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数理解析研究所講究録 404
実験配置の理論と応用
京都大学数理解析研究所
1980 年 11 月
RZMS Ko fe yafto te” 404
Tneory of Experimental Layouts and
Its Applications
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27O4248
pt t21
21kLJeganfrutrsin
November, l980
Research Tnstitute for Mathematical Sciences
Kyoto University, Kyoto, Japan
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SYMPOSIUM ON THEORY OF EXPERIMENTAL LAYOUTS AND
ZTS APPL:CATZONS
PIace : Research Tnstitute for Mathematical Sciences r
Kyoto University, Kyoto, Japan
Date :
July l4- l6, l980
Organizex : Sanpei Kageyama, Department oE Mathematics,
Faculty of School Education,
Hiroshima University
PROGRAM AND ABSTRACT
i. S. Kageyama
(Hiroshima University)
On 5-designs
Abstract: We show that an inequahty b) v(v-l)
holds for a 5-(vrkrXs) design. Furthermore, a Steinesc
system S(5e6,l2) is shown to be the unique 5-design
with b = v(v-l)r up to complementation.
N
2. M. Yoshizawa
(Kelo Unwerslty)
Block intersectzon nurabers of block deszgns
Abstract: The following results are gwen:
[rheorem l. For each n ) l and A ) l7
(a) there exist at most finxtely many block-schematic
t-(v,k,X) designs with k-t=n and t ) 3, and
(b) if also X l 2, there exist at most finitely many
block-schematzc t-(v,k,X) designs wzth k·-t=n and t l 2.
tV
Theorem 2. A Steiner system S(t,t+i,v) is block-gchematic
if and only if one o￡ the followmg hoids: (i) t=2, (u)
t=3, v=8, (iu) t=4, v=ll, (iv) t=5, v=l2.
3e
N. !to and H. Kimura
(Umv. of Zllinois and Hokkaido Univ.)
Hadamard rnatrices with 2-transitive automorphism groups
Abstract: We consider Hadamard matrices with
2-transitwe automorphism groups not containing regular
normal subgroups. Under some assumptions we have some
results. For example (l) the degrees of Hadamard
matrices are square; (2) automorphism groups are non-
solvabie and7 (3) they are not 3-transitive.
4.
K. Takeuchi
(Umversity of Tokyo)
Randomzzation design Revisited
t
Abstract: RandomzzaUon Design, in which factor levels
are randomizedr was discussed by several authors includzng
Taguchi, Satterthwaite, some twenty years agor but has been
since nearly forgotten. Zn 1958 Kiefer proved that randomly
balanced extremely unbaZanced designs are optimum in terms
of the local power of the test of null hypothesisr which
￡act has never been further analyzed. The author once wrote
a series of papers on this topic, and now wants to revitalxze
interests m this problem, and discusses its basic features
taking the szmplest case of comparmg means of severai normal
populations for illustration.
V
5.
K. Suda
(National Gunma Technicai College)
An automatical design and analysis system for orthogonal
expenments
Abstract: The use of fractional factorial designs
has now become widely accepted as an efficient way to carry
out experiments in 9uality Control. Howeverr one of the
main difficulties with the fracttonal factorial designs
involving rnany different factors is how to construct an
orthogonal design which can estimate various effects without bezng confounded. Tn such situations, we developed the
program that can automatically construct an orthogonal
design and compute the estimates of the main effects and
interactions for any given model for micro-cornputer. We
show this automatical design and analysis system has wide
applications for improving product qua”ty.
6.
T. Shirakura
(Kobe University)
Norm of alias matrices for (2+l)–factor interactions in
balanced fractional 2M factorial designs of resolution 22+l
Abstract: Consider a balanced fractional 2M factoriai
design of resolution 22+l derived from a balanced array of
strength 22+l. consider the norm llAll = {tr(A’A)}1/2 of
alias matrzx A for (￡+1)-factor interactions m this design
This norm can be used as a rneasure for selecting a design.
rn this paper, an expliczt expression for llAll is gwen
by using algebraic structures o￡ a balanced design. By
vE
this expressioni designs of resolution V(2==2) which
mmimize llAH are presented for any fixed assembhes
satisfying (i) m == 5, 16 S N s 32, (ii) m = 6, 22 S N S 32,
and (ui) m= 7, 29 SNS 50·
7.
