DEFORMATIONS OF ASSOCIATIVE ALGEBRAS WITH INNER PRODUCTS
|
|
- Claire Manning
- 5 years ago
- Views:
Transcription
1 Homology, Homotopy nd Applitions, vol. 8(2), 2006, pp DEFORMATIONS OF ASSOCIATIVE ALGEBRAS WITH INNER PRODUCTS JOHN TERILLA nd THOMAS TRADLER (ommunited y Jim Stsheff) Astrt We develop the deformtion theory of A lgers together with -inner produts nd identify differentil grded Lie lger tht ontrols the theory. This generlizes the deformtion theories of ssoitive lgers, A lgers, ssoitive lgers with inner produts, nd A lgers with inner produts. 1. Introdution A nturl onsidertion for n lgeri struture in topology is whether it is homotopy invrint. The C struture on the ohins of spe is lssi exmple. While mnifolds re distinguished y the inner produt fforded y Poinré dulity, n inner produt is not homotopy invrint onept. The right mening the homotopy roust onept is n -inner produt s introdued in [12]. In lgeri generlity, n -inner produt is defined in the setting of n A lger. In this pper, we desrie the deformtion theory of A lgers together with -inner produts y giving ontrolling differentil grded Lie lger. An pplition tht we hve in mind involves string topology. It is known tht if X nd Y hve the sme homotopy type, then they hve the sme string topology opertions [1]. One my ssign n A lger A X with n -inner produt I X to Poinre dulity spe X. Bsed on results in [11, 13], it is resonle to think tht if the two differentil grded Lie lgers ontrolling the deformtions of (A X, I X ) nd (A Y, I Y ) re qusi-isomorphi, then X nd Y hve the sme string topology opertions. One my speulte tht the qusi-isomorphism lss of the differentil grded Lie lger ontrolling the deformtions (A X, I X ) determines the string topology type of the spe X (muh the sme wy tht the C struture on the ohins on spe determines the rtionl homotopy type of spe; see [10]). In ny event, it would e interesting to proe this ontrolling differentil grded Lie lger for its invrints. The uthors would like to thnk Jim Stsheff, Mrtin Mrkl, nd Dennis Sullivn for mny helpful disussions. Reeived Mrh 12, 2006, revised April 7, 2006, July 20, 2006; pulished on August 16, Mthemtis Sujet Clssifition: 55P10, 16S10, 16S80. Key words nd phrses: homotopy, inner produt, deformtion theory. Copyright 2006, Interntionl Press. Permission to opy for privte use grnted.
2 Homology, Homotopy nd Applitions, vol. 8(2), Let us review the si ide of deformtion theory governed y differentil grded Lie lger [9, 3, 5, 6]. Fix ground field k of hrteristi 0. For ny differentil grded Lie lger (g = i g i, d, [, ]) over k, one n onsider deforming the differentil d in the diretion of n inner derivtion. Informlly, suh deformtion is given y n (equivlene lsses of) α mking d α := d + d(α) into differentil. The mp d α is lwys derivtion nd the ondition tht d 2 α = 0 trnsltes into the Muer Crtn eqution: dα + 1 [α, α] = 0. 2 The deformed differentil d α my involve prmeters from the mximl idel m of Z grded Artin lol ring: α (g k m) 1. If m is the mximl idel of lol Artin ring R nd α (g k m) 1 is solution to the Muer Crtn eqution, then one my ll d α deformtion of d over R. A ring mp R S will trnsport deformtion of d over R to deformtion of d over S. More formlly, one hs funtor Def g from the tegory of Z grded Artin lol rings with residue field k to the tegory of sets, ssigning to suh ring R with mximl idel m the set Def g (R) = {α (g k m) 1 : dα + 1 [α, α] = 0}/. 2 Here, is the equivlene reltion determined y the tion of the guge group, whih we now rell. Sine R is n Artin ring, m is nilpotent lger, nd (g k m) 0 g k m is nilpotent Lie lger. Therefore, there exists group G = {exp β : β (g k m) 0 }, lled the guge group, with multiplition defined y the Bker Cmpell Husdorff formul. The tion of e β G on n element α (g k m) 1 is determined y the infinitesiml tion: This tion stisfies α β α = [β, α] dβ, α (g m) 1, β (g m) 0. e d β d α e d β = d eβ α, nd preserves the set of solutions to the Murer Crtn eqution. In this pper, we work with A lgers equipped with inner produts. One hs the notion of deformtion of n A lger with n inner produt over ring R, nd there is nturl equivlene on the set of deformtions. A ring mp R S trnsports deformtions over R to deformtions over S. The ssoition R { deformtions of the A lger with the inner produt over R } /{ equivlent deformtions defines ovrint deformtion funtor. We onstrut differentil grded Lie lger (h = i h i, d, [, ]) ssoited to n A lger with n inner produt, nd prove tht the funtor desried ove is isomorphi to Def h. This is the preise mthemtil ontent of the sttement the differentil grded Lie lger (h, [, ], d) ontrols the deformtions of the A lger with n inner produt. }
3 Homology, Homotopy nd Applitions, vol. 8(2), Definitions of A lgers nd inner produts We now review the onept of n inner produt on n A lger [12], [11]. The onepts of A lgers, A imodules, A imodule mps, nd A inner produts re generliztions of the usul onepts of ssoitive lgers, imodules, imodule mps, nd invrint inner produts A lgers Let V = j Z V j e grded module over ring S. Rell tht the suspension V [1] of V is defined to e V [1] = j Z (V [1])j with (V [1]) j := V j 1. For grded S-module A, we denote y T A the tensor lger of the suspended spe A[1], T A = S A[1] A[1] An A lger over S is defined to e pir (A, D) where A is grded S module nd D Coder(T A) of degree 1 with D 2 = 0. In ddition, we require the no homotopy unit onvention tht D hs no omponent S T A. Suppose tht (A, D) nd (A, D ) re A lgers over S. Then, n A mp from (A, D ) to (A, D) is mp λ : T A T A stisfying λ D = D λ A imodules Let (A, D) e n A lger over S, nd let M e grded S module. Let T M A denote the tensor iomodule T M A := k,l 0 A[1] k M[1] A[1] l of M[1] over T A. An A imodule struture on M over A is defined to e oderivtion D M Coder D (T M A, T M A) over D of degree 1 with (D M ) 2 = 0. Let (M, D M ) nd (N, D N ) e A imodules over A. Let Comp(T M A, T N A) denote the mps F : T M A T N A stisfying T M A (T A T M A) (T M A T A) M F (Id F ) (F Id) T N A (T A T M A) (T M A T A) N The spe Comp(T M A, T N A) rries differentil defined y δ M,N (F ) := D N F ( 1) F F D M. In this se, n A imodule mp from M to N is defined to e n element F Comp(T M A, T N A) of degree 0 with δ M,N (F ) = 0, i.e. D N F = F D M inner produts For ny f Coder(T A), there re indued oderivtions f A Coder f (T A A, T A A) nd f A Coder f (T A A, T A A), where A = hom S (A, S) denotes the dul of A.
