Fixed points of IA-endomorphisms of a free metabelian Lie algebra

Size: px
Start display at page:

Download "Fixed points of IA-endomorphisms of a free metabelian Lie algebra"

Transcription

1 Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ 2 1 Department of Mathematcs, Çukurova Unversty, Adana, Turkey 2 Department of Mathematcs, Melkşah Unversty, Kayser, Turkey Emal: nekc@cu.edu.tr; dsonmez@melksah.edu.tr MS receved 16 November 2009; revsed 21 June 2011 Abstract. Let L be a free metabelan Le algebra of fnte rank at least 2. We show the exstence of non-trval fxed ponts of an IA-endomorphsm of L and gve an algorthm detectng them. In partcular, we prove that the fxed pont subalgebra Fx ϕ of an IA-endomorphsm ϕ of L s not fntely generated. Keywords. Free metabelan Le algebra; fxed pont. 1. Introducton One of the mportant problem n the theory of Le algebras s to determne the non-trval fxed ponts of endomorphsms of free Le algebras. The most mportant results about fxed ponts of a fnte group actng on a free algebra were obtaned by Formanek [5]. Smlar results for free Le algebras were proved by Bryant [2] and Drensky [4]. They showed that f F s a free Le algebra of fnte rank n (n 2) and G s a non-trval fnte group of automorphsms of F then under some assumptons the fxed pont subalgebra F G = {u F : ug = u for all g G} of F s not fntely generated. In [3], Bryant and Papstas have extended these results. Some of the results about fxed ponts of a fnte group of automorphsms of a free algebra can also apply to fxed ponts for a sngle endomorphsm. In [8], Shplran has gven a matrx characterzaton of the fxed ponts of an IA-endomorphsm of a free metabelan group. Matrx methods have been used by a number of authors to get nterestng results on endomorphsms of free metabelan Le algebras and groups. In ths paper we obtan a crteron for detectng non-trval fxed ponts of an IA-endomorphsm of a free metabelan Le algebra. We use the method of [8] to obtan a matrx characterzaton of IA-endomorphsms of a free metabelan Le algebra wth non-trval fxed ponts. We also prove that the non-trval fxed pont subalgebra of an IA-endomorphsm of a free metabelan Le algebra s not fntely generated. In our proof we use the method developed n [3]. 2. Prelmnares Let K be a feld and let F be a free Le algebra (over K ) freely generated by the set X = {x 1,...,x n } wth n 2. We use commutator notaton [ f, g] to denote the product of elements f and g of F, whle [ f 1,..., f n ] denotes the rght-normed products of elements 405

2 406 Name Ekc and Demet Parlak Sönmez f 1,..., f n of F. We denote by F and F the derved algebra [F, F] and the second derved algebra [[F, F], [F, F]] respectvely. We dentfy a free metabelan Le algebra L of rank n wth F/F n the usual way. An endomorphsm of F whch nduces the dentty on F/F s called an IAendomorphsm of the free Le algebra F. We wrte U (F) for the unversal envelopng algebra of the free Le algebra F.LetP be the polynomal algebra K [x 1,...,x n ] and let V be the subspace spanned by {x 1,...,x n }. We regard P K V (tensor product taken over K )asaleftp-module n the obvous way. Clearly t s a free P-module wth {1 x : x X} as a free generatng set. It s easy to verfy that the derved subalgebra L of the free metabelan Le algebra L may be vewed as a left P-module n whch the mage of an element g of L under the acton of a monomal y 1...y m of P s the rght-normed Le product [y 1,...,y m, g]. Note that P may be regarded as U(F/F ).Forg L, u U(F/F ) we wrte u g to denote the mage of g under the module acton of u. We denote by x,1 n the left Fox dervatves [6,7]. The operators x : U (F) U (F) are lnear mappngs such that x j x = δ j (Kronecker delta), (u+v) x = x u + x v, (uv) x = x u ε (v) + u x v, where ε : U (F) K s the augmentaton homomorphsm defned as ε (x ) = 0 for all = 1,..., n. By we denote the kernel of the augmentaton homomorphsm ε. The deal s a free left U (F)-module wth bass {x 1, x 2,..., x n }. Thus any element u can be unquely wrtten n the form u = n u x x. The Jacoban matrx J ϕ of an endomorphsm ϕ of F s defned as J ϕ = ( ) ϕ(x ) x j x j, where denotes partal Fox dervaton wth respect to x j n the unversal envelopng algebra U (F) (see [7] for detals). If R = F s an deal of F then we 1, j n denote by R the deal of U (F) generated by R. In [6], Fox has gven a detaled account of the dfferental calculus n a free group rng. Snce any assocatve algebra s naturally embedded n a free group algebra, most of the techncal results reman vald for free assocatve algebras (see [7] for detals). Throughout ths paper, we need the followng lemmas. Lemma 1. Let J be an arbtrary deal of F and let u. Then u J f and only f u x j J for each, 1 n. Proof. Let u = vw J,where v J, w. The element w of can be wrtten n the form w = n w j=1 x j x j. Therefore x u = v x w J for each, 1 n. The proof of the other part of the lemma s obvous. The proof of the next lemma can be found n [9]. Lemma 2. Let R be an deal of F and let u F. Then u R f and only f u R. For any endomorphsm ϕ of a Le algebra G we wrte Fx ϕ = {u G : ϕ (u) = u}. Thus Fx ϕ s the fxed pont subalgebra of G.LetH be any subset of a vector space over K (K -space). We wrte H for the K -subspace spanned by H.