Re NiShli
(Hiroshima University)
On fractional factonal designs with orthogonal structure
Abstract: Fold-over designs have been discussed as
they give the orthogonality between estimate of odd
parameter and one of even parameter. Here level-symrnetric
designs are defined, which aye given by generalization of
the concept of fold-over designs. Those designs are proved
to have the structure that any odd parameter and any even
parameter can be estimated uncorrelatedly, and it is proved
that designs with this structure must be level-symmetric
designs.
8.
i
M. Kuwada
(Maritzme Safety Academy)
On an alias relation to 3-factor interactions in balanced
fractionaZ 3M factonal designs derivabie from balanced
arrays of strength 5
Abstract: Consider a balanced fractional 3M factorial
design T denvable from a balanced array of strength 5.
By use of the multidimensional relationship and its algebra,
we wiil present an explicit expressxon for the norm of A[Dr
x.e., IIATll = {tr(AT’AT)}l/2, where AT is the ahas matnx.
ct
vs”
9.
Y. Ohashi
(University of Tokyo)
An application o￡ ”sub-sampiing” to the robust estimation
of u (o2)
Abstract: !n this paper, a procedure utilizing B!BD
is proposed for the robust estimation of u(u2) from a normai
sample possibly with a few outiiers. An original sampie is
divided into m subsamples; m sample variances are calculated
and frorn them a robust estimate Ui (here, a wtnsorized mean)
is obtamed. Although 3i zs robust, its efficiency is
considerabiy low as compared with the usual estimatorr that
is, the sample variance, so ”sub–sampling” is repeated r
txmes, gxving 5;,”’,5i and the final estimate 52:=(2i3;)/r.
It is shown that much higher efficiency is achieved by the
systematic repetition scheme utilizing B!BD than by the
random repetition or other non-systematic schemes. Aleans
and MsE’s of 32 and 5 = conser/52 are numericany compared
with those of familiar alternatwes such as linear estimators
under the ”shppage” modei. The above procedure is applicable
to the estimation of error variance in a ”near modelr and
an applzcation to ”]ack-knife” is suggested.
10.
1. Takahashi
(Unwersity of Tsukuba)
The least Hamming distance method applied to a file construction
Abstract: R.C.Bose and others introduced balanced fiies
based on k-dzmensionai subspaces S with strength t in GF(q)M.
We propose new filing scheme based on a subspace S fitted on
e)
V”s
the set R of given records. ”i”he cnterion of the fitness
is the least Hamming distance and fitttng algorith is
essentially the same as decoding method of Reed Muller codes.
Il. M. Yamada
(Tokyo Woman’s Christian CoUege)
On the Wiliiarnson matrices of Turyn’s type
Abstract:
In l972 Turyn found an infinite family of Wilhamson matrices.
Namely if q = 2n-1 is a prime power E l (mod 4), then there exxsts
a Wilhamson matnx of order 4n. Whiteman gave a new proof by
2
using the trace from GF(q ) to GF(q) in 1973. In this paper we
interpret this theorem in terms of the theory of the Gauss sum
over a finite field.
we let: E = GF(q2), F = GF(q), q = pt :’ l (mod 4) and n = (q+1)/2,
SE = trace from E, SF = trace from Ft SE/F = trace from E to F,
2Xi/p
je:a generator of E*, gn : an n-th root of unityr l;p =e ·
X4 : the character of E such that X4(S) = i,
nl
Xn : the character of E such that Uh(g) = Sn and Xn
= l,
,C= (h·Ai) = the character of E which gives the Legendre symbo1
wh en re stncted to F, ur = SI + Y-nr (r = lr ···r (n-l) /2),
s ct
?E(23) = liitllE 7((ot) ISpE = the Gauss sum over E,
sS = the Gauss sum over F.
T, (X) = cliil I. X( o() gp
Put Qu= rE(PC)/leF(X)· Then we have
eh = .(..:Etii :. F..x(o()51(sE/Foc)
n-l
= l.Z=o (-i)r{ U(sE/Fse2r) + i”x(sE/F}2r+n)} gkr
= (-,)(n+i)/2(-i+i){igi + i?l.i, ,B.gkr)·
n-1
lx
where ,gS, = (-i) -in {x( sE/F S;2 r) + i” u( sE /F s;2 r+” )} (r = i, ...,
(n+l)/2 + r
we know that ,Br is +l, -l, +i, or -i, and ./3n-r = JBr· Further zf
we put Kine” (-i)(.+i9X/2(-i+.) = iltlZ + lil.llil Askr then 2Kx2-Kx= 2q·
Therefore let A+, A-r B+, B- be a partitzon of R= {1,...,(n-1)/2}
for ,jBr = lr -lr ir -z respectively, then the Willxamson equation
4n = 2 + 2Kx2 Kx
= l22
+ 1 + (l+2Z ur-2:iE ur)2 + (1+2E: ur-2Z ur)2r
r6A +
riA- reB+ reB-
is true for every n-th root of unity.