4 Homology, Homotopy nd Applitions, vol. 8(2), One lso hs n indued mp δ f : Comp(T M A, T N A) Comp(T M A, T N A) given y δ f (F ) = f A F ( 1) f F F f A. Note tht, in prtiulr, if (A, D) is n A lger, then A nd A hve A imodule strutures given y D A nd D A. Definition 2.1. Let (A, D) e n A lger over S. We define n inner produt on A over S to e n A imodule mp I from A to A. Equivlently, n inner produt is n element I Comp(T A A, T A A) stisfying δ D (I) = D A I I D A = 0. Every inner produt, : A A S defines n element I Comp(T A A, T A A). In this se, the ondition D A I I D A = 0 is equivlent to D( 1,..., n ), n+1 = ± 1, D( 2,..., n+1 ). See the ppendix for dditionl illustrtions Indued mps Rell tht if λ : A A is n lger mp etween two ssoitive lgers, then every module over A is lso module over A, nd similrly for module mps. Also, λ : A A nd λ : A (A ) will e module mps over A. Here we give the orresponding homotopy generliztions. Suppose tht λ is n A mp from (A, D ) to (A, D). First, every A imodule (M, D M ) over A is lso n A imodule over A, whose struture mp is determined y the lowest omponents (whih re mps T M A M) (D M ) λ ( 1,..., k, m, k+1,..., k+l) = ±pr M D M (λ( 1,...),..., λ(..., k), m, λ( k+1,...),..., λ(..., k+l)). Here, pr M denotes the projetion onto M. The signs re given y the usul sign rule, nmely introduing sign ( 1) α β, whenever α jumps over β. The relevnt degrees re the degrees given in T M A. Also, ny A imodule mp F : T M A T N A over A indues n A imodule mp F λ : T M A T N A over A given y F λ ( 1,..., k, m, k+1,..., k+l) = ±pr N F (λ( 1,...),..., λ(..., k), m, λ( k+1,...),..., λ(..., k+l)). Furthermore, λ indues the two A imodule mps over A defined y the omponents nd λ : T A A T A A nd λ : T A A T (A ) A λ( 1,..., k+l+1) = pr A λ( 1,..., k+l+1) ( λ( 1,...,,..., k+l))( ) = ± (pr A λ( k+1,..., k+l,, 1,..., k)).
5 Homology, Homotopy nd Applitions, vol. 8(2), Deformtions of A lgers nd inner produts Before we define the speifi differentil grded Lie lger (h, d, [, ]) tht ontrols the deformtions of A strutures nd inner produts, we disuss simple exmple, whih is relevnt to our setting, nd mke remrk. Exmple 3.1. Any grded ssoitive lger g eomes Lie lger y defining the rket to e the usul ommuttor. An element α g 1 stisfying α 2 = 0 is sometimes lled polriztion. With polriztion α g 1, g eomes differentil grded Lie lger y setting the differentil to e δ = d(α). With δ so defined, the Muer Crtn eqution eomes 0 = δ(γ) [γ, γ] = 1 [α + γ, α + γ]. 2 In other words, γ g 1 stisfies the Muer Crtn eqution if nd only if α + γ is nother polriztion. Now let S e grded ring nd onsider g defined y {( ) 0 g =, S} with the rket defined s the usul grded ommuttor of mtrix multiplition: [( ) ( )] ( ) 0 0 [, ] 0, =. d [, ] + [, d] [, ] ( ) D 0 Then, g I D 1 is polriztion if nd only if 0 = [D, D] = 2 D 2 nd 0 = [D, I] + [I, D] = 2 [D, I]. ( ) D 0 Hving hosen polriztion P =, the formul for δ = d(p ) is given y I D ( ) [( ) ( )] ( ) f 0 D 0 f 0 [D, f] 0 δ =, =. i f I D i f [D, i]+[f, I] [D, f] Now we look t the guge equivlene. First of ll, the guge group G = exp(g 0 ) is the Lie group onsisting of mtries of the form e A, for ny A g 0. The guge tion of G on g is then determined y e d(a) B = Ad(e A )(B) = e A Be A. A omputtion shows tht ( ) ( f 0 e f 0 exp = i f x e f ), where x = n 1 Then the guge equivlene summrizes s ( ) ( e A D 0 e A = I D This onludes the exmple. 1 n! k+l=n 1 f k i f l. e f De f 0 e f Ie f + [e f De f, xe f ] e f De f Remrk 3.2. Let N e grded olger over S. Then hom(n, N) will e grded ssoitive lger y omposition of liner mps nd Lie lger with the ).
