AFormula for N-Row Macdonald Polynomials
|
|
- Melanie Hawkins
- 5 years ago
- Views:
Transcription
1 Journa of Agebraic Combinatorics, 21, , 25 c 25 Springer Science + Business Media, Inc. Manufactured in The Netherands. AFormua for N-Row Macdonad Poynomias ELLISON-ANNE WILLIAMS North Caroina State University kingpuck2@yahoo.com Received February 13, 23; Revised October 3, 23; Accepted October 11, 24 Abstract. We derive a formua for the n-row Macdonad poynomias with the coefficients presented both combinatoricay and in terms of very-we-poised hypergeometric series. Keywords: Macdonad poynomias, symmetric functions, hypergeometric series 1. Introduction Denote the ring of symmetric functions over the fied F as F and et n F denote its nth graded space. The space n F consists of a symmetric functions of tota degree n Z, indexed by the partitions λ λ 1,...,λ k for which i λ i n. There are four Z-bases and one Q-basis of n. The Q-basis consists of the power sum symmetric functions p n i x i n, where p λ p λ1 p λk, and the four Z-bases are: The monomia symmetric functions m λ i 1 < <i k x λ 1 i 1...x λ k i k, the eementary symmetric functions e n i 1 < <i n x i1...x in, where e λ e λ1 e λk, the compete symmetric functions h λ i 1... i k x λ 1 i 1 the Schur functions s λ x 1,...,x k detx λ j +k j i 1 i, j k /detx k j i....x λ k i k, and Let H Q, t bethe fied of rationa functions in and t. In1988, Macdonad introduced a new cass of two-parameter symmetric functions P λ, t, over the ring H, which generaize severa casses of symmetric functions. In particuar, taking t we obtain the Schur functions, setting t 1wehave the monomia symmetric functions, and etting gives the Ha-Littewood functions. We know from [4] that the P λ are a basis of n H. Further, with respect to the scaar product: p λ, p µ δ λ,µ i λ i m i λ j m i! 1 t λ j j1 we have that P λ, P µ if λ µ,
2 112 WILLIAMS where m i denotes the mutipicity of i in λ and λ denote the ength of λ. Weaso know that for each λ, there exists a uniue P λ, t such that: P λ m λ + µ<λ c λµ m µ where c λµ Q, t. Define: Q λ P λ P λ, P λ. Then, the bases P λ and Q λ of n H are dua to each other, Q λ, P µ δ λ,µ, and from [4], for λ n: Q n 1 λ 1 t λ j i λ n i m i mi! λ p j λ. j1 For partitions of the form λ λ 1,...,λ k, we wi derive a formua for the Macdonad poynomias Q λ with coefficients presented first combinatoricay in Section 3 and then as very-we-poised hypergeometric series in section 5. A formua for Q λ1,λ 2 can be found in [2] and foowing the competion of this work, the author discovered the paper [3] which gives a formua, without proof, for Q λ1,λ 2,λ Preiminaries Let λ, µ be partitions such that µ λ; the diagram of µ being contained in the diagram of λ. For compactness, we wi denote the skew diagram λ µ as λ\µ. Moreover, we wi view the diagram of a partition with row one being the argest and row n begin the smaest Engish notation. The diagram of λ\µ is said to be a horizonta r-strip if the number of bocks contained in λ\µ euas r and, of the remaining r bocks, there is at most one in each coumn of λ\µ. For each bock d found in the diagram of λ, et: ed t sd+1 if d λ b λ d ed+1 t sd 1 if d / λ, 1 where ed denotes the number of bocks to the east of d and sd denotes the number of bocks to the south of d in the diagram of λ [4]. Let R λ\µ denote the union of the rows and et C λ\µ denote the union of the coumns which intersect the diagram of λ\µ.
3 N-ROW MACDONALD POLYNOMIALS 113 Using a Pieri-type formua [4], for λ, µ such that µ λ is a horizonta r-strip, we have: Q µ Q r λ d b µ d b λ d Q λ 2 for d R λ\µ C λ\µ. Let r r 1,...,r n and r r 1,...,r n 1 bepartitions with the ength of row i eua to r i.inorder to derive a formua for Q r1,...,r n,webegin with Q r1,...,r n 1. To utiize 2, we must obtain a partitions ω w 1,...,w n such that r ω and ω\r is a horizonta r n -strip. Denote this set as. 3. Construction of Ω Begin with the desired partition r r 1,...,r n 1. For a ω,ω\r is a horizonta r n -strip and therefore we must add exacty r n bocks to r r 1,...,r n 1. To this end, we strategicay enarge the ength of the rows, in some cases creating an nth row, by adding the appropriate number of bocks to row r i, and beow row r n 1, for a i. Let ji denote the number of bocks added to row r i.wemay add a maximum of r n bocks to row r 1 and thus: j 1,...,r n. Conseuenty, the number of bocks added to a other rows of r is first dependent on j 1. Second, beginning with row r 1 and adding bocks to r in order of row succession, we see that the number of bocks which can be added to row r i is depedent upon the number of bocks added to rows r k for 1 k < i. Finay, since we may add at most one bock anywhere in r in order to create any new coumn which appears in ω, the number of bocks added to row r m is dependent on the difference between its ength and the ength of the preceeding row, r m 1 r m. Assimiating this, for m {1,...,n 1}: m 1,..., r n jm j s,...,r m 1 r m if if m 1 r n r n Lasty, for row w n ω, where w n r n, set: n 1 w n r n j m. js m 1 js r m 1 r m > r m 1 r m. Taking a possibe combinations of j m,beginning with m 1 and ending with m n 1, we generate {ω w 1,...,w n }.
4 114 WILLIAMS Exampe 3.1 Let r 5, 4, 3 and r 4 2. Then, j1 {, 1, 2}, j 2 {, 1}, and j3 {, 1}. Thus: {5, 4, 3, 2, 5, 4, 4, 1, 5, 5, 3, 1, 6, 4, 3, 1, 5, 5, 4, 6, 4, 4, 6, 5, 3, 7, 4, 3}. 4. Construction of the coefficients For the partition λ λ 1,...,λ n, define: v i {p where p max{1,...,n} such that λ p i}. Proposition 4.1 Let λ λ 1,...,λ n be a partition with row i, λ i.foreach row λ m and each bock d k λ m, 1 k λ m, numbered eft to right, sd k v k m. Proof: Let λ denote the conjugate partition of λ, the partition whose diagram is the transpose of the diagram of λ. Let d k be the kth bock of row λ m. Then, d k is the mth bock of the kth row of λ. Since row λ m of λ is eua to coumn m of λ, it foows that the number of bocks in row λ m is eua to p where: p max{1,...,n} such that λ p m. This impies that the number of bocks to the east of d k is the diagram of λ is eua to p m.taking the transpose, it foows that the number of bocks south of d k in the diagram of λ euas p m. Remark 4.1 As demonstrated by the proof of Proposition 4.1, for λ λ 1,...,λ n,v i λ i.aswesha see, the desirabiity of using v i rather than the conjugate partition is evidenced by the considerabe faciity that it gives to the impimentation of the principa formua. For exampe, when λ 6, 4, 2, 2 and i 3, we may simpy observe from λ that v i 2. Proposition 4.2 Let λ λ 1,...,λ n be a partition with row i, λ i.foreach row λ m, et {d k }, 1 k λ m, denote the set of bocks which compose λ m, numbered eft to right. Then: b λm b λ {d k } λ m 1 i λ m i+1 t v i+1 m 1 λ m i t v i+1 m Proof: For the product imits {i,...,λ m 1}, et bock k, d k, for 1 k λ m, correspond to i k 1. Then, {i,...,λ m 1} corresponds directy to {d k }, where 1 k λ m. It is easiy seen that the number of bocks east of bock d k in the diagram of λ is eua to λ m k. By Proposition 4.1, we know that the number of bocks south of d k in the diagram
5 N-ROW MACDONALD POLYNOMIALS 115 of λ is eua to v k m. Therefore: λ m k t v k m 1 b λ d k λ m k+1 t. v k m Shifting bock b k to bock b i where i k 1, we have: λ m i+1 t v i+1 m 1 b λ d k b λ d i λ m i t. v i+1 m Taking the product over {i,...,λ m 1} to encompass {d k }, 1 k λ m, yieds the desired resut. Let ω. Inorder to utiize the Pieri-type formua, we must construct the reuired coefficent for each Q ω : d b r d b ω d d R ω\r C ω\r. For m {1,...,n 1}, wecompute these coefficients according to row, r m, and according to the number of bocks added to r m, j m, where ω m r m + j m. Proposition 4.3 Let ω and r r 1,...,r n 1. For row w m r m + j m, 1 m n 1, such that d w m and d R ω\r C ω\r, d b rm d b wm d T j m, T j m, j k, m < k n 1 3 where T j m, r m 1 ir n n 1 j s r m i+1 t v i+1 m 1 r m+ j m i t v i+1 m r m i t v i+1 m r m+ j m i+1 t v i+1 m 1 j m 1 j m 4 and r k + j k 1 rm i t k m 1 rm+ jm i+1 t k m j T j m, jk, ir k r m i+1 t k m r m+ jm i t k m 1 m and j k 1 otherwise 5 restricting v i+1 to r ; v i+1 {p where p max{1,...n 1} such that r p i + 1}.
