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I ~ I I I I I I I A Necessary Condition for the Existence of Regular and Symmetrical PBIB Designs of T Type m by Motoyasu Ogasawara Nihon University, Tokyo, Japan. Institute of Statistics Mimeo Series No. 418 February 1965 --I I I I I I I. 1 J This research was supported by Air Force Office of Scientific Research Grant No. 84-64. DEPARTMENT OF STATISTICS UNIVERSITY OF NORTH CAROLINA. Chapel Hill, N. C. I ..I I I I I I I --I I I I I I I. I I A Necessary Condition for the Existence of ReGular and Symmetrical PBIB Designs of T Type m By MOto~~su Ogasawara Dept. of Math., Collece of Sci. and Eng., Nillon Univ. §l. Introduction and summary. A necessary condition for regular and symmetrical PBIB desiGns in terms of the iant 118.S tl~ existence of Hasse-~uru:owski p-invar- been obtained, for trianrrular type by J. Ogawa [5], for T type by 3 K. Kusumoto [4], both basing upon tile work of L.C.A. Corsten [1] concerning the proper space related to PBIB designs of triangular tYI>e. In this article, the author introduces an association of T type as an m extension of the type of association stated above, and determines the proper spaces related to PBIB designs of this type, along the line of Corsten's work. Non-existence criteria of PBIB designs of Tm examples. Hence, the present \'TOrJ~ t~me olr are also r,iven with some . is a generalization of those by J. Ogawa [5] and K. Kusumoto [4]. In tIle SUbsequent section, definition of the association of T type is m given and tIle corresponding association algebra is discussed. In section 3 we discuss tile proper spaces related to PBIB designs of Tu type. m Section l~ is devoted to the derivation of a set of necessary conditions for the existence of PBIB designs of T type, and in tIle final section, sone examples ~11 of non-existent PBIB designs of T"'l " t~'Pe 1 are given. I ..I I I I I I I --I I I I I I I. I I Association of T type and tIle corresponding association algebra. m §2. association of T m type is defined as follows: positive integers such that 1 ~ j) Let n ~d J:l and 2m _< n, and let v An be an:)' given = (11). m Let us take (~) different subsets, {r , ••• ; roJls, of {l, ••• , nJ, and ~'re associate to l each of tllese subsets one of one way. t~le treatments, CPl' ••• , CPv' in any but one-to- Two treatments, CPi and C{)j' which correspond to (rl , ••• , rm) and r'} respectively, are said to be the u-th associated if and only if m [rl , • , rm} and (ri' ••• , r~} contain exactly m-u integers in cormnon, u = 0, 1; ••• , me Hence the number of the u-th associates of eac1'l treatment is given by n = (m ) (nu-m), u u (2.1) ( u = 0, 1, ••• , m) • Parameters characterizing the association are t _ Psu - m=t ~ ~ a=o (m-t) ( t )( t )( n-m-t ) t a m-s-a m-u-a s+u-m+a' u,s,' Tile association matrices A (u u = 0,1, ... , m) = 0, 1 , ••• ,m. generate a linear comrrru.tative algebratT over the field of all rational numbers and it is called the association algebra. by the