6 Homology, Homotopy nd Applitions, vol. 8(2), rket defined y the grded ommuttor of omposition. The spe Coder(N) hom(n, N) is not n ssoitive sulger, ut it is Lie sulger. In prtiulr, for ny vetor spe A, Coder(T A) is grded Lie lger. An A struture on A onsists of n element D Coder(T A) stisfying D 2 = 0. Thus, one n sy tht n A struture on A is hoie of polriztion D Coder(T A). Hene, if (A, D) is n A lger, Coder(T A) rries differentil δ : Coder(T A) Coder(T A) defined y δ(f) := [D, f] = D f ( 1) f f D. The omplex (Coder(T A), δ) is lled the Hohshild ohin omplex of A. Together with the rket from hom(t A, T A), it is differentil grded Lie lger tht ontrols the deformtions of the A lger (A, D). In order to mke this sttement preise, we rell the deformtion theory of A lgers (see for exmple [2]). As first oservtion, one my note tht γ is solution to the Muer Crtn eqution in the Hohshild differentil grded Lie lger if nd only if D + γ is nother polriztion in Coder(T A); i.e., nother A struture on A Deformtions of A lgers Let A e grded vetor spe over field k of hrteristi zero nd let R e grded Artin lol lger with residue field k. Let m denote the mximl idel of R. We hve the deomposition R R/m m k m nd the projetion pr k : R k, hene the deomposition A R A (A m) nd the projetion pr A : A R A. For definiteness, the reder my hve the onrete exmple R = k[t]/t l+1 in mind. In this exmple, the mximl idel is m = tk[t]/t l+1, A R A + At + At At l (with the tensor signs suppressed) nd the nturl projetion pr A mps t + 2 t l t l 0. Let (A, D) e n A lger over k. A deformtion of (A, D) over R is n A lger (A R, D ) over R with the property tht the projetion pr : T (A R) T A R T A is morphism of A lgers over k. This mens tht pr D = D pr. Suppose, tht D is deformtion of (A, D) over R. Vi ny mp R S, one n view A R s n S module nd (A R, D ) s deformtion of (A, D) over S. Let π hom(r R, R) denote the multiplition in R. Let D R denote the A struture D π on A R. The A lger (A R, D R ) is the model for trivil deformtion of (A, D). Tht is, (A R, D ) is trivil deformtion if it is isomorphi to (A R, D R ) s n A lger. This mens tht there is n utomorphism λ : T (A R) T (A R) stisfying λ D = D R λ. Two deformtions re equivlent if nd only if they differ y trivil one Deformtions of A lgers with inner produts Definition 3.3. Let A e grded vetor spe over field k. We define the grded Lie lger (h = i h i, [, ]) y h i = Coder(T A) i Comp(T A A, T A A) 1 i (1)
7 Homology, Homotopy nd Applitions, vol. 8(2), nd [(f, i), (g, j)] = ([f, g], δ f (j) ( 1) f g δ g (i)) = (fg ( 1) f g gf, f A j ( 1) f j jf A ( 1) f g g A i + ( 1) g ( f + i ) ig A ). (2) The skew-symmetry nd Joi identity of [, ] re strightforwrd to hek fter one noties tht δ f δ g ( 1) f g δ g δ f = δ f g ( 1) f g g f. Proposition 3.4. A pir (D, I) h is n A struture with inner produt on A if nd only if [(D, I), (D, I)] = 0. Proof. This is immedite: 0 = [(D, I), (D, I)] 0 = [D, D] = 2 D 2 nd 0 = 2 δ D (I) = 2(D A I I D A ). The ondition D 2 = 0 mens tht D defines n A struture on A nd the ondition D A I I D A = 0 mens tht I defines omptile -inner produt. Now fix n A struture together with n inner produt, whih is to sy, fix pir (D, I) h with [(D, I), (D, I)] = 0. Then, define d : h h y The triple (h, d, [, ]) is differentil grded Lie lger. d(f, i) = [(D, I), (f, i)]. (3) Definition 3.5. A deformtion of n A lger with inner produt (A, D, I) over R is n A lger over R with inner produt (A R, D, I ), suh tht the projetion pr : T (A R) T A is morphism of A lgers over k omptile with the -inner produts. Comptiility with the inner produt mens tht the following digrm of A - imodule mps over k is ommuttive: T A R (A R) T A (A R) pr k I T (A R) (A R) T A (A R) pr fpr I pr Here, the -inner produt I on A R over R indues n -inner produt on A R over k y omposing with the mp indued y the projetion hom R (A R, R) hom k (A R, k), f pr k f. There is nturl extension of I to n -inner produt I R = I π on (A R, D R ).
