Some necessary and sufficient conditions for two variable orthogonal designs in order 44
|
|
- Dorcas Howard
- 5 years ago
- Views:
Transcription
1 University of Wollongong Reserch Online Fculty of Informtics - Ppers (Archive) Fculty of Engineering n Informtion Sciences 1998 Some necessry n sufficient conitions for two vrile orthogonl esigns in orer 44 Christos Koukouvinos M. Mitrouli Jennifer Seerry University of Wollongong, jennie@uow.eu.u Puliction Detils Koukouvinos C, Mitrouli M n Seerry J, Some necessry n sufficient conitions for two vrile orthogonl esigns in orer 44, Journl of Comintoril Mthemtics n Comintoril Computing, 28, (1998), Reserch Online is the open ccess institutionl repository for the University of Wollongong. For further informtion contct the UOW Lirry: reserch-pus@uow.eu.u
2 Some necessry n sufficient conitions for two vrile orthogonl esigns in orer 44 Astrct We give new lgorithm which llows us to construct new sets of sequences with entries from the commuting vriles 0, ±, ±, ±c, ± with zero utocorreltion function. We show tht for twelve cses if the esigns exist they cnnot e constrcte using four circulnt mtrices in the Goethls-Seiel rry. Further we show tht the necessry conitions for the existence of n OD(44;s 1,s 2 ) re sufficient except possily for the following 7 cses. (7,32) (8,31) (9,30) (9,33) (11,30) (13,29) (15,26) which coul not e foun ecuse of the lrge size of the serch spce for complete serch. These cses remin open. In ll we fin 398 cses, show 67 o not exist n estlish 12 cses cnnot e constructe using four circulnt mtrices. We give new construction for OD(2n) n OD(n+1) from OD(n). The full OD(44;s 1,s 2,s 3,44-s 1 -s 2 -s 3 ) given in this pper yiel t lest 68 equivlence clsses of Hmr mtrices. Disciplines Physicl Sciences n Mthemtics Puliction Detils Koukouvinos C, Mitrouli M n Seerry J, Some necessry n sufficient conitions for two vrile orthogonl esigns in orer 44, Journl of Comintoril Mthemtics n Comintoril Computing, 28, (1998), This journl rticle is ville t Reserch Online:
3 Necessry n sucient conitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deicte to Professor Anne Penfol Street Astrct We give new lgorithm which llows us to construct new sets of sequences with entries from the commuting vriles 0 c with zero utocorreltion function. We show tht for twelve cses if the esigns exist they cnnot e constrcte using four circulnt mtrices in the Goethls-Seiel rry. Further we show tht the necessry conitions for the existence of n OD(44 s 1 s 2 ) re sucient except possily for the following 7 cses: (7 32) (8 31) (9 30) (9 33) (11 30) (13 29) (15 26) which coul not e foun ecuse of the lrge size of the serch spce for complete serch. These cses remin open. In ll we n 398 cses, show 67 o not exist n estlish 12 cses cnnot e constructe using four circulnt mtrices. We give new construction for OD(2n) nod(n + 1) from OD(n). The full OD(44 s 1 s 2 s 3 44 ; s 1 ; s 2 ; s 3 )given in this pper yiel t lest 68 equivlence clsses of Hmr mtrices. Key wors n phrses: Autocorreltion, construction, sequence, orthogonl esign. AMS Suject Clssiction: Primry 05B15, 05B20, Seconry 62K05. 1 Introuction Throughout this pper we will use the enition n nottion of Koukouvinos, Mitrouli, Seerry n Krels [2]. We note from [3] tht we hve totest 1 4 n2 = 484 cses. We n 398 cses, show 67o not exist n estlish 12 cses cnnot e constructe using four circulnt mtrices. There re 7 open cses which coul not e foun ecuse of the lrge size of the serch spce for complete serch. 2 New orthogonl esigns Theorem 1 An OD(44 s 1 s 2 ) cnnot exist for the following 2;tuples (s 1 s 2 ): Deprtment of Mthemtics, Ntionl Technicl University ofathens, Zogrfou 15773, Athens, Greece. y Deprtment of Mthemtics, University ofathens, Pnepistemiopolis 15784, Athens, Greece. z School of IT n Computer Science, University ofwollongong, Wollongong, NSW, 2522, Austrli. 1
4 (1 7) (1 15) (1 23) (1 28) (1 31) (1 39) (1 42) (2 14) (2 30) (3 5) (3 13) (3 20) (3 21) (3 29) (3 37) (3 40) (4 7) (4 15) (4 23) (4 28) (4 31) (4 39) (5 11) (5 12) (5 19) (5 27) (5 35) (6 10) (6 26) (7 9) (7 16) (7 17) (7 25) (7 28) (7 33) (7 36) (8 14) (8 30) (9 15) (9 23) (9 28) (9 31) (10 17) (10 22) (10 24) (11 13) (11 16) (11 20) (11 21) (11 29) (12 13) (12 15) (12 20) (12 21) (12 29) (13 19) (13 27) (14 18) (15 16) (15 17) (15 20) (15 25) (16 19) (16 23) (16 28) (17 23) (19 20) (19 21) Proof. These cses re eliminte y the numer theoretic necessry conitions given in [1] or[2, Lemm 3]. Exmple. To illustrte how we use the numer theoretic conitions to estlish the nonexistence of n OD(4n 11 20) we consier the prouct = now thisis numer of the form 4 (8 + 7) which cnnot e written s the sum of three squres n hence n OD(4n 11 20) cnnot exist. Remrk. A computer serch, which we elieve ws exhustive, ws crrie out which les us to elieve tht 1. there re no 4-NPAF(7 19) sequences of length there re no 4-NPAF(3 31), 4-NPAF(5 30), 4-NPAF(6 29) n 4-NPAF(8 27) sequences of length 9. This mens tht there re lso no 4-NPAF(1 5 30), 4- NPAF(1 6 29) n 4-NPAF(1 8 27) of length there re no 4-NPAF(2 41) sequences of length 11. This mens tht there re lso no 4-NPAF(1 2 41) sequences of length there re no 4-NPAF(6 37) sequences of length 11. Lemm 1 OD( ) n n OD( ) o not exist (this is prove theoreticlly). The Germit-Verner Theorem sys tht if n OD( ) exists then n OD( ) will exist, n if n OD( ) exists then n OD( ) will exist. Hence the OD( ) n OD( ) o not exist. Lemm 2 The following OD( ; ) n OD(44 43 ; ) cnnot e constructe using four circulnt mtrices in the Goethls-Seiel rry: (6 37) (1 6 37) (10 33) ( ) (12 31) ( ) (13 30) ( ) (14 29) ( ) (16 27) ( ) (19 24) ( ) (20 23) ( ) Proof. By the Germit-Verner theorem if n orthogonl esign OD(n x 1 x 2 x u;1 x u ) with u i=1 x i = n ; 1 exists, n 0(mo 4), then n OD(n 1 x 1 x 2 x u;1 x u ) exists. Now for ech of the cses in this lemm we hve nod(44 43 ; ) n tht is y the Germit-Verner theorem n OD( ; ). Using the sum-ll mtrix metho we write 1 = , = n 43; = We require the sum-ll mtrix to e 34 orthogonl mtrix with the rst row contining n 0 the secon row contining n 4 in some orer n the thir row contining n 4 in some orer. 2