3 Free metabelan Le algebra Matrx characterzaton of fxed ponts Let K be a feld and F be a free Le algebra freely generated by the set X = {x 1,...,x n }, n 2, over K and let L = F/F. We denote the elements of the free Le algebra F and ther mages n L by the same letters. When u F or u U (F) we denote by ū the mage of u n the free abelan Le algebra F/F or n the algebra U(F/F ) respectvely. Let ϕ be an IA-endomorphsm of L defned as ϕ : x x + u, where u F, 1 n. Then the Jacoban matrx J ϕ of ϕ can be wrtten as J ϕ = I + D ϕ (u 1,...,u n ), where I s the dentty matrx and D ϕ (u 1,...,u n ) = u 1 u 1 x n x u n x 1... u n x n The mage of D ϕ (u 1,...,u n ) over the abelanzed algebra U(F/F ) wll be denoted by D ϕ (u 1,...,u n ).Bytherank of a matrx A over a commutatve rng R we mean the maxmal number of rows of A ndependent over R.. PROPOSITION 3 Let ϕ be an IA-endomorphsm of L defned by ϕ : x x + u, u F, 1 n. Then the columns of the matrx D ϕ (u 1,...,u n ) are dependent over U(F/F ). Proof. Let ϕ : L L defned by ϕ : x x + u, where u F,1 n. Now consder the mage D ϕ (u 1,...,u n ) of the matrx D ϕ (u 1,...,u n ) over U(F/F ). Then there s the followng matrx equaton: D ϕ (u 1,...,u n ) x 1.. = x n = u 1 u 1 x n x u n x 1 n u 1 x n. u n x u n x n. x.. x x 1.. x n

4 408 Name Ekc and Demet Parlak Sönmez Indeed n u j x x = n u j x x = u j. Snce u j F, u j = 0nU(F/F ), 1 j n. Ths yelds the matrx equalty D ϕ (u 1,...,u n ) x 1. = x n 0. 0 n U(F/F ). Hence the columns of the matrx U(F/F ). D ϕ (u 1,...,u n ) are dependent over Our frst man result guarantees the exstence of the non-trval fxed ponts of an IAendomorphsm of a free metabelan Le algebra. Theorem 4. Let ϕ be an IA-endomorphsm gven by ϕ : x x + u, where u L, 1 n. Then f rank D ϕ (u 1,...,u n ) n 2 then ϕ has a non-trval fxed pont nsde L. Proof. Let ϕ : L L be an endomorphsm gven by ϕ : x x + u, u L, 1 n. For any element f F we denote by f the mage of f n the algebra F/F. If rank D ϕ (u 1,...,u n ) n 2 then the rows of the matrx D ϕ (u 1,...,u n ) are dependent over the envelopng algebra U(F/F ). We wrte ths dependence as n u a = 0, j = 1,...,n (1) x j n U(F/F ) for some a s U(F/F ), not all of them zero. Wrte system (1) as ( n ) a u = 0, j = 1,...,n. (2) x j The system (2) s equvalent to the relaton n a u = 0 by Lemmas 1 and 2. It s clear that the matrx D ϕ (u 1,...,u n ) s the mage of the Jacoban matrx of the endomorphsm ϕ : x u, = 1,...,n of L n U(F/F ),.e. J ϕ = D ϕ (u 1,...,u n ). If rank J ϕ = k n 2, then the maxmal U(F/F )-ndependent subset of the set {u 1,...,u n } has k elements. Let {u 1,...,u k } be a maxmal ndependent set. Therefore u 1,...,u k generate a free submodule of the U(F/F )-module L. Hence for any j k+1 we have a relaton of the form k a j u j = b j u, where b j U ( F/F ). (3)

5 Free metabelan Le algebra 409 Let f L be an element n the form f = v 1 [x 1, x 2 ]+v 2 [x 2, x 3 ]+ +v n 1 [x n 1, x n ], where v U(F/F ), = 1,...,n 1. We are gong to construct elements v 1,...,v n 1 of U(F/F ) such that f = v 1 [x 1, x 2 ]+v 2 [x 2, x 3 ]+ +v n 1 [x n 1, x n ] s a non-trval fxed pont of the endomorphsm ϕ. Assume that ϕ ( f ) = f. We have v 1 [x 1, x 2 ]+ +v n 1 [x n 1, x n ] = v 1 [x 1 + u 1, x 2 + u 2 ]+ +v n 1 [x n 1 + u n 1, x n + u n ]. From ths equalty we get v 1 ([x 1, u 2 ]+[u 1, x 2 ]) + +v n 1 ([x n 1, u n ]+[u n 1, x n ]) = 0. (4) For j k + 1 multply both sdes of (4) by a j, where a j come from (3). Replace every a j u j n (4) wth k b j u. Ths gves a relaton of the form w 1 u 1 + +w k u k = 0, (5) where each w s a U(F/F )-lnear combnaton of elements v 1,v 2,...,v n 1. Snce u 1,...,u k generate a free U(F/F )-submodule, the relaton (5) leads a system of equatons w j = 0, 1 j k (6) wth unknowns v 1,v 2,...,v n 1. Snce k n 2, ths system has a non-trval soluton (z 1,...,z n 1 ) over U(F/F ). Hence f = z 1 [x 1, x 2 ]+z 2 [x 2, x 3 ]+ +z n 1 [x n 1, x n ] s a fxed pont of the endomorphsm ϕ whch s contaned n L. Now we wll consder the case rank D ϕ (u 1,...,u n ) = n 1. In ths case an IAendomorphsm may not have non trval fxed ponts. We are gong to consder an element g of L,ntheform g = a 1 [x 1, x 2 ] + a 2 [x 2, x 3 ] + +a n 1 [x n 1, x n ], where a k U ( F/F ), 1 k n 1. Lemma 5. For an element h L, there exst elements z, a 1,...,a n 1 of U(F/F ) such that z h = a 1 [x 1, x 2 ] + a 2 [x 2, x 3 ] + +a n 1 [x n 1, x n ]. Proof. Rght normed basc monomals [x r,...,x 2, x 1 ], where r 2 form a bass of the derved algebra L (see [1] for detals). Let P be the polynomal algebra K [x 1,...,x n ]. We clam that () the elements [x 1, x 2 ], [x 2, x 3 ],...,[x n 1, x n ] generate a free P-submodule of rank n 1, () the P-module L has no submodules of rank n.