6 116 WILLIAMS Proof: First, note that within this proof, we use the expressions b r m and b w m to identify incompete stages in the deveopment of b rm and b wm. For row r m,weincude ony the d r m for which d R ω\r C ω\r. Since row w n r n n 1, it foows that, for each row of ω and hence of r, the bocks {d 1,...,d rn n 1 } are not incuded in R ω\r C ω\r. Therefore, by Proposition 4.2: b r m r m 1 i r n n 1 j m r m i+1 t v i+1 m 1 r m i t v i+1 m. 6 Simiary, for row w m r m + j m : b w m r m 1 i r n n 1 j m r m + j m i+1 t v i+1 m 1 r m + j m i t v i+1 m 7 where we restrict v i+1 to r in order to excude any bocks d / r which ie beneath row r m. If j m, we do not incude bocks from r m in 3, and thus from 6 and 7: b r m T b w j m, m r m 1 r m i+1 t v i+1 m 1 r m+ jm i t v i+1 m ir n n 1 r m i t v i+1 m r m+ j j s m i+1 t j v i+1 m 1 m 1 jm. However, we do not incude any d w m such that d C ω\r ; therefore, we must get rid of any terms corresponding to these bocks in the uotient 4. These bocks, which generate the unwanted terms, directy correspond to the jk, for m + 1 k n 1, which were added beow them on row r k to create row w k.inorder for these terms to appear in 4, we must have jm and j k. Therefore, to correct this in order to incude ony the d w m such that d R ω\r C ω\r,wemutipy 4 by: r k + jk 1 r m i t k m 1 r m+ jm i+1 t k m T j m, jk, ir k r m i+1 t k m r m+ jm i t k m 1 and j k 1 otherwise, yeiding the desired resut.
7 N-ROW MACDONALD POLYNOMIALS Construction of the principa formua Let r r 1,...,r n beapartition. For {1,...,r n 1}, et j1 r,..., n 1 n. js c c And, for m {2,...,n 1}, et n 1,..., r n c n 1 1 m 1 js c m 1 if r n js c js r m 1 r m + jm c c,..., r m 1 r m + j c m 1 j c m c n 1 m 1 if r n js c js > r m 1 r m + c j s c j c m 1 jm c j c m 1 jm c. Remark 5.1 The purpose of the ji, 1 i n 1, is to aow us to systematicay reduce row w n r n n 1 i1 j i tozero. To this end, we first must compute the ji in order to generate. Then, restricting the ji such that n 1 i1 j i, < n 1 i1 j i r n n 1 1 c j s c, we are abe to achieve the desired resut. Further, restrict the ji such that n 1 i1 j i, excuding the origina partition r r 1,...,r n from this reduction. We must now add at east one bock to the partition r, and to any subseuenty created partitions i.e., ω ; thus, the maximum number of times that we may need to repeat this reduction process is r n, yieding r n n 1 rn 1 i1 ji. Exampe 5.1 Buiding upon Exampe 3.1, for the partition ω 6, 4, 3, 1, j1 1, j2, and j 3, we desire to reduce row w 4 to zero. Using the restriction given in Remark 5.1, we have j1 1 {, 1}, j 2 1 {, 1}, j 3 1 {, 1} where < Taking a possibe combinations of the jm 1 dictated by the j 1 1, j 2, and j 3, 1 m 3, and appying them to ω, yieds the set of three-row partitions {7, 4, 3, 6, 5, 3, 6, 4, 4}. Given jm, 1 m n 1, 4 becomes: r m + 1 c c 1 T j m, i r n n 1 c j s c r m + 1 c c i+1 t v i+1 m 1 r m+ c c i t v i+1 m r m + 1 c c i t v i+1 m r m+ c c i+1 t v i+1 m 1 j m 1 jm 8
8 118 WILLIAMS and, 5 becomes: r k + c j k c 1 i r k + 1 c T j m, jk, j k c r m + 1 c c i t k m 1 r m+ c c i+1 t k m r m + 1 c c i+1 t k m r m+ c c i t k m 1 j m and j k 1 otherwise 9 where for, 1 c c and where v i+1 is restricted to the partition r c j 1 c,...,r n c j n 1 c ; v i+1 {p where p max{1,...,n 1} such that r p + 1 c j p c i + 1}. Theorem 5.1 For the partition r r 1,...,r n, m {1,...,n 1}, and {1,..., r n 1},wehave Q r1,...,r n Q r1,...,r n 1 Q rn + r n 1 i 1 n 1 1 i jm, j i1 m Q r 1 + i 1 j 1,...,r n 1+ i 1 j n 1 + j m, j m n 1 1 jrn m 1 r n r n 1 T j m, Q n 1 Q r 1 + rn 1 j 1,...,r n 1+ rn 1 j n 1 n 1 r n n 1 T j m, km+1 i 1 j m T j m, j k, n 1 km+1 T j m, j k, where jm, signifies to sum over a possibe combinations of the vaues of j j m 1 and such that < n 1 m2 j 1 + r n and a possibe combinations of the vaues jm dictated by the jm such that n 1 j m. Proof: We want to show that: Q r1,...,r n Q r1,...,r n 1 Q rn r n 1 i 1 n 1 1 i j1, i1 Q r 1 + i 1 j 1,...,r n 1+ i 1 j n 1 T j m, Q n 1 km+1 r n n 1 i 1 j m T j m, j k,
9 N-ROW MACDONALD POLYNOMIALS 119 j 1, n 1 1 jrn m 1 r n r n 1 n 1 Q r 1 + rn 1 j 1,...,r n 1+ rn 1 j n 1 Using 2, for n 1 j m, we have: T j m, n 1 km+1 Q r1,...,r n 1 Q rn Q r1,...,r n + T j m, j1, Q r 1 + j1,...,r n 1+ jn 1,r n n 1 T j m, j k,. 1 n 1 Substituting 11 into 1, we need ony to show: j 1, j m n 1 T j m, n 1 km+1 r n 1 i 1 1 i + j1, i1 Q r 1 + i 1 j 1,...,r n 1+ i 1 j n 1 n 1 j 1, 1 jrn m 1 r n 1 n 1 km+1 T j k, j k, T j k, jk, Q r 1 + j1,...,r n 1+ jn 1,r n n 1 n 1 r n 1 Q r 1 + rn 1 j 1,...,r n 1+ rn 1 j n 1 T j m, Q n 1 n 1 r n n 1 T j m, km+1 i 1 j m. 11 T j m, j k, n 1 km+1 Note that each Pieri-formua expansion via 2 and 11 of j 1, j m n 1 T j m, n 1 km+1 T j m, j k,. 12 T j k, jk, Q r 1 + j1,...,r n 1+ jn 1,r n n 1 13 yieds two terms. To show our desired resut, we wi show that with each successive numbered Pieri-formua expansion of 13, the term which contains the product of two Macdonad poynomias cances with the correspondingy numbered term i in r n 1 i 1 1 i j1, i1 n 1 T j m, Q r 1 + i 1 j 1,...,r n 1+ i 1 j n 1 n 1 Q km+1 r n n 1 T j m, j k, i 1 14 j m