5 However, s we illustrte for OD( ), this is not possile for the cses mentione in the enuncition. Using the sum-ll mtrix metho for OD( ), 1 = , 20 = n 23 = (;1) There is no wy to form n orthogonl mtrix unless oth 20 n 23 cn e written s the sum of 3 squres. 2 Theorem 2 Therere OD(44 s 1 s 2 s 3 44;s 1 ;s 2 ;s 3 ) constructe using four sequences to otin four circulnt mtrices for use in the Goethls-Seiel rry for the following 2;tuples: Corollry 1 By suitly choosing the vriles of the known OD(44 s 1 s 2 s 3 44 ; s 1 ; s 2 ; s 3 ) to e replce y1 these le to t lest 36 lgericlly inequivlent Hmr mtrices of orer 44. Bysuitlychoosing the vriles of the known OD(44 s 1 44 ; s 1 ) to e replce y 1 these le to t lest 12 more lgericlly inequivlent Hmr mtrices of orer 44. Corollry 2 By suitly choosing the vriles of the known OD(44 1 s 1 35 ; s 1 ) to e replce y 1 we otin t lest 20 lgericlly inequivlent skew-hmr mtrices of orer 44. The numer epens on whether ech skew-hmr mtrix is equivlent to its trnspose or not. 3 New Algorithm The lgorithm previously use to n OD vi four sequences of length t 10 ws prohiitively slow for length 11. Hence we trie new lgorithm, which epene on the previous lgorithm, to n rst W (4t k) me with four sequences of length t with PAF =0orNPAF = 0. In the new lgorithm MAT LAB ws use to set up series of equtions to e solve for ech iniviul k n then ll solutions to these equtions were foun. Exmple. We illustrte the lgorithm y trying to construct the OD( ). We rst notice tht 11 hs unique ecomposition into squres 11 = , while 27 hs three ecompositions into four squres. All three cn e use in this construction s they must e le to e use in n integer mtrix (the sum-ll mtrix) which is orthogonl. Hence we use27= = = Sowehve the mtrices " ;1 5 # " ;1 4 ;1 3 # or " ;3 3 We now ll ech of the positions which re represente y 0y one of 17 vriles x 1 x 2 x 17. Wenow use MATLAB to expn the rst rows to mke four circulnt mtrices with row inner prouct zero: this correspons to forming four sequences with PAF = 0. The equtions will e those tht involve somex j,1j 17 with, n those which hve no terms in. This gives t most 6 inepenent equtions. A serch isnow me through the 17 vriles, llowing them to ssume the vlues 0 1, where six of them must lwys e zero, n using the extr constrints tht 3 #
6 3X i=1 x i = ;1 5X i=4 x i = ;1 X11 i=6 x i =3 X17 i=12 x i =0: We strt with the following four sequences of length 11 n PAF = We replce the 1 yvrile such s n we replce the 17 zeros y the vriles. Thus we hve the sequences ; ; x 1 x 2 x 3 ; ; ; ; ; x 4 x 5 x 6 x 7 x 8 x 9 ; ; x 10 x 11 ; x 12 x 13 x 14 x 15 x 16 x 17 We then use MATLAB to set up series of equtions, tht when solve, yiel, mong others, the following solution: x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x 15 x 16 x 17 0 ; 0 0 ; ; 0 ; ; 0 0 We now replce the vriles in the originl four sequences y these solutions to otin the OD( ). 2 Remrk. Using this lgorithm we teste ll unknown two vrile cses n foun 7 cses which we were unle to resolve ue to the extremely lrge serch spce. We estimte tht complete serch for the OD( ) using this lgorithm requires 2 37 opertions. 2 4 New Results Theorem 3 Write X( ) = fe 1 x 1 e 2 x 2 e n;1 x n;1 e n x n g, Y ( ) = ff 1 y 1 f 2 y 2 f n;1 y n;1 f n y n g for the sequences of length n, NPAF=0,where e i n f i re chosen from where, re commuting vriles n x i, y i hve elements 0 1 n the sequences X(1 1) n Y (1 1) hve NPAF = 0. Suppose occurs totl of s 1 times n totl of s 2 times then we sy the two sequences we hve re 2-NPAF(n s 1 s 2 ). Write i = if e i = n i = if e i = for i = 1,...,n, n similrly, i = if f i = n i = if f i = for i = 1,...,n. Then (i) X( ) Y ( ) n Y ( ) X (; ;) where Z enotes the reverse of the sequence Z or n fe 1 x 1 e 2 x 2 e n;1 x n;1 e n x n n y n n;1 y n;1 2 y 2 1 y 1 g ff 1 y 1 f 2 y 2 f n;1 y n;1 f n y n ; n x n ; n;1 x n;1 ; 2 x 2 ; 1 x 1 g re two sequences with elements f0 g with NPAF = 0. These sequences re 2-NPAF(2n 2s 1 2s 2 ). 4
7 n (ii) If x n;1 n y n;1 re oth zero then the sequences fe 1 x 1 e 2 x 2 n y n e n x n n;2 y n;2 2 y 2 1 y 1 g ff 1 y 1 f 2 y 2 ; n x n f n y n ; n;2 x n;2 ; 2 x 2 ; 1 x 1 g re two sequences with elements f0 g with NPAF = 0. These sequences re 2-NPAF(2n ; 2 2s 1 2s 2 ). (iii) Similrly with 4-NPAF(n s 1 s 2 ), X( ), Y ( ), Z( ) n W ( ) we hve X( ) Y ( ) Y ( ) X (; ;) Z( ) W ( ) n W ( ) Z (; ;) where Z enotes the reverse of the sequence Z re 4-NPAF(2n 2s 1 2s 2 ). (iv) Similrly with 4-NPAF(n s 1 s 2 ),ifthesecon lst element of ech of the four sequences is zero thenproceeing s in (ii) we otin 4-NPAF(2n ; 2 2s 1 2s 2 ). (v) Similrly if there re 4-NPAF(n s 1 s 2 ),nthesecon lst element of two of the sequences is zero n the lst element of two of the sequences is zero then comining the methos of (ii) n (iii) we cn get 4-NPAF(2n ; 2 2s 1 2s 2 ). Proof. The proof follows y writing out the sequences n checking the NPAF. Exmple. We use to men ; n c to men ;c. To illustrte prt (v) of the theorem we note tht c c c 0 c 0 c c c 0 c 0 c 0 c 0 c 0 c c 0 c 0 c 0 c n c c c c 0 c c c c 0 c 0 c 0 c 0 c c 0 c 0 c 0 c re 4-NPAF(7 2 16) n 4-NPAF(7 4 16), respectively. In fct we note c c c 0 c 0 c c c 0 c 0 c 0 c c 0 c c 0 c 0 c 0 c n c c c c 0 c c c c 0 c 0 c c 0 c c 0 c 0 c 0 c re 4-NPAF( ) n 4-NPAF( ), respectively. We lso note tht c c c c c c 0 c 0 c c c c c c c 0 c 0 c c c c c c c c 0 c c c c c c c c 0 c n c c c c c c c c c c c c c c c c c 0 c 0 c c c c c 0 c c 0 c 0 c c c c c 0 c re 4-NPAF( ) n 4-NPAF( ), respectively. c c c c c c 0 c 0 c c c c c c c 0 c 0 c c c c c c c 0 c 0 c c c c c c c 0 c 0 c n c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 5