6 410 Name Ekc and Demet Parlak Sönmez Proof of (). Let V be the subspace spanned by the set {x 1,...,x n }. Form the tensor product P K V. It s shown n Lemma 4.1() of [3] that there s a P-module embeddng θ : L P K V n whch [v r,...,v 2,v 1 ] v r...v 3 v 2 v 1 v r...v 3 v 1 v 2 for all r 2 and all v 1,...,v r V. Suppose that f 1 [x 1, x 2 ]+ f 2 [x 2, x 3 ]+ + f n 1 [x n 1, x n ]=0 wth f 1,..., f n 1 P. Applyng θ to the above equaton, we get f 1 x 1 (1 x 2 ) f 1 x 2 (1 x 1 ) + f 2 x 2 (1 x 3 ) f 2 x 3 (1 x 2 ) + + f n 1 x n 1 (1 x n ) = 0. Snce {1 x 1,...,1 x n } s a free generatng set of P K V as P-module, we obtan f 1 x 2 = 0, f 1 x 1 f 2 x 3 = 0, f 2 x 2 f 3 x 4 = 0,. f n 1 x n 1 = 0. Snce P s a prncpal deal doman (PID), we have f 1 = = f n 1 = 0 and so, we have the requred result. Proof of (). To get a contradcton we assume that there exsts a P-submodule M of L wth rank n. Snce P K V s a free P-module and P s a PID, we obtan θ(m) s a free P-module and rank (θ(m)) n. Snce θ s P-module embeddng, we get rank (θ(m)) = n. Therefore θ s onto whch s a contradcton. (For example, for any u L,θ(u) = x 1 x 1.) We now proceed wth the proof of the lemma. For any par k, l we have b k,l [x k, x l ]=c 1 [x 1, x 2 ]+c 2 [x 2, x 3 ]+ +c n 1 [x n 1, x n ] (7) for some b k,l, c j U(F/F ). Therefore any k-lnear combnaton of elements of the form (7) s n the form a 1 [x 1, x 2 ]+a 2 [x 2, x 3 ]+ +a n 1 [x n 1, x n ], where a k U(F/F ). Ths completes the proof. The followng examples wll show that the stuaton s very subtle n the case rank D ϕ (u 1,...,u n ) = n 1. In ths case, anythng can happen. PROPOSITION 6 Let n 3 and let ϕ be an endomorphsm of L defned by ϕ : x x + u, 1 n 2 x n 1 x n 1 + u, x n x n + u,

7 Free metabelan Le algebra 411 where u, u L and the elements u, uareu(f/f )-ndependent, 1 n 2. Then ϕ has no non-trval fxed ponts nsde L. Proof. For any element f L there exst an element z U(F/F ) such that z f = v 1 [x 1, x 2 ] + +v n 1 [x n 1, x n ], where v U(F/F ). Suppose ϕ ( f ) = f. Multplyng ths equalty wth z from left, we get ϕ (z f ) = z f. Then we have (cf. (4)) v 1 (x 1 u 2 + u 1 x 2 ) + +v n 2 (x n 2 u + u n 2 x n 1 ) + v n 1 (x n 1 u + ux n ) = 0. Snce u, u 1,...,u n 2 are ndependent, we get the system v 1 x 2 = 0, v j x j v j+1 x j+2 = 0, j = 1,...,n 3 v n 2 x n 2 + v n 1 (x n 1 x n ) = 0 (8) n U(F/F ). Snce U(F/F ) s an ntegral doman, all of the elements v 1,...,v n 1 are equal to zero. Hence system (8) has no non-zero solutons. Ths completes the proof. PROPOSITION 7 Let M be a free metabelan Le algebra freely generated by {x 1, x 2 } and ϕ be an endomorphsm of M defned by ϕ : x 1 x 1 + [x 1, u], x 2 x 2 + u, for some non-zero u M. Then ϕ has no non-trval fxed ponts. Proof. Frst we prove that ϕ has no non-trval fxed ponts n M. Let h M and ϕ (h) = h. Snce h has the form h = w [x 1, x 2 ] for some w U(F/F ), we have It follows that w [x 1, x 2 ] = w [x 1 + [x 1, u], x 2 + u]. w (x 1 x 2 x 1 ) u = 0. Hence w (x 1 x 2 x 1 ) = 0nU(F/F ). Ths s possble only f w = 0. Thus h = 0. Now we wll show that ϕ has no non-trval fxed ponts outsde M.Letm be an arbtrary element of M. Then m has the form ax 1 + bx 2 + w [x 1, x 2 ] for some w U(F/F ) and a, b K. Suppose ϕ (m) = m. Then we have It follows that ax 1 + bx 2 + w [x 1, x 2 ] = a (x 1 + [x 1, u]) + b (x 2 + u) + w [x 1 + [x 1, u], x 2 + u]. (ax 1 + b + w (x 1 x 2 x 1 )) u = 0.