10 12 WILLIAMS and that, in the fina Pieri expansion of 13, we wi be eft ony with r n 1 1 r n 1 j 1, j m n 1 1 jrn m n 1 Q r 1 + rn 1 j 1,...,r n 1+ rn 1 j n 1 T j m, n 1 km+1 T j m, j k,. 15 To this end, we wi induct on i. Consder i 1. Competing the first Pieri expansion of 13, and computing i 1 from 14, 12 becomes: j 1, j m n 1 n 1 T j m, km+1 1 n 1 T jm, j1, Q r j 1, j m n 1 T j k, j k, Q r 1 + j1,...,r n 1+ j Q n 1 r n n 1 n 1 km+1 T j m, j k, j 1,...,r n 1+ 1 j n 1,r n n 1 1 j m T j m, r n 1 i 1 1 i + j1, i2 Q r 1 + i 1 1 n 1 j1, n 1 km+1 n 1 j 1,...,r n 1+ i 1 j n 1 T j m, T j m, T j m, j k, Q n 1 km+1 n 1 km+1 r n n 1 i 1 j m T j m, j k, Q r j 1,...,r n 1+ 1 j n 1,r n n 1 1 j m + r n 1 i 1 n 1 n 1 1 i T j m, j1, i2 Q r 1 + i 1 j 1,...,r n 1+ i 1 j n 1 Q Q r 1 + j1,...,r n 1+ j Q n 1 r n n 1 km+1 T j m, j k, r n n 1 i 1 j m T j m, j k, Therefore, in the first Pieri expansion of 13, the term containing the product of two Macdonad poynomias canceed with the correspondingy numbered i 1 in14, as desired. Assume that with each further numbered Pieri expansion of 13, up to r n 2, the term containing a product of two Macdonad poynomias cances with the correspondingy numbered i in 14. We wi show the resut for r n 1..
11 N-ROW MACDONALD POLYNOMIALS 121 For i r n 2, we have that 12 euas: j 1, j m 1 r n 3 r n 3 Q r 1 + rn r n 2 j1, Q r 1 + rn r n 2 j1, Q r 1 + rn r n 1 j1, Q r 1 + rn 2 1 r n 2 j1, Q r 1 + rn r n 1 j1, n 1 T j m, j1,...,r n 1+ rn 3 jn 1 r n 2 n 1 n 1 km+1 Q T j m, T j m, j k, r n n 1 n 1 km+1 j1,...,r n 1+ rn 2 jn 1,r n n 1 rn 2 jm r n 3 n 1 T j m, j1,...,r n 1+ rn 3 jn 1 r n 2 n 1 Q T j m, j1,...,r n 1+ rn 2 jn 1 r n 2 n 1 T j m, Q n 1 km+1 r n n 1 n 1 km+1 r n n 1 n 1 km+1 j1,...,r n 1+ rn 2 jn 1,r n n 1 rn 2 jm r n 2 n 1 T j m, Q r 1 + rn 2 j1,...,r n 1+ rn 2 jn 1 Q r n n 1 rn 2 j. m n 1 km+1 rn 3 jm T j m, j k, T j m, j k, rn 3 j m T j m, j k, rn 2 j m T j m, j k, T j m, j k, Performing the Pieri expansion on 13 a fina time, 12 is eua to: r n 2 1 r n 2 j1, Q r 1 + rn r n 1 j1, n 1 T j m, j1,...,r n 1+ rn 2 jn 1 r n 1 n 1 n 1 km+1 Q T j m, Q r 1 + rn 1 j 1,...,r n 1+ rn 1 j n 1 T j m, j k, r n n 1 n 1 km+1 rn 2 jm T j m, j k,
12 122 WILLIAMS + 1 r n 1 j1, Q r 1 + rn 2 j 1, n 1 1 jrn m r n 2 n 1 T j m, j1,...,r n 1+ rn 2 jn 1 1 r n 1 r n 1 Q n 1 Q r 1 + rn 1 j 1,...,r n 1+ rn 1 j n 1 n 1 km+1 r n n 1 T j m,. T j m, j k, rn 2 jm n 1 km+1 T j m, j k, Since, by definition of jm,wemust have n 1 r n 1 r n jm. Exampe 5.2 We sha buid upon Exampe 3.1 to cacuate the expansion of the Macdonad poynomia Q 5,4,3,2. Given the nature of the partition r 5, 4, 3, 2, it is fitting and instructive to use the foowing notation: n 1 n 1 km+1 T j m, T j 1,..., j n 1 T jm, T j k,,..., jm+1,..., j n 1 jm, j m+i, 1 i n m 1 For 2 m 3, we have: j1 {, 1, 2} j 1 {,..., 2 m 1,..., 2 jm j s,...,r m 1 r m 3 js c c m 1 if 2 if 2 js m 1 js }, r m 1 r m > r m 1 r m,
13 N-ROW MACDONALD POLYNOMIALS 123 jm,..., if 2 2,..., if 3 c 1 m 1 js c 3 m 1 js c c js 1 j s r m 1 r m + j c m 1 jm c 2 c 3 m 1 js c c j s r m 1 r m + c c j c m 1 jm c > r m 1 r m + j c m 1 jm c. We begin with jm since these vaues determine the subseuent choices for j m.wehave: j1 {, 1, 2} j 1 1 {, 1} j2 {, 1} j 2 1 {, 1} j3 {, 1} j 3 1 {, 1}. Taking a possibe combinations of the jm, {, 1}, such that < 3 2 and < 3 1 1, we have: Q 5,4,3,2 Q 5,4,3 Q T j m, T j m, jk, j1, km+1 Q 5+ j Q 1,4+ j 2,3+ j T j m, T j m, jk, Q 5+ 1 j1, km+1 j 1,4+ 1 j 2,3+ 1 j 3 Q 5,4,3 Q 2 + T1,, T 1,, 1 T 2,, Q7,4,3 + T1,, T,1, 1 + T,1, T 1,, 1 T 1,1, T,1, Q6,5,3 + T 1,, T 1,,1 + T,,1 T 1 1,, T 1,,1 T,,1 Q6,4,4 + T,1, T 1,,1 + T,,1 T 1,1, T,1,1 T,,1 Q5,5,4 T 1,, Q 6,4,3 Q 1 T,1, Q 5,5,3 Q 1 T,,1 Q 5,4,4 Q Very-we-poised hypergeometric series We may express the coefficients T j m, for m {1,...,n 1}, and {,...,r n 1 }, in terms of very-we-poised hypergeometric series. We wi use the foowing notations. a; 1 n 1 a; n 1 a i i
14 124 WILLIAMS a; 1 a i i a; n a; a n ; a 1,...,a m ; n a 1 ; n a m ; n a 1,...,a m ; a 1 ; a m ; The basic hypergeometric series φ r+1,r is of the form: [ ] a1, a 2,..., a r+1 b 1, b 2,..., b r, z where if φ r+1,r n a 1,...,a r+1 ; n b 1,...,b r ; n ; n z n. We say that φ r+1,r is very-we-poised, denoted W r+1,r a 1 ; a 4,...,a r+1 ;, z, a 1 a 2 b 1... a r+1 b r and a 2 a 1 2 1, a 3 a From [1], we have: W 6,5 a; b, c, d;, a bcd a, a bc, a bd, a a b, a c, a d, a bcd ; cd ;. 16 The Coefficients T j m, and T j m,j k, We first compute the coefficient T j m in terms of very-we-poised hypergeometric series. Beginning with row r m,weidentify the indentions which ie under it in the diagram of r.inother words, we identify a rows r i, m < i n 1, such that r i < r i+1. For k 1 {1,...,n m}, et r n k1 be the argest indexed row of r such that r n k1 r m. Proposition 6.1 Suppose r n k1 r m.itfoows that k 1 n m and r m r n 1. Then: W 6,5 r m r n + n 1 j s t n m 1 ; t 1,, r n 1 r n + n 1 j s ;, jm t n m jm T j m, 1 jm.