8 re 4-NPAF( ) n 4-NPAF( ), respectively. Lemm 3 If there exist 2-NPAF(n s 1 s 2 ) then there exist 4-NPAF(n s 1 2s 2 ). Corollry 3 Since there exist2-npaf(n s 1 s 2 ) for the vlues liste in the tle we get the corresponing lrger 4-NPAF(n s 1 2s 2 ). 2-NPAF(n s 1 s 2 ) ) 4-NPAF(n s 1 2s 2 ) (9 13) (10 2,2,26) (11 13) (12 2,2,26) (14 17) (15 2,2,34) (18 25) (19 2,2,50) (4 4,4) (5 2,2,8,8) (6 2,8) (7 2,2,4,16) (6 5,5) (7 2,2,10,10) (8 8,8) (9 2,2,16,16) (10 10,10) (11 2,2,20,20) (14 13,13) (15 2,2,26,26) Corollry 4 Using the previous theorem we see tht 4-NPAF(n s 1 s 2 ) ) 4-NPAF(2n 2s 1 2s 2 ) NPAF(5 1,18) NPAF(5 1,19) NPAF(5 2,17) NPAF(5 2,18) NPAF(5 3,17) NPAF(7 3,18) NPAF(5 4,16) NPAF(7 4,17) NPAF(7 4,18) NPAF(5 5,14) NPAF(5 5,15) NPAF(7 5,16) NPAF(7 5,17) NPAF(7 5,18) NPAF(5 6,14) NPAF(7 6,16) NPAF(7 7,14) NPAF(7 7,15) NPAF(5 8,11) NPAF(5 8,12) NPAF(5 9,10) NPAF(5 9,11) NPAF(7 9,12) NPAF(10 2,36) NPAF(10 2,38) NPAF(10 4,34) NPAF(10 4,36) NPAF(10 6,34) NPAF(14 6,36) NPAF(10 8,32) NPAF(14 8,34) NPAF(14 8,36) NPAF(10 10,28) NPAF(10 10,30) NPAF(14 10,32) NPAF(14 10,34) NPAF(14 10,36) NPAF(10 12,28) NPAF(14 12,32) NPAF(14 14,28) NPAF(14 14,30) NPAF(10 16,22) NPAF(10 16,24) NPAF(10 18,20) NPAF(10 18,22) NPAF(14 18,24) Theorem 4 The sequences given in the Appenices cn e use to construct the pproprite esigns to estlish tht the necessry conitions for the existence ofnod(44 s 1 s 2 ) re sucient, except possily for the following 12 cses which cnnot e constructe from four circulnt mtrices: 6
9 (5 38) (6 37) (8 35) (10 33) (12 31) (13 30) (14 29) (15 28) (16 27) (19 24) (20 23) (21 22): n the following 7 cses which re unecie: (7 32) (8 31) (9 30) (9 33) (11 30) (13 29) (15 26) Remrk. There re 484 possile 2;tuples. Tle 1 lists the 398 which correspon to esigns which exist in orer 44: 67 2-tuples correspon to esigns eliminte y numer theory (NE). For 12 cses, if the esigns exist, they cnnot e constructe using circulnt mtrices (Y). 7 cses remin unecie. P inictes tht 4-PAF sequences with length 11 exist n inictes 4-NPAF sequences with length n exist. 7
10 NE NE NE NE NE NE NE NE NE P NE NE NE 3 21 NE NE NE NE NE NE NE NE NE P NE NE 5 12 NE NE NE P 5 35 NE P 5 38 Y NE NE P P P Y NE NE 7 17 NE NE NE P NE 7 34 P 7 35 P 7 36 NE NE P P 8 30 NE P Y NE NE Tle 1: The existence of OD(44 s 1 s 2). 8
11 NE 9 29 P NE 9 32 P P 9 35 P NE P NE NE P P P Y NE NE NE NE P P P NE P P P NE NE NE NE NE P P NE P Y NE P P P P NE P Y P P P NE P P P P P Y P NE NE NE P P P NE P Y P NE NE P Y NE P P NE P P P P P P P P P P NE NE P P Y P P Y Y P Tle 1(Cont): The existence of OD(44 s 1 s 2). References [1] A.V.Germit, n J.Seerry, Orthogonl esigns: Qurtic forms n Hmr mtrices, Mrcel Dekker, New York-Bsel, [2] C.Koukouvinos, M.Mitrouli, J.Seerry, n P.Krels, On sucient conitions for some orthogonl esigns n sequences with zero utocorreltion function, Austrls. J. Comin., 13, (1996), [3] C.Koukouvinos n Jennifer Seerry, New orthogonl esigns n sequences with two n three vriles in orer 28, Ars Comintori, (to pper). 9
12 A1 A2 A3 A4 Appenix A: Orer 40 (Sequences with zero non-perioic utocorreltion function) Design ) (1 (1 4 32) ( ) (2 2 34) (2 4 32) ( ) ( ) ( ) (2 35) (3 31) (3 34) notinyet ; ; ; ; ; ; ; ; ; ; ; c ;c ; c ;c c c c c ;c ; ;c c ;c ;c ;c c c ;c ;c c ;c ;c c c c c ;c ;c c ;c c ; ;c c ; ; ; ; ; ; ; ; ; c ;c ; c ; c c ;c ; ; ;c ;c ; ;c c ; ;c ; ;c ;c c ; ; ; c c c c 0 c ;c ; c c c 0 c c ; ; ;c ; ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; c ; ; ; 0 ; 0 ; 0 ; ;c ; c ; ;c ; ; ; ; c c c 0 c ;c c ;c c c c c 0 c ;c c c ; ;c c ; 0 c ; ; 0 ;c ; c ;c ;c ;c ; c ;c ; c ; c ; c ; ; c ; ; ; ; c ;c 0 ;c ;c c ;c c ;c 0 ;c c ; ;c ; c ; 0 ; 0 ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; 0 ; 0 ; 0 10
13 A1 A2 A3 A4 Appenix A(cont): Orer 40 (Sequences with zero non-perioic utocorreltion function) Design ) (4 ( ) ( ) ( ) ( ) (5 30) (5 33) (6 31) (7 31) ( ) ( ) notinyet c ;c ; ;c ; c c ;c ; c ;c ; c ; ;c ; ; c ; ; c c ;c ; ; c ;c ; ; ; c ;c ; ; c ;c ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; c 0 ;c c c ; c 0 ;c ; ;c ; ;c ; ; c ;c ; ; ; ;c c c ; c ;c ;c ; c ; c ; c ; c c ; c ; ;c c c ; c ; ; c ; ;c ; 0 ; c 0 c ; 0 ; ; ;c 0 ;c c c ; ;c ; c c ; ; ; c ; 0 ; ; ; ; 0 ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; 0 ; ;c ; 0 ; ; ; c ; c c ; ;c ; c c ; c ; ; ; 11
14 A1 A2 A3 A4 Appenix B: Orer 44 (Sequences with zero non-perioic utocorreltion function) Design ) (1 (1 30) (1 34) (1 35) (1 37) (1 38) (1 40) (1 41) ( ) ( ) ( ) ( ) 0 ;c c ; 0 ; ;c c 0 ;c 0 ;c 0 c c 0 0 ; ; ; ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; c ; ; ; ; c ; ;c ; ; ;c ; ; ; c ; ; ; ;c ; 0 0 ;c c ;c ; 0 0 ;c ; ;c ; c ; c 0 c ; ; c c ; c 0 ;c ; ;c c ; ; 0 ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; 0 ; 0 ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ;c ; ; ; ; c ; ; c ; ;c ; ; ;c ; ; ; ; ; c ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; 0 ;c c ; c 0 ; 0 ;c ; ;c ;c ; 12