8 412 Name Ekc and Demet Parlak Sönmez Hence ax 1 + b + w (x 1 x 2 x 1 ) = 0 (9) n U(F/F ). Ths s possble only f a = b = w = 0. Ths completes the proof. Now we are gong to prove that an IA-endomorphsm of L has non-zero fxed ponts n a certan case. Frst we need some observatons. Let ϕ be an IA-endomorphsm of L defned by ϕ (x ) = y = x + u, where u L, 1 n. If rank D ϕ (u 1,...,u n ) = n 1 then the maxmal U(F/F )-ndependent subset of {u 1,...,u n } has n 1 elements. Let {u 1,...,u n 1 } be a maxmal ndependent set. Therefore u 1,...,u n 1 generates a free submodule of the U(F/F )-module L. Hence we have a relaton of the form n 1 a n u n = a u, (10) for some a j U(F/F ). PROPOSITION 8 Let ϕ be an IA-endomorphsm of L defned by ϕ (x ) = x +u, where u L, 1 n and rank D ϕ (u 1,...,u n ) = n 1. Then ϕ has a non-trval fxed pont n L f and only f the system v 1 x 2 a n + v n 1 x n 1 a 1 = 0, v x a n v +1 x +2 a n + v n 1 x n 1 a +1 = 0, 1 n 2 wth unknowns v 1,...,v n 1, has a non-zero soluton n U(F/F ). Proof. Let ϕ (h) = h for any element h of L. By Lemma 5 there exsts an element z U(F/F ) such that z h = v 1 [x 1, x 2 ]+v 2 [x 2, x 3 ]+ +v n 1 [x n 1, x n ] for some elements v j U(F/F ). Apply ϕ to ths equalty: ϕ (z h) = v 1 [y 1, y 2 ]+v 2 [y 2, y 3 ]+ +v n 1 [y n 1, y n ]. (11) Multplyng the equalty ϕ (h) = h wth z from left we get ϕ (z h) = z h = v 1 [x 1, x 2 ]+v 2 [x 2, x 3 ]+ +v n 1 [x n 1, x n ]. (12) Combnng (12) wth (11) we obtan Equalty (13) gves v 1 [x 1, x 2 ]+ +v n 1 [x n 1, x n ]=v 1 [y 1, y 2 ]+ +v n 1 [y n 1, y n ]. (13) n 2 v 1 x 2 u 1 + (v x v +1 x +2 ) u +1 + v n 1 x n 1 u n = 0. (14)

9 Free metabelan Le algebra 413 Multplyng both sdes of (14) by a n and replacng a n u n n (14) wth n 1 a u, we obtan asystemof n 1 equatons v 1 x 2 a n + v n 1 x n 1 a 1 = 0, v x a n v +1 x +2 a n + v n 1 x n 1 a +1 = 0, 1 n 2 (15) wth unknowns v 1,...,v n 1. Ths completes the proof. The f part s obvous. Theorem 9. There s an algorthm for detectng non-trval fxed ponts of an arbtrary IA-endomorphsm of the free metabelan Le algebra L. Proof. Let ϕ be an IA-endomorphsm of L defned by ϕ : x x + u, u F, 1 n. Frst we compute the rank of the matrx D ϕ (u 1,...,u n ): Indeed by usng elementary transformatons of the rows of the matrx D ϕ (u 1,...,u n ), we can construct a bass of the free left U(F/F )-submodule of the free left U(F/F )-module (U(L)) n, and compute ts rank. We have two cases for rank D ϕ (u 1,...,u n ). Case I. If rank D ϕ (u 1,...,u n ) n 2, then we refer to Theorem 4. Case II. If rank D ϕ (u 1,...,u n ) = n 1, we wll consder two cases: (a) We fnd f there s a non-trval fxed ponts nsde L : We consder a system (15) as n the proof of Proposton 8. Ths s a system of n 1 homogeneous U(F/F )-lnear equatons n n 1 unknowns v 1,...,v n 1. Now we check the dependence of the rows of the coeffcent matrx. If they are dependent then ϕ has a non-trval fxed pont nsde L. If these rows are ndependent then ϕ has no non-trval fxed ponts nsde L. Then we consder the case (b). (b) To fnd out f there s a non-trval fxed pont of ϕ outsde L we proceed as n the proof of Proposton 7, but nstead of havng just one equaton of the form (9), we wll have a system of n 1 U(F/F )-lnear equatons n n 1 unknowns. Snce U(F/F ) s somorphc to a polynomal algebra, usng algorthms n polynomal algebras we can solve the system. Ths completes the proof. 4. The fxed pont subalgebra Let K be a feld and let L be the free metabelan Le algebra generated by the set X over K. In ths secton we recall some defntons of [2]. Le monomals of L are defned n the usual way as non-zero Le products of elements of X. The degree of a monomal s the length of ths product. For each postve nteger n we wrte L n for the K -subspace spanned by the monomals of degree n. Thus L s a K -space drect sum L = L 1 L 2.