15 N-ROW MACDONALD POLYNOMIALS 125 Proof: Let r n k1 r m ; k 1 n m and r m r n 1. There are two cases: and jm. For jm, we have that T jm, 1byProposition 4.3. For jm, using properties of hypergeometric series and 16, 4 becomes: T j m, r n 1 1 ir n n 1 j s rm i+1 t n m rm+ j m i t n m 1 rm i t n m 1 rm+ j m i+1 t n m t n m, rm rn 1+ j rm rn 1 m +1 t n m 1 ; rn 1 r n 1 n+ j s rm rn 1+1 t n m 1, rm rn 1+ t n m ; rn 1 r n 1 n+ j s rm rn 1 t n m, rm+ rn 1+1 t n m 1, rm rn+ n 1 j s +1 t n m 1, rm+ rn+ n 1 j s t n m ; rm rn+ n 1 j s t n m, rm+ rn+ n 1 j s +1 t n m 1, rm rn 1+1 t n m 1, rm+ rn 1 t n m ; W 6,5 rm rn+ n 1 j s t n m 1 ; t 1,, r n 1 n 1 r n+ j s ;, r m r n 1+ jm t n m W 6,5 rm rn+ n 1 j s t n m 1 ; t 1,, r n 1 n 1 r n+ j s ;, jm t n m. Remark 6.1 Let r m r n k1.itfoows that for a m < k n 1, we have j k, and thus T j m, j k, 1. Consider the case when r n k1 r m. Let r n k2 be the argest indexed row of r r 1,...,r n 1 such that r n k1 r n k2.ifr n k2 r m, et r n k3 be the argest indexed row of r for which r n k2 r n k3, ect. Continue this process unti one reaches row r n k f for which r n k f r m, creating the chain: r n k1 < r n k2 < < r n k f r m for k i {1,...,n m}, 1 i f. Proposition 6.2 Suppose r n k1 r m. Then, 4 becomes: T j m, W 6,5 r m r n + n 1 j s t n m k 1 ; t 1, j m, r n k1 r n + n 1 j s ;, r m r n k1 + j m t n m k 1 +1 f 1 h2 W 6,5 r m r n kh+1 + j m t n k h+1 m ; t 1,, r n kh r n kh+1 ;, r m r n kh t n m k h+1 +1 jm 1 jm. Proof: Using our chain, we are abe to identify the indentions under row r m in the diagram of r ; thus, we obtain a intervas of bocks d r m on which sd changes. Using properties of hypergeometric series and 16, we decompose 4 as foows:
16 126 WILLIAMS T j m, rn k 1 1 irn n 1 f 1 h2 j s rn k h+1 1 irn k h rm i+1 t n m k1+1 rm + j m i t n m k1 rm i t n m k1 rm + j m i+1 t n m k1+1 rm i+1 t n m kh+1+1 rm + i t n m kh+1 rm i t n m kh+1 rm + jm i+1 t n m kh+1+1 rm rn k 1 t n m k1+1, rm + rn k 1 +1 t n m k1 ; rn k 1 rn+ n 1 j s rm + jm rn k 1 t n m k1+1, rm rn k 1 +1 t n m k1 ; rn k1 rn+ n 1 j s f 1 h2 rm rn k h+1 t n m kh+1+1, rm + rn k h+1 +1 t n m kh+1 ; rm rn k h+1 +1 t n m kh+1, rm rn k h+1 t n m kh+1+1 ; rn k h+1 rn k h rn k h+1 rn k h rm rn k 1 t n m k1+1, rm + rn k 1 +1 t n m k1, rm rn+ + n 1 j s t n m k1+1, rm rn+ n 1 j s +1 t n m k1 ; rm + rn k 1 t n m k1+1, rm + rn k 1 t n m k1, rm rn+ n 1 j s t n m k1+1, rm rn+ n 1 j s + +1 t n m k1 ; f 1 h2 P Q W 6,5 rm rn+ n 1 j s t n m k1 ; t 1, j m, rn k 1 rn+ n 1 j s ;, rm + j m rn k 1 t n m k1+1 f 1 W 6,5 rm rn k h+1 + j m t n kh+1 m ; t 1,, rn k h rn k h+1 ;, rm rn k h t n m kh+1+1. h2 Where: P r m r n kh+1 t n m k h+1 +1, r m+ jm r n k +1 h+1 t n m k h+1, r m r n kh +1 t n m k h+1, r m r n kh + jm t n m k h+1 +1 ; Q r m r n kh+1 +1 t n m k h+1, r m+ jm r n k h+1 t n m k h+1 +1, r m r n kh t n m k h+1 +1, r m r n kh + jm +1 t n m k h+1 ;. Proposition 6.3 For r n k1 r m, the coefficient T j m, j k, becomes: T j m, j k, W 6,5 r m r k j k t k m 1 ; j k, t 1, j m ;, r m r k + j m t k m. Proof: Using properties of hypergeometric series and 16, we have: T j m, j k, r k + j k 1 rm i t k m 1 r m + jm i+1 t k m ir k r m i+1 t k m r m+ jm i t k m 1 r m r k jk 1 t k m 1, r m r k + jm j k t k m ; jk r m r k jk t k m, r m r k + jm j k +1 t k m 1 ; jk
17 N-ROW MACDONALD POLYNOMIALS 127 r m r k j k +1 t k m 1, r m r k + j m j k t k m, r m r k t k m, r m r k + j m +1 t k m 1 ; r m r k j k t k m, r m r k + j m j k +1 t k m 1, r m r k +1 t k m 1, r m r k + j m t k m ; W 6,5 r m r k j k t k m 1 ; j k, t 1, j m ;, r m r k + j m t k m. The Coefficients T j m, and T j m,j k,. For {1,...,r n 1}, wecompute the coefficients T j m, and T j m, j k, in terms of verywe-poised hypergeometric series. To this end, fix and carry out the procedure previousy described to obtain the chain: r n k1 + jn k c 1 < < r n k f + jn k c f r m + c Notation 6.1 Set: c c j c m. c n 1 c jn k c i c j c m J m js c J s J n ki Proposition 6.4 Suppose that r n k1 + J n k1 r m + J m.itfoows that r m + J m r n 1 + J n 1 and 8 becomes: W 6,5 r m r n +J m +J s t n m k 1 ; t 1, j m, r n k1 +J n k1 r n J s ;, r m r n k1 J n k1 +J m + jm t n m k 1 W 6,5 r m r n ki J n ki +J m t n k i m+1 ; T j m, h2 t 1, j m, r n ki+1 r n ki J n ki +J n ki+1 ;, r m r n ki+1 J n ki+1 +J m + jm t n m k i jm 1 jm. Proof: Let r n k1 + J n k1 r m + J m r m + J m r n 1 + J n 1. There are two cases: jm and j m. For jm, by Proposition 4.3, we have that T jm, 1. f 1