15 A1 A2 A3 A4 Appenix B(cont): Orer 44 (Sequences with zero non-perioic utocorreltion function) Design ) (2 (2 37) (2 39) (3 35) (3 36) (3 38) (3 39) (3 41) (4 35) (5 36) (5 39) (7 37) c ;c ; ;c 0 ;c c ;c ; ; c 0 c ; 0 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; 0 0 ; ; 0 0 ; 0 ; ; 0 ; 0 ; ; 0 0 ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; 0 0 ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;c ;c ; ; ; c ;c ; ;c ;c ; ; ;c ; c ; 0 0 ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 13
16 A1 A2 A3 A4 Appenix C: Orer 44 (Sequences with zero perioic utocorreltion function) Design 9 34) (1 ( ) ( ) ( ) ( ) (2 41) (4 37) (5 34) (5 37) (6 29) (6 33) (6 35) (7 30) ; ; c ; ; ; ; ; ; ; ; ; ; ; c ; ; ; ; ; ; ; ; ; c ; ; ; ; ; ; ; ; ; c ; ; ; ; ; ; ; ; ; 0 ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 0 ; ; ; ; 0 0 ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; c 0 ; ; ;c 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; 0 0 ; 0 ; 0 ; ; 0 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; 0 ; ; 14
17 A1 A2 A3 A4 Appenix C(cont): Orer 44 (Sequences with zero perioic utocorreltion function) Design 34) (7 (7 35) (8 27) (8 29) (8 33) (9 32) (10 31) (11 27) (11 28) (11 31) (12 25) (12 26) (12 30) ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; 0 0 ; ; ; 0 ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 0 ; ; 0 ; 0 ; ; ; 0 ; ; ; 0 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; 0 ; ; ; ; 0 ; ; ; 0 ; ; ; 0 0 ; ; ; ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; 0 ; ; 0 0 ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; ; 0 ; 15
18 A1 A2 A3 A4 Appenix C(cont): Orer 44 (Sequences with zero perioic utocorreltion function) Design 22) (13 (13 24) (13 25) (13 26) (13 28) (13 31) (14 15) (14 17) (14 23) (14 24) (14 25) (14 28) ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 ; 0 0 ; 0 ; ; ; ; ; ; ; ; 0 ; 0 ; 0 0 ; ; ; ; ; 0 ; 0 ; ; ; 0 ; ; ; 0 0 ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; 0 ; ; 0 ; ; ; ; 0 ; ; 0 ; ; ; ; ; 0 0 ; 0 ; 0 ; ; ; ; ; 0 ; ; 0 ; ; ; 0 ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; 0 ; 0 ; 0 ; ; ; 0 ; ; ; ; 0 ; ; 0 ; 0 ; ; ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; 16
19 A1 A2 A3 A4 Appenix C(cont): Orer 44 (Sequences with zero perioic utocorreltion function) Design 30) (14 (15 22) (15 23) (15 24) (15 27) (15 29) (17 21) (17 22) (17 24) (17 25) (18 19) (18 21) ; ; ; ; ; ; ; ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; 0 ; 0 ; ; 0 ; ; 0 ; ; 0 0 ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; 0 ; ; 0 ; ; 0 ; ; ; ; 0 0 ; 0 ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 0 ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; 0 ; ; 0 ; 0 ; ; ; 17
20 A1 A2 A3 A4 Appenix C(cont): Orer 44 (Sequences with zero perioic utocorreltion function) Design 23) (18 (19 22) (19 23) (20 21) (21 23) ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 18
MA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork
MA1008 Clculus nd Liner Algebr for Engineers Course Notes for Section B Stephen Wills Deprtment of Mthemtics University College Cork s.wills@ucc.ie http://euclid.ucc.ie/pges/stff/wills/teching/m1008/ma1008.html
More informationIf you are at the university, either physically or via the VPN, you can download the chapters of this book as PDFs.
Lecture 5 Wlks, Trils, Pths nd Connectedness Reding: Some of the mteril in this lecture comes from Section 1.2 of Dieter Jungnickel (2008), Grphs, Networks nd Algorithms, 3rd edition, which is ville online
More informationMTH 146 Conics Supplement
105- Review of Conics MTH 146 Conics Supplement In this section we review conics If ou ne more detils thn re present in the notes, r through section 105 of the ook Definition: A prol is the set of points
More informationMATH 25 CLASS 5 NOTES, SEP
MATH 25 CLASS 5 NOTES, SEP 30 2011 Contents 1. A brief diversion: reltively prime numbers 1 2. Lest common multiples 3 3. Finding ll solutions to x + by = c 4 Quick links to definitions/theorems Euclid
More informationA dual of the rectangle-segmentation problem for binary matrices
A dul of the rectngle-segmenttion prolem for inry mtrices Thoms Klinowski Astrct We consider the prolem to decompose inry mtrix into smll numer of inry mtrices whose -entries form rectngle. We show tht
More informationA Tautology Checker loosely related to Stålmarck s Algorithm by Martin Richards
A Tutology Checker loosely relted to Stålmrck s Algorithm y Mrtin Richrds mr@cl.cm.c.uk http://www.cl.cm.c.uk/users/mr/ University Computer Lortory New Museum Site Pemroke Street Cmridge, CB2 3QG Mrtin
More informationF. R. K. Chung y. University ofpennsylvania. Philadelphia, Pennsylvania R. L. Graham. AT&T Labs - Research. March 2,1997.
Forced convex n-gons in the plne F. R. K. Chung y University ofpennsylvni Phildelphi, Pennsylvni 19104 R. L. Grhm AT&T Ls - Reserch Murry Hill, New Jersey 07974 Mrch 2,1997 Astrct In seminl pper from 1935,
More informationCS311H: Discrete Mathematics. Graph Theory IV. A Non-planar Graph. Regions of a Planar Graph. Euler s Formula. Instructor: Işıl Dillig
CS311H: Discrete Mthemtics Grph Theory IV Instructor: Işıl Dillig Instructor: Işıl Dillig, CS311H: Discrete Mthemtics Grph Theory IV 1/25 A Non-plnr Grph Regions of Plnr Grph The plnr representtion of
More informationTries. Yufei Tao KAIST. April 9, Y. Tao, April 9, 2013 Tries
Tries Yufei To KAIST April 9, 2013 Y. To, April 9, 2013 Tries In this lecture, we will discuss the following exct mtching prolem on strings. Prolem Let S e set of strings, ech of which hs unique integer
More informationPointwise convergence need not behave well with respect to standard properties such as continuity.