10 414 Name Ekc and Demet Parlak Sönmez The degree of any element f of L, denoted by deg f, s the smallest value of n such that f L 1 L 2 L n. For each postve nteger m we have γ m (L) = L m L m+1, where γ m (L) s the m-th term of the lower central seres of L. Thus, L s resdually nlpotent. Let x X and for = 1,...,n let L,n be the K -subspace spanned by all Le monomals whch have degree n x. Then, for each n 0, we can wrte L n = L 0,n L 1,n L n,n. Let L (x) denote the subspace of L spanned by all monomals whch have at least one factor from X\ {x}.thus L(x) = L 0,1 (L 0,2 L 1,2 ) (L 0,n L n 1,n ). Note that for all n 2, we have L n,n = {0}, and L (x) = L 0,1 L. Let q be a real number such that 0 q 1. Defne L (x, q) to be the subspace of L, spanned by all subspaces L,n, wth n 0 and qn. We can wrte L = L (x, 1) and L (x) = 0 q<1l (x, q). The followng lemma whch was proved by Bryant and Papstas gves a useful necessary condton for a subalgebra of a free Le algebra to be fntely generated. Lemma 10 [3]. () For each 0 q 1, L(x, q) and L (x) s a subalgebra of L. () Let S be a fntely generated subalgebra of L such that S L (x). Then S L (x, q) for some q wth 0 q < 1. The followng result s proved n [3] for resdually nlpotent Le algebras. Lemma 11 [3]. Let ϕ be a non trval IA-endomorphsm of a resdually nlpotent Le algebra G. Then Fx ϕ + G = G. Lemma 12 [3]. If u s a non-zero element of L and v s a non-zero element of U(L/L ) then v u = 0. Now we are gong to show that the fxed pont subalgebra of an IA-endomorphsm of a free metabelan Le algebra s not fntely generated. Our proof s a mnor adaptaton of the arguments gven n [3]. Theorem 13. Let ϕ be a non trval IA-endomorphsm of L such that Fx ϕ L = {0}. Then Fx ϕ s not fntely generated. Proof. Snce L s resdually nlpotent by Lemma 11, Fx ϕ + L = L and L = X L. Hence we have Fx ϕ < X\ {x} > L = L (x). To get a contradcton assume that Fx ϕ s fntely generated. By Lemma 10(), there exsts a real number q wth 0 q 1

11 Free metabelan Le algebra 415 such that Fx ϕ L (x, q). Snce Fx ϕ L = {0} there exsts a non-zero element g of Fx ϕ L. Assume that deg g = n. Wrte g = g g n, where g L for = 2,...,n. Snce ϕ(g j ) L j for j = 2,...,n, from ϕ (g) = g we obtan ϕ(g j ) = g j, where j = 2,...,n. Hence wthout loss of generalty, we can replace g by g n. Wrte g n = n =0 w,n, where each w,n L,n for = 0,...,n. Now consder the element h =[x,...,x, g }{{} n ] of L. We clam that h / L (x, q). Snce m-tmes ϕ (g n ) = g n, ths mples ϕ (h) =[x,...,x,ϕ(g n )]=[x,...,x, g n ]=h. Thus h Fx ϕ. Consder the element h = n [x,...x,w,n ], =0 where w,n L,n for = 1,...,n. Then deg([x,...x,w,n ]) = n + m. Now choose k maxmal such that w k,n = 0 and choose m such that m > q(n + m). Then [x,...x,w,n ] = 0 and h = 0 by Lemma 12. Hence by the choce of m we have k + m > m > q(n + m) and h >q(n+m) L,n+m. Thus h / L(x, q). But h Fx ϕ L(x, q). Ths contradcton completes the proof. COROLLARY 14 Let ϕ be a non-trval IA-endomorphsm of L. If Fx ϕ L = {0} then t s not fntely generated. Proof. Let the element h and q be as n the proof of the Theorem 13. Snce Fx ϕ L Fx ϕ L (x) L(x, q) the result follows. Acknowledgement We would lke to thank the referee for helpful comments and for suggestng sgnfcant refnements of Lemma 5. The second author gratefully acknowledges the fnancal support for her doctoral research from the Scentfc and Techncal Research Councl of Turkey (Tübtak-Bdeb). References [1] Bahturn Yu A, Identcal relatons n Le algebras (VNU Scence Press) (1987) [2] Bryant R M, On the fxed ponts of a fnte group actng on a free Le algebra, J. London Math. Soc. 43(2) (1991) [3] Bryant R M and Papstas A I, On the fxed ponts of a fnte group actng on a relatvely free Le algebra, Glasgow Math. J. 42 (2000) [4] Drensky V, Fxed algebras of resdually nlpotent Le algebras, Proc. Am. Math. Soc. 120 (1994) [5] Formanek E, Noncommutatve nvarant theory, n: Group actons on rngs, Contemp. Math. 43 (1985)