8 Homology, Homotopy nd Applitions, vol. 8(2), Definition 3.6. We sy tht (D, I ) is trivil deformtion of (D, I) provided the triple (A R, D, I ) is isomorphi to (A R, D R, I R ) s A lgers with inner produts. Tht is, if there exists n utomorphism nd omp stisfying (i) λ D = D R λ, λ : T (A R) T (A R) ρ : T A R (A R) T (A R) (A R) (ii) I λ (I R ) λ λ = D (A R) ρ + ρ D A R. It my e helpful to think of the seond ondition in Definition 3.6 s sying I equls I R under hnge of oordintes (given y λ) up to homotopy (given y ρ). Tht is, the following digrm ommutes, up to homotopy defined y ρ Comp(T A R (A R)). T A R λ (A R) T A R (A R) I (I R) λ T (A R) (A R) T (A R) (A R) Two deformtions re equivlent if nd only if they differ y trivil one. Now, the onlusion: eλ Theorem 3.7. Let (A, D) e n A lger nd let I e n -inner produt. Then the differentil grded Lie lger (h, d, [, ]) defined y equtions (1), (2) nd (3) ontrols the deformtions of the A lger with inner produt (A, D, I). Proof. The ontent of this theorem is summrized in the following two sttements. Deformtions, over R, of the (A, D, I) orrespond to solutions to the Muer Crtn eqution in h m, nd equivlent deformtions orrespond to guge equivlent solutions. First we prove the first sttement. Let α = (f, i) (h m) 1. Oserve tht dα [α, α] = [(D R, I R ), (f, i)] + 1 [(f, i), (f, i)] 2 = 1 2 [(D R + f, I R + i), (D R + f, I R + i)]. Then, Proposition 3.4 proves tht dα [α, α] = 0 if nd only if (A R, D R + f, I R + i) is deformtion of (A, D, I). It is immedite tht ny (A R, D, I ) tht is deformtion of (A, D, I) must stisfy [(D, I ), (D, I )] = 0 h R. The ft tht pr : T (A R) T A is mp of A lgers with inner produts implies tht D = D R + f nd I = I R + i for some (f, i) h m.
9 Homology, Homotopy nd Applitions, vol. 8(2), Now we prove the seond sttement. Let α = (f, i) (h m) 0. The guge tion for h eomes e d(f,i) (D R, I R ) = d(f, i) n (D R, I R ). n! n 0 It follows from δ f ( δd(f) r (D R)((δ f ) s (i)) ) = δ d(f) r+1 (D R)((δ f ) s (i)) + δ d(f) r (D R)((δ f ) s+1 (i)), tht d(f, i) n (D R, I R ) is given y ( d(f) n (D R ), (δ f ) n (I R ) k+l=n 1 1 (l+1)! (δ f ) l (i). Then for the uto- Now define λ 1 = e f = k 0 1 k! f k nd ρ = l 0 morphism λ nd the homotopy ρ, we hve d(f, i) n (D R, I R ) n! n 0 = d(f) n (D R ), (δ f ) n n! n! n 0 n 0 ( = λ 1 D R λ, λ(i ) R ) λ λ + δ λ 1 D Rλ(ρ). ) n! k!(l + 1)! δ d(f) k (D R) (δ f ) l (i). (I R )+δp d(f) k k 0 k! (D R) ( l 0 ) 1 (l + 1)! (δ f ) l (i) This proves tht e d(f,i) (D R, I R ) is trivil deformtion of (D, I). It is not hrd to see tht every trivil deformtion of (D, I) rises from n element guge equivlent to the identity. The ondition tht the A lger mp λ : T (A R) T (A R) is n utomorphism implies tht λ = e f for some f (Coder(T A) m) 0. Also, sine ρ = i 1 2 δ f (i), the mp i ρ(i) = l 0 1 (l + 1)! (δ f ) l (i) is invertile. So one n otin ny homotopy ρ, y hoosing suitle element i = m 0 m (δ f ) m (ρ) (h m) 0 with ρ(i) = ρ. 4. Moduli, infinitesiml deformtions, nd reltionship to yli ohomology Let us return riefly to generl deformtion theory in order to review the notions of infinitesiml deformtions nd moduli spe. Let (g, d, [, ]) e differentil grded Lie lger nd ssume tht Ker(d)/ Im(d) =: H(g) = m i= m Hi (g) is finite dimensionl. Consider the (grded version of the) ring of dul numers R = k[t m,..., t m ]/t i t j. Here deg(t i ) = i 1 nd the mximl idel of R is m = i t i R. From solution (γ j t j ) (g m) 1 to the Muer Crtn eqution, one my produe the mp d + t j d(γ j ) : g k[t m,..., t m ] g k[t m,..., t m ] whih
10 Homology, Homotopy nd Applitions, vol. 8(2), stisfies ( d + t j d(γ j )) 2 = 0 modulo ti t j. One refers to γ = γ j s n infinitesiml deformtion. One n redily hek tht Def g (R) = Ker(d)/ Im(d) = H(g). Suppose Def g is prorepresentle. Tht is, there exists projetive limit of (grded) lol Artin rings O nd n equivlene of the funtors Def g ( ) hom(o, ). In the se tht O = O M is the ring of lol funtions t the se point of pointed Z grded spe M, then M is the lol moduli spe for Def g. Denote the se point of M y p. One n hek tht T p (M) hom(o M, R). It follows tht the grded tngent spe to the moduli spe t the se point is isomorphi to the ohomology of (g, d): T p (M) H(g). Now, let (A, D) e n A lger nd let I e n inner produt on (A, D). Theorem 3.7 sys tht the differentil grded Lie lger ontrolling deformtions of (A, D, I) is with rket nd differentil h = Coder(T A) Comp(T A A, T A A) [(f, i), (g, j)] = ([f, g], δ f (j) ( 1) f g δ g (i)) d(f, i) = [(D, I), (f, i)]. Thus follows the expeted infinitesiml sttement: Corollry 4.1. The grded tngent spe to the moduli spe of A strutures with inner produts is isomorphi to H(h). As finl remrk, we mention some onnetions etween the ohomology H(h) nd ouple of its ousins. If (A, D, I) is n A lger with -inner produt, we hve the Hohshild differentil grded Lie lger (Coder(T A), δ, [, ]) nd the su differentil grded Lie lger of yli Hohshild ohins Coder(T A) Cyli, defined y Coder(T A) Cyli = {f Coder(T A) : δ f (I) = 0}. If I onsists of n ordinry symmetri inner produt I =,, then the ondition δ f (I) = f A I I f A = 0 is equivlent to f( 1,..., n ), n+1 = ± 1, f( 2,..., n+1 ).
