Download Ogasawara, M.; (1965)A necessary condition for the existence of regular and symmetrical PBIB designs of T_M type."

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A Necessary Condition for the Existence of Regular
and Symmetrical PBIB Designs of T Type
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by
Motoyasu Ogasawara
Nihon University, Tokyo, Japan.
Institute of Statistics Mimeo Series No. 418
February
1965
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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.
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A Necessary Condition for the Existence of ReGular
and Symmetrical PBIB Designs of T Type
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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
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given and tIle corresponding association algebra is discussed.
In section 3
we discuss tile proper spaces related to PBIB designs of Tu type.
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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.
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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
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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
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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.
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Fron
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(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.
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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
~
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= 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
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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
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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,
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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
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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
<)
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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.
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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
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~
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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,
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",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
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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.