We have several results on e5;. By the Davenport-Hasse theorem
we have
ei - J(x, if),
where J(X,.afI) is the Jacobi sum. And by Stickeiberger’s theoxem we
have the factozazation
e,, ·v ser e–li-r Bl=
.. dcr
( ¡- 2il ’- llTt) ) ¿ O
-l
ceZ*(4ne)
where 8 zs the prime ideal zn the cyclotomic field 9(S4nv) Such
‘f
that Xip, f is the smallest positive integer which sat-sfies
p :. 1
(mod 4n’), Z*(4n’) is the multiplicative group of Z/4n’Z, eC! an
automorphzsm SZIn,e-¿12n, of Z*(4n’), and Bl(x) = x-l/2 zs the
Bernoulli polynomial of degree i.
!t is not settied whether we can get a new famzly of Willzamson
matrices by developing our xnteypretation.
×
n-i).
12.
K. Sawada
(Nagoya !nstitute of Technology)
The Williamson matrices of a special form
Abstract: We consider Willzamson equations of the
followmg type:
12”12”(1’2-EA.Uv-2v=tA-Uv)2”(1’2v=tB.Uv-2v=tB-Uv)2=4n’
n-l
where A+, Ae, B+, B- is a partition Of {l,2,·e·r 2 },
corresponding to the decomposition 4n = l2+l2+a+4wl)2+(l+
2
4W2) e An infinite class of the Williamson matrices found
by Turyn belongs to this class. In this paper it is shown
that #A+, #Am, #B+, #B- are explicitly determined in terms
of Wl and W2. We have found that there are no Wilhamson
matrices of the above form, except for those due to Turyn,
￡or n¡ 37 and for n= 61.
l3.
J. Kinoshita
(Hokkaido University)
Networks and Matroids
Abstract: This paper explains a max-flow and min-cut
theorem concerning networks with matroid restrictions in
capacity.
l4e
K. Ushio
(Nnhama Technical College)
On bipartite decomposztion of a complete bipartzte graph
Abstract: A complete biPartXte graPh Knl,n2 (nl =¡ n2)
is said to have a bipartite decomposition if zt can be
decomposed into a union of line-dis)oint subgraphs each
xi
isomorphic to a complete bipartite graph K
kl,k2 (kl E k2)·
ln this papert a necessary condition for a complete bipartite graph K
nl,n2 (nl E n2) to have a bipartite de-
composition is given. And several theorems which state
that the necessary condition is also sufficient in many
cases are glven.
15. S. Tazawa
(Hiroshima College of Economics)
On claw-decomposition of a complete multi-partite graph
Km(nl,n2,”’vnm)
Abstract:
Let Vl, V2,”’, Vm be point sets with nl, n2,e.e, nm point$ each. A graph
is said to be complete m-partite graph, denoted by Km(nl, n2,e”, nm), if no 1ine
joins two points in the same point set and if each point in Vi is adjacent to aLl
points of sets other than Vi. A complete bipartite graph K2(1, c) is, in particuLar, called a claw of degree c. In this paper two theorems are given, provideC
m-1 m
nl, n2,eee, nm are positive integers satisfying iil j.21+lninj/C = integer and nl :E
m
n2 E ’.’ :EL nm: (a) The case N-nm ¡ c, where N= 2 ni. Km(nl, n2,”’, nm) is
1=l
decomposed into a union of line-disjoint claws of degree c each if and only if
(N-nm)IIIIal!t -¡-. Ie.l/l j.:+ininJ/c s I:.l/i niIN’cn2tl· Here LrJ is the greatest integer not
ex￠eeding r and rr;1 is the smaUest integer not less than r. (b) [[[he case
N’nm ) C· If Km(nl, n2,·”, nm) is decomposed into a union of line-disjomt claws
m-l m
of degree c each, then i;.i ji+ininj ) N’nm’
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