Chpter 3 Uniform Convergence Lecture 9 Sequences of functions re of gret importnce in mny res of pure nd pplied mthemtics, nd their properties cn often be studied in the context of metric spces, s in Exmples
More informationWhat are suffix trees?
Suffix Trees 1 Wht re suffix trees? Allow lgorithm designers to store very lrge mount of informtion out strings while still keeping within liner spce Allow users to serch for new strings in the originl
More informationCS321 Languages and Compiler Design I. Winter 2012 Lecture 5
CS321 Lnguges nd Compiler Design I Winter 2012 Lecture 5 1 FINITE AUTOMATA A non-deterministic finite utomton (NFA) consists of: An input lphet Σ, e.g. Σ =,. A set of sttes S, e.g. S = {1, 3, 5, 7, 11,
More informationGraphs with at most two trees in a forest building process
Grphs with t most two trees in forest uilding process rxiv:802.0533v [mth.co] 4 Fe 208 Steve Butler Mis Hmnk Mrie Hrdt Astrct Given grph, we cn form spnning forest y first sorting the edges in some order,
More information1.5 Extrema and the Mean Value Theorem
.5 Extrem nd the Men Vlue Theorem.5. Mximum nd Minimum Vlues Definition.5. (Glol Mximum). Let f : D! R e function with domin D. Then f hs n glol mximum vlue t point c, iff(c) f(x) for ll x D. The vlue
More information12-B FRACTIONS AND DECIMALS
-B Frctions nd Decimls. () If ll four integers were negtive, their product would be positive, nd so could not equl one of them. If ll four integers were positive, their product would be much greter thn
More informationLily Yen and Mogens Hansen
SKOLID / SKOLID No. 8 Lily Yen nd Mogens Hnsen Skolid hs joined Mthemticl Myhem which is eing reformtted s stnd-lone mthemtics journl for high school students. Solutions to prolems tht ppered in the lst
More informationLecture 7: Integration Techniques
Lecture 7: Integrtion Techniques Antiderivtives nd Indefinite Integrls. In differentil clculus, we were interested in the derivtive of given rel-vlued function, whether it ws lgeric, eponentil or logrithmic.
More informationCS143 Handout 07 Summer 2011 June 24 th, 2011 Written Set 1: Lexical Analysis
CS143 Hndout 07 Summer 2011 June 24 th, 2011 Written Set 1: Lexicl Anlysis In this first written ssignment, you'll get the chnce to ply round with the vrious constructions tht come up when doing lexicl
More informationa(e, x) = x. Diagrammatically, this is encoded as the following commutative diagrams / X
4. Mon, Sept. 30 Lst time, we defined the quotient topology coming from continuous surjection q : X! Y. Recll tht q is quotient mp (nd Y hs the quotient topology) if V Y is open precisely when q (V ) X
More information9 Graph Cutting Procedures
9 Grph Cutting Procedures Lst clss we begn looking t how to embed rbitrry metrics into distributions of trees, nd proved the following theorem due to Brtl (1996): Theorem 9.1 (Brtl (1996)) Given metric
More informationApproximation by NURBS with free knots
pproximtion by NURBS with free knots M Rndrinrivony G Brunnett echnicl University of Chemnitz Fculty of Computer Science Computer Grphics nd Visuliztion Strße der Ntionen 6 97 Chemnitz Germny Emil: mhrvo@informtiktu-chemnitzde
More informationPremaster Course Algorithms 1 Chapter 6: Shortest Paths. Christian Scheideler SS 2018
Premster Course Algorithms Chpter 6: Shortest Pths Christin Scheieler SS 8 Bsic Grph Algorithms Overview: Shortest pths in DAGs Dijkstr s lgorithm Bellmn-For lgorithm Johnson s metho SS 8 Chpter 6 Shortest
More informationCOMP 423 lecture 11 Jan. 28, 2008
COMP 423 lecture 11 Jn. 28, 2008 Up to now, we hve looked t how some symols in n lphet occur more frequently thn others nd how we cn sve its y using code such tht the codewords for more frequently occuring
More informationThe Complexity of Nonrepetitive Coloring
The Complexity of Nonrepetitive Coloring Dániel Mrx Deprtment of Computer Science nd Informtion Theory Budpest University of Technology nd Econonomics Budpest H-1521, Hungry dmrx@cs.me.hu Mrcus Schefer
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationAlgorithm Design (5) Text Search
Algorithm Design (5) Text Serch Tkshi Chikym School of Engineering The University of Tokyo Text Serch Find sustring tht mtches the given key string in text dt of lrge mount Key string: chr x[m] Text Dt:
More informationIn the last lecture, we discussed how valid tokens may be specified by regular expressions.
LECTURE 5 Scnning SYNTAX ANALYSIS We know from our previous lectures tht the process of verifying the syntx of the progrm is performed in two stges: Scnning: Identifying nd verifying tokens in progrm.
More informationDynamic Programming. Andreas Klappenecker. [partially based on slides by Prof. Welch] Monday, September 24, 2012
Dynmic Progrmming Andres Klppenecker [prtilly bsed on slides by Prof. Welch] 1 Dynmic Progrmming Optiml substructure An optiml solution to the problem contins within it optiml solutions to subproblems.
More informationUnit #9 : Definite Integral Properties, Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties, Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationNOTES. Figure 1 illustrates typical hardware component connections required when using the JCM ICB Asset Ticket Generator software application.
ICB Asset Ticket Genertor Opertor s Guide Septemer, 2016 Septemer, 2016 NOTES Opertor s Guide ICB Asset Ticket Genertor Softwre Instlltion nd Opertion This document contins informtion for downloding, instlling,
More informationNotes for Graph Theory
Notes for Grph Theory These re notes I wrote up for my grph theory clss in 06. They contin most of the topics typiclly found in grph theory course. There re proofs of lot of the results, ut not of everything.
More informationSummer Review Packet For Algebra 2 CP/Honors
Summer Review Pcket For Alger CP/Honors Nme Current Course Mth Techer Introduction Alger uilds on topics studied from oth Alger nd Geometr. Certin topics re sufficientl involved tht the cll for some review
More informationHomework. Context Free Languages III. Languages. Plan for today. Context Free Languages. CFLs and Regular Languages. Homework #5 (due 10/22)
Homework Context Free Lnguges III Prse Trees nd Homework #5 (due 10/22) From textbook 6.4,b 6.5b 6.9b,c 6.13 6.22 Pln for tody Context Free Lnguges Next clss of lnguges in our quest! Lnguges Recll. Wht
More informationBruce McCarl's GAMS Newsletter Number 37
Bruce McCrl's GAMS Newsletter Number 37 This newsletter covers 1 Uptes to Expne GAMS User Guie by McCrl et l.... 1 2 YouTube vieos... 1 3 Explntory text for tuple set elements... 1 4 Reing sets using GDXXRW...