12 416 Name Ekc and Demet Parlak Sönmez [6] Fox R H, Free dfferental calculus I, Dervaton n the free group rng, Ann. Math. 57 (1953) [7] Mkhalev A A, Shplran V and Yu J T, Combnatoral methods, Free groups, polynomals and free algebras (Sprnger-Verlag) (2004) [8] Shplran V, Fxed ponts of endomorphsms of a free metabelan group, Math. Proc. Cambrdge Phlos. Soc. 123(1) (1998) [9] Yagzhev V, Endomorphsms of free algebras, Sbrsk. Math. Zh. 21 (1980) ; Englsh Transl. Sberan Math. J. 21 (1980)

12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product

12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product 12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton

More information

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

An Introduction to Morita Theory

An Introduction to Morita Theory An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory

More information

FINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN

FINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There

More information

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

MTH 819 Algebra I S13. Homework 1/ Solutions. 1 if p n b and p n+1 b 0 otherwise ) = 0 if p q or n m. W i = rw i

MTH 819 Algebra I S13. Homework 1/ Solutions. 1 if p n b and p n+1 b 0 otherwise ) = 0 if p q or n m. W i = rw i MTH 819 Algebra I S13 Homework 1/ Solutons Defnton A. Let R be PID and V a untary R-module. Let p be a prme n R and n Z +. Then d p,n (V) = dm R/Rp p n 1 Ann V (p n )/p n Ann V (p n+1 ) Note here that

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space. Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +

More information

A New Refinement of Jacobi Method for Solution of Linear System Equations AX=b

A New Refinement of Jacobi Method for Solution of Linear System Equations AX=b Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,

More information

where a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets

where a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets 5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws

Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac

More information

SL n (F ) Equals its Own Derived Group

SL n (F ) Equals its Own Derived Group Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu

More information

NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules

NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules EVAN WILSON Quantum groups Consder the Le algebra sl(n), whch s the Le algebra over C of n n trace matrces together wth the commutator

More information

On Finite Rank Perturbation of Diagonalizable Operators

On Finite Rank Perturbation of Diagonalizable Operators Functonal Analyss, Approxmaton and Computaton 6 (1) (2014), 49 53 Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: http://wwwpmfnacrs/faac On Fnte Rank Perturbaton of Dagonalzable

More information

where a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets

where a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets 11. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of

More information

DISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization

DISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.

More information

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system. Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and

More information

An application of non-associative Composition-Diamond lemma

An application of non-associative Composition-Diamond lemma An applcaton of non-assocatve Composton-Damond lemma arxv:0804.0915v1 [math.ra] 6 Apr 2008 Yuqun Chen and Yu L School of Mathematcal Scences, South Chna Normal Unversty Guangzhou 510631, P. R. Chna Emal:

More information

Math 101 Fall 2013 Homework #7 Due Friday, November 15, 2013

Math 101 Fall 2013 Homework #7 Due Friday, November 15, 2013 Math 101 Fall 2013 Homework #7 Due Frday, November 15, 2013 1. Let R be a untal subrng of E. Show that E R R s somorphc to E. ANS: The map (s,r) sr s a R-balanced map of E R to E. Hence there s a group

More information

ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION

ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION

More information

Lecture 14 - Isomorphism Theorem of Harish-Chandra

Lecture 14 - Isomorphism Theorem of Harish-Chandra Lecture 14 - Isomorphsm Theorem of Harsh-Chandra March 11, 2013 Ths lectures shall be focused on central characters and what they can tell us about the unversal envelopng algebra of a semsmple Le algebra.

More information

2 More examples with details

2 More examples with details Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and

More information

ALGEBRA MID-TERM. 1 Suppose I is a principal ideal of the integral domain R. Prove that the R-module I R I has no non-zero torsion elements.

ALGEBRA MID-TERM. 1 Suppose I is a principal ideal of the integral domain R. Prove that the R-module I R I has no non-zero torsion elements. ALGEBRA MID-TERM CLAY SHONKWILER 1 Suppose I s a prncpal deal of the ntegral doman R. Prove that the R-module I R I has no non-zero torson elements. Proof. Note, frst, that f I R I has no non-zero torson

More information

DIFFERENTIAL FORMS BRIAN OSSERMAN

DIFFERENTIAL FORMS BRIAN OSSERMAN DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introduction to Cluster Algebra Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented

More information

Polynomials. 1 More properties of polynomials

Polynomials. 1 More properties of polynomials Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

LECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.

LECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture. LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

arxiv: v1 [math.co] 1 Mar 2014

arxiv: v1 [math.co] 1 Mar 2014 Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest

More information

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples

More information

a b a In case b 0, a being divisible by b is the same as to say that

a b a In case b 0, a being divisible by b is the same as to say that Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :

More information

Maximizing the number of nonnegative subsets

Maximizing the number of nonnegative subsets Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum

More information

A Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras"

A Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras Chn. Ann. Math. 27B(4), 2006, 419{424 DOI: 10.1007/s11401-005-0025-z Chnese Annals of Mathematcs, Seres B c The Edtoral Oce of CAM and Sprnger-Verlag Berln Hedelberg 2006 A Note on \Modules, Comodules,

More information

P.P. PROPERTIES OF GROUP RINGS. Libo Zan and Jianlong Chen

P.P. PROPERTIES OF GROUP RINGS. Libo Zan and Jianlong Chen Internatonal Electronc Journal of Algebra Volume 3 2008 7-24 P.P. PROPERTIES OF GROUP RINGS Lbo Zan and Janlong Chen Receved: May 2007; Revsed: 24 October 2007 Communcated by John Clark Abstract. A rng