18 128 WILLIAMS For jm, using properties of hypergeometric series, we have: r n 1 +J n 1 1 r m +J m i+1 t n m r m+j m + jm i t n m 1 T j m, r m +J m i t n m 1 r m+j m + jm i+1 t n m ir n J s r m +J m r n 1 J n 1 t n m, r m+j m + jm r n 1 J n 1 +1 t n m 1 ; r n 1+J n 1 r n +J s r m +J m J n 1 r n 1 +1 t n m 1, r m r n 1 J n 1 +J m + jm t n m ; r n 1 r n +J s +J n 1 P Q Where: W 6,5 r m r n +J s +J m t n m 1 ; t 1, j m, r n 1 r n +J s +J n 1 ;, r m r n 1 +J m J n 1 + jm t n m. P r m r n 1 +J m J n 1 t n m, r m r n 1 +J m J n 1 + jm +1 t n m 1, r m r n +J s +J m +1 t n m 1, r m r n +J s +J m + jm t n m ; Q r m r n +J s +J m t n m, r m r n +J s +J m + jm +1 t n m 1, r m r n 1 +J m J n 1 t n m+1, r m r n 1 +J m J n 1 + jm t n m ;. Remark 6.2 Since r m + J m r n 1 + J n 1 jk have that T j m, jk, 1. for a m < k n 1, we Proposition 6.5 Suppose that r n k1 + J n k1 r m + J m. Obtaining the chain rn k1 + J n k1 < < rn k f + J n k f rm + J m, we have that: W 6,5 r m r n +J s +J m t n m k 1 ; t 1, j m, r n k1 r n +J n k1 +J s ;, r m r n k1 J n k1 +J m + jm t n m k 1 +1 W 6,5 r m r n kh+1 +J m + jm J n k h+1 h2 T j m, t n k h+1 m ; t 1, j m, r n kh r n kh+1 +J n kh J n kh+1 ;, r m r n kh J n kh +J m t n m k h+1 jm 1 jm. Proof: For r n k1 + J n k1 r m + J m, using properties of hypergeometric series, 8 becomes: r n k1 +J n k1 1 T j m, ir n J s f 1 rn kh+1 +J n kh+1 1 h2 f 1 r m +J m i+1 t n m k 1+1 r m+j m + j m i t n m k 1 r m +J m i t n m k 1 r m +J m + j m i+1 t n m k 1+1 ir n kh +J n kh r m +J m i+1 t n m k h+1+1 r m+j m + jm i t h+1 n m k r m +J m i t n m k h+1 r m +J m + jm i+1 t n m k h+1+1
19 N-ROW MACDONALD POLYNOMIALS 129 r m r n k1 +J m J n k1 t n m k1+1, r m r n k1 +J m J n k1 + jm +1 t n m k 1 ; r n k1 r n+j s +J n k1 r m r n k1 +J m J n k1 + jm t n m k 1 +1, r m r n k1 +J m J n k1 t n m k 1 ; f 1 P 1 Q h2 1 P f 1 2 Q 2 h2 P 3 Q 3 W 6,5 r m r n +J m +J s t n m k 1 ; t 1, j m, r n k1 +J n k1 r n +J s ;, r m r n k1 J n k1 +J m + jm t n m k Where: f 1 i2 W 6,5 r m r n kh+1 +J m + j m J n k h+1 t n k h+1 m ; r n k1 r n +J n k1 +J s t 1, j m, r n kh r n kh+1 +J n kh J n kh+1 ;, r m r n kh J n kh +J m t n m k h+1. P 1 r m r n kh+1 +J m J n kh+1 t n m k h+1 +1, r m r n kh+1 +J m J n kh+1 + j m +1 t n m k h+1 ; r n kh+1 r n kh +J n kh+1 J n kh Q 1 r m r n kh+1 +J m J n kh+1 +1 t n m k h+1, r m r n kh+1 +J m J n kh+1 + j m t n m k h+1+1 ; r n kh+1 +J n kh+1 r n kh J n kh P 2 r m+j m r n k1 J n k1 t n m k 1 +1, r m+j m + jm r n k 1 J n k1 +1 t n m k 1, r m r n +J s +J m + jm t n m k 1 +1, r m r n +J s +J m +1 t n m k 1 ; Q 2 r m r n k1 +J m J n k1 + jm t n m k 1 +1, r m r n k1 +J m J n k1 + jm t n m k 1, r m r n +J s +J m t n m k1+1, r m r s +J n +J m + jm +1 t n m k 1 ; P 3 r m r n kh+1 +J m J n kh+1 t n m k h+1 +1, r m r n kh+1 +J m J n kh+1 + jm +1 t n m k h+1, r m r n kh +J n kh +J m +1 t n m k h+1, r m r n kh J n kh +J m + jm t n m k h+1 +1 ; Q 3 r m r n kh+1 J n kh+1 +J m +1 t n m k h+1, r m+j m + jm r n k h+1 J n kh+1 t n m k h+1 +1, r m r n kh +J m J n kh t n m k h+1 +1, r m r n kh J n kh +J m + jm +1 t n m k h+1 ;. Proposition 6.6 For T j m, jk,, m < k n 1, we have: W 6,5 r m r k J k +J m jk t k m 1 ; T j m, jk, j k, t 1, j m ;, r m r k +J m +J k + jm t k m jm and j k 1 otherwise.
20 13 WILLIAMS Proof: Using properties of hypergeometric series, 9 becomes: T j m, j k, Where: r k +J k + jk 1 rm+jm i t k m 1 r m +J m + jm i+1 t k m ir k +J k r m +J m i+1 t k m r m+j m + jm i t k m 1 r m r k +J m J k jk +1 t k m 1, r m r k +J m J k + jm j k t k m ; jk r m r k J k +J m jk t k m, r m r k +J m J k + jm j k +1 t k m 1 ; jk P Q W 6,5 r m r k J k +J m jk t k m 1 ; j k, t 1, j m ;, r m r k +J m +J k + jm t k m. P r m r k J k +J m jk +1 t k m 1, r m r k +J m J k + jm j k t k m, r m r k J k +J m t k m, r m r k +J m J k + jm +1 t k m 1 ; Q r m r k J k +J m jk t k m, r m r k +J m J k + jm j k +1 t k m 1, r m r k J k +J m +1 t k m 1, r m r k +J m J k + jm t k m ;. Acknowegdments I woud ike to thank Professor Ernest L. Stitzinger and Professor Naihuan Jing for their professiona support and encouragement during the course of this work and for their hepfu comments and suggestions regarding this manuscript. I am gratefu to Prof. Jing for introducing me to Macdonad poynomias and for his suggestion of this probem. I woud aso ike to thank my husband, Michae Wiiams, for his comments and suggestions concerning this text. References 1. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge N. Jing and T. Jozefiak, A Formua for two-row macdonad functions, Duke Math , M. Lassae, Expication des poynomes de Jack et de Macdonad en oungueur trois, C.R. Acad. Sci. Paris. 333I 21, I.G. Macdonad, Symmetric Functions and Ha Poynomias, second edition, Oxford University Press, Oxford, 1995.
Problem set 6 The Perron Frobenius theorem.