11 Homology, Homotopy nd Applitions, vol. 8(2), We hve the following mps of differentil grded Lie lgers: (Coder(T A) Cyli, δ, [, ]) (h, d, [, ]) nd (h, d, [, ]) (Coder(T A), δ, [, ]). (4) The first mp is the injetion f (f, 0) h, whih is ohin mp d(f, 0) = ([D, f], ±(f A I I f A )) = (δf, 0), euse elements of the domin re yli. The indued mp in ohomology desries sttement from [7], nmely tht the first order deformtions of D omptile with the inner produt re lssified y yli ohomology. We do not know under wht onditions the mp f (f, 0) h indues n isomorphism in ohomology. The seond mp in (4) is simply the projetion Coder(T A) Comp(T A A, T A A) Coder(T A) nd the indued mp in ohomology desries the simple sttement tht ny infinitesiml deformtion of the pir (D, I) gives n infinitesiml deformtion of D. Appendix A. Expliit formuls of δ f (i) Let f Coder(T A) nd i Comp(T A A, T A A). We wnt to desrie the term δ f (i) = f A i ( 1) f i i f A Comp(T A A, T A A) more expliitly. Here, f : k 1 A k A nd i : k,l 0 A k A A l A S hve the omponents k f k : A k A f k ( 1,..., k ) k i k,l = k,l : A k A A l A S k+1 k+l+2 k+2... k+l+1 By onvention, the inputs re lwys inserted using the ounterlokwise diretion. Then f A i ( 1) f i i f A is given y inserting f into i in ll possile omintions.
12 Homology, Homotopy nd Applitions, vol. 8(2), ± ± ± ± First, here re some exmples of how these digrms re to e red.,,, d 2,0 d,,, d, e, f, g, h, i 3,4 d i e f g h f 2 (f 2 (, ), f 2 (d, e)), f 2 (f, ) 0,0 f d e
13 Homology, Homotopy nd Applitions, vol. 8(2), ,, f 3 (, d, f 2 (e, f)), g, f 2 (h, i)) 1,2 i d g h e f, f 2 (d, e), f 2 (f 2 (f, g), h), i, f 4 (j, k,, ) 2,1 f e d g k h i j Here re the terms of δ f (i) = f A i ( 1) f i i f A up to sign, when they re eing pplied to elements from A k A A l A: k = 0, l = 0: f 1 (), 0,0 ±, f 1 () 0,0 ± k = 1, l = 0: f 1 (),, 1,0 ±, f 1 (), 1,0 ±,, f 1 () 1,0 ± f 2 (, ), 0,0 ±, f 2 (, ) 0,0
14 Homology, Homotopy nd Applitions, vol. 8(2), ± ± ± ± k = 0, l = 1: f 1 (),, 0,1 ±, f 1 (), 0,1 ±,, f 1 () 0,1 ± f 2 (, ), 0,0 ±, f 2 (, ) 0,0 ± ± ± ± k = 2, l = 0: f 1 (),,, d 2,0 ±, f 1 (),, d 2,0 ±,, f 1 (), d 2,0 ±,,, f 1 (d) 2,0 ± f 2 (, ),, d 1,0 ±, f 2 (, ), d 1,0 ±,, f 2 (d, ) 1,0 ± f 3 (,, ), d 0,0 ±, f 3 (d,, ) 0,0 Note tht for exmple the term,, f 2 (, d) 2,0 does not pper, euse nd d re the two speil elements of d A 2 A A 0 A, whih re put on the horizontl line of the digrm. The two speil elements from A k A A l A n never e inside ny f n.
15 Homology, Homotopy nd Applitions, vol. 8(2), k = 0, l = 2: f 1 (),,, d 0,2 ±, f 1 (),, d 0,2 ±,, f 1 (), d 0,2 ±,,, f 1 (d) 0,2 ± f 2 (, ),, d 0,1 ±, f 2 (, ), d 0,1 ±,, f 2 (, d) 0,1 ± f 3 (,, ), d 0,0 ±, f 3 (,, d) 0,0 The speil elements re nd d from d A 0 A A 2 A. k = 1, l = 1: f 1 (),,, d 1,1 ±, f 1 (),, d 1,1 ±,, f 1 (), d 1,1 ±,,, f 1 (d) 1,1 ± f 2 (, ),, d 0,1 ±,, f 2 (d, ) 0,1 ±, f 2 (, ), d 1,0 ±,, f 2 (, d) 1,0 ± f 3 (,, ), d 0,0 ±, f 3 (, d, ) 0,0 The speil elements re nd d from d A 1 A A 1 A.