More information

DIFFERENTIAL SCHEMES

DIFFERENTIAL SCHEMES DIFFERENTIAL SCHEMES RAYMOND T. HOOBLER Dedcated to the memory o Jerry Kovacc 1. schemes All rngs contan Q and are commutatve. We x a d erental rng A throughout ths secton. 1.1. The topologcal space. Let

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41, The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson

More information

On functors between module categories for associative algebras and for N-graded vertex algebras

On functors between module categories for associative algebras and for N-graded vertex algebras On functors between module categores for assocatve algebras and for N-graded vertex algebras Y-Zh Huang and Jnwe Yang Abstract We prove that the weak assocatvty for modules for vertex algebras are equvalent

More information

D.K.M COLLEGE FOR WOMEN (AUTONOMOUS), VELLORE DEPARTMENT OF MATHEMATICS

D.K.M COLLEGE FOR WOMEN (AUTONOMOUS), VELLORE DEPARTMENT OF MATHEMATICS D.K.M COLLEGE FOR WOMEN (AUTONOMOUS), VELLORE DEPARTMENT OF MATHEMATICS SUB: ALGEBRA SUB CODE: 5CPMAA SECTION- A UNIT-. Defne conjugate of a n G and prove that conjugacy s an equvalence relaton on G. Defne

More information

Differential Polynomials

Differential Polynomials JASS 07 - Polynomals: Ther Power and How to Use Them Dfferental Polynomals Stephan Rtscher March 18, 2007 Abstract Ths artcle gves an bref ntroducton nto dfferental polynomals, deals and manfolds and ther

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

Affine and Riemannian Connections

Affine and Riemannian Connections Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons

More information

Lecture 10 Support Vector Machines II

Lecture 10 Support Vector Machines II Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Poisson brackets and canonical transformations

Poisson brackets and canonical transformations rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order

More information

Graph Reconstruction by Permutations

Graph Reconstruction by Permutations Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer

More information

5 The Rational Canonical Form

5 The Rational Canonical Form 5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces

More information

First day August 1, Problems and Solutions

First day August 1, Problems and Solutions FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve

More information

SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION

SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776

More information

On intransitive graph-restrictive permutation groups

On intransitive graph-restrictive permutation groups J Algebr Comb (2014) 40:179 185 DOI 101007/s10801-013-0482-5 On ntranstve graph-restrctve permutaton groups Pablo Spga Gabrel Verret Receved: 5 December 2012 / Accepted: 5 October 2013 / Publshed onlne:

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

Modelli Clamfim Equazione del Calore Lezione ottobre 2014

Modelli Clamfim Equazione del Calore Lezione ottobre 2014 CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g

More information

Restricted Lie Algebras. Jared Warner

Restricted Lie Algebras. Jared Warner Restrcted Le Algebras Jared Warner 1. Defntons and Examples Defnton 1.1. Let k be a feld of characterstc p. A restrcted Le algebra (g, ( ) [p] ) s a Le algebra g over k and a map ( ) [p] : g g called

More information

REDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1].

REDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1]. REDUCTION MODULO p. IAN KIMING We wll prove the reducton modulo p theorem n the general form as gven by exercse 4.12, p. 143, of [1]. We consder an ellptc curve E defned over Q and gven by a Weerstraß

More information

The Order Relation and Trace Inequalities for. Hermitian Operators

The Order Relation and Trace Inequalities for. Hermitian Operators Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence

More information

Convexity preserving interpolation by splines of arbitrary degree

Convexity preserving interpolation by splines of arbitrary degree Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete

More information

Smarandache-Zero Divisors in Group Rings

Smarandache-Zero Divisors in Group Rings Smarandache-Zero Dvsors n Group Rngs W.B. Vasantha and Moon K. Chetry Department of Mathematcs I.I.T Madras, Chenna The study of zero-dvsors n group rngs had become nterestng problem snce 1940 wth the

More information

Self-complementing permutations of k-uniform hypergraphs

Self-complementing permutations of k-uniform hypergraphs Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty

More information

Randić Energy and Randić Estrada Index of a Graph

Randić Energy and Randić Estrada Index of a Graph EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL

More information

Bernoulli Numbers and Polynomials

Bernoulli Numbers and Polynomials Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that

More information

Linear Approximation with Regularization and Moving Least Squares

Linear Approximation with Regularization and Moving Least Squares Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...

More information

Perron Vectors of an Irreducible Nonnegative Interval Matrix

Perron Vectors of an Irreducible Nonnegative Interval Matrix Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of

More information

Week 2. This week, we covered operations on sets and cardinality.