It can be sho"m [6] the regular representation of (( is generated l~lB.ppings (3.2) Transforming P 's by a non-singular and rational matrix u z (2.4) C = II -1!! II , n (s, t = 0, 1, ••• , m) t with 2 I ~ I I I I I I I I4t I I I I I I I. I I we get ° Zoo C P1.1 C- l (2.6) Z = 11.1 • , 1.1 = 0,1, ••• ,:n • Z mu ° and conseCl1.1entlJr we obtain the following (m+l) mutually ort}logonal idempotent matrices belonging to m a =~ ( Z JJ!1.1 V - Zua n a=o 1.1 = O,l, ••• ,m a wi th respective rank # (2.8) au = tr. Au ( n = (n) 1.1 - 1.1-1) = n+1-21.1 (n) n+l-1.1 1.1' 1.1 = 0,1, ••• ,D4 Let II be the incidence matrix of the design, then it is irell-known that m NN' = Z ?\A uu 1.1=0 = m E 1.1=0 where m (2.10) p 1.1 = l:z a=o ua A 1.1 's. = O,l, ••• ,m • T.le latent vector of NN' corresponding to the c11aracteristic root is evidently JI = (1,1, ... ,1). -v p o = r1: From (2.9), it can be seen that the latent vectors of NN' corresponding to pu are a 1.1 collunn vectors of Ail, '''hicl1 are U linearl:,r independent. §3. They are all rational vectors. Sor,Je ;properties of proper space related to PBIBD of Tm t;lJ;>e. to L.e.A. CorSten [1], we conceive P = NN' as the matrix of t:le linear transformation P on a vector space.c consisting of vectors ~' into itself, "There the coordinate x t According = (~,~, .•. ,xv) corresponds to the t-tll treatment. 3 Fron I ~ I I I I I I I --I I I I I I ..I I m (2.9) the t-tll coordinate Yt in if. = ~ is equal to the SttO E A 3 , 1'Jhere 3 designate a a a of the coordinates of 2£ corresponding to tIle a-til associates of tIle treatment qt' J.;. 0.=0 = (1,1, ... ,1) is the latent vector of P ",Titll tIle proper value m E A n = rl~. aa lie consider the (v-l) dimensional subspace .ttl of r. ortllOgonal a=o to J.v ' Tllen, for every vector 2£ in s.tl, we have the fol1mdnG relation X + 3 + ••• + 3 m t 1 = O. Let us construct a set of (n) vectors of dimension v,{ c. ~1, U . ••• ,1u I (i , •• • , i } c {l, ••• , n }}, "Thicll are contained by the vector space 1. in the u l folloi'rlnc lla~r: let the t-th conwonent ~ be 1 or 0, accordinc; as [il , ••• , i u } c (r , ••• ,rm) or [i , ••• ,1u ) 1 1 ¢ {rl, ••• ,r ill ), where (rl, ••• ,rI1 } is a set of integers corresponding to the t-tll treatment CPt' t=l, ••• ,v. Let the (~) dimensional subspace of .\: spanned by those (n) linearly independent vectors u c. ,···,J..' ,~( 0) bei~ -1 • s be .'I':, (u) , ' t ',len t'!le space u--1 , ••• ,1;1, the one dimensional space spanned by ~ (u) conta:Lns · t-ue space J.-... (u-l) , iv' Hence, tl1ere exists au dimensional subspace ot"(U) of tlle space .t(u) orthogonal to .<;(u-1), u=lj'."m. . lI(u) .' ) Z {l 'J7i . 1n c't, . c. ( )l'.·.'J.u C , ••• ,n 1,···,1u ~l'."'3.u the t-tll coordinate x:'CU is equal to (.J. , .