Probem set 6 The Perron Frobenius theorem. Math 22a4 Oct 2 204, Due Oct.28 In a future probem set I want to discuss some criteria which aow us to concude that that the ground state of a sef-adjoint operator
More informationTHINKING IN PYRAMIDS
ECS 178 Course Notes THINKING IN PYRAMIDS Kenneth I. Joy Institute for Data Anaysis and Visuaization Department of Computer Science University of Caifornia, Davis Overview It is frequenty usefu to think
More informationInvestigation on spectrum of the adjacency matrix and Laplacian matrix of graph G l
Investigation on spectrum of the adjacency matrix and Lapacian matrix of graph G SHUHUA YIN Computer Science and Information Technoogy Coege Zhejiang Wani University Ningbo 3500 PEOPLE S REPUBLIC OF CHINA
More informationPartial permutation decoding for MacDonald codes
Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics
More informationVI.G Exact free energy of the Square Lattice Ising model
VI.G Exact free energy of the Square Lattice Ising mode As indicated in eq.(vi.35), the Ising partition function is reated to a sum S, over coections of paths on the attice. The aowed graphs for a square
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More informationGeneralized Bell polynomials and the combinatorics of Poisson central moments
Generaized Be poynomias and the combinatorics of Poisson centra moments Nicoas Privaut Division of Mathematica Sciences Schoo of Physica and Mathematica Sciences Nanyang Technoogica University SPMS-MAS-05-43,
More informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationSimple Algebraic Proofs of Fermat s Last Theorem. Samuel Bonaya Buya*
Avaiabe onine at www.peagiaresearchibrary.com eagia Research Library Advances in Appied Science Research, 017, 8(3:60-6 ISSN : 0976-8610 CODEN (USA: AASRFC Simpe Agebraic roofs of Fermat s Last Theorem
More information(f) is called a nearly holomorphic modular form of weight k + 2r as in [5].
PRODUCTS OF NEARLY HOLOMORPHIC EIGENFORMS JEFFREY BEYERL, KEVIN JAMES, CATHERINE TRENTACOSTE, AND HUI XUE Abstract. We prove that the product of two neary hoomorphic Hece eigenforms is again a Hece eigenform
More informationResearch Article On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation
Appied Mathematics and Stochastic Anaysis Voume 007, Artice ID 74191, 8 pages doi:10.1155/007/74191 Research Artice On the Lower Bound for the Number of Rea Roots of a Random Agebraic Equation Takashi
More informationAn explicit Jordan Decomposition of Companion matrices
An expicit Jordan Decomposition of Companion matrices Fermín S V Bazán Departamento de Matemática CFM UFSC 88040-900 Forianópois SC E-mai: fermin@mtmufscbr S Gratton CERFACS 42 Av Gaspard Coriois 31057
More informationFOURIER SERIES ON ANY INTERVAL
FOURIER SERIES ON ANY INTERVAL Overview We have spent considerabe time earning how to compute Fourier series for functions that have a period of 2p on the interva (-p,p). We have aso seen how Fourier series
More informationTheory of Generalized k-difference Operator and Its Application in Number Theory
Internationa Journa of Mathematica Anaysis Vo. 9, 2015, no. 19, 955-964 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ijma.2015.5389 Theory of Generaized -Difference Operator and Its Appication
More informationA Brief Introduction to Markov Chains and Hidden Markov Models
A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,
More informationLECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,
More informationThe Group Structure on a Smooth Tropical Cubic
The Group Structure on a Smooth Tropica Cubic Ethan Lake Apri 20, 2015 Abstract Just as in in cassica agebraic geometry, it is possibe to define a group aw on a smooth tropica cubic curve. In this note,
More informationMARKOV CHAINS AND MARKOV DECISION THEORY. Contents
MARKOV CHAINS AND MARKOV DECISION THEORY ARINDRIMA DATTA Abstract. In this paper, we begin with a forma introduction to probabiity and expain the concept of random variabes and stochastic processes. After
More informationTHE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE
THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationDo Schools Matter for High Math Achievement? Evidence from the American Mathematics Competitions Glenn Ellison and Ashley Swanson Online Appendix
VOL. NO. DO SCHOOLS MATTER FOR HIGH MATH ACHIEVEMENT? 43 Do Schoos Matter for High Math Achievement? Evidence from the American Mathematics Competitions Genn Eison and Ashey Swanson Onine Appendix Appendix
More informationCE601-Structura Anaysis I UNIT-IV SOPE-DEFECTION METHOD 1. What are the assumptions made in sope-defection method? (i) Between each pair of the supports the beam section is constant. (ii) The joint in
More informationFRIEZE GROUPS IN R 2
FRIEZE GROUPS IN R 2 MAXWELL STOLARSKI Abstract. Focusing on the Eucidean pane under the Pythagorean Metric, our goa is to cassify the frieze groups, discrete subgroups of the set of isometries of the
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationReichenbachian Common Cause Systems
Reichenbachian Common Cause Systems G. Hofer-Szabó Department of Phiosophy Technica University of Budapest e-mai: gszabo@hps.ete.hu Mikós Rédei Department of History and Phiosophy of Science Eötvös University,
More informationSome Measures for Asymmetry of Distributions
Some Measures for Asymmetry of Distributions Georgi N. Boshnakov First version: 31 January 2006 Research Report No. 5, 2006, Probabiity and Statistics Group Schoo of Mathematics, The University of Manchester
More informationHonors Thesis Bounded Query Functions With Limited Output Bits II
Honors Thesis Bounded Query Functions With Limited Output Bits II Daibor Zeený University of Maryand, Batimore County May 29, 2007 Abstract We sove some open questions in the area of bounded query function
More informationSUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS
ISEE 1 SUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS By Yingying Fan and Jinchi Lv University of Southern Caifornia This Suppementary Materia
More informationORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION
J. Korean Math. Soc. 46 2009, No. 2, pp. 281 294 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy,
More informationXSAT of linear CNF formulas
XSAT of inear CN formuas Bernd R. Schuh Dr. Bernd Schuh, D-50968 Kön, Germany; bernd.schuh@netcoogne.de eywords: compexity, XSAT, exact inear formua, -reguarity, -uniformity, NPcompeteness Abstract. Open
More informationThe EM Algorithm applied to determining new limit points of Mahler measures
Contro and Cybernetics vo. 39 (2010) No. 4 The EM Agorithm appied to determining new imit points of Maher measures by Souad E Otmani, Georges Rhin and Jean-Marc Sac-Épée Université Pau Veraine-Metz, LMAM,
More informationEfficient Generation of Random Bits from Finite State Markov Chains
Efficient Generation of Random Bits from Finite State Markov Chains Hongchao Zhou and Jehoshua Bruck, Feow, IEEE Abstract The probem of random number generation from an uncorreated random source (of unknown
More informationLecture 9. Stability of Elastic Structures. Lecture 10. Advanced Topic in Column Buckling
Lecture 9 Stabiity of Eastic Structures Lecture 1 Advanced Topic in Coumn Bucking robem 9-1: A camped-free coumn is oaded at its tip by a oad. The issue here is to find the itica bucking oad. a) Suggest
More information#A48 INTEGERS 12 (2012) ON A COMBINATORIAL CONJECTURE OF TU AND DENG
#A48 INTEGERS 12 (2012) ON A COMBINATORIAL CONJECTURE OF TU AND DENG Guixin Deng Schoo of Mathematica Sciences, Guangxi Teachers Education University, Nanning, P.R.China dengguixin@ive.com Pingzhi Yuan
More informationUniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly conp-complete
Uniprocessor Feasibiity of Sporadic Tasks with Constrained Deadines is Strongy conp-compete Pontus Ekberg and Wang Yi Uppsaa University, Sweden Emai: {pontus.ekberg yi}@it.uu.se Abstract Deciding the feasibiity
More information2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationChemical Kinetics Part 2
Integrated Rate Laws Chemica Kinetics Part 2 The rate aw we have discussed thus far is the differentia rate aw. Let us consider the very simpe reaction: a A à products The differentia rate reates the rate
More informationThe ordered set of principal congruences of a countable lattice
The ordered set of principa congruences of a countabe attice Gábor Czédi To the memory of András P. Huhn Abstract. For a attice L, et Princ(L) denote the ordered set of principa congruences of L. In a
More informationDiscrete Techniques. Chapter Introduction
Chapter 3 Discrete Techniques 3. Introduction In the previous two chapters we introduced Fourier transforms of continuous functions of the periodic and non-periodic (finite energy) type, we as various
More informationRestricted weak type on maximal linear and multilinear integral maps.