More informationTheory of Computation CSE 105
$ $ $ Theory of Computtion CSE 105 Regulr Lnguges Study Guide nd Homework I Homework I: Solutions to the following problems should be turned in clss on July 1, 1999. Instructions: Write your nswers clerly
More information2 Computing all Intersections of a Set of Segments Line Segment Intersection
15-451/651: Design & Anlysis of Algorithms Novemer 14, 2016 Lecture #21 Sweep-Line nd Segment Intersection lst chnged: Novemer 8, 2017 1 Preliminries The sweep-line prdigm is very powerful lgorithmic design
More informationMa/CS 6b Class 1: Graph Recap
M/CS 6 Clss 1: Grph Recp By Adm Sheffer Course Detils Adm Sheffer. Office hour: Tuesdys 4pm. dmsh@cltech.edu TA: Victor Kstkin. Office hour: Tuesdys 7pm. 1:00 Mondy, Wednesdy, nd Fridy. http://www.mth.cltech.edu/~2014-15/2term/m006/
More informationSection 3.1: Sequences and Series
Section.: Sequences d Series Sequences Let s strt out with the definition of sequence: sequence: ordered list of numbers, often with definite pttern Recll tht in set, order doesn t mtter so this is one
More informationThe Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 80Q Multiply two frctions or frction nd whole numer. (QT N ) Excerpted from: The Mth Lerning Center PO Box 99, Slem, Oregon 9709 099 www.mthlerningcenter.org
More informationThe Complexity of Nonrepetitive Coloring
The Complexity of Nonrepetitive Coloring Dániel Mrx Institut für Informtik Humoldt-Universitt zu Berlin dmrx@informtik.hu-erlin.de Mrcus Schefer Deprtment of Computer Science DePul University mschefer@cs.depul.edu
More informationLU Decomposition. Mechanical Engineering Majors. Authors: Autar Kaw
LU Decomposition Mechnicl Engineering Mjors Authors: Autr Kw Trnsforming Numericl Methods Eduction for STEM Undergrdutes // LU Decomposition LU Decomposition LU Decomposition is nother method to solve
More informationP(r)dr = probability of generating a random number in the interval dr near r. For this probability idea to make sense we must have
Rndom Numers nd Monte Crlo Methods Rndom Numer Methods The integrtion methods discussed so fr ll re sed upon mking polynomil pproximtions to the integrnd. Another clss of numericl methods relies upon using
More informationRETRACTS OF TREES AND FREE LEFT ADEQUATE SEMIGROUPS
Proceedings of the Edinurgh Mthemticl Society (2011) 54, 731 747 DOI:10.1017/S0013091509001230 RETRACTS OF TREES AND FREE LEFT ADEQUATE SEMIGROUPS MARK KAMBITES School of Mthemtics, University of Mnchester,
More informationMa/CS 6b Class 1: Graph Recap
M/CS 6 Clss 1: Grph Recp By Adm Sheffer Course Detils Instructor: Adm Sheffer. TA: Cosmin Pohot. 1pm Mondys, Wednesdys, nd Fridys. http://mth.cltech.edu/~2015-16/2term/m006/ Min ook: Introduction to Grph
More informationThe Greedy Method. The Greedy Method
Lists nd Itertors /8/26 Presenttion for use with the textook, Algorithm Design nd Applictions, y M. T. Goodrich nd R. Tmssi, Wiley, 25 The Greedy Method The Greedy Method The greedy method is generl lgorithm
More informationFig.25: the Role of LEX
The Lnguge for Specifying Lexicl Anlyzer We shll now study how to uild lexicl nlyzer from specifiction of tokens in the form of list of regulr expressions The discussion centers round the design of n existing
More informationCSCE 531, Spring 2017, Midterm Exam Answer Key
CCE 531, pring 2017, Midterm Exm Answer Key 1. (15 points) Using the method descried in the ook or in clss, convert the following regulr expression into n equivlent (nondeterministic) finite utomton: (
More informationWelch Allyn CardioPerfect Workstation Installation Guide
Welch Allyn CrdioPerfect Worksttion Instlltion Guide INSTALLING CARDIOPERFECT WORKSTATION SOFTWARE & ACCESSORIES ON A SINGLE PC For softwre version 1.6.6 or lter For network instlltion, plese refer to
More informationDefinition of Regular Expression
Definition of Regulr Expression After the definition of the string nd lnguges, we re redy to descrie regulr expressions, the nottion we shll use to define the clss of lnguges known s regulr sets. Recll
More informationAn Efficient Divide and Conquer Algorithm for Exact Hazard Free Logic Minimization
An Efficient Divide nd Conquer Algorithm for Exct Hzrd Free Logic Minimiztion J.W.J.M. Rutten, M.R.C.M. Berkelr, C.A.J. vn Eijk, M.A.J. Kolsteren Eindhoven University of Technology Informtion nd Communiction
More informationAn Expressive Hybrid Model for the Composition of Cardinal Directions
An Expressive Hyrid Model for the Composition of Crdinl Directions Ah Lin Kor nd Brndon Bennett School of Computing, University of Leeds, Leeds LS2 9JT, UK e-mil:{lin,brndon}@comp.leeds.c.uk Astrct In
More informationA Fixed Point Approach of Quadratic Functional Equations
Int. Journl of Mth. Anlysis, Vol. 7, 03, no. 30, 47-477 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.988/ijm.03.86 A Fixed Point Approch of Qudrtic Functionl Equtions Mudh Almhlebi Deprtment of Mthemtics,
More informationGENERATING ORTHOIMAGES FOR CLOSE-RANGE OBJECTS BY AUTOMATICALLY DETECTING BREAKLINES
GENEATING OTHOIMAGES FO CLOSE-ANGE OBJECTS BY AUTOMATICALLY DETECTING BEAKLINES Efstrtios Stylinidis 1, Lzros Sechidis 1, Petros Ptis 1, Spiros Sptls 2 Aristotle University of Thessloniki 1 Deprtment of
More informationORDER AUTOMATIC MAPPING CLASS GROUPS. Colin Rourke and Bert Wiest
PACIFIC JOURNAL OF MATHEMATICS ol. 94, No., 000 ORDER AUTOMATIC MAPPING CLASS GROUPS Colin Rourke nd Bert Wiest We prove tht the mpping clss group of compct surfce with finite numer of punctures nd non-empty
More informationMisrepresentation of Preferences
Misrepresenttion of Preferences Gicomo Bonnno Deprtment of Economics, University of Cliforni, Dvis, USA gfbonnno@ucdvis.edu Socil choice functions Arrow s theorem sys tht it is not possible to extrct from
More informationCOMMON FRACTIONS. or a / b = a b. , a is called the numerator, and b is called the denominator.