•• ,1. ) CE { r , ••• , r "l~ Y.J. ' ••• , J.. • U ;;1u 1 1 1 It is noted that, if 1fe rep1a.ce the scalar .,. i b Jc the vector c. . ].1' • • • , ~1' • • • , 3. . (ul 't'" u t' 100._, 11 i n t ile e~CJ?ress1on X -(. . } LJ ( 7. i 1 Ilen we uClove a t ].l'···'].u c r 1 ,···,rm) ].1'···' U vector in £(U) such as c\U) = . . ~ c.. • .::.t {].l' • .. ,]. ) c {r1 ,···, r , } ~1'···']. (k) U(O) In u li01'T, since <~ II is ortlloc;onal to s, 1.:s k .:s m, it holds tllat For a.n~r vector x (u) = {. < "Thicll i1'1J)lies tJ18.t (3.2) ' i }E { } 7i . {].1'···' k c 1, ••• ,n 1,···,J.k 4 = 0, k = l ..••• ,rn. I ~ I I I I I I I aI I I I I I ..I I 1~ For any 11 and k, (11 = 1, ••• : :.1-1, = 1, ••• , m) the ilmer proc1uct of .sill) and X(::) becomes (11)' (k) .£t ~ n). (11:) = (11 ..i:t (m-1)s(lC) + ... + (m-(n-l1»s(k) + 11 · wi tIl tIle convent J.on that (m-1~)'f h-1: = 0 J. 11 1 = til' II • ,~} ~ c.I {rl' -:1.1 " " , II • ~l' ,rl') . x (Ie) II ., n-11..... = ( JLn-11 "'"'l.l'··· ,~- r . ~1 of the expression ~l j (n-h-l) + m-11-1 L:'l J. , • • • ,J• _ , j 1I k l l j l' • • ., k n-h-l\:.) + ••• + ( I:r'l ra-hk J l ,'" i·there, the swmnation sign of '£r.. Jl . rn-h < k• In tile case when h > k, since, for c. - 11 ., .. ,Jl~ ., desir,nates the sun of ,···,J 1\,-u - , Jl,···,J u "I.. .• .' I . , for all [j" ••• , jl , j • , ••• , j • ) sue!l t:;lat {j1' • •• , jl- } J 1 ,"" J;:_u' J l ,···, J u \:.-u 1 u '.-u i , ... ,i }', it folloi-TS from (3.2) c {i , ... ,i;1} and (ji, ... ,j~} C (l, ... ,n} n1 l 1 , c. . x -J.1 ,···,J.11- (::) h = 11-11-1) _ (n-I1-1e)]<:" + [( . 1 . 1 ~"I. . ., l,l-nm-H- ~ J l' • • • ,JJ:_l" J 1 ..... _,' [(n-]l-(l~-l»_ (n-h-k)]~ .. (,.x:-1) . k ...."1.Jl,Jl, • • ••• ;J• • _ U-J1m-l1-' k 1 B:,' tilC rclation , . J:..,.J . 2:-U) + ••• + ( t.-h l 5 .• .' I , • • ., J 1':.-ll ,J , ••• ,.J u l I ~ I I I I I I I I_ = E'7. .' ., j' Jl,···,J,· ..-u ,J l ,···, U 1.fl1ere, i:' designates the sUl11nation for all {jl, ••• ,jlc-u ,jl', •••.' ;i'} n suell that I. I I J, {l, ... , n} 1jl"'" jlr.-... tIle above hmer product can be rem'itten as c i, - 1"'" . x \1- (k)_- (n-il-k)~, - 1 r. 40 l:1.-m- + ., ., J l , J l ,···, J1<~_l ( l1-l1-l~)~, 2~r.· j' ., J l , J 2 , 1"'" J 1:_ 2 n-l11- ••••• vTllere "tre llave tIle equaJ.ity ~_ (k-(k-u»(n-ll-::) = (n-l1-(k-U», u-a a=o I I I I I I n {jl' • • • ,j,. } c {i ,· .. , i n } and (jl', ••• , j U') .,-U l n-m-a On tlie other band, in tile case "then 11 o- c' (U n-l11 < = 1, ••• ,1: ) • k, since ... (1:) = ( n-k-h)"" .... n-m- 1 - -i , ••• , i. ~ 11 l r j l' J., , ••• , ;)1:_1 ., < l n-k-11) .... , + ( 2 ~ 7. .~, ~, J , J 2 , J l , • •• ,J 1:_ 2 l n-rl1- ..... + ( n-k-11 . )Z'y. .., ., n-m-ll Jl' ••• 'Jil'Jl' ..... Jk_l1 ' it fol101/S that 0, 6 U = 1, ••• ,k-1, I ~ I I I I I I I I_ I I I I I I I. I I where, II ~ 1.. .., I • I J1' • ••, ,J .. u U,J l ,···, J l.-.. denotes the su.nnnation for all {j', ••• , jk' } 1 -u n 1jl' .•• ' ju 1 and (j1'''.' j.) such tl"lS;'c {ji'.'" jk-u} c (1, ... , n} Tl1erefore (3.4) ,·re being fixed. have cI x (1:) .::.:1.1 ' ••• ' ~- = and 11ance (11) I ~ -'" (k) _ (m-I:) (n-Il-Ie) (k) n-m-k X t · x - 11-;: I - Therefore, b;y' (3.3) and (3.5), we IlB.ve the relation ehm) Xt(k) (3.6) + (m-,l)C1(J:) . ;1 °1 11 1- ••• = 0, ... , 1, m-1, from lThich, lTe c;et the follouinc e'lualities 1Su(l;:) -- Zku x{.!:) t ,., HOlT, b~l u - 1, . • • , m. the argument in tIle preceding section, it is seen tilat the coordina.te Y of p!" ! being a vector belonging to !