Restricted weak type on maxima inear and mutiinear integra maps. Oscar Basco Abstract It is shown that mutiinear operators of the form T (f 1,..., f k )(x) = R K(x, y n 1,..., y k )f 1 (y 1 )...f k (y
More informationA NOTE ON QUASI-STATIONARY DISTRIBUTIONS OF BIRTH-DEATH PROCESSES AND THE SIS LOGISTIC EPIDEMIC
(January 8, 2003) A NOTE ON QUASI-STATIONARY DISTRIBUTIONS OF BIRTH-DEATH PROCESSES AND THE SIS LOGISTIC EPIDEMIC DAMIAN CLANCY, University of Liverpoo PHILIP K. POLLETT, University of Queensand Abstract
More informationIntegrating Factor Methods as Exponential Integrators
Integrating Factor Methods as Exponentia Integrators Borisav V. Minchev Department of Mathematica Science, NTNU, 7491 Trondheim, Norway Borko.Minchev@ii.uib.no Abstract. Recenty a ot of effort has been
More informationGOYAL BROTHERS PRAKASHAN
Assignments in Mathematics Cass IX (Term 2) 14. STATISTICS IMPORTANT TERMS, DEFINITIONS AND RESULTS The facts or figures, which are numerica or otherwise, coected with a definite purpose are caed data.
More informationLECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October 7, 202 Prof. Aan Guth LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationCONSISTENT LABELING OF ROTATING MAPS
CONSISTENT LABELING OF ROTATING MAPS Andreas Gemsa, Martin Nöenburg, Ignaz Rutter Abstract. Dynamic maps that aow continuous map rotations, for exampe, on mobie devices, encounter new geometric abeing
More informationProblem Set 6: Solutions
University of Aabama Department of Physics and Astronomy PH 102 / LeCair Summer II 2010 Probem Set 6: Soutions 1. A conducting rectanguar oop of mass M, resistance R, and dimensions w by fas from rest
More informationGauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law
Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s
More informationEfficiently Generating Random Bits from Finite State Markov Chains
1 Efficienty Generating Random Bits from Finite State Markov Chains Hongchao Zhou and Jehoshua Bruck, Feow, IEEE Abstract The probem of random number generation from an uncorreated random source (of unknown
More informationSmall generators of function fields
Journa de Théorie des Nombres de Bordeaux 00 (XXXX), 000 000 Sma generators of function fieds par Martin Widmer Résumé. Soit K/k une extension finie d un corps goba, donc K contient un éément primitif
More informationQUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3
QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3 JEREMY LOVEJOY AND ROBERT OSBURN Abstract. Recenty, Andrews, Hirschhorn Seers have proven congruences moduo 3 for four types of partitions using eementary
More informationWeek 6 Lectures, Math 6451, Tanveer
Fourier Series Week 6 Lectures, Math 645, Tanveer In the context of separation of variabe to find soutions of PDEs, we encountered or and in other cases f(x = f(x = a 0 + f(x = a 0 + b n sin nπx { a n
More informationSmoothness equivalence properties of univariate subdivision schemes and their projection analogues
Numerische Mathematik manuscript No. (wi be inserted by the editor) Smoothness equivaence properties of univariate subdivision schemes and their projection anaogues Phiipp Grohs TU Graz Institute of Geometry
More informationMAT 167: Advanced Linear Algebra
< Proem 1 (15 pts) MAT 167: Advanced Linear Agera Fina Exam Soutions (a) (5 pts) State the definition of a unitary matrix and expain the difference etween an orthogona matrix and an unitary matrix. Soution:
More informationChemical Kinetics Part 2. Chapter 16
Chemica Kinetics Part 2 Chapter 16 Integrated Rate Laws The rate aw we have discussed thus far is the differentia rate aw. Let us consider the very simpe reaction: a A à products The differentia rate reates
More informationPrimal and dual active-set methods for convex quadratic programming
Math. Program., Ser. A 216) 159:469 58 DOI 1.17/s117-15-966-2 FULL LENGTH PAPER Prima and dua active-set methods for convex quadratic programming Anders Forsgren 1 Phiip E. Gi 2 Eizabeth Wong 2 Received:
More informationA natural differential calculus on Lie bialgebras with dual of triangular type
Centrum voor Wiskunde en Informatica REPORTRAPPORT A natura differentia cacuus on Lie biagebras with dua of trianguar type N. van den Hijigenberg and R. Martini Department of Anaysis, Agebra and Geometry
More informationStat 155 Game theory, Yuval Peres Fall Lectures 4,5,6
Stat 155 Game theory, Yuva Peres Fa 2004 Lectures 4,5,6 In the ast ecture, we defined N and P positions for a combinatoria game. We wi now show more formay that each starting position in a combinatoria
More informationOn the Goal Value of a Boolean Function
On the Goa Vaue of a Booean Function Eric Bach Dept. of CS University of Wisconsin 1210 W. Dayton St. Madison, WI 53706 Lisa Heerstein Dept of CSE NYU Schoo of Engineering 2 Metrotech Center, 10th Foor
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
odue 2 naysis of Staticay ndeterminate Structures by the atri Force ethod Version 2 E T, Kharagpur esson 12 The Three-oment Equations- Version 2 E T, Kharagpur nstructiona Objectives fter reading this
More informationHigher dimensional PDEs and multidimensional eigenvalue problems
Higher dimensiona PEs and mutidimensiona eigenvaue probems 1 Probems with three independent variabes Consider the prototypica equations u t = u (iffusion) u tt = u (W ave) u zz = u (Lapace) where u = u
More informationCourse 2BA1, Section 11: Periodic Functions and Fourier Series
Course BA, 8 9 Section : Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins 9 Contents Periodic Functions and Fourier Series 74. Fourier Series of Even and Odd Functions...........
More informationBiometrics Unit, 337 Warren Hall Cornell University, Ithaca, NY and. B. L. Raktoe
NONISCMORPHIC CCMPLETE SETS OF ORTHOGONAL F-SQ.UARES, HADAMARD MATRICES, AND DECCMPOSITIONS OF A 2 4 DESIGN S. J. Schwager and w. T. Federer Biometrics Unit, 337 Warren Ha Corne University, Ithaca, NY
More informationT.C. Banwell, S. Galli. {bct, Telcordia Technologies, Inc., 445 South Street, Morristown, NJ 07960, USA
ON THE SYMMETRY OF THE POWER INE CHANNE T.C. Banwe, S. Gai {bct, sgai}@research.tecordia.com Tecordia Technoogies, Inc., 445 South Street, Morristown, NJ 07960, USA Abstract The indoor power ine network
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationDiscrete Techniques. Chapter Introduction
Chapter 3 Discrete Techniques 3. Introduction In the previous two chapters we introduced Fourier transforms of continuous functions of the periodic and non-periodic (finite energy) type, as we as various
More informationProduct Cosines of Angles between Subspaces
Product Cosines of Anges between Subspaces Jianming Miao and Adi Ben-Israe Juy, 993 Dedicated to Professor C.R. Rao on his 75th birthday Let Abstract cos{l,m} : i cos θ i, denote the product of the cosines
More informationLecture 17 - The Secrets we have Swept Under the Rug
Lecture 17 - The Secrets we have Swept Under the Rug Today s ectures examines some of the uirky features of eectrostatics that we have negected up unti this point A Puzze... Let s go back to the basics
More informationEfficient Algorithms for Pairing-Based Cryptosystems
CS548 Advanced Information Security Efficient Agorithms for Pairing-Based Cryptosystems Pauo S. L. M. Barreto, HaeY. Kim, Ben Lynn, and Michae Scott Proceedings of Crypto, 2002 2010. 04. 22. Kanghoon Lee,
More informationJoel Broida UCSD Fall 2009 Phys 130B QM II. Homework Set 2 DO ALL WORK BY HAND IN OTHER WORDS, DON T USE MATHEMAT- ICA OR ANY CALCULATORS.