16 Homology, Homotopy nd Applitions, vol. 8(2), i =, 0,0 for ny k, l: Assume tht i =, 0,0 hs only lowest omponent, ut f hs ll higher omponents. We pply f A i ( 1) f i i f A to the element 1... k k+1 k+2... k+l+1 k+l+2 A k A A l A to get f( 1,..., k+l+1 ), k+l+2 0,0 ± k+1, f( k+2,..., k+l+2, 1,..., k ) 0,0 ± Referenes [1] R. Cohen, J. Klein nd D. Sullivn. The homotopy invrine of the string topology loop produt nd string rket. mth.gt/ , [2] A. Filowski nd M. Penkv. Deformtion theory of infinity lgers. Journl of Alger 255, 2002, [3] W.M. Goldmn nd J.J. Millson. The deformtion theory of representtions of fundmentl groups of ompt Kähler mnifolds. Pul. Mth IHES 67, IHES, 1988, [4] H. Kjiur. Nonommuttive homotopy lgers ssoited with open strings. mth.qa/ , [5] M. Kontsevih. Deformtion quntiztion of Poisson mnifolds, I. Letters in Mthemtil Physis 66, no 3, Springer, Netherlnds, 2003, [6] M. Mnetti. Deformtion theory vi differentil grded Lie lgers. Seminri di Geometri Algeri, Suol Normle Superiore, 1999, [7] M. Penkv. Infinity lgers, ohomology nd yli ohomology, nd infinitesiml deformtions. mth.qa/ , 2001.
17 Homology, Homotopy nd Applitions, vol. 8(2), [8] M. Shlessinger. Funtors of Artin rings. Trns. Am. Mth. So. 130, 1968, [9] M. Shlessinger nd J. Stsheff. The Lie lger struture of tngent ohomology nd deformtion theory. J. Pure nd Applied Alger 38, 1985, [10] D. Sullivn. Infinitesiml omputtions in topology. Pulitions Mthémtiques de l IHÉS 47 (1977), [11] T. Trdler nd M. Zeinlin. Poinré dulity t the hin level. mth.at/ , [12] T. Trdler. Infinity inner produts on A-infinity lgers. mth.at/ , [13] T. Trdler. The BV lger on Hohshild ohomology indued y infinity inner produts. mth.qa/ , John Terill john@q.edu Deprtment of Mthemtis Queens College of the City University of New York Kissen Blvd. Flushing, NY Thoms Trdler ttrdler@ityteh.uny.edu Deprtment of Mthemtis College of Tehnology of the City University of New York 300 Jy Street Brooklyn, NY This rtile is ville t
Pre-Lie algebras, rooted trees and related algebraic structures
Pre-Lie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004 Definition 1 A pre-lie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers
More informationOn Implicative and Strong Implicative Filters of Lattice Wajsberg Algebras
Glol Journl of Mthemtil Sienes: Theory nd Prtil. ISSN 974-32 Volume 9, Numer 3 (27), pp. 387-397 Interntionl Reserh Pulition House http://www.irphouse.om On Implitive nd Strong Implitive Filters of Lttie
More informationFigure 1. The left-handed and right-handed trefoils
The Knot Group A knot is n emedding of the irle into R 3 (or S 3 ), k : S 1 R 3. We shll ssume our knots re tme, mening the emedding n e extended to solid torus, K : S 1 D 2 R 3. The imge is lled tuulr
More informationLecture 1 - Introduction and Basic Facts about PDEs
* 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More information#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS
#A42 INTEGERS 11 (2011 ON THE CONDITIONED BINOMIAL COEFFICIENTS Liqun To Shool of Mthemtil Sienes, Luoyng Norml University, Luoyng, Chin lqto@lynuedun Reeived: 12/24/10, Revised: 5/11/11, Aepted: 5/16/11,
More informationarxiv: v1 [math.ca] 21 Aug 2018
rxiv:1808.07159v1 [mth.ca] 1 Aug 018 Clulus on Dul Rel Numbers Keqin Liu Deprtment of Mthemtis The University of British Columbi Vnouver, BC Cnd, V6T 1Z Augest, 018 Abstrt We present the bsi theory of
More informationElectromagnetism Notes, NYU Spring 2018
Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system
More informationLecture 8: Abstract Algebra
Mth 94 Professor: Pri Brtlett Leture 8: Astrt Alger Week 8 UCSB 2015 This is the eighth week of the Mthemtis Sujet Test GRE prep ourse; here, we run very rough-n-tumle review of strt lger! As lwys, this
More informationChapter 3. Vector Spaces. 3.1 Images and Image Arithmetic
Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationLearning Objectives of Module 2 (Algebra and Calculus) Notes:
67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under
More informationMatrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
More informationTechnische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:
More informationDiscrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α
Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
More informationMAT 403 NOTES 4. f + f =
MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd
More informationILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS
ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu
More informationReference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.