COMMON FRACTIONS BASIC DEFINITIONS * A frtion is n inite ivision. or / * In the frtion is lle the numertor n is lle the enomintor. * The whole is seprte into "" equl prts n we re onsiering "" of those
More informationSolving Problems by Searching. CS 486/686: Introduction to Artificial Intelligence Winter 2016
Solving Prolems y Serching CS 486/686: Introduction to Artificil Intelligence Winter 2016 1 Introduction Serch ws one of the first topics studied in AI - Newell nd Simon (1961) Generl Prolem Solver Centrl
More informationAgenda & Reading. Class Exercise. COMPSCI 105 SS 2012 Principles of Computer Science. Arrays
COMPSCI 5 SS Principles of Computer Science Arrys & Multidimensionl Arrys Agend & Reding Agend Arrys Creting & Using Primitive & Reference Types Assignments & Equlity Pss y Vlue & Pss y Reference Copying
More informationPosition Heaps: A Simple and Dynamic Text Indexing Data Structure
Position Heps: A Simple nd Dynmic Text Indexing Dt Structure Andrzej Ehrenfeucht, Ross M. McConnell, Niss Osheim, Sung-Whn Woo Dept. of Computer Science, 40 UCB, University of Colordo t Boulder, Boulder,
More informationCompression Outline :Algorithms in the Real World. Lempel-Ziv Algorithms. LZ77: Sliding Window Lempel-Ziv
Compression Outline 15-853:Algorithms in the Rel World Dt Compression III Introduction: Lossy vs. Lossless, Benchmrks, Informtion Theory: Entropy, etc. Proility Coding: Huffmn + Arithmetic Coding Applictions
More informationUnit 5 Vocabulary. A function is a special relationship where each input has a single output.
MODULE 3 Terms Definition Picture/Exmple/Nottion 1 Function Nottion Function nottion is n efficient nd effective wy to write functions of ll types. This nottion llows you to identify the input vlue with
More informationPipeline Example: Cycle 1. Pipeline Example: Cycle 2. Pipeline Example: Cycle 4. Pipeline Example: Cycle 3. 3 instructions. 3 instructions.
ipeline Exmple: Cycle 1 ipeline Exmple: Cycle X X/ /W X X/ /W $3,$,$1 lw $,0($5) $3,$,$1 3 instructions 8 9 ipeline Exmple: Cycle 3 ipeline Exmple: Cycle X X/ /W X X/ /W sw $6,($7) lw $,0($5) $3,$,$1 sw
More informationCS201 Discussion 10 DRAWTREE + TRIES
CS201 Discussion 10 DRAWTREE + TRIES DrwTree First instinct: recursion As very generic structure, we could tckle this problem s follows: drw(): Find the root drw(root) drw(root): Write the line for the
More informationAI Adjacent Fields. This slide deck courtesy of Dan Klein at UC Berkeley
AI Adjcent Fields Philosophy: Logic, methods of resoning Mind s physicl system Foundtions of lerning, lnguge, rtionlity Mthemtics Forml representtion nd proof Algorithms, computtion, (un)decidility, (in)trctility
More informationMatrices and Systems of Equations
Mtrices Mtrices nd Sstems of Equtions A mtri is rectngulr rr of rel numbers. CHAT Pre-Clculus Section 8. m m m............ n n n mn We will use the double subscript nottion for ech element of the mtri.
More informationMath 142, Exam 1 Information.
Mth 14, Exm 1 Informtion. 9/14/10, LC 41, 9:30-10:45. Exm 1 will be bsed on: Sections 7.1-7.5. The corresponding ssigned homework problems (see http://www.mth.sc.edu/ boyln/sccourses/14f10/14.html) At
More informationAVolumePreservingMapfromCubetoOctahedron
Globl Journl of Science Frontier Reserch: F Mthemtics nd Decision Sciences Volume 18 Issue 1 Version 1.0 er 018 Type: Double Blind Peer Reviewed Interntionl Reserch Journl Publisher: Globl Journls Online
More informationThe Structure of Forward, Reverse, and Transverse Path Graphs in The Pattern Recognition Algorithms of Sellers
The Structure of Forwrd, Reverse, nd Trnsverse Pth Grhs in The Pttern Recognition Algorithms of Sellers Lewis Lsser Dertment of Mthemtics nd Comuter Science York College/CUNY Jmic, New York 11451 llsser@york.cuny.edu
More informationSection 10.4 Hyperbolas
66 Section 10.4 Hyperbols Objective : Definition of hyperbol & hyperbols centered t (0, 0). The third type of conic we will study is the hyperbol. It is defined in the sme mnner tht we defined the prbol
More informationsuch that the S i cover S, or equivalently S
MATH 55 Triple Integrls Fll 16 1. Definition Given solid in spce, prtition of consists of finite set of solis = { 1,, n } such tht the i cover, or equivlently n i. Furthermore, for ech i, intersects i
More informationSingle-Player and Two-Player Buttons & Scissors Games
Single-Plyer nd Two-Plyer Buttons & Scissors Gmes The MIT Fculty hs mde this rticle openly ville. Plese shre how this ccess enefits you. Your story mtters. Cittion As Pulished Pulisher Burke, Kyle, et
More informationIf f(x, y) is a surface that lies above r(t), we can think about the area between the surface and the curve.
Line Integrls The ide of line integrl is very similr to tht of single integrls. If the function f(x) is bove the x-xis on the intervl [, b], then the integrl of f(x) over [, b] is the re under f over the
More informationUT1553B BCRT True Dual-port Memory Interface
UTMC APPICATION NOTE UT553B BCRT True Dul-port Memory Interfce INTRODUCTION The UTMC UT553B BCRT is monolithic CMOS integrted circuit tht provides comprehensive MI-STD- 553B Bus Controller nd Remote Terminl
More informationLecture 10 Evolutionary Computation: Evolution strategies and genetic programming
Lecture 10 Evolutionry Computtion: Evolution strtegies nd genetic progrmming Evolution strtegies Genetic progrmming Summry Negnevitsky, Person Eduction, 2011 1 Evolution Strtegies Another pproch to simulting
More informationZZ - Advanced Math Review 2017
ZZ - Advnced Mth Review Mtrix Multipliction Given! nd! find the sum of the elements of the product BA First, rewrite the mtrices in the correct order to multiply The product is BA hs order x since B is
More informationThe dictionary model allows several consecutive symbols, called phrases
A dptive Huffmn nd rithmetic methods re universl in the sense tht the encoder cn dpt to the sttistics of the source. But, dpttion is computtionlly expensive, prticulrly when k-th order Mrkov pproximtion
More informationINTRODUCTION TO SIMPLICIAL COMPLEXES
INTRODUCTION TO SIMPLICIAL COMPLEXES CASEY KELLEHER AND ALESSANDRA PANTANO 0.1. Introduction. In this ctivity set we re going to introduce notion from Algebric Topology clled simplicil homology. The min
More informationGrade 7/8 Math Circles Geometric Arithmetic October 31, 2012
Fculty of Mthemtics Wterloo, Ontrio N2L 3G1 Grde 7/8 Mth Circles Geometric Arithmetic Octoer 31, 2012 Centre for Eduction in Mthemtics nd Computing Ancient Greece hs given irth to some of the most importnt
More informationEngineer To Engineer Note
Engineer To Engineer Note EE-186 Technicl Notes on using Anlog Devices' DSP components nd development tools Contct our technicl support by phone: (800) ANALOG-D or e-mil: dsp.support@nlog.com Or visit
More informationRevisiting the notion of Origin-Destination Traffic Matrix of the Hosts that are attached to a Switched Local Area Network
Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 Revisiting the notion of Origin-Destintion Trffic Mtrix of the Hosts tht re ttched to Switched Locl Are Network Mondy
More informationToday. CS 188: Artificial Intelligence Fall Recap: Search. Example: Pancake Problem. Example: Pancake Problem. General Tree Search.
