton s Laws Units Th basis of Nwtonian Mchanics D lmbrt s Principl 1 Scinc of Mchanics: It is concrnd with th motion of matrial bodis. odis hav diffrnt
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More information3 Finite Element Parametric Geometry
3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics,
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationCOUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM
COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,
More informationInjective topological fibre spaces
Topology and its pplications 125 (2002) 525 532 www.lsvir.com/locat/topol Injctiv topological ibr spacs F. Cagliari a,,s.mantovani b a Dipartimnto di Matmatica, Univrsità di Bologna, Piazza di Porta S.
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More informationUNTYPED LAMBDA CALCULUS (II)
1 UNTYPED LAMBDA CALCULUS (II) RECALL: CALL-BY-VALUE O.S. Basic rul Sarch ruls: (\x.) v [v/x] 1 1 1 1 v v CALL-BY-VALUE EVALUATION EXAMPLE (\x. x x) (\y. y) x x [\y. y / x] = (\y. y) (\y. y) y [\y. y /
More informationCLONES IN 3-CONNECTED FRAME MATROIDS
CLONES IN 3-CONNECTED FRAME MATROIDS JAKAYLA ROBBINS, DANIEL SLILATY, AND XIANGQIAN ZHOU Abstract. W dtrmin th structur o clonal classs o 3-connctd ram matroids in trms o th structur o biasd graphs. Robbins
More informationDerivation of Electron-Electron Interaction Terms in the Multi-Electron Hamiltonian
Drivation of Elctron-Elctron Intraction Trms in th Multi-Elctron Hamiltonian Erica Smith Octobr 1, 010 1 Introduction Th Hamiltonian for a multi-lctron atom with n lctrons is drivd by Itoh (1965) by accounting
More information2.3 Matrix Formulation
23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,
More informationBrief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationPROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS
Intrnational Journal Of Advanc Rsarch In Scinc And Enginring http://www.ijars.com IJARSE, Vol. No., Issu No., Fbruary, 013 ISSN-319-8354(E) PROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS 1 Dharmndra
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationAbstract Interpretation: concrete and abstract semantics
Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationOn Certain Conditions for Generating Production Functions - II
J o u r n a l o A c c o u n t i n g a n d M a n a g m n t J A M v o l 7, n o ( 0 7 ) On Crtain Conditions or Gnrating Production Functions - II Catalin Anglo Ioan, Gina Ioan Abstract: Th articl is th scond
More informationREPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS
REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS JIAQI JIANG Abstract. This papr studis th rlationship btwn rprsntations of a Li group and rprsntations of its Li algbra. W will mak th corrspondnc in two
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationMCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems
MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a
More informationEstimation of apparent fraction defective: A mathematical approach
Availabl onlin at www.plagiarsarchlibrary.com Plagia Rsarch Library Advancs in Applid Scinc Rsarch, 011, (): 84-89 ISSN: 0976-8610 CODEN (USA): AASRFC Estimation of apparnt fraction dfctiv: A mathmatical
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More informationWhere k is either given or determined from the data and c is an arbitrary constant.
Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationHomotopy perturbation technique
Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,
More informationBackground: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.
Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby
More informationDeift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
More informationData Assimilation 1. Alan O Neill National Centre for Earth Observation UK
Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
More informationu r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C
Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin
More informationInternational Journal of Scientific & Engineering Research, Volume 6, Issue 7, July ISSN
Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July-25 64 ISSN 2229-558 HARATERISTIS OF EDGE UTSET MATRIX OF PETERSON GRAPH WITH ALGEBRAI GRAPH THEORY Dr. G. Nirmala M. Murugan
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
More informationSundials and Linear Algebra
Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationCharacterizations of Continuous Distributions by Truncated Moment
Journal o Modrn Applid Statistical Mthods Volum 15 Issu 1 Articl 17 5-016 Charactrizations o Continuous Distributions by Truncatd Momnt M Ahsanullah Ridr Univrsity M Shakil Miami Dad Coll B M Golam Kibria
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by Addison-Wsly.
More informationGROUP EXTENSION HOMOMORPHISM MATRICES. A. M. DuPre. Rutgers University. October 1, 1993
GROUP EXTENSION HOMOMORPHISM MATRICES A. M. DuPr Rutgrs Univrsity Octobr 1, 1993 Abstract. If i1 j1 1! N 1?! G 1?! H 1! 1 i2 j2 1! N 2?! G 2?! H 2! 1 f ar short xact squncs of groups, thn w associat to
More informationGEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia
GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* Rostov-on-Don, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationperm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l
h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional
More informationCalculus concepts derivatives
All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving
More informationThus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.
Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationFirst order differential equation Linear equation; Method of integrating factors
First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationA Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction
Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P.
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationContent Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles.
Tachr: CORE TRIGONOMETRY Yar: 2012-13 Cours: TRIGONOMETRY Month: All Months S p t m b r Angls Essntial Qustions Can I idntify draw ngativ positiv angls in stard position? Do I hav a working knowldg of
More informationSelf-interaction mass formula that relates all leptons and quarks to the electron
Slf-intraction mass formula that rlats all lptons and quarks to th lctron GERALD ROSEN (a) Dpartmnt of Physics, Drxl Univrsity Philadlphia, PA 19104, USA PACS. 12.15. Ff Quark and lpton modls spcific thoris
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationWhat are those βs anyway? Understanding Design Matrix & Odds ratios
Ral paramtr stimat WILD 750 - Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow. 000. Applid logistic rgrssion. John Wily & ons Inc.
More informationME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or
More informationThe Equitable Dominating Graph
Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay
More information