Week 2. This week, we covered operations on sets and cardinality. Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from

More information

Kuroda s class number relation

Kuroda s class number relation ACTA ARITMETICA XXXV (1979) Kurodas class number relaton by C. D. WALTER (Dubln) Kurodas class number relaton [5] may be derved easly from that of Brauer [2] by elmnatng a certan module of unts, but the

More information

General viscosity iterative method for a sequence of quasi-nonexpansive mappings

General viscosity iterative method for a sequence of quasi-nonexpansive mappings Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,

More information

Genericity of Critical Types

Genericity of Critical Types Genercty of Crtcal Types Y-Chun Chen Alfredo D Tllo Eduardo Fangold Syang Xong September 2008 Abstract Ely and Pesk 2008 offers an nsghtful characterzaton of crtcal types: a type s crtcal f and only f

More information

Ideal Amenability of Second Duals of Banach Algebras

Ideal Amenability of Second Duals of Banach Algebras Internatonal Mathematcal Forum, 2, 2007, no. 16, 765-770 Ideal Amenablty of Second Duals of Banach Algebras M. Eshagh Gord (1), F. Habban (2) and B. Hayat (3) (1) Department of Mathematcs, Faculty of Scences,

More information

GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n

GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n KANG LU FINITE DIMENSIONAL REPRESENTATIONS OF gl n Let e j,, j =,, n denote the standard bass of the general lnear Le algebra gl n over the feld of

More information

ON THE JACOBIAN CONJECTURE

ON THE JACOBIAN CONJECTURE v v v Far East Journal of Mathematcal Scences (FJMS) 17 Pushpa Publshng House, Allahabad, Inda http://www.pphm.com http://dx.do.org/1.17654/ms1111565 Volume 11, Number 11, 17, Pages 565-574 ISSN: 97-871

More information

Weyl group. Chapter locally finite and nilpotent elements

Weyl group. Chapter locally finite and nilpotent elements Chapter 4 Weyl group In ths chapter, we defne and study the Weyl group of a Kac-Moody Le algebra and descrbe the smlartes and dfferences wth the classcal stuaton Here because the Le algebra g(a) s not

More information

Finding Primitive Roots Pseudo-Deterministically

Finding Primitive Roots Pseudo-Deterministically Electronc Colloquum on Computatonal Complexty, Report No 207 (205) Fndng Prmtve Roots Pseudo-Determnstcally Ofer Grossman December 22, 205 Abstract Pseudo-determnstc algorthms are randomzed search algorthms

More information

On the partial orthogonality of faithful characters. Gregory M. Constantine 1,2

On the partial orthogonality of faithful characters. Gregory M. Constantine 1,2 On the partal orthogonalty of fathful characters by Gregory M. Constantne 1,2 ABSTRACT For conjugacy classes C and D we obtan an expresson for χ(c) χ(d), where the sum extends only over the fathful rreducble

More information

Ballot Paths Avoiding Depth Zero Patterns

Ballot Paths Avoiding Depth Zero Patterns Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,

More information

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0 Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the

More information

THE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION

THE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton

More information

ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES. 1. Introduction

ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES. 1. Introduction ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES TAKASHI ITOH AND MASARU NAGISA Abstract We descrbe the Haagerup tensor product l h l and the extended Haagerup tensor product l eh l n terms of

More information

Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )

Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 ) Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often

More information

Dirichlet s Theorem In Arithmetic Progressions

Dirichlet s Theorem In Arithmetic Progressions Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons,

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

9 Characteristic classes

9 Characteristic classes THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct

More information

Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0

Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0 Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of

More information

Advanced Quantum Mechanics

Advanced Quantum Mechanics Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton

More information

Complete subgraphs in multipartite graphs

Complete subgraphs in multipartite graphs Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G

More information

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16 STAT 39: MATHEMATICAL COMPUTATIONS I FALL 218 LECTURE 16 1 why teratve methods f we have a lnear system Ax = b where A s very, very large but s ether sparse or structured (eg, banded, Toepltz, banded plus

More information

Pacific Journal of Mathematics

Pacific Journal of Mathematics Pacfc Journal of Mathematcs IRREDUCIBLE REPRESENTATIONS FOR THE ABELIAN EXTENSION OF THE LIE ALGEBRA OF DIFFEOMORPHISMS OF TORI IN DIMENSIONS GREATER THAN CUIPO JIANG AND QIFEN JIANG Volume 23 No. May

More information

STEINHAUS PROPERTY IN BANACH LATTICES

STEINHAUS PROPERTY IN BANACH LATTICES DEPARTMENT OF MATHEMATICS TECHNICAL REPORT STEINHAUS PROPERTY IN BANACH LATTICES DAMIAN KUBIAK AND DAVID TIDWELL SPRING 2015 No. 2015-1 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookevlle, TN 38505 STEINHAUS

More information

princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg

princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there

More information

SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) M(B) := # ( B Z N)

SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) M(B) := # ( B Z N) SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) S.BOUCKSOM Abstract. The goal of ths note s to present a remarably smple proof, due to Hen, of a result prevously obtaned by Gllet-Soulé,

More information

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)? Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2

More information

REAL ANALYSIS I HOMEWORK 1

REAL ANALYSIS I HOMEWORK 1 REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α

More information

Modulo Magic Labeling in Digraphs

Modulo Magic Labeling in Digraphs Gen. Math. Notes, Vol. 7, No., August, 03, pp. 5- ISSN 9-784; Copyrght ICSRS Publcaton, 03 www.-csrs.org Avalable free onlne at http://www.geman.n Modulo Magc Labelng n Dgraphs L. Shobana and J. Baskar

More information

International Journal of Algebra, Vol. 8, 2014, no. 5, HIKARI Ltd,

International Journal of Algebra, Vol. 8, 2014, no. 5, HIKARI Ltd, Internatonal Journal of Algebra, Vol. 8, 2014, no. 5, 229-238 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/ja.2014.4212 On P-Duo odules Inaam ohammed Al Had Department of athematcs College of Educaton

More information