CS 88: Artificil Intelligence Fll 00 Lecture : A* Serch 9//00 A* Serch rph Serch Tody Heuristic Design Dn Klein UC Berkeley Multiple slides from Sturt Russell or Andrew Moore Recp: Serch Exmple: Pncke
More informationSpring 2018 Midterm Exam 1 March 1, You may not use any books, notes, or electronic devices during this exam.
15-112 Spring 2018 Midterm Exm 1 Mrch 1, 2018 Nme: Andrew ID: Recittion Section: You my not use ny books, notes, or electronic devices during this exm. You my not sk questions bout the exm except for lnguge
More informationBefore We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1):
Overview (): Before We Begin Administrtive detils Review some questions to consider Winter 2006 Imge Enhncement in the Sptil Domin: Bsics of Sptil Filtering, Smoothing Sptil Filters, Order Sttistics Filters
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Bnch J. Mth. Anl. 4 (2010), no. 1, 170 184 Bnch Journl of Mthemticl Anlysis ISSN: 1735-8787 (electronic) www.emis.de/journls/bjma/ GENERALIZATIONS OF OSTROWSKI INEQUALITY VIA BIPARAMETRIC EULER HARMONIC
More informationCSCI 104. Rafael Ferreira da Silva. Slides adapted from: Mark Redekopp and David Kempe
CSCI 0 fel Ferreir d Silv rfsilv@isi.edu Slides dpted from: Mrk edekopp nd Dvid Kempe LOG STUCTUED MEGE TEES Series Summtion eview Let n = + + + + k $ = #%& #. Wht is n? n = k+ - Wht is log () + log ()
More information4452 Mathematical Modeling Lecture 4: Lagrange Multipliers
Mth Modeling Lecture 4: Lgrnge Multipliers Pge 4452 Mthemticl Modeling Lecture 4: Lgrnge Multipliers Lgrnge multipliers re high powered mthemticl technique to find the mximum nd minimum of multidimensionl
More informationEfficient K-NN Search in Polyphonic Music Databases Using a Lower Bounding Mechanism
Efficient K-NN Serch in Polyphonic Music Dtses Using Lower Bounding Mechnism Ning-Hn Liu Deprtment of Computer Science Ntionl Tsing Hu University Hsinchu,Tiwn 300, R.O.C 886-3-575679 nhliou@yhoo.com.tw
More information10.2 Graph Terminology and Special Types of Graphs
10.2 Grph Terminology n Speil Types of Grphs Definition 1. Two verties u n v in n unirete grph G re lle jent (or neighors) in G iff u n v re enpoints of n ege e of G. Suh n ege e is lle inient with the
More informationDistributed Systems Principles and Paradigms
Distriuted Systems Principles nd Prdigms Chpter 11 (version April 7, 2008) Mrten vn Steen Vrije Universiteit Amsterdm, Fculty of Science Dept. Mthemtics nd Computer Science Room R4.20. Tel: (020) 598 7784
More informationA Heuristic Approach for Discovering Reference Models by Mining Process Model Variants
A Heuristic Approch for Discovering Reference Models by Mining Process Model Vrints Chen Li 1, Mnfred Reichert 2, nd Andres Wombcher 3 1 Informtion System Group, University of Twente, The Netherlnds lic@cs.utwente.nl
More informationProduct of polynomials. Introduction to Programming (in C++) Numerical algorithms. Product of polynomials. Product of polynomials
Product of polynomils Introduction to Progrmming (in C++) Numericl lgorithms Jordi Cortdell, Ricrd Gvldà, Fernndo Orejs Dept. of Computer Science, UPC Given two polynomils on one vrile nd rel coefficients,
More informationCSCI 3130: Formal Languages and Automata Theory Lecture 12 The Chinese University of Hong Kong, Fall 2011
CSCI 3130: Forml Lnguges nd utomt Theory Lecture 12 The Chinese University of Hong Kong, Fll 2011 ndrej Bogdnov In progrmming lnguges, uilding prse trees is significnt tsk ecuse prse trees tell us the
More informationSubtracting Fractions
Lerning Enhncement Tem Model Answers: Adding nd Subtrcting Frctions Adding nd Subtrcting Frctions study guide. When the frctions both hve the sme denomintor (bottom) you cn do them using just simple dding
More informationOUTPUT DELIVERY SYSTEM
Differences in ODS formtting for HTML with Proc Print nd Proc Report Lur L. M. Thornton, USDA-ARS, Animl Improvement Progrms Lortory, Beltsville, MD ABSTRACT While Proc Print is terrific tool for dt checking
More informationON THE DEHN COMPLEX OF VIRTUAL LINKS
ON THE DEHN COMPLEX OF VIRTUAL LINKS RACHEL BYRD, JENS HARLANDER Astrct. A virtul link comes with vriety of link complements. This rticle is concerned with the Dehn spce, pseudo mnifold with oundry, nd
More informationON SOME GRÜSS TYPE INEQUALITY IN 2-INNER PRODUCT SPACES AND APPLICATIONS. S.S. Kim, S.S. Dragomir, A. White and Y.J. Cho. 1.
ON SOME GRÜSS TYPE INEQUALITY IN 2-INNER PRODUCT SPACES AND APPLICATIONS S.S. Kim, S.S. Drgomir, A. White nd Y.J. Cho Abstrct. In this pper, we shll give generliztion of the Grüss type inequlity nd obtin
More informationDetermining Single Connectivity in Directed Graphs
Determining Single Connectivity in Directed Grphs Adm L. Buchsbum 1 Mrtin C. Crlisle 2 Reserch Report CS-TR-390-92 September 1992 Abstrct In this pper, we consider the problem of determining whether or
More informationSpectral Analysis of MCDF Operations in Image Processing
Spectrl Anlysis of MCDF Opertions in Imge Processing ZHIQIANG MA 1,2 WANWU GUO 3 1 School of Computer Science, Northest Norml University Chngchun, Jilin, Chin 2 Deprtment of Computer Science, JilinUniversity
More informationHyperbolas. Definition of Hyperbola
CHAT Pre-Clculus Hyperols The third type of conic is clled hyperol. For n ellipse, the sum of the distnces from the foci nd point on the ellipse is fixed numer. For hyperol, the difference of the distnces
More informationOnline Portal Guide. Access your policy information, documentation, claim forms and claims history easily and securely.
Online Portl Guide Access your policy informtion, documenttion, clim forms nd clims history esily nd securely. version dte: 12/2017 YOUR ONLINE PORTAL ACCESS URL & REGIONAL CONTACTS HONG KONG SINGAPORE
More information