(u), is t (~ \Zua)Xt • Therefore a=o Pu' (u = 1, ••• ,r:t). HOlT, · b eJ.ng let us consider t 1 , ••• , v. St("l-U) ,-= sJKu) is a J)roper space of NU' llitl1 J?roJ)er value . a r.w,.lc ru C (m-u) of order v, w'hose COl1..1.1nrl m ....nen,: 'J.t can b e seen tlla;t {m-u) _II (n-u) (m-u) (m-u) C - £1 '.£e ,. •., £v = (m-u m) II A + (m-l) A +. •• + (m-u) A 0 m-u 1 1a-u u where, A. 's are association matrices. J. e'~ual to Hence we get vectors I ~ I I I I I I I I_ I I I I I I I. I I C(m-a)A # k = (n-l:)(n-m-k+a) a a. AIi k' and l1encc, from uJ!iCll ue obtain (3.7) ~~u T~le .~_1I(1) ,• = u E (-1) n-a(n-a) (m-k) (n-m-k+a) • n-u a=o a Grw·:rl.a...'1 P. of the basic vectors of •••... ,_ .~.II J. a .~ (i), (i), and ,~(o) (~1) _ ... :: , ••• ,TIl are, the join of proper spaces n01'T, "I;:.r 0) 1 t[unea " , easJ. as follOlis: In order to calculate P;, ire J. Si~lrply need tIle inner products of the vectors -J1, c. ••• ,J' I s, (j 1, ••• ,J'} c t~l , ••• ,n. } i i rna trix Ir( i) I iil10se COh11lU1 vectors are c, I nd ee d , .~ J..L iTe consJ.'der t",le . 's, then Pi is c:iven -J 1 ,···,J i b~i (3.8) It Silou.ld be noted that tJle natrix N(i) is the incidence ;;atrix of the PBIB desicn of T, t;ype with paraneters J. n-i-a) ( ) (Lli -a ' a.=O, 1, • • • ,i , and thell, it is i'1ell-l:nown tlm-G n <) I ~ I I I I I I I from 'It,dell it follm'ls that (3.11) It is also seen fron (2.8) or (3.10) that T'llerefore, the determinant of P. is given by J. I_ I I I I I I I. I I i=l, ••• , m. Nou, let 0: linearly indepeDde.nt column vectors of A# be u ' u (u) !l anci. (u) (u) '!e ' ... '!au ' :put t11en o SiS = • o • • (m) Q \'There 9 u=l, • • • }m I ~ I I I I I I I I_ I I I I I I I. I I Q(U) = ll~iu), ... ,!(u)1I 'II !.i u ), ••• ,! (u)lI. u U Moreover, it is clear tIlat pv o (1) 0 P1Q. (3.16) SINN'S = • f\ o Pm"'" (m) Since . v (1) 0 Q i = 1, ••• , m.• o It is shovm easily that , i=1, ••• ,m, "ritl1 Ipo I = v. Hence, by using (3.13) ~re Get IQ () i i-I INn [(i-j)(n-i+1-j)] a j=o a j[(n~:i)] i, i=l, ••• ,m llllere, as before, U §1.~. = 1, ••• , m. Non-existence criteria of regular and symmetrical PBIB design of Tm t;;pe. In the present section we Give a set of necessary conditions for the existence of reG"Ular and symmetrica.l PBIBD's of Tm type, i.e. v =b and hence r = k and (n+1) characteristic roots of mit are given by Po =r 2 , Pu = m E Z a=o ua 10 \ > 0, U = 1, ... ,m, I ~ I I I I I I I I_ I I I I I I I. I I ",there, as before, (h.2) zus s = ~ a=o (_l)s-a(m-a)(m-u)(n-m-u+a) m-s a a • From (3.15), (3.16) and (4.1), it follows that Q(l) 0 "P Q(l) 1 • o Q,(m.) 0 • • P Q(m) o m Therefore, by the Hasse theorem [5], it fo11mrs ti.lat ex m II p u=l u u ,.." 1 and m Ct (Y- II [( -1, (p ' u u=l !Q(u) l)p)J n_l ( P ' ;)11 u p -, u II - l<u<ll<m for all primes p, where, as before IQ (!1.