Joe Broida UCSD Fa 009 Phys 30B QM II Homework Set DO ALL WORK BY HAND IN OTHER WORDS, DON T USE MATHEMAT- ICA OR ANY CALCULATORS. You may need to use one or more of these: Y 0 0 = 4π Y 0 = 3 4π cos Y
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f( q ) = exp ( βv( q )), which is obtained by summing over
More informationLimited magnitude error detecting codes over Z q
Limited magnitude error detecting codes over Z q Noha Earief choo of Eectrica Engineering and Computer cience Oregon tate University Corvais, OR 97331, UA Emai: earief@eecsorstedu Bea Bose choo of Eectrica
More informationK a,k minors in graphs of bounded tree-width *
K a,k minors in graphs of bounded tree-width * Thomas Böhme Institut für Mathematik Technische Universität Imenau Imenau, Germany E-mai: tboehme@theoinf.tu-imenau.de and John Maharry Department of Mathematics
More informationPattern Frequency Sequences and Internal Zeros
Advances in Appied Mathematics 28, 395 420 (2002 doi:10.1006/aama.2001.0789, avaiabe onine at http://www.ideaibrary.com on Pattern Frequency Sequences and Interna Zeros Mikós Bóna Department of Mathematics,
More informationDiscrete Bernoulli s Formula and its Applications Arising from Generalized Difference Operator
Int. Journa of Math. Anaysis, Vo. 7, 2013, no. 5, 229-240 Discrete Bernoui s Formua and its Appications Arising from Generaized Difference Operator G. Britto Antony Xavier 1 Department of Mathematics,
More informationarxiv:math/ v2 [math.ag] 12 Jul 2006
GRASSMANNIANS AND REPRESENTATIONS arxiv:math/0507482v2 [math.ag] 12 Ju 2006 DAN EDIDIN AND CHRISTOPHER A. FRANCISCO Abstract. In this note we use Bott-Bore-Wei theory to compute cohomoogy of interesting
More informationHow many random edges make a dense hypergraph non-2-colorable?
How many random edges make a dense hypergraph non--coorabe? Benny Sudakov Jan Vondrák Abstract We study a mode of random uniform hypergraphs, where a random instance is obtained by adding random edges
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationScott Cohen. November 10, Abstract. The method of Block Cyclic Reduction (BCR) is described in the context of
ycic Reduction Scott ohen November, 99 bstract The method of ock ycic Reduction (R) is described in the context of soving Poisson's equation with Dirichet boundary conditions The numerica instabiityof
More informationChapter 4. Moving Observer Method. 4.1 Overview. 4.2 Theory
Chapter 4 Moving Observer Method 4.1 Overview For a compete description of traffic stream modeing, one woud reuire fow, speed, and density. Obtaining these parameters simutaneousy is a difficut task if
More informationTHE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS
ECCM6-6 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Sevie, Spain, -6 June 04 THE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS M. Wysocki a,b*, M. Szpieg a, P. Heström a and F. Ohsson c a Swerea SICOMP
More information8 Digifl'.11 Cth:uits and devices
8 Digif'. Cth:uits and devices 8. Introduction In anaog eectronics, votage is a continuous variabe. This is usefu because most physica quantities we encounter are continuous: sound eves, ight intensity,
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationMat 1501 lecture notes, penultimate installment
Mat 1501 ecture notes, penutimate instament 1. bounded variation: functions of a singe variabe optiona) I beieve that we wi not actuay use the materia in this section the point is mainy to motivate the
More informationMIXING AUTOMORPHISMS OF COMPACT GROUPS AND A THEOREM OF SCHLICKEWEI
MIXING AUTOMORPHISMS OF COMPACT GROUPS AND A THEOREM OF SCHLICKEWEI KLAUS SCHMIDT AND TOM WARD Abstract. We prove that every mixing Z d -action by automorphisms of a compact, connected, abeian group is
More informationHow the backpropagation algorithm works Srikumar Ramalingam School of Computing University of Utah
How the backpropagation agorithm works Srikumar Ramaingam Schoo of Computing University of Utah Reference Most of the sides are taken from the second chapter of the onine book by Michae Nieson: neuranetworksanddeepearning.com
More informationSrednicki Chapter 51
Srednici Chapter 51 QFT Probems & Soutions A. George September 7, 13 Srednici 51.1. Derive the fermion-oop correction to the scaar proagator by woring through equation 5., and show that it has an extra
More informationAST 418/518 Instrumentation and Statistics
AST 418/518 Instrumentation and Statistics Cass Website: http://ircamera.as.arizona.edu/astr_518 Cass Texts: Practica Statistics for Astronomers, J.V. Wa, and C.R. Jenkins, Second Edition. Measuring the
More informationLecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String
ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary
More informationBALANCING REGULAR MATRIX PENCILS
BALANCING REGULAR MATRIX PENCILS DAMIEN LEMONNIER AND PAUL VAN DOOREN Abstract. In this paper we present a new diagona baancing technique for reguar matrix pencis λb A, which aims at reducing the sensitivity
More informationApproximation and Fast Calculation of Non-local Boundary Conditions for the Time-dependent Schrödinger Equation
Approximation and Fast Cacuation of Non-oca Boundary Conditions for the Time-dependent Schrödinger Equation Anton Arnod, Matthias Ehrhardt 2, and Ivan Sofronov 3 Universität Münster, Institut für Numerische
More informationHaar Decomposition and Reconstruction Algorithms
Jim Lambers MAT 773 Fa Semester 018-19 Lecture 15 and 16 Notes These notes correspond to Sections 4.3 and 4.4 in the text. Haar Decomposition and Reconstruction Agorithms Decomposition Suppose we approximate
More informationResearch Article Numerical Range of Two Operators in Semi-Inner Product Spaces
Abstract and Appied Anaysis Voume 01, Artice ID 846396, 13 pages doi:10.1155/01/846396 Research Artice Numerica Range of Two Operators in Semi-Inner Product Spaces N. K. Sahu, 1 C. Nahak, 1 and S. Nanda
More informationDistributed average consensus: Beyond the realm of linearity
Distributed average consensus: Beyond the ream of inearity Usman A. Khan, Soummya Kar, and José M. F. Moura Department of Eectrica and Computer Engineering Carnegie Meon University 5 Forbes Ave, Pittsburgh,
More informationTHE PARTITION FUNCTION AND HECKE OPERATORS
THE PARTITION FUNCTION AND HECKE OPERATORS KEN ONO Abstract. The theory of congruences for the partition function p(n depends heaviy on the properties of haf-integra weight Hecke operators. The subject
More informationarxiv:math/ v2 [math.pr] 6 Mar 2005
ASYMPTOTIC BEHAVIOR OF RANDOM HEAPS arxiv:math/0407286v2 [math.pr] 6 Mar 2005 J. BEN HOUGH Abstract. We consider a random wa W n on the ocay free group or equivaenty a signed random heap) with m generators
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More information