I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the
More informationarxiv: v1 [math.ct] 8 Sep 2009
On the briding of n Ann-tegory rxiv:0909.1486v1 [mth.ct] 8 Sep 2009 September 8, 2009 NGUYEN TIEN QUANG nd DANG DINH HANH Dept. of Mthemtis, Hnoi Ntionl University of Edution, Viet Nm Emil: nguyenqung272002@gmil.om
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationON LEFT(RIGHT) SEMI-REGULAR AND g-reguar po-semigroups. Sang Keun Lee
Kngweon-Kyungki Mth. Jour. 10 (2002), No. 2, pp. 117 122 ON LEFT(RIGHT) SEMI-REGULAR AND g-reguar po-semigroups Sng Keun Lee Astrt. In this pper, we give some properties of left(right) semi-regulr nd g-regulr
More informationHyers-Ulam stability of Pielou logistic difference equation
vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationMath 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)
Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
More informationApril 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
More informationCS 573 Automata Theory and Formal Languages
Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple
More informationLecture Notes No. 10
2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite
More informationAlgebra in a Category
Algebr in Ctegory Dniel Muret Otober 5, 2006 In the topos Sets we build lgebri strutures out o sets nd opertions (morphisms between sets) where the opertions re required to stisy vrious xioms. One we hve
More informationNON-DETERMINISTIC FSA
Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is
More informationDiscrete Structures Lecture 11
Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.
More informationSymmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationKENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS CLASS - XII MATHEMATICS (Relations and Functions & Binary Operations)
KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS 6-7 CLASS - XII MATHEMATICS (Reltions nd Funtions & Binry Opertions) For Slow Lerners: - A Reltion is sid to e Reflexive if.. every A
More informationProperties of Different Types of Lorentz Transformations
merin Journl of Mthemtis nd ttistis 03 3(3: 05-3 DOI: 0593/jjms03030303 roperties of Different Types of Lorentz Trnsformtions tikur Rhmn izid * Md hh lm Deprtment of usiness dministrtion Leding niversity
More informationFunctions. mjarrar Watch this lecture and download the slides
9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides
More informationPOSITIVE IMPLICATIVE AND ASSOCIATIVE FILTERS OF LATTICE IMPLICATION ALGEBRAS
Bull. Koren Mth. So. 35 (998), No., pp. 53 6 POSITIVE IMPLICATIVE AND ASSOCIATIVE FILTERS OF LATTICE IMPLICATION ALGEBRAS YOUNG BAE JUN*, YANG XU AND KEYUN QIN ABSTRACT. We introue the onepts of positive
More informationThe University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS
The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their
More informationTOPIC: LINEAR ALGEBRA MATRICES
Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED
More informationLecture 6: Coding theory
Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those
More information1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationCS 491G Combinatorial Optimization Lecture Notes
CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,
More informationGeometrically Realizing Nested Torus Links
Geometrilly Relizing Nested Torus Links mie ry ugust 18, 2017 strt In this pper, we define nested torus links. We then go on to introdue ell deomposition of nested torus links. Using the ell deomposition
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationSEMI-EXCIRCLE OF QUADRILATERAL
JP Journl of Mthemtil Sienes Volume 5, Issue &, 05, Pges - 05 Ishn Pulishing House This pper is ville online t http://wwwiphsiom SEMI-EXCIRCLE OF QUADRILATERAL MASHADI, SRI GEMAWATI, HASRIATI AND HESY
More informationGeneralized Kronecker Product and Its Application
Vol. 1, No. 1 ISSN: 1916-9795 Generlize Kroneker Prout n Its Applition Xingxing Liu Shool of mthemtis n omputer Siene Ynn University Shnxi 716000, Chin E-mil: lxx6407@163.om Astrt In this pper, we promote
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationCo-ordinated s-convex Function in the First Sense with Some Hadamard-Type Inequalities
Int. J. Contemp. Mth. Sienes, Vol. 3, 008, no. 3, 557-567 Co-ordinted s-convex Funtion in the First Sense with Some Hdmrd-Type Inequlities Mohmmd Alomri nd Mslin Drus Shool o Mthemtil Sienes Fulty o Siene
More informationCan one hear the shape of a drum?
Cn one her the shpe of drum? After M. K, C. Gordon, D. We, nd S. Wolpert Corentin Lén Università Degli Studi di Torino Diprtimento di Mtemti Giuseppe Peno UNITO Mthemtis Ph.D Seminrs Mondy 23 My 2016 Motivtion:
More informationBisimulation, Games & Hennessy Milner logic
Bisimultion, Gmes & Hennessy Milner logi Leture 1 of Modelli Mtemtii dei Proessi Conorrenti Pweł Soboiński Univeristy of Southmpton, UK Bisimultion, Gmes & Hennessy Milner logi p.1/32 Clssil lnguge theory
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationGeneralization of 2-Corner Frequency Source Models Used in SMSIM
Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville
More informationElectromagnetic-Power-based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors
LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 1 Eletromgneti-ower-bsed Modl Clssifition Modl Expnsion nd Modl Deomposition for erfet Eletri Condutors Renzun Lin Abstrt Trditionlly
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationQualgebras and knotted 3-valent graphs
Qulgers nd knotted 3-vlent grphs Vitori Leed To ite this version: Vitori Leed. Qulgers nd knotted 3-vlent grphs. 2014. HAL Id: hl-00951712 https://hl.rhives-ouvertes.fr/hl-00951712 Sumitted
More informationWord-Induced Measures on Compact Groups
Word-Indued Mesures on Compt roups ene S. Kopp nd John D. Wiltshire-ordon Februry 23, 2011 rxiv:1102.4353v1 [mth.r] 21 Feb 2011 Abstrt Consider group word w in n letters. For ompt group, w indues mp n
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationDescriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata
Desriptionl Complexity of Non-Unry Self-Verifying Symmetri Differene Automt Lurette Mris 1,2 nd Lynette vn Zijl 1 1 Deprtment of Computer Siene, Stellenosh University, South Afri 2 Merk Institute, CSIR,
More informationCS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6
CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized
More informationLecture 2: Cayley Graphs
Mth 137B Professor: Pri Brtlett Leture 2: Cyley Grphs Week 3 UCSB 2014 (Relevnt soure mteril: Setion VIII.1 of Bollos s Moern Grph Theory; 3.7 of Gosil n Royle s Algeri Grph Theory; vrious ppers I ve re
More informationA Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version
A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment
More informationQubit and Quantum Gates
Quit nd Quntum Gtes Shool on Quntum omputing @Ygmi Dy, Lesson 9:-:, Mrh, 5 Eisuke Ae Deprtment of Applied Physis nd Physio-Informtis, nd REST-JST, Keio University From lssil to quntum Informtion is physil
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationAbstraction of Nondeterministic Automata Rong Su
Astrtion of Nondeterministi Automt Rong Su My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 1 Outline Motivtion Automton Astrtion Relevnt Properties Conlusions My 6, 2010 TU/e Mehnil Engineering,
More informationNecessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 )
Neessry n suient onitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deite to Professor Anne Penfol Street Astrt We give new lgorithm whih llows us
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd h : I C writing h = u + iv where u, v : I C, we n extend ll lulus 1 onepts
More informationNew Perspectives in Arakelov Geometry
Centre de Reherhes Mthémtiques CRM Proeedings nd Leture Notes Volume 36, 2004 New Perspetives in Arkelov Geometry Cterin Consni nd Mtilde Mrolli Astrt. In this pper we give unified desription of the Arhimeden
More informationarxiv: v1 [math.gr] 11 Jan 2019
The Generlized Dehn Property does not imply liner isoperimetri inequlity Owen Bker nd Timothy Riley rxiv:1901.03767v1 [mth.gr] 11 Jn 2019 Jnury 15, 2019 Astrt The Dehn property for omplex is tht every
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationSystem Validation (IN4387) November 2, 2012, 14:00-17:00
System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise
More informationwhere the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}
More informationSEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II: A MULTIPLICATION FORMULA
SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II: A MULTIPLICATION FORMULA CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Abstrt. Let n be mximl nilpotent sublgebr of omplex symmetri K-Moody Lie lgebr.
More informationSolutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More informationThe study of dual integral equations with generalized Legendre functions
J. Mth. Anl. Appl. 34 (5) 75 733 www.elsevier.om/lote/jm The study of dul integrl equtions with generlized Legendre funtions B.M. Singh, J. Rokne,R.S.Dhliwl Deprtment of Mthemtis, The University of Clgry,
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationA STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS
Commun. Koren Mth. So. 31 016, No. 1, pp. 65 94 http://dx.doi.org/10.4134/ckms.016.31.1.065 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS Hrsh Vrdhn Hrsh, Yong Sup Kim, Medht Ahmed Rkh, nd Arjun Kumr Rthie
More informationThe Word Problem in Quandles
The Word Prolem in Qundles Benjmin Fish Advisor: Ren Levitt April 5, 2013 1 1 Introdution A word over n lger A is finite sequene of elements of A, prentheses, nd opertions of A defined reursively: Given
More informationProportions: A ratio is the quotient of two numbers. For example, 2 3
Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)
More informationA Functorial Query Language
A Funtoril Query Lnguge Ryn Wisnesky, Dvid Spivk Deprtment of Mthemtis Msshusetts Institute of Tehnology {wisnesky, dspivk}@mth.mit.edu Presented t Boston Hskell April 16, 2014 Outline Introdution to FQL.
More informationarxiv: v1 [math.qa] 27 Apr 2017
SIMPLY-LACED QUANTUM CONNECTIONS GENERALISING KZ rxiv:1704.08616v1 [mth.qa 27 Apr 2017 Astrt. We onstrut new fmily of flt onnetions generlising the KZ onnetion, the Csimir onnetion nd the dynmil onnetion.
More informationRelations between a dual unit vector and Frenet vectors of a dual curve
Kuwit J. Si. 4 () pp. 59-69, 6 Burk Şhiner *, Mehmet Önder Dept. of Mthemtis, Mnis Cell Byr University, Murdiye, Mnis, 454, Turkey * Corresponding uthor: burk.shiner@bu.edu.tr Abstrt In this pper, we generlize
More informationUniversity of Warwick institutional repository: A Thesis Submitted for the Degree of PhD at the University of Warwick
University of Wrwik institutionl repository: http://go.wrwik..uk/wrp A Thesis Sumitted for the Degree of PhD t the University of Wrwik http://go.wrwik..uk/wrp/3645 This thesis is mde ville online nd is
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationGRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames
Visuliing Quternions Andrew J. Hnson Computer Siene Deprtment Indin Universit Siggrph Tutoril GRAND PLAN I: Fundmentls of Quternions II: Visuliing Quternion Geometr III: Quternion Frmes IV: Clifford Algers
More information8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
More informationDorf, R.C., Wan, Z. T- Equivalent Networks The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
orf, R.C., Wn,. T- Equivlent Networks The Eletril Engineering Hndook Ed. Rihrd C. orf Bo Rton: CRC Press LLC, 000 9 T P Equivlent Networks hen Wn University of Cliforni, vis Rihrd C. orf University of
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or
More informationT b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
More informationConneted sum of representations of knot groups
Conneted sum of representtions of knot groups Jinseok Cho rxiv:141.6970v4 [mth.gt] 3 Mr 016 November 4, 017 Abstrct When two boundry-prbolic representtions of knot groups re given, we introduce the connected
More information