• 6) () u I,.." u-1 IT j=o [(u-j)(n-u+1-j)] ex j[(n-2u) m-u ~ ,u=l, ••• ,m. lTit~.1 ~lese are necessary conditions for the existence of regular and S:'1:1netrical PBIBD' s of Tn t~'J?e. In cases '\-,hen m=2 and n=5, these coincide lTitll conditions obtained l)y J. Ogawa [5] and K. Kusumoto §5. [!~] respective1::,r. Exa."gP1es of non-existent reeuJ.ar a.nd symmetrical PBIBD's of Tm ~. The folloin.ng designs can be seen to be non-existent b:r the criteria (4.4) and 11 I ..I I I I I I I I_ I I I I I I I. I I n=3, n=7, v=b=35, r=k=12, \ =h, ~=4, ~=3, P1=11, P2=6, ~=9 u=5, n=19, v=b=969, r=1I:=5'"(, \=9, ~=3, ~=3, P1=228, P2=126, p.;=36. r.J.=!~, \=4, ~=2, ,,=1, \=4, P1=33, P2="'"i v:b=70, r=k=16, \=6, ~=3, ~=2, \=4, =18, p';=4, f\=6. ~=44, P0 t:. m=4, n=9, v:b=70, r=1I:=17, \=5, ~=4, ,,=3, \=0, P1 =33, p2=9, n=8, v=b=70, r=1I:=13, udj., n=8 , m,=q.: n=9, v=b=126, r=k=10, '1=1, m::!J., n=9, v-b=126, r=1I:=15, .111="~ , , f'1=4, ~=1, ~=O, p.;=3, f\=9. f.\-() -",. p';=13, \=2, P1 =19, P2=12, p.;=2, P2~=14. ~=1, ,,=1, \=6, p1 =27, p2=41, ~=1, f).r ='7. n=9, v=b=126, r=k=15, \=2, ~=2, ~=1, \=2, P1 =27, P2=13 , p;=8, P1/.=17• l~=h, n=9, v=b=126, r=1I:=16, '1=3, ~=2, ~=1, \=4, p1 =31, md~, n=10, v=b=210, r=1I:=20, '1=5, \=2, ~=1, \=0, P0=2l~, '- p.; =l~, p,,=16. ",' n P1 =100, P =28, f=3=16, P1j. =0. 2 n=h, n=10, v=b=210, r=1I:=23 , '1=4, ~=3, ~=1, \=4, p1 =6h, p2 =40, P:3=1, p],=25, T L-r, ')-" j l=)~, n=10, v=b=210, r=k=30, \=5, '2=5, n=10) v=b=210, r=1~=5h, \=13, \,=5, c:. AC;:110il1ec1Gements. The author e~':presses ~=3, :; \=4, l =75, P2 =27, P:3=12, PP!'-:. =x:. P ~=3, ~.j.=.s, ~=151, :.J P2 =103, Po;. =):') ~) ;', PJ'~4 =('. ids apprecia'i;ion to Prof. s J. 0C0.iro.. a.mi. S. Ikeda for their interests and helpful discussions c:iven to this 'lTorJ:. T;la..'11:s are also due to Pro:,:'. S. Yamamoto and Mr. Y. Fujii for their Idndncss in d.l~alrll1G the author's attention to Mr. Hamada t s 11Or1: [2], vTllich eave tile sa.ne result as (3.7) in t"ua paper, but by somei·Tlla.t different method. References [1] Corsten, L. C.A., Prcper space related to trianGUlar PBIB designs, Ann. l~th. Stat., Vol. 31 (1960) 498-501. Hamada, N ., On the cor,tposition of the triangular association all3ebra. TalJ~ given at t11e o.nnual meeting of Ja.pan MatIl. Society held at vTaseda Uni v., June 27, 19611-. 12 I ~ I I I I I I I I_ I I I I I I I. I I [3] • ~ .. .1 Hasse, H., Uber die AtllUi'm.l.enz quadratisl1er Fornell. 1n Korper der rationalen zalllen, Crelle 152 (1923) 205-221~. K., A necessary condition for existence of reGular and s:'1:nnetrica.l PBIB desiens of T type. KUSl~~to, 3 [5] Oca;ua, J., A necessar:.' condition for existence of reC'll.lar and symmetrical experimental c.le::;:i<.J1s of triangular type, i,lith partiall~l balanced incomplete blocks, Ann. J:v1ntll. Stat., Vol. 30 (195S'), 1063-1071. [6] O[';a'';1a, J. and Ishii, G., On the a.na.lysis of partially balanced incomplete blocl~ desic;r...s j.n tIle regular case, suomitted